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This is to certify that the
dissertation entitled
Equilibrium Shocks in a Directionally Reinforced Neo-Hookean
Material Under Plane Deformation

presented by

Jose Merodio

has been accepted towards fulfillment
of the requirements for

Ph.D. Engineering Mechanics

degree in

 

 

M ajO/rb‘rbfessor

 

Date 141/ 35, /‘7‘;7

MSU i: an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

LIBRARY

Michigan State
University

 

 

 

PLACE IN RETURN Box:c . .. . .- _ “' herkm Ir from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLH) with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

71m elem-4:659."

 

 

EQUILIBRIUM SHOCKS IN A DIRECTIONALLY REINFORCED NEO-HOOKEAN
MATERIAL UNDER PLANE DEFORMATION

By

Jose Merodio

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Materials Science and Mechanics

1999

 

ABSTRACT

EQUILIBRIUM SHOCKS IN A DIRECTIONALLY REINFORCED NEO-HOOKEAN
MATERIAL UNDER PLANE DEFORMATION

By

Jose Merodio

The purpose of this work is the analysis of discontinuous deformation gradients in
NonLinear Elasticity for a transversely isotropic material under plane deformation. In
particular, the material model is an augmented neo—Hookean base with a simple
unidirectional reinforcement characterized by a single parameter. The existence of these
solutions is related to material instabilities and is associated to a shear band-type failure
mode. The loss of ellipticity of the governing differential equations is a necessary condition
to get these material instabilities.

After a description of the problem, which includes a brief introduction of the theory
of finite elastostatics in Section 2, the analysis of the material model in Section 3 and a
complete kinematics analysis of the equilibrium shocks in section 4, it is shown in Section 5
that the neo-Hookean material does not support equilibrium shocks in the present
circumstances. In Section 6, 7 and 8 the plane shock formulation for the Reinforced Neo-
Hookean model is analyzed. The kinking angle associated to the any elastostatic shock is
given. The correlation between the weak shocks and the loss of ellipticity is presented. In
Section 9 the main features of the shocks obtained in the previous sections are given. In
particular, it is shown that the existence of these equilibrium shocks involve fiber
contraction. Further, it is shown that the loss of ellipticity is a necessary condition for the
existence of strong shocks. In Section 10, an energetic analysis that accounts for the
existence of these equilibrium shocks is carried out. The purpose of the analysis is to rule
out as inadmissible some of the elastostatic shocks solutions obtained in Section 6. The use
of a dissipativity inequality yields two important results for the Reinforced neo-Hookean.

First, quasi-static shocks can only advance toward the side with a lower hydrostatic pressure

 

(in analogy with fully dynamical shocks in gases). Second, the stable side of the shock is
that one with either elongation or less fiber contraction (in analogy with the fiber kinking
mechanism). Therefore, the energetically admissible shocks move simultaneously towards
the side with lower hydrostatic pressure and higher fiber contraction. Furthermore, the
energetically admissible elastostatic shocks are viewed as a bifurcation away from a simple
homogeneous deformation. At last, maximization of the dissipative inequality is proposed as
the condition to single out a physically admissible elastostatic shock among all the
energetically admissible elastostatic shocks. It is shown that the elatostatic shocks
associated with this maximal dissipative inequality follow characteristics similar to the ones
observed during kink-band formation. In particular, the results obtained for the kinking
angle and the shock angle associated to maximally dissipative shocks are shown to be close

to experimental results of fiber kinking in composite materials during kink-band formation.

ACKNOWLEDGEMENTS

My gratitude to my advisor Dr. Pence for his dynamic involvement at every step of
my graduate program and throughout the writing of this dissertation.

I also want to thank the members of my committee Dr. Altiero, Dr. Tsai, Dr. Yan
and Dr. Zhou for the interest and time they have shown not only in this work but also in the

many courses I have taken with them.

TABLE OF CONTENTS

List of Figures ..................................................................................... vii
1. Introduction ................................................................................. 1
2. Preliminaries on Finite Incompressible Elastostatics .............................. 6
3. The Reinforced Neo-Hookean Material Model ....................................... 7
4. Elastostatic Shocks ........................................................................ 8
4.1 Piecewise Homogeneous Plane Elastostatic Shocks ............................ 9
4.2. Specialization to Diagonal Deformation Gradients on one Side
of the Elastostatic Shock .......................................................... 15
5. Neo-Hookean Material .................................................................. l6
6. The Plane Elastostatic Shock Formulation for Reinforced Neo-Hookean
Materials .................................................................................... 19
7. Parameter Space Representation for Mechanically Consistent Elastostatic
Shocks ....................................................................................... 27
7.1. The discriminant condition for k—limit points ................................... 28
7.2. Analysis in the (at, 0t)-Plane ...................................................... 32
8. The Elastostatic Shock Manifold E(y) ............................................... 42
8.1. Analysis in The (A, k)—P1ane ...................................................... 43
8.2. Constant A Cross-Sections of E(y) .............................................. 50
8.3. Shock Angle Multiplicity .......................................................... 58
9. Qualitative Features of the Mechanically Consistent Elastostatic
Shocks ....................................................................................... 67
9.1. Fiber Contraction ................................................................... 69
9.2. Ellipticity in 11+ and rr ............................................................ 70
10. Energetics ................................................................................... 80
10.1 Favorability of 1T versus IT ...................................................... 82
10.2 Maximally Dissipative Elastostatic Shock Motion ............................. 87
10.3 Evolution of the Kink Angle ...................................................... 95
10.4 Discussion .......................................................................... 98
Summary or Conclusions ...................................................................... 102
Appendix
Appendix A. Kinematics of plane elastostatic shocks for isochoric
Deformations ............................................................... 106
Appendix B. Components of the tensor G in the coordinate system Y .............. 108
Appendix C. Principal directions for a plane deformation gradient F ............... 110
Appendix D. Conditions for the Loss of Ellipticity in plane strain deformation
for the reinforced Neo-Hookean ......................................... 112

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LIST OF FIGURES

Figure 1. Kinematics of plane piecewise homogeneous elastostatic shocks.
The undeforrned configuration is described by the rectangular coordinate
system X = (x1, x2), while the deformed body is referred to the rectangular
frame Y = (yl, yz). The elastostatic shock separates the plane into two open
half planes: a “+” side with a deformation gradient F” and a ”-” side with a
deformation gradient F. The normal vector to the elastostatic shock, denoted
as N in the undeforrned configuration (n in the deformed one) is taken to
point into the “+” side .................................................................. 11

Figure 2. Kinematics of the fiber reinforcement in the presence of an elastostatic shock.
The orientation of the fiber reinforcement is preserved in 1'13.L due to a
diagonal defamation gradient F‘ aligned with the fiber. The deformation
in I'll, characterized by F", does not preserve the orientation of the fiber, and
gives a kinking angle denoted by ¢. The angles a and 45 can not coincide ......... 21

Figure 3. Values (2», k) corresponding to shock angles a = if for the values of the

reinforcing parameter y :3, 100. The basic form of these curves is the same
regardless of the value of the reinforcing parameter y. The curves are
symmetric with respect to k = 0, as given by (6.15). In particular at k = 0,

A = j. (y) E ’1 - 2—1— , for each y-value. These orthogonal shocks involve
Y

fiber contraction on both sides of the elastostatic shock. Further, as the

fiber is contracted on IT’, the fiber on IT keeps the constant stretch given by

ll (3! ) , eventhough rotating with k through an angle 4) = tan'1 (—k) from

its initial “unkinked” direction 4) = 0 2’5

Figure 4. Angles 0t that give the orientation of the spatial mechanically consistent
elastostatic shocks at the k-limit points. The fiber reinforcement is kept
parallel to the yl—axes on In“ . For a given A—value that permit mechanically
consistent elastostatic shock existence, there are 2 of such mechanically
consistent elastostatic shocks, which in turn give 4 unit vectors n. In

general, the kinking angle (0 at 0 so that the orientation of the fiber will
not be preserved on IT: ................................................................ 30

Figure 5. Kinematics of the fiber reinforcement at the k-limit points. The
cases depicted correspond to cases 2 and 3 of Figure 4. The other
two cases give a symmetric state of affairs with respect to the angle %, i.e.,

the kinematics of case 4 (respectively case 1) is equivalent to the
kinematics of case 2 (respectively case 3), with HI located at the right
of the mechanically consistent elastostatic shock .................................... 31

Figure 6. The curves, denoted as 8 Pk (7), give values (7t, Ct) at the k-limit points.

Each curve is symmetric with respect to a = 1;. The interior

to each 8 Pk (y), denoted by F k (7), represent pairs (2., or) consistent
with elastostatic shock existence. Points 0», 0t) outside F k (y) are not
consistent with the existence of such a shock Lines of constant at

intersect 8 Pk (y) at most twice, giving the upper and lower A-values

able to sustain that particular shock direction. The curves 8 Pk (y) are nested
with respect to v. For values % < y S 3.283 ,8 Pk (3!) involves A-values

satisfying 0 S )1 S 1: (y ). For values 7 > 3.283, 8 Pk (y) involves

values 2» >1: (y ) , as shown in the figure for the particular
values y = 6, 15. The curves 8 Pk (y)can not involve values CL = 0 or 1: ............ 34

Figure 7. The curve represents the maximum A—values, called Amm of each
curve 8 Pk (y) in Figure 6. This curve is monotone increasing, which follows

from the nesting property of the curves 8 l" k (y), with respect to y. Note

1 . . . .
for y S — that there are no A—values consrstent With elastostatic shock exrstence.

At y = 3.283 the formula defining this curve shifts from (6.16) to (7.11). Also,
Am(8)= 1, which indicates that y > 8 gives the range of y that supports
mechanically consistent elastostatic shocks with fiber extension

on IT, i.e., A > 1. There is a vertical asymptote given by

4
lim lmax (y) = J:=1.155 ........................................................... 39
r—w 3

Figure 8. Values 0t associated to KMW). The curve is symmetric with respect to o: = g.
The left side of the curve is related to the lower half of the curves 8 Pk (y) in
Figure 6, while the right side of the curve is related to the upper half of the
curves 8Fk (y). For g < y S 3.283,the curve indicates that this angle

is a = % since Am(y)=/l (y) ........................................................ 40

Figure 9. Values on vs 7 obeying (7.13) define two regions: the region interior to the curve
involves values a that support mechanically consistent elastostatic shocks,
while the region outside the curve involves values on that are not admissible

for the particular y. The curve is symmetric with respect to a = %.The left side

of the curve is related to the lower half of the curves 8 Pk (y) in Figure 6,

therefore, giving the minimum oc-value for each 7, while the right side is related
to the upper half of the curves 8 Pk (1!), hence giving the maximum (it-value

for each y ................................................................................. 41

Figure 10. The plot shows the k-limit points in the (A, 0t)—plane or8 Pk (y) curve, as given
in Figure 6, along with the weak shocks curve (k = O) fory = 15. Both curves
are symmetric with respect to a = E .The curve k = O is inside the curve

8 Pk (y), as it corresponds to values (A, 0L) 6 I‘k(y). At the k—limit points, for

shock angles 0 < a < 325, (A, 0t)-values correspond to k > 0, while for shock

angles % < a < 7! , (A, (JO-values on the k-limit points correspond to
k < 0 42
Figure 11. Projection of the k-limit points into the (A, k)-plane for the values of the

reinforcing parameter 7 = 3, 6, 15. The curves are symmetric with respect
to k = 0. Values k > 0 in each curve are associated to shock angles

0 < a < %,while values k < O in each curve are associated to shock angles

E < a < 7: . The curves are similar to the curves in the (A, 0t)-plane shown in
Figure 6. In particular, for each 7, points of horizontal and vertical tangency in
each curve correspond with points of vertical and horizontal tangency in each

curve of Figure 6. Further, (A, k) = (0, 0) and (A, k) = (A (y ) , 0) correspond to
(A, on) = (0, g) and (A, CL) = ( A (y),%) respectively for each y .................. 44

Figure 12. The plots give the (A, k) values at the a—limit point curves and the projection
of the k-limit points curves into the (A, k)-plane, for y = 3, 6, 15, 50. All the
curves are symmetric with respect to k = 0. The a—limit points are
(A, k)e 8 l10’)and bound all the pairs (A, k)E Pa (1!). The points (A, k) =

(A (y ) , 0) related to the angle a = g and (A, k) = (Amw), k), related to a

unique angle shown in Figure 8, satisfy simultaneously the oc-limit point curves
and the k-lirnit point curves. Otherwise, the projection of the k-limit points curves
are inside the oc-limit points 48

 

Figure 13. For the particular value of the reinforcing parameter 7 = 15, the k-limit points
and the (A, (JO-curve associated to the a-limit points are shown. Both curves are
symmetric with respect to a = E. The k-limit points bound all the (A, 0t)-points
consistent with shock existence. The curves coincide at locations of vertical
tangency where A = A (y) and for y > 3.283, A = Amax (y ). Otherwise, the
a-limit points are interior to the k-limit points in the (A, a)—plane ................. 49

Figure 14. Different cross sections of the E(y) manifold for y = 15 in the (a, k)-plane

shows features representative of values y > 3.283. For values Amw) > A > A (y)
the 13(7) cross sections will involve two symmetric components as shown by the

Figure 15.

Figure 16.

Figure 17.

cross section at the upper left comer (A = 1.02). In particular, at the maximal
value A: Am(y), the “cross section” first emerges as the two separate points
(a, k) = (am, km") and (a, k) = (fit—am, -km) where kw follows from (8.1)

and am follows from (7.5). At the value A = A (7) these two components
coalesce at the unique point (A, on) = (A (y ) ,g). As A decreases below A (y)

the cross-sections of E(y) develop following the patterns shown by the two lower
figures. For y = 15, each cross section associates at most two values k for each
angle or, and at most two values 0t for each k. The k-limit points are related in

 

each cross section to points obeying g—k' = 00 ,while the (x-limit points are
a

related in each cross section to points obeyingdili = 0. ........................... 52
a

Different cross sections of the of the E(y) manifold for y = 3 in the
(Qt, k)—plane, at the same A—values displayed by the two lower plots in
Figure 14, i.e., A = 0.8, 0.3. Recall that for y < 3.283, the E(y) manifold

does not include values A > A (y ) , and the cross section of E(y)

at A = A (y) includes a unique point (A, on) = (A (y ) ’g)' In each cross section
for y = 3 there exist at most two values k for any 0t and two values oz

for any k .................................................................................. 54
Different cross sections of the of the E(y) manifold for y = 1000 in the (on, k)—
plane, at the same A-values that in Figure 14, i.e., A = 1.02, A (y), 0.8, 0.3. Note

that for y = 1000, A (y) = 0.9997. The character of the E(y) manifolds as shown
in Figure 14 for y = 15 and Figure 16 for y = 1000 are similar. Nevertheless, in
the lower right cross section of the E(y) manifold for y = 1000, the horizontal
dashed line k = 3 intersects the E(y) cross section four times. This indicates

the existence of four real solutions at for some (A, k)-pairs, as opposed to

Figure 14 where pairs (A, k) are associated with two real solutions at ............ 54

The plot shows the or-limit points and the (A, k)-values related to 0L :3; for y =
15, partitioning Pa (3!) in three regions: the upper region (wholly contained in k >

0) involves two shock angles 0t <g; the middle region containing the A—axis
involves two shocks, one on <%, and the other or >%; the lower region (wholly

contained in k < 0) involves two shock angles or >%. Similar plots will follow for

values 7 obeying 'y < y*. In the plot the symmetry has been considered to draw
the direction of the mechanically consistent elastostatic shocks with respect to the
fiber on IT”. Recall that in the event of elastostatic shock existence, the fiber will
not keep the horizontal direction on the IT side of the mechanically consistent
elastostatic shock, giving rise to a kinking angle 56

 

Figure 18. The plot represents the k—limit points and the (A, (Jo-values associated to k = 0
(weak shocks) for y = 15. Both curves are symmetric with respect to or = g, and

partition Fk (1!) into three regions. The upper region is associated to two negative

values k, the middle region to one positive and one negative value k, while
the lower region to two positive values k. These features are representative for
any y—value, since any (A, or) 6 Pk (y) is associated at most with two values

k 57

Figure 19. Values (A, k) 6 Fa (y) for y = 410 depicted here is qualitatively similar to
previous Figure 17 for y = 15. However, since 410 > 7* = 403 certain (A, k) are
associated with four real solutions of (8.4). The (A, k)-pairs occupy a small
region that appears as two small (symmetric) line segments. The line segment
endpoints (see points T1 and T2 in the blow-up) are triple roots of (8.4).

For 7 >> 7* the four solution region more fully unfolds as clearly seen in
Figure 20 65

 

Figure 20. Values (A, k)E D10!) for y = 1000. The unfolding of the region given by (A,
k)—pairs sustaining four shock angles a is more fully developed compared to
Figure 19. The (A, k)-points T1, T2, T3 and T4 are associated with triple roots of
(8.4). T1 and T2 (cases k > 0) are related to one angle 0t triple root of (8.4)

obeying 0t >§ and one angle 0: obeying at <%. T3 and T4 (cases k < 0) are
related to one angle or triple root of (8.4) obeying or <§ and one angle on
obeying or >%. For k > 0, pairs (A, k) on the boundary of these unfolding
regions are associated with one angle at double root of (8.4) obeying 0t > %, one

angle on obeying at >5 and one angle on obeying a < E. For k < 0, pairs (A, k)
on the boundary of these unfolding regions are associated with one angle or

double root of (8.4) obeying 0t <gone angle at obeying at >% and one angle 0:

obeying at <% (this is appreciated in the lower right plot of Figure 16). For k >
0, pairs (A, k) inside these regions are associated with three angles obeying 0t

>—;£and one angle obeying 0t <§ (Theorem 3, case a.2 < 0 and a0 < 0). For k < 0,
pairs 0,, k) inside these regions are associated with three angles a obeying a <§

and one angle or obeying at >% ........................................................ 66

Figure 21. Different cross sections of E(y) and E00 manifolds for y :15. The manifold
H7) is related to values k, while the manifold FM is related to values k. Both

_ _ 3 n
manifolds intersect at k = k = 0, and k = k = -2——2— (the k-limit points). Points
n

(A, or, k) E E(y) are associated with a non-elliptic deformation on IT if there exist
points (A, or,k,)€ F01) and (A, 0t,k2)E Fm such that k, < k < E.
Otherwise, the ellipticity status of (A, a, k) E E(y) is checked by means of (9.3).
The complete analysis of the ellipticity status of (A, 0t, k) E E(y) is shown in
Figures 22 and 23. From the cross sections, it is easy to verify that there exists at
least one E obeying 0 S lkl S |k|. This establishes that a mechanically consistent
elastostatic shock involves the loss of ordinary ellipticity at some deformation on
the particular path connecting F’ and F that is parametrized by E. Further, the
direction of every consistent elastostatic shock is a characteristic lin_e, since
values or as obtained from E(y) are also values or as obtained fromE(y). . . . . .....76

Figure 22. For values k > 0, the plot gives the projection into the (A, or)-plane of the triplets
(A, on, k) E E(y) elliptic on IT when y = 15. Every point (A, or) inside the IT
elliptic region is associated with a unique point of the E(Y) manifold. Recall
Figure 18, pairs (A, (1)6 F k (y) inside the curves k = 0 correspond to a unique

point (A, or, k)E E(y) such that k > 0. Pairs (A, (1)6 P k (3!) between the (lower)

dashed line k: 0 and the k—limit points correspond to two points (A, or, k) E E(y)
such that k > 0. In the latter region, the (A, (JO-pair associated with an elliptic

deformation on IT is the one with the greater value k. This is further appreciated
in Figure 23. The k-limit points are associated with non-elliptic deformations on

H'. Points (A, or) = (A,£) are associated with elliptic deformations on IT. The

closeness of some of the boundaries of the H‘ elliptic regions with the k-limit
points is fully appreciated in Figure 23. Values k < 0 involve a symmetric state of

affairs about or :3, namely, if (A, or) is associated with an elliptic (respectively,

non-elliptic) deformation on IT then (A, “rt—0t) is associated with an elliptic
(respectively, non-elliptic) deformation on IT ....................................... 78

Figure 23. Different cross-sections of the E(Y) manifold in the (a, k)-plane, for y = 15,
showing the IT ellipticity status of points (A, 0t, k) E E(y). Each cross—section
involves four segments: two segments giving rise to (A, a, k) e E(y) non-elliptic
on IT and two segments giving rise to (A, or, k) E E(y) elliptic on IT. The plots at
the left corner depict the closeness of the k-limit points and the transition between
pairs (or, k) associate with elliptic and non-elliptic deformations on IT. In each

cross section the central symmetry about (at, k) = (%,0) is kept, namely if pairs

(or, k) define an elliptic (respectively, non-elliptic) deformation on IT then pairs
(rt—0t, -k) define an elliptic (respectively, non-elliptic) deformation on IT. The l-
1 correspondence of the projection into the (A, 0t)-plane of the E(y) manifold
involving deformations elliptic on IT and either values k > 0 (Figure 22) or
values k < 0 is obvious from each cross section ..................................... 79

Figure 24. Projection of the E('y) manifold satisfying S > 0 on the (A, or)-plane for y = 15
and k > 0 is contained within the solid curve. This bounding curve is defined by a

Figure 25.

Figure 26.

k = 0 segment (above) and an S = 0 segment (below). The (A, 00—pairs obeying S
> 0 are inside the k-lirnit points. Further, and in correlation with Figure 22, there
is a 1-1 correspondence between (A, 006 I‘k (y) obeying S > 0 and points (A, or,
k)6 E(y), as can be seen in Figure 27. All the points (A, 00 obeying S > 0 satisfy

A <A (y ) , therefore, the shocks associated to S > 0 lose ellipticity on IT. For
values k < 0, there is a symmetric state of affairs about or = E, namely, if

(A, (06 Pk (y) obeys S > 0 (respectively S < 0), then (A, n—006 Pk (y) obeys

S > 0 (respectively S < 0) ............................................................... 86

In addition to the correlation with Figure 24, pairs (A, 006 Sm are shown for
values k > 0, for the particular value of the reinforcing parameter 7 = 15. The

picture shows that Sum involves elastostatic shocks obeying on < E. In fact, this
remains for any value 7. Values k < 0 involve a symmetric state of affairs with

respect to %. Namely, for values k < 0, Sm involves elastostatic shocks obeying

7t
or > —. This is clearly appreciated in Figure 24. Even more, correlation with

Figure 22 indicates that the shocks associated to Sm involve elliptic
deformations on IT ..................................................................... 89

Here we show the only two plots of Figure 15 which involve A <A (y) as

only these cross sections give values S > 0. For values A <A (y ) , each E(y) cross
section is separated into four segments according to the value of S: two segments
in which S > 0, and two segments in which S < 0. The symmetry of each segment
in each cross section is as follows: if pairs (or, k)6 E(y) satisfy S > 0, then (rt—0t,

-k)6 E(Y) and S > 0, and if pairs (on, k) 6 E(y) satisfy S < 0, then (rt—0t, -k)6 E(Y)

and S < 0. Therefore, each segment keeps the central symmetry about (0t, k)=(£,

0). Correlation with Figure 23 shows that Sm involves elliptic deformations on2
If, while pairs obeying S > 0 may involve either elliptic or non-elliptic
deformations on IT ..................................................................... 90

Figure 27. Cross section of the E(y) manifold for y = 15 at A = 0.8. Points A—H have

Figure 28.

special significance in the k 20 portion of this cross section. The point B is the
generator of the or-limit points and the point G is the generator of the k-limit
points. The segment AD gives (0t, k)-pairs obeying S > 0 and the segment DH
gives points (or, k) obeying S < 0. The segments AB and FH are associated with
(or, k)-pairs giving rise to non-elliptic deformations on IT, while the segment BF
is associated with (or, k)—pairs giving rise to elliptic deformations on IT. The
point C is associated with Sm ......................................................... 92

Cross section of the E(y) manifold fory = 15 at A = 0.3. Points A-H maintain
the same significance as in Figure 22. Note however that point D has migrated
around points E and F. Once again, the point B is the generator of the a-limit
points and the point G is the generator of the k-lirnit points. The segments AB
and FH are associated with (or, k)-pairs giving rise to non-elliptic deformations

Figure 29.

Figure 30.

Figure 31.

Figure 32.

on IT, while the segment BF is associated with (or, k)-pairs giving rise to elliptic
deformations on IT.The segment AD gives (0L, k)-pairs obeying S > 0 and the
segment DH gives points (or, k) obeying S < 0. The point C is associated with

m ........................................................................................ 92
Cross section of the E(y) manifold for y = 15 at A = 1.02. The points E and I
are the generators of the (Jr-limit points and the points G and J are the generators
of the k-limit points. In correlation with Figures 27 and 28 the segment BF is
associated with (or, k)-pairs giving rise to elliptic deformations on IT. Note

that the points A, H and D do not appear on the plot because S < 0 everywhere
on the cross section. It follows that C can not appear on the plot either ......... 94

Cross section of the E(y) manifold for y = 15 at A =A (y )= 0.983. The

points E and G are the generators of the (Jr—limit points and the k-limit points
respectively. In correlation with Figure 27 and 28 at the point A coincide

B, C, D and H. Thus, the (or, k)—pairs in the cross section obey S < 0 except

at A where S = 0. Further, the segment AF is associated with (or, k)-pairs

giving rise to elliptic deformations on IT. In correlation with Figure 29, at

the point A also coincide the points I and J .......................................... 94

Curves aSm, ‘9th (13:0 and ¢S=o that give the shock angle or and the
kinking angle 0 at Smax and S = 0 respectively for the particular
values 7 = 3, 15, 50, 150. The values k associated to the curves are values k < 0.

. . . 7:
The curves 013m and (13:0 rnvolvrng values k > 0, are symmetric about or = —,

while the curves 05m and 05:0 involving values k > O, are symmetric about 4) = 0.
Note that the curves (ISM and 05m given by Sm do not intersect nor do the
curves 0L5:o and 03:0 given by S = 0. This is in agreement with our analysis of
(6.5), where it was shown that the angles or and d can not coincide. As 7
increases, for fixed values A the plots depict ever greater values of (l), i.e., the more
anisotropic is the material the more rapid is the rotation of the kinked fiber.
Further, for each y-value, both 05m or 05:0 show a first stage at which, small
changes of A cause a very rapid rotation of the kinked fiber. This is followed by a
more steady state, in which the kinking angle 4) does not vary much with

A. Nevertheless, the behavior of both curves departures as A —> 0 , since 05m

—) g and ¢5=0—) 0. Note also that the curve 015:0 involves angles on greater

than otSm ................................................................................. 96

Fiber stretch A on IT vs. the fiber stretch J? on IT as given by
Sm fory = 3,15, 50,150.1n all cases A s A (y) <1, but «fit: may

be either «ff < 1 or s/E > 1. For each curve, «[1? > A, indicating
that the elongation on IT is greater than the elongation on IT. This is in

agreement with our discussion of (10.9). Values J? > 1 follow immediately
after initial kinking for values Y sufficiently large. Note that for each curve,

as). —) Othen WA A (1!). Further, for each curve «[1? > A (1!), therefore,
maximally dissipative elastostatic shocks keep an elliptic deformation on IT.

The dashed line represents values A = JI? , so that it corresponds to values

of the fiber stretch A on 11“ vs. values of the fiber stretch «FIE on IT as given
by S = 0 .................................................................................. 98

Figure 33. These plots are just a magnification of the plots in Figure 31 in a 5-10% of
admissible deformation. In all cases, although it is further appreciated as 7
increases, the angle 0, either for 03“,, or (95:0, as A decreases (compressive
load), shows a first stage of slow rotation, followed by a second stage of rapid

rotation, and a third stage in which the rotation is much slower than in the
previous stages ......................................................................... 100

Figure 34. Kinematics relation between the vectors L and N defining respectively the
shock direction and its normal in the undeformed configuration and the

vectors land n defining respectively the shock direction and its normal
in the undeformed configuration .................................................... 106

Figure 35. Coordinate systems Y and 2 used to represent the tensor G ..................... 108

1. INTRODUCTION

Fiber reinforced materials exhibit nonlinear behavior during service due to the
properties of the constituents phases, as well as the interaction between these phases. A
mathematical formulation that might address this nonlinearity is called hyperelasticity, which
is the theory of non-linear elasticity for hyperelastic materials whose elastic potential energy
is described by a strain energy function. The strain energy function serves as the
constitutive equations, i.e., as the mechanical behavior model of the material.

During the last few years the mechanical behavior of non-linear anisotropic
materials has been of special interest, due to the increasing use of these materials, not only
in the aerospace industry, but also in the automotive industry and the modeling of human
body organs. One of the most important analysis in these researches concern the study of
instability of non-linear orthotropic materials subjected to loads aligned with their principal
directions. It has been observed that most of these solids fail at different strain levels leading
to a sudden structural rearrangement. In particular, there is a great deal of interest in
determining the instability of non-linear transversely isotropic materials subjected to axial
loads, since most of the applications involve these configurations.

The loss of stability can either be of a local or a global character. A local instability,
also called a material instability, refers to one that occurs at a point in the body once a
certain critical state of stress or strain has been reached. It is associated to the so called
shear band-type failure mode. The loss of ellipticity of the governing differential equations
is a necessary condition to get these instabilities. A global instability, also called a
geometrical one, is associated with a buckling type of failure which depends on the details
of the boundary conditions.

Here, our concern focuses on material instabilities for a kind of incompressible
hyperelastic material: reinforced neo-Hookean. The reinforcement is chosen so as to render

the material a transversely isotropic character.

As it has been mentioned, material instabilities are known as shear band type
instabilities. These zones of localized deformation, in the form of narrow shear bands, are a
common failure mechanism under compressive principal stresses. That localization can be
seen as an instability in the macroscopic constitutive description of the material. Whence,
instability is understood in the sense that the constitutive relations allow the homogeneous
deformation of an initially uniform material lead to a bifurcation point at which non—uniform
deformation can be created in a band under conditions of equilibrium. The general
theoretical framework for these shear bands was given by Hill [22], who investigates them
in the particular case of a stationary acceleration wave. With the purpose to obtain a realistic
model able to predict the critical bifurcation stress (strain) into a shear band, different
constitutive equations were analyzed. The conditions for bifurcation into a shear band were
studied for isotropic materials in a variety of contexts. Isotropic elastic solids and elastic-
plastic with a smooth yield surface were predicted not to develop shear band instabilities
Rice [23]. Within the theory of plasticity with kinematics hardening, Hutchinson &
Tvergaard [27] and Anand & Spitzig [28] in plane strain; and Hill and Hutchinson [26], in
plane strain uniaxial tension, made the most important theoretical contributions. In the
context of inelastic behavior, applied to overconsolidated clay soils, the investigation was
carried out by Rudnicki and Rice [24], and Rice[25].

All these approaches to the characterization of shear band formations were based on
isotropic behavior of the material model. Clearly this is not a phenomena found only in
those materials. For a fiber reinforced material this instability is usually called fiber kinking.
In the early works, Rosen [29] studied the discontinuity as an internal buckling of the fiber
reinforced, i.e., the kinking of the fiber. The mechanism of kinking was faced as an elastic
shear buckling giving results not meaningful to most composites. Later fiber misalignment
and plastic yielding of the matrix [30], were incorporated to the analysis. The model was
further improved to incorporate combined stress loading [31],[32]. This has been a major

contribution in the understanding of fiber kinking mechanism, and has allowed to predict

realistic ranges for kink band angles: [33], and [34] in fiber carbon epoxy composites, [35]
in ductile matrix fiber composites, [36] in wood at the fiber level, and [37], [38] for an
advanced fiber reinforced composite. In particular these experimental studies focus not only
in the prediction of compressive strength, but on the propagation of the kink band.
Nevertheless, whether a kink band starts from a well-defined initial band of wavy fibers
(imperfection due to misalignment) or from a deformation induced band (localization) is not
clear. In an attempt to provide a theoretical framework that gives the features of kink band
formation, the kink band has been dealt as a localized deformation, allowing for the use of
standard elastic-plastic constitutive equations of the constituents [39], [40] [41].

The approach to these instabilities has not had the continuum mechanical treatment
of non-linear elasticity. In this case fiber kinking, shear band formations and elastostatic
shocks are equivalent failure modes that develop due to the loss of ellipticity of the
incremental equations describing homogeneous deformations. In connection with
asymptotic studies of crack problems, Knowles and Stemberg (1975) found that the field
equations of non-linear elastostatics may suffer a loss of ellipticity in the presence of severe
local deformations. It was shown for a general compressible hyperelastic material in plane
deformations that this loss of ellipticity is a necessary condition for the emergence of
solution fields lacking the smoothness required by the differential equations [5]. Abeyaratne
showed it for an incompressible one in plane deformations too [8]. The analogous three-
dimensional results were established by Simpson & Spector [10] for compressible isotropic
solids, and by Zee & Stenberg [7] for incompressible isotropic ones. Since then,
constitutive equations that suffer a loss of ellipticity have been studied in a variety of
contexts [3], [4], [6], [14], [15]. In particular, the loss of ellipticity for the reinforced neo-
Hookean material was analyzed by Qiu & Pence [2].

Among the new solutions that emerge due to the loss of ellipticity, of particular
interest are those that involve a continuous displacement field but a discontinuous

deformation gradient. These solutions are called in the context of non-linear elasticity

elastostatic shocks (or elastostatic two-phase states). The term elastostatic shock is used in
analogy to the discontinuities found in fluid mechanics, denoted in that context by shocks,
although here the elastostatic shocks do not move as waves. The related problem of
existence of discontinuous deformation gradients involves two issues. The first focuses on
the characterization of the elastic potentials exhibiting more than one phase. The second
deals with the elastostatic shocks given by a particular elastic potential.

Both issues have been successfully analyzed in the case of isotropic incompressible
materials in plane deformations [9]. Necessary and sufficient conditions were established in
terms of the so called shear stress fimction. It was shown that an elastostatic shock is to
exist when the elastic potential, as a function of a defined amount of shear, losses convexity.

Rosakis deals with these issues for a compressible hyperelastic material in three-
dimensional deformations [ll]. Necessary and sufficient conditions are derived for the
elastic potential to sustain elastostatic shocks in a special case in which the deformation
gradient is a symmetric positive-definite tensor. The elastic potential is expressed as a
function of kinematics variables which arise from the directional resolution of the assumed
deformation gradient. The concept of ellipticity is related to the invertibility of this particular
form of the elastic potential. Further, this loss of invertibility is related to the existence of the
elastostatic shocks sustainable by the elastic potential. Rosakis and Jiang [12] look upon
this result for compressible isotropic materials in plane deformations.

In a different approach, based on a variational point of view, Ball [16] establishes
that the class of elastic potentials capable of sustaining elastostatic shocks are the so called
strictly rank-one convex stored energy functions. He shows that the elastic potential is
strictly rank-one convex if and only if various conditions hold, one of which is that the
elastic potential does not sustain two-phase states.

We are studying an incompressible anisotropic finite elastic material, extended from
the neo-Hookean one, by accounting for the effect of uniaxial reinforcement in what will be

referred to as the fiber direction. The neo-Hookean model gives rise to partial differential

equations that are elliptic at all deformations, therefore, no “weak” solutions, i.e., solutions
in which the smoothness requirements are released, are to be expected. In recent
papers[1],[2] the reinforced neo-Hookean material was analyzed. In [1], the mechanical
response of this material was considered under homogeneous deformations. In [2], the
conditions under which the governing differential equations lose ordinary ellipticity were
established in the context of plane strain homogeneous deformations. It was shown that the
loss of ellipticity requires contraction in the fiber direction.

In the present study we focus on the local issues related to the so called elastostatic
shocks, i.e., on the local structure of such shocks. After a brief description of the theory of
finite elastostatics for incompressible materials in Section 2, and of the material model in
Section 3, a complete kinematics analysis of a plane elastostatic shock is made in Section 4.
In Section 5, the case of neo-Hookean materials is analyzed and it is shown that no
elastostatic shocks are to be found. The existence of a plane elastostatic shock is analyzed
for the reinforced neo—Hookean under plane deformations in section 6, 7 and 8 . Some
general results are established based on the parameter which characterizes the anisotropy of
the augmented neo-Hookean model. The kinking of the fiber associated to any elastostatic
shock direction is presented. In Section 9 the characteristics of these elastostatic shocks are
studied. In particular, it is shown that the existence of such elastostatic shocks require fiber
contraction. Further, it is shown that the necessary condition to obtain such elastostatic
shocks is the loss of ordinary ellipticity at some deformation. In Section 10, an energetic
analysis that accounts for the existence of these elastostatic shocks is carried out. The use
of a “dissipativity inequality” derived in [7] allows to rule out as inadmissible some of the
elastostatic shocks obtained in Section 6. In analogy with the shocks in gases, it is shown
that the side of the elastostatic shock with the higher hydrostatic pressure remains stable,
i.e., the elastostatic shock always moves with respect to the material, from regions of higher
hydrostatic pressure to regions with a lower hydrostatic pressure. Furthermore, in analogy

with the fiber kinking mechanism, it is shown that the energetically favored state is that one

which involves either elongation or less fiber contraction. In addition, maximization of the
“dissipativity inequality” is proposed as the supplementary condition to obtain one
physically admissible solution among the many possible ones. This extra condition would
allow to model the kink—band broadening mechanism.

Our discussion in this work is related to the local existence of homogeneous
discontinuous deformation gradients. There is a great deal of interest in determining the
global existence of these solutions and here we just want to denote some important results.
In particular, Gurtin [17] proves that a non-trivial pairwise-homogeneous defamation of a
bounded region can not have homogeneous boundary-values. Hence, if the boundary-values
of a two-phase deformation are homogeneous, then the deformation itself must be
complicated. Further, Pitteri [18] has shown that for a half plane under uniaxial loading and
a free lateral surface, such pairwise-homogeneous deformation do not exist. The argument
is based on the impossibility of satisfying simultaneously the traction-free lateral surface

and traction continuity across the shock.

2. PRELIMINARIES ON FINITE INCOMPRESSIBLE ELASTOSTATICS
Let R be the region occupied by a body in its undeformed configuration. A defor-
mation of that body is represented by a sufficiently smooth and invertible transformation
y=y(x)=x+u(x) on R, (2.1)
mapping R onto a domain R*, where R* represents the deformed configuration of the
body. Therefore, x is the position vector of a particle in the undeformed configuration, y is
the position vector of that particle in the deformed configuration and u is the displacement

vector of the particle. The deformation gradient tensor F is given by

= % on R (2.2)

Since the material is incompressible, the deformation is subject to:

det F :1 on R . (2.3)

Denote 1: as the Cauchy stress tensor accompanying the deformation. In the absence of
body forces, the equations of equilibrium are

divy t = o 1: = J on R *, (2.4)
the Piola-Kirchhoff stress tensor, 6, corresponding to t is

o = 1 FT, (2.5)
and the equilibrium equations in terms of this stress tensor are

div o=0 0FT=F0T on R . (2.6)
If the material at hand is homogeneous and elastic, with an elastic potential W = W (F),
then the Piola—Kirchhoff stress tensor is:

o = w F(F) — p F'T (2.7)

in which p(x) is the scalar pressure field arising from the incompressibility constraint and

awar)

WF(F)= (9F

3. THE REINFORCED NEO-HOOKEAN MATERIAL MODEL
Consider an incompressible anisotropic material model extended from a neo-
Hookean one by accounting for the effect of uniaxial reinforcement. The associated strain

energy density per unit undeformed volume is taken to be given by:
1 Y 2)
W: —I—3 +—K—1 3.1
u (2( i ) 2( ) ( )

where 1,: tr (C), C = FT F, «fl—(— gives the fiber stretch. For instance; if i1 gives the
direction of the reinforcement in the undeformed configuration, then

K =Fi, o Fil = in Ci1=C11
Herep > 0 is the shear modulus of the base neo-Hookean response and y > 0 is the
reinforcing strength parameter in the fiber direction. The principal stretches will be denoted

by Ai (i = l, 2, 3), and the fiber stretch s/E may or may not be a principal stretch.

Let A give the orientation of the fiber direction in an undeformed reference
configuration. The Cauchy stress corresponding to (3.1) is given by

t=-pI+pFFT+2uy(FA-FA-1)FA®FA. (3.2)

Note that y = 0 retrieves the neo—Hookean response, while the limit y ——> 00 corresponds to

an inextensible material. A detailed analysis of the strain energy (3.1) that defines the

behavior of this transversely isotropic incompressible nonlinear elastic is given in [1].

4. ELASTOSTATIC SHOCKS

In the derivation of the field equations of incompressible elastic materials, the
displacement field u(x), the pressure field p(x) and the stress field 6(x) (or t(x)) are
assumed to satisfy certain smoothness conditions. If these conditions are released, then the
possibility to find solutions satisfying the governing partial differential equations with
different smoothness properties is opened. These new solutions may involve a continuous
displacement field but a discontinuous deformation gradient, i.e., discontinuities in the first
derivatives of the displacement field. These solutions are of physical interest since it has
been observed that the failure of many materials is related to these configurations. For
instance, one of the most important failures in fiber reinforced composite materials involves
the kinking of the reinforcing fibers. This phenomenon has been observed at different
conditions of loading. To determine these states for a material obeying (3.1) is a primary
goal of this analysis.

The mathematical formulation of this problem is as follows: suppose there exists a
solution to the system of differential equilibrium equations (2.3) and (2.6) such that p, F
and o satisfy the usual differentiability requirements, except at a smooth surface S in R
across which some or all of these quantities are discontinuous. The displacement field u is

assumed to be continuous across S. The image of S in the deformed configuration R* will

be denoted by S*. Let p+, 0+ and F+ give field values on one side of S,

 

and p—, 0'— and F- give the field values on the other side. Global equilibrium of forces
gives:
div 0 = 0 onR \S (4.1)

[01:N = o onS (4.2)

in which [0]: = (0)+ - (0)_ is the discontinuity suffered by the variable upon
traversing the surface S, and N is the normal vector to S. From the equilibrium of moments:
y(x)x[o]:’N=0 (4.3)
is trivially satisfied if (4.2) is satisfied.
The surface carrying the discontinuities of the pressure field p, the deformation
gradient F, and the stress tensor 6 is called a strong elastostatic shock. We will refer to it as
a spatial elastostatic shock when viewed with respect to the deformed configuration, and as a

material elastostatic shock if it is viewed with respect to the undeformed state.

There is also a notion of a weak elastostatic shock. This involves discontinuities in

2
a—anda u

a; 32;

otherwise, the term elastostatic shock shall refer to a strong elastostatic shock. In addition, a

where C is a coordinate that is in the direction of N. Unless stated

 

notion of shock strength will be introduced such that strong elastostatic shocks tend to a

weak elastostatic shock limit as the shock strength tends to zero.

4.1 PIECEWISE HOMOGENEOUS PLANE ELASTOSTATIC SHOCKS

In the local study of the existence of an elastostatic shock, it is sufficient to consider
the case in which the surface S is a plane in R and both the deformation gradient and the
pressure are constant on each side of the shock plane. We restrict attention to plane

deformations and follow [7] in our formulation.

Since the theory of finite elasticity involves a distinction between the geometry
of the deformed and undeformed configuration, let us introduce some notation to
characterize both states. Let X = (XI, x,) be a fixed rectangular coordinate system describing
the plane deformation and {i,,i2} be the corresponding orthonormal basis. The out-of-plane
direction is described by the coordinate x3 with unit basis vector i3, where i3 lies in the
shock plane S.

The plane deformations of interest involve displacement fields of the form:

u = ul (x1, x2) il+ u2 (x,, X) i2
whence, by (2.1) and (2.2)

F13 = F23 = F31: F32 = 0, F33 =1 (4.4)
The intersection of the elastostatic shock plane S with the (x1, x2)-plane of deformation
defines a shock line in the undeformed configuration. Let L be the unit vector along this
line. Let N be the unit normal to S obtained by a counterclockwise rotation of L (see Figure
1), then:

L = L,i1 + in2 and N = Nlil + Nzi2 = -inl + Lli2

L1 = cos (p and L2 = sin (p. (4.5)
Let IT and IT give the two open half planes into which L divides the (x,, x2)-plane, where N

points from IT to IT. Consider the piecewise homogeneous plane deformation:

y=F+x on 11+

y = F_ x on H—
where F‘ and F‘ are each subject to (4.4) and (2.3). Further, let p” and p' be the associated

pressures on IT and II" respectively.

10

 

X2 y2

A L
Deformation
———>

+ material
II shock

  

fiber

 

 

 

 

 

Figure 1. Kinematics of plane piecewise homogeneous elastostatic shocks. The
undeformed configuration is described by the rectangular coordinate system X = (x,, x2),
while the deformed body is referred to the rectangular frame Y = (y,, yz). The elastostatic
shock separates the plane into two open half planes: a “+” side with a deformation
gradient F‘ and a "-" side with a deformation gradient F'. The normal vector to the
elastostatic shock, denoted as N in the undeformed configuration (1) in the deformed one)
is taken to point into the “+" side.

The deformed configuration of the body is referred to the frame Y = (y,, y), where
the direction of the shock is defined by the unit vector l, and its normal given by the unit
vector n. Considering the orthonormal basis {e,, e2}, 1 and n can be expressed:

l= llel + lze2 and n = n1e1+ nze2 = -lzel +1,e2

11 = cos 0t and 12 = sin (X. (4.6)
The kinematics study of the elastostatic shock for incompressible materials gives a relation
between the normal directions to the shock in the undeformed and deformed states
(Appendix A):

l

= F'TN or N: | FL | FTn (4.7)
l FL |

 

which in turn gives:

11

 

F21 + F22 tan (0

tana = .
F11+ Flztantp

(4.8)

Here the values F2" F22, FH and F12 can be evaluated (as a set) on either H” or IT. The

Piola-Kirchhoff stress tensor on IT and IT are
o+ = w F (17*) - p+ F“T on 11*, (4.9)

o‘ = WF (F') - p‘ F“T on H‘. (4.10)
The displacement continuity requirement implies:

F+x = F‘ x. (4-11)
Since (4.11) is enforced along the shock defined by L, its unit vector in the undeformed
configuration, it follows that

F+L=F'L= IF+LII= IF'Lllsh (4.12)
where l is given in (4.6). Note that (4.12) with (4.6) implies that (4.7) is well defined with
respect to both F = F‘ and F = I". The traction continuity requirement by means of (4.2),

(4.9) and (4.10) gives in indicial notation:

+ _
aw (F )_ 1,.th NB: ”WV—(Fl- p—FE; Nfl, ;a,f3=1,2; (4.13)

or in component notation, in the i1 direction:

aw F+ - aw F+ -
( )'P+F1+11Nl+ ( )-p+F2+11N2=
3 Fll
‘2 (4.14)
3W(F‘) _ _-1 aW(F') _ _-1
= _—'P F11 N1+ —'P F21 N2
8F 8F

11 12

and in the 12 direction:

aw F+ - aw F+ -
( )'P+F1+21N1+ ( )-p+F2+21N2=
8 F21 22
(4.15)
aW(F‘) _ _-1 awar‘) _ _-1
= —'P F12 N1+ _'P F22 N2:
81:21 81:22

Traction continuity in the i3 direction perpendicular to the (x1, x2)-plane is trivially
satisfied if the strain energy W(F) ensures
a w (F) =
3 F3 0

0 0t: 1, 2; (4.16)

for F obeying (4.4). In the study of plane elastostatic shocks (4.16) is taken as an
assumption for planar F. It holds identically true for isotropic materials, and further, for the
strain energy (3.1).

Let us focus on the displacement continuity requirement. Since det F+= detF‘: I,

introduce G such that:

G = F‘ F+-l or F‘: G F“ 2 det G=1 (4.17)
The continuity requirement (4.12) is:

F+L= GF+L or h=Gh (4.18)
Consider a coordinate system located at the elastostatic shock in the deformed
configuration, i.e., one whose axes coincide with the direction of the shock I and the normal
n. Since the vector h is aligned with the elastostatic shock, (4.18) can be expressed in this

new coordinate system as:
bl = Gll G12 h1 (419)
0 G21 G22 0 '

from which it follows:

0: 02, hl, thus 02,: 0.

Further, since det G = 1, we conclude that G22 = 1. Hence, the form of the matrix G in this

particular frame is:

_ 1 G12
G- [0 1 ]. (4.20)

By (4.17) the deformation on IT, defined by E, can be viewed as the deformation of IT,
defined by F’, followed by a simple shear of amount G12 parallel to the elastostatic shock in
the deformed configuration. In view of this physical interpretation G12 is now rewritten as k,
commonly used in [l], [5], [7], [8] to denote amount of shear. Note that G is generally
expressed as (see Appendix B):

G = (1+ kl ® 11). (4.21)
Since k is a measure of the difference between both deformation gradients, k is also called

the strength of the shock. Strong elastostatic shocks become weak elastostatic shocks in the

k—>0 limit, in the sense that the discontinuities in the pressure and deformation gradients

. . . . . . 8 8 2 u
are not sustained, but discontinuities 1n _p_ and

at «92;

are sustainable in this limit.

 

Regard now the state on IT as fixed so that the constant tensor F‘ and the constant
scalar p+ are given. The problem of whether or not this state is consistent with a piecewise
homogeneous elastostatic shock can be postulated as follows: given F’ and p”, one seeks
three scalar values p', or and k, such that the traction continuity requirement (4.13) is met.
Here or is the angle that defines the direction of the elastostatic shock in the deformed
configuration (see Figure 1), and k determines the jump in the deformation gradients across
the shock. The deformation gradient F" will be given by (4.17) and can be written as:

F‘ =(I + kl® n)F+ (4.22)
Triplets {p—, 0t, k} with k ¢0, that satisfy (4.13) provide a mechanically consistent
elastostatic shock state for the given F’ and p”. Since the traction continuity requirement

constitutes two equations for three unknowns (p‘, or, and k) the existence of a mechanically

consistent elastostatic shock will generally ensure the existence of a one—parameter family of
such states [8].

We henceforth restrict attention to deformation gradients F+ that are diagonal in the
{i,, 12, i3} frame. The extent to which this enables more general conclusions is the subject of

Appendix C.

4.2 SPECIALIZATION TO DIAGONAL DEFORMATION GRADIENTS ON ONE
SIDE OF THE ELASTOSTATIC SHOCK
It is assumed that the deformation gradient F“ and the pressure p+ are known on IT,

where it is further assumed that F+ is diagonal:

A 0 0
17+: 0 r1 0. (4.23)
o o 1

The incompressibility constraint (2.3) has been invoked to express F” in terms of the single

stretch A. The components of the deformation gradient F given by (4.22) and (4.23) are
Fl‘l =A (1+kll n1)=A (1 - kcosa sin or)
F1} = A "1 kllnz = A '1k(cos 0:)2
F51 =A k12n1= - A k(sin()t)2
F23 = A -1(1+k12 n2): A —1 (1 + ksin a cosa). (4.24)

Note that F" can not be diagonal in the {i,, i2, i3} frame unless k = 0, in which case F‘ = F

(a weak shock). Equation (4.7) yields

 

 

. N1
nl=—srna= 2 2 2 2
,1 Jr N2 + 71' N1
1N2
n2= cosa= (4-25)

22 -22
(p N2+A N1

15

 

where N, = — sin (p and N2 = cos (p. According to (4.8) the angles (p and or are related
through A via
tan or = A I 2 tan 0). (4.26)

The right Cauchy-Green strain tensor C = FT F on IT is

712 0 0
0+ = 0 71—2 0, (4.27)
o o
andonII‘is
22(1+2kn,n2+k2nf) 1t(n§.n,2)+1t2n,n2 0
C‘= k(n§—n%)+k2n,n2 A_2(1-2kn,n2+k2n%) 0. (4.28)
0 o 1

Thus, although {i,, i2, i3} is a principal frame vector basis for IT, it is generally not such a

basis for IT.

5. NEO-HOOKEAN MATERIAL

In this section it is shown that a neo—Hookean material can not sustain elastostatic
shocks in planar deformation. This in fact is well known, and Abeyaratne and Knowles have
an elegant characterization in terms of the so called shear stress response function to
determine whether or not elastostatic shocks can be sustained in an arbitrary isotropic
incompressible material [8]. Their procedure could be applied to the neo-Hookean model to
verify shock nonexistence. Nevertheless and with the purpose of having a complete
understanding of the problem we are facing, we will develop a solution following the
procedure explained in Section 4.1 applied to a diagonal F”.

Recall that the neo-Hookean model involves a strain energy of the form (see Section

3):

w =§(1,—3) (5.1)

 

The strain energy associated with F+ from (5.1) and (4.23) is
W F+ =fl 12+r-2—2 5.2
( ) 2( ) < >
the Piola-Kirchhoff stress tensor associated with this strain energy, by (4.9) and (4.23):

01+1= 11A -p+A'l

 

0'22: [AA—l -p+A
01+2 = 0'21: 0 (5.3)

Consider now the other side of the elastostatic shock, in which the deformation gradient is

F'. Recall that it has the form:

F,‘2 0

F = F2, F22 0 .
0 0 1

(5.4)

Its inverse, using the incompressibility requirement for this deformation, is:
-1 F23 -F,3 0
F = — F2, F,l 0 .
O 0 1

(5.5)

The strain energy associated to this side of the elastostatic shock, by (5.1) and (5.4):
w- (F‘) =% (Fl—,2 + F132 + F2‘12 + 13;,2 — 2) (5.6)
so the Piola-Kirchhoff stress tensor, by (4.10) using (5.4) and (5.5):
01—1 = F‘ Fr—r ' P_ F22
“1’2 = 1‘ Ffz + P‘ F21
02—1 = 1‘ 1221* P_F1—2
02—2 = [1 F23 - p‘ Ff, (5.7)
The traction continuity requirement along the elastostatic shock, given in indicial

notation by (4.13) defines two equations, corresponding to the two in-plane axes of the

coordinate system X in the undeformed configuration. The equation (4.14) for the direction
x, is:

( “1+1' 01—1)N1' 01—2 N2=0 (5-8)
and (4.15) for the direction x2 is:

Substituting from (5.3) and (5.7) into (5.8) and (5.9) gives respectively:

([1 A - p‘LA'l -(p F,_, - p—FfiD N1 — ([1 F13 - p‘F§,)N2 = 0, (5.10)
(it 1-1— n+1 — (it 132—2 — p‘FfiDNZ — (it Ff, - p"F,'2)N1= o. (5.11)
Upon use of (4.7) and (4.24), it is found that (5.10) and (5.11) reduce to:
_ + 2 2 —2 2
(p-p )n,-rtkl,(A n,+A n2) =0 (5.12)
(- +) kl 7127- 71—22 —0 513
p-p n2-” 2 n,+ n2 —. (-)

Since I and n are perpendicular unit vectors, the sum of the product of (5.12) with n, and
(5.13) with n2 shows that:

p‘ = p+. (5.14)
Whence equations (5.12) and (5.13) yield:

hk1,(/tznf+/1'2n§) =0 (5.15)

nk12(22nf+71‘2n§) =0 (5.16)

Clearly, since u it 0 and l is a unit vector, satisfaction of both equations (5.15) and
(5.16) require that k must vanish. Thus, we conclude that neo-Hookean materials can not

support elastostatic shocks in the present circumstances.

- .9
.' ‘

6. THE PLANE ELASTOSTATIC SHOCK FORMULATION FOR REINFORCED
NEO-HOOKEAN MATERIALS
We now turn to (3.1), the strain energy associated with this material. The effect of

directional reinforcement is our concern in the analysis of the possible existence of

elastostatic shocks for this model. Recall that s/E in (3.1) gives the fiber stretch, and it may
or may not be a principal stretch. If we are to assume, as was done in the neo-Hookean case,
that the known deformation gradient F” is in its principal directions, they may or may not
coincide with the direction of the fiber. Therefore, to face this new problem we should
include the angle that the fiber makes with the principal directions of the deformation.

We only consider the case where the fiber direction is given by i,. This is a principal

direction for the deformation gradient F’ on the IT side of the elastostatic shock. On this IT
side the fiber is not rotated and the fiber stretch s/K is given via:
. . . . 2
K+ =F+110F+1, = 110Cf11, =Cf,=A (6.1)
The strain energy associated to this deformation in II” by (3.1) , (4.23) and (6.1) is
2
W+(F+)=%(A2+A—2—2) +121(,12—1) (6.2)
the Piola—Kirchhoff stress tensor by (2.5), (3.2) and (4.23) is
01+1= it A - pJ'A‘1 + 2rtyA (A2 -1)
03-2: M 27.1- p+l
of} = 03’, = 0 (6.3)
In general, the direction 1, will not be a principal direction in II'. Since the direction
of the fiber in the undeformed configuration is given by i,, the value of K on IT is given via

K- = F'i, . F'i, = i, . Cf, i, = C1]: F,-,2+ F232 =
(6.4)
= A2(1+ 2kn,n2+k2nf)=A2(l — 2ksinacosa+ kzsinza).

l9

Further, let (1) be the angle that the fiber makes with the direction y, in II; due to the

deformation gradient (4.24). This kinking angle is given by (Figure 2)

F‘ ' 2
tan¢ =_Z-i=_____k(s1na). . (6.5)
F,, l- kcosasrna
The strain energy on IT is
_-#_2 -2 -2 -2 #Y _2 -2 2
= %(712(1 + ran? + 2kn1n2)+A'2(1 + kzng — 2kn,n2)— 2) + (6.6)

+%(A2(l + kznf + 2kn1n2)-1)2.

Using (2.5), (3.2), (5.4) and (5.5) the components of the Piola—Kirchhoff stress tensor on
IT are
_ _ _ _ _ _2 _2
c’1‘2 = # 1:12 + 9‘ F23
_ _ _ _ _ _2 _2

022 = # F22 ' P" Fri (6-7)

The traction continuity requirement across the elastostatic shock again gives
equations (5.8) and (5.9). The parameter y, which characterizes the transverse isotropic
material at hand, appears only in the 0",], 02—,, of“, components of the IT and II‘ Piola-
Kirchhoff stress tensor. Note that these particular components drop out of equations (5.8)
and (5.9) in the event that N1 = 0. Therefore if N,=0, then (5.8) and (5.9) collapse to the
conditions governing the neo—Hookean case which has been shown to be incapable of
sustaining such elastostatic shocks. Whence, in the present circumstances a necessary

condition for the existence of an elastostatic shock is that Nl ¢ 0. Since N represents the

20

 

normal to the elastostatic shock in the undeformed configuration, we can conclude that the
elastostatic shock can not contain the fiber, i.e., it can not be aligned with the fiber in the
undeformed configuration. Furthermore, the kinematics relation (4.7) shows that a vector
aligned with the x2 axes (undeformed) will, under the deformation described by F*, deform
into a vector aligned with y2 axes (deformed). Therefore N, r: 0 implies n, at 0. Now on
III the fiber is kept in the e, direction, hence the elastostatic shock in the deformed

configuration can not contain the fiber either on the II: side.

 

   

   

 

material Spatial
shock shock
II _
II- * Hart
fiber fiber
reinforcement reinforcement ¢
undeformed deformed

 

 

 

Figure 2. Kinematics of the fiber reinforcement in the presence of an elastostatic shock.
The orientation of the fiber reinforcement is preserved in II: due to a diagonal
deformation gradient F“ aligned with the fiber. The deformation in HI, characterized by

F“, does not preserve the orientation of the fiber, and gives a kinking angle denoted by 0.
The angles 01 and 0 can not coincide.

Analysis of (6.5) shows that the elastostatic shocks can not involve the fiber on IT, Le.

a ¢ d). To see this note from (6.5) that if k = 0 then 6 = 0, in agreement with a state in

21

which both deformation gradients F and F” are equal and given by (4.23). More generally,
setting 00:4) in (6.5) yields tan or = 0, so that or = 0, which has been ruled out. Thusa at o.
This result agrees with [2] where it was shown that “weak” elastostatic shocks (i.e.,
k —) 0) can not involve a normal perpendicular to the fiber. Now a similar result has been
shown to hold for “strong” elastostatic shocks provided that Fw is of the diagonal form
(4.23). Therefore, the direction of the shock can not coincide with the direction of the fiber
either on III or on III. This kinematics result limits the range of the angles or and (b in the
event of elastostatic shock existence.

After this observation, and as was done in the neo—Hookean case, in the equations
given by the traction continuity requirement, let us introduce the values of the components
of the different Piola-Kirchhoff stress tensor in terms of the different components of both

deformation gradients. In the direction x,, (4.14) gives:
(it A - p+A'1+ ZuyA (A2 -1))N,—
—(,uF,_,-p‘F2—2+2uyF,—,(F,_,2+F2_,2- 1))N,— (6.8)
-(/.t Ff, + p'Fz_,)N2= 0

while in the x2 direction (4.15) gives:
(uA-l_p+A-(pF2_2-p"Ffi))N2- 69
-(MFZ‘,+p‘F,‘2+2rtyF2‘,(F,',2+F2’,2-1))N,=0 (J

These equations generalize (5.10) and (5.11), and reduce to them when y = 0.
Introducing now the relations (4.24) and using (4.6) to express the equations in the
deformed configuration, the equation (6.8) becomes:
(p'-p+)n,- rtkl,(A2 nf+A'2 n%)- (610)
-2 2 - 2 2 2 2 3 2 2 _ '
-2pyA n, (k1, (3,1 —1) +k A n,(1+21,)+ k ,1 n, 1,) _ 0

and the equation (6.9) yields:

22

(p‘-p+)n2- uk12(A2 n,2+/'L'2 n2)- 611
-2)”, 712 n,2 (k12 (,1 '2 —1) +k22A2n, 1, 12+ 1822 nf12)= 0 (' )
These equations are the y at 0 generalizations of (5.12) and (5.13).

Now multiply (6.10) by n,, multiply (6.11) by 112 and add these results. Then, using

the geometry relations (4.6), and since I and n are perpendicular unit vectors, one draws:
p—-p+=2iiy/t4nf(2kn2 +k2 n,) (6.12)
This expression relates the pressure on both sides of the elastostatic shock. If we regard p”

and A as given, then to determine p' it is still necessary to know the elastostatic shock
direction 11 (via 00 and the strength k of the shock. Note also that (6.12) in the weak shock
limit k—)0 gives p‘ = p+ + 4p)! A 4 n? n2k+O (k2). This in agreement with [7]
where it was found that the pressure jump across a weak shock is linear in k to leading
order.

Substituting from (6.12) into (6.11), eliminating u, and expressing the result in
terms of the components of the normal vector to the elastostatic shock in the deformed
configuration via (4.25) yields:

kn,(A2 n,2+ A'2 11%) -
2 3 2 2 2 2 2 3 2 2 _

-2yA n,(k (-21 2—2. +l)-k 3,1 n,n2—kA n,)—0
One solution of this cubic equation in k is immediate; namely k = 0. This is a trivial solution
as it corresponds to the absence of an elastostatic shock. Factoring out this root and
grouping the result as a quadratic equation in k, gives

P(A,n,,n2, k, y)=0, n,=-sin 0t, n2=cos01 (6.13)
where

P(A,n,,n2,k,y)= kzn, + k3n2 -

-W(2y12nf(-212n§— 712 +1)—(7t2 nf+ 14%)). (“4)
H1

23

Here division by n, is allowed since the possibility n, = 0 has been ruled out. In summary:
for fixed material parameter 1!, the deformation on IT " characterized by A via ( 4.23) gives
rise to a mechanically consistent elastostatic shock state if and only if there exists real
numbers k and a such that (6.13) is satisfied for n, = -sin a and n, = cos 0!.

Before embarking on a more general discussion we consider two special cases. The
. 7t .
first case 1S that of an orthogonal shock (a = $3). The second case 1s that of a “weak”

shock (k —) 0).

Consider first the case of an orthogonal shock meaning that the elastostatic shock in

the deformed configuration is orthogonal to the fiber (or = :25). This, in turn, implies that

(p = i; by (4.26). Introducing n,= i1 and n2: 0 into (6.13) and (6.14), one finds that

the strength of the shock k is given by
1 1 7:
k: i — 1—— —l, a=i—. 6.15
A2[ 2y] [ 2) ( )
. . l
Th1s requires 3! > §,and

0 <1. 3 ,i (y), A (y); 1-——. (6.16)

By virtue of (6.5), the kinking angle (I) is 4) = tan'1 k. The deformation gradient on IT by

(4.24) is
A (1) 0
F' = — k A — 0
A
0 0 I

and the fiber stretch on IT given byVK‘ in (6.4), upon use of (6.15), yields K":

1/C‘ = l- L = A (y). In particular, an orthogonal shock involves variable fiber
11 2 Y

24

contraction A on IT obeying (6.16) and constant fiber contractionA (y) on IT. Note that
although the contraction on IT is independent of A (and hence k), the kink angle 4) will

indeed vary with A (and hence k). Furthermore, by (6.5) the kinking angle yields d) = tan'1 (-

k). Whence, values k > 0 are related to angles 0 obeying —% < 4) < 0 while values k < 0 are

 

related to angles 4) obeying 0 < (i) < %. In particular, (1)) < |a| = % since by our discussion

of (6.5) the angles or and t) can not coincide. Figure 3 represents the values A vs. k given by

equation (6.15) for y =3 and y = 100.

 

 

 

I:
Figure 3. Values (A, k) corresponding to shock angles a = i;- for the values of the

reinforcing parameter 7 =3, 100. The basic form of these curves is the same regardless of
the value of the reinforcing parameter y. The curves are symmetric with respect to k = 0,

. 1
as given by (6.15). In particular at k = 0, A = A (y) :- 1-2— , for each y-value.
1’

These orthogonal shocks involve fiber contraction on both sides of the elastostatic shock.
Further, as the fiber is contracted on IT. the fiber on IT keeps the constant stretch given
byA (y), eventhough rotating with k through an angle 4) = tan" (-k) from its initial
“unkinked” direction 0 = 0.

25

We now turn to consider the separate case of a weak shock. Letting k —90 in

equation (6.13), yields
2 2 2 2 2 2 2 ——2 2 _

2M n,(-2A 2—71 +1)—(,1 n,+A n2)—0. (6.17)
Equation (6.17) upon use of (4.25) shows
71 2N§+2y (3714-2 2)NfN§ + A 4N? + 2y (1-). -2)N;‘ = 0. (6.18)
This equation coincides with equation (4.6) in [2] - particularized for (4.27) - which gives
the loss of ordinary ellipticity for the governing partial differential equations in global plane
strain on IT'. Again, this is in agreement with [7], where it was shown that a necessary
condition for the existence of a weak elastostatic shock is the loss of ellipticity of the
displacement equations of equilibrium at the given constant tensor F+and constant scalar

p+. For the reinforced neo-Hookean material under study here, it follows from [2] that

ordinary ellipticity is lost if and only if y > g, and O < A S A (y). These are the same

conditions that were obtained for the case of an orthogonal shock (a = i?) of arbitrary

strength k. To obtain the orientation of these weak shocks, group (6.17) as a quadratic

. . 2 .
equation 1n n, .

4yA4nf+(—6yA4+2yA2—A2+A‘2)nf—A‘2=0. (6.19)
This equation for each (A, y) has exactly one solution 0 < n? S1 if and only if y > g,

which is seen after grouping (6.18) as a cubic equation in A2 and applying Descartes rule of

signs. The restriction 0 < A S A (y) follows by considering the discriminant of (6.19).

The unique positive solution 11% of (6.19), in turn gives 112 = i 1— n? , and so defines two

possible weak shocks. These two weak shocks are symmetric with respect to g, i.e., one

26

 

involves a shock anglea > g (112 < 0), while the other involves a shock angle a < 3;— (112 >
0). In particular n2 =1 ((1 = 125) if A = A (y) and as A —> 0. However, if

1

0 < A < A (y) then a :t i-ni, i.e., the weak shock is distinct from the orthogonal shock.

7. A PARAMETER SPACE REPRESENTATION FOR MECHANICALLY
CONSISTENT ELASTOSTATIC SHOCKS

l
The previous analysis establishes that if y > — then for each A obeying
2
O < A S A (y) there exists both an orthogonal shock (a = 325) and a weak shock (k —>0)
. . . . 1 . 1 . c
solution to (6.13). However, no such solut1ons exrst 1f y S -2—, or if}! > — wrth A > A (y).
2
l ..
Further, fory > —2- the weak shock and orthogonal shock solutions coincide at A = A (y)<

l
1. Thus if y > -—, then a “compressive loading program” of decreasing A, first permits such
2

solutions at A = A (y). Upon decreasing through A (y) this single special weak shock

solution splits into four special elastostatic shock solutions: two of which are shear

 

symmetric orthogonal shocks (k = i JAL2—[l_—21—) —l ,a = g) and two of which are
1’

angle symmetric weak shocks (k = 0, a =12r—iarcsin(,/1-n,2), where 1112 is the unique

solution of (6.19) obeying 0 < n,2 S 1).

27

7.1 THE DISCRIMINANT CONDITION FOR K-LIMIT POINTS
We now broaden our inquiry so as to examine more general mechanically consistent

elastostatic shock solutions (pairs (or, k) satisfying (6.13) such that either the restriction
or = g or the restriction k = 0 need not hold). To this end, we note that if y, A and or are

given, then two formal expressions for k follow from the quadratic formula as applied to
(6.13). These will either be real or else will be a complex conjugate pair. Only real values of

k are relevant, in which case the discriminant of (6.13) must be nonnegative. Hence, it is

requiredthat
D (A, -sin a, y) 20, (7.1)
where
4 2 2 2
D A,n , y a —n2+——+——— 3-———— —. (7.2)
( 1 ) 1 22 yA6 yiénf yA2

Here n2 has only appeared as a square term, allowing easy elimination in terms of 11, via 11?

= 1 - 11:. The shock strength k, from (6.14), is given by

 

 

k = - 3 n2 i\/D (A, n,, Y) = 3 cos a$\/D(A, -sin a, y). (7.3)
Zn, 2s1n a

 

 

Let us analyze the physical meaning of (6.13) and (7.1). The material characterized

by strength of reinforcing y is subject locally to a planar deformation with fiber stretch A ,
and in-plane stretch % transverse to the fiber. A goal is to determine those deformations

(values of A) that admit mechanically consistent elastostatic shocks. If we carry this idea to
equation (6.13), i.e., fix the parameter 7 and consider deformations (4.23) as characterized
by A, our interest focuses in determining values for n,=- sin or such that the condition (7.1)

is satisfied. In this regard, the k-limit points , defined as those (A, on, 7) that satisfy

D (A, n,, y): o, n,=- sin o (7.4)

28

are useful for organizing the inquiry. The shock strength at the k-limit points is a double

root to (6.13), given by

3
k = -—"l=§cota, (7.5)
2nl

so that by (7.5) the components of n obey

2
= __:‘L_ (7.6)

9+41<2

9
n,2=——— and n

9+4k2

NM

at k-limit points.
Given 7 and A, suppose that D (A, m, y) = 0 for some m obeying 0 < m _<_ 1. This

ensures that D (A, -m, y) = 0 which in turn indicates the existence of 4 possible unit vectors

n consistent with k-limit points for this 7 and A, namely (see Figure 4)

 

 

 

 

(l) n = me, +We2 => k=-3 ' 2mm2 = geota (--;£< a $0)
(2)n=-me,+\/T-—-m_2_e2=>k=3';;nm2 =§cota (0<aS:2t-)
(3) n =-me, -We2 = k=-3 ' :{nmz = geota (-%< or SO)
(4)n=me,-\/—:?e2=>k=3';I—nm2 geota (O<aS-7,;:-).

With reference to Figure 4 the cases (1) and (3) correspond to a mechanically consistent

elastostatic shock where S* passes through the second and fourth quadrants, while (2) and
(4) correspond to a symmetric state of affairs (or -—> -a) where S * resides in the first and
third quadrants. Physically, the sign of k reflects the particular shear direction on I11

associated with the mechanically consistent elastostatic shock orientation. Figure 4 shows

the correspondence between or and k along with the proper +/- sign dependence of k upon

0t as given by (7.5). Namely, the k-limit points in the (A, 00-plane with "725 <0: <0

correspond to k < 0, while the k-limit points with O < a < 3;- correspond to k > 0.

 

 

11 case (2)

A spatial shock, related to
11 case (1) values of k positive
fiber reinforcement direction y,

4) K spatial shock, related to
n case (3) values of k negative

Figure 4. Angles or that give the orientation of the spatial mechanically consistent
elastostatic shocks at the k-limit points. The fiber reinforcement is kept parallel to the y,-
axes on II: . For a given A-value that permit mechanically consistent elastostatic shock

 

 

n points into I'I,,.

 

 

 

existence, there are 2 of such mechanically consistent elastostatic shocks, which in turn
give 4 unit vectors n. In general, the kinking angle 4? ¢ 0 so that the orientation of the

fiber will not be preserved on II; .

The kinking angle o, given by (6.5), when evaluated at the k-limit points using (7.5), gives

3 .
tan d = - “0.“: 5‘" a .In particular, a = 35 gives 4) = 0 at the k-limit points. Further, a
3 sm (1 -l 2

 

right kink angled) = i1:- is associated with a = T sin-(7%). Hence, at the k-limit points

we conclude that :
rt
k>0=> O<a<—2- and a-n<¢<0,
rt
k<0 => ——2—<a<0 and O<¢<7r+a.

Figure 5 summarizes these results.

30

 

 

k>0

 
 
 
 

 

 

spatial Case 2 in Figure 4
shock
_ spatial
11,, (I [1* [1,:
fiber + fiber
reinforcement reinforcement
(11 _

sin‘l[—l—) < at < E- 0 < a < sin'l[-1—)
73 2 J3

 

 

  
 
 
 

  
 

 

 

k < 0
Case 3 in Figure 4
spatial
shock spatial
shock ¢ _
Us
fiber fiber
reinforcement reinforcemen
Ii...+ 0‘ HS

7: . -1 1 _ ° ’1(_1_)
__<a<_S]n _ Sln <a<0
2 (73] i3

 

Figure 5. Kinematics of the fiber reinforcement at the k-limit points. The cases depicted
correspond to cases 2 and 3 of Figure 4. The other two cases give a symmetric state of

It
affairs with respect to the angle —, i.e., the kinematics of case 4 (respectively case 1) is
2

equivalent to the kinematics of case 2 (respectively case 3), with 1'1: located at the right
of the mechanically consistent elastostatic shock.

31

 

 

Grouping equation (7.4) as a polynomial in A yields.

 

A6(-n,2—3) + 714(4-3) +G—yiz) = 0 (7.7)
1

By apphcatron of Descartes rule of srgns, the condrtrony > 5 1s a necessary condrtron to
get positive roots A of (7.7). Therefore we can conclude that a mechanically consistent

I
elastostatic shock will exist for F” given by (4.23) only if y > 2' Notice, if n 1 ——> 0 (ie.

 

a —> 0) then (7.7) gives —3 A 6 — ~ 0 to leading order, confirming the absence of

2
Y n,
pos1t1ve roots A2 for n, suffic1ently small. Notrce also that substrtutmg n,2= 1 (re. or = i-2—)

in (7.7) yields

—4716+ 14(4—3)=o,
r

giving A: O and A = A (y) where A (y) was given previously in (6.16). This is
consistent with our previous result that a mechanically consistent elastostatic shock

orthogonal to the fiber can be obtained if and only if A obeys 0 < A S A (y ).

7.2 ANALYSIS IN THE (A, 00-PLANE
Equation (7.7) allows us to construct a diagram relating shock angles or to the range
of permissible values A for different values of the reinforcing parameter y. The construction

has the additional benefit of determining the range of Athat is consistent with a

1
mechanically consistent elastostatic shock for the values y > —. The construction procedure
2

is as follows. If y >% and n12=sin2a are fixed, then (7.7) provides a third order

32

 

polynomial equation forA 2. Since n,enters the analysis only via nil, it follows that the

roots A are preserved uhder n <——>- n interchange. Thus it suffices to take or in the interval

1 1

7r . . . .
0 < or S ‘2" The number of posrtrve roots A 2 then give the number of A > O consrstent

with (7.4). These A-root values turn out to bound all the A—values consistent with an
elastostatic shock of orientation or, i.e., even A-values that are not associated with k-limit

point solutions.

We have developed a numerical procedure to accomplish this A root extraction, and

find in general that there will exist either zero, one or two roots A 2 0 to (7.4) for given 7

> g and 01 obeying 0 < a S 225. The plot of (A, 00-values consistent with (7.4) on the

domain A> O, O S a S 71’ is shown in figure 6 for various y > 212—. Each such (A, 00-plot on
this domain is a simple closed curve denoted by8I‘k (y). That is (A, 006 8I‘k(y) ©D(A,

i sin a, y) = O.Each curve81‘k(y) is symmetric about the horizontal linea =% and

includes exactly two points on this line of symmetry: (A, 00 = (0,?) and (A, 00 =

A

(A OIL-725). The region that is interior to8I‘k(y) will be denoted by I‘k(y). The

closure81‘,c (y) U I‘ k (y) represent values (A, 00 that support mechanically consistent
elastostatic shocks (discriminant nonnegative, i.e., (7.1) is satisfied) for the given y. In
particular if (A, 006 8 I), (y) then (6.13) yields a unique (double root) k for such a shock.
However, if (A, 006 F,,(y), then (6.13) yields two real roots k for such a shock. The
region exterior to 17,0!) is inconsistent with such a shock (discriminant negative, i.e., (7.1)

is not satisfied).

33

175 *
150 *
125
100 :
75 *
50
25 :

0

 

 

 

. i . 1 . . A

0.2 0.4 0.6 0.8 1

Figure 6. The curves, denoted as 81”,, (y), give values (A, 00 at the k-limit points.
it

Each curve is symmetric with respect toa = —. The interior to each 8Fk(y), denoted
2

by I“, (y), represent pairs (A, or) consistent with elastostatic shock existence. Points (A,
00 outside I‘k (y) are not consistent with the existence of such a shock Lines of constant
or intersect 8I‘k(y) at most twice, giving the upper and lower A-values able to sustain
that particular shock direction. The curves 81‘k(y) are nested with respect to 7. For

1 ..
values -2- < y S 3.283, 8Fk(y) involves A-values satisfying 0 S A S A (y). For values

7 > 3.283, 8 Pk (y) involves values A > A (y), as shown in the figure for the particular
values 7 = 6, 15. The curves 8 I‘k(y) can not involve values a = O or 1:.

The curves 81‘ k(y) show differing qualitative behavior as determined by the values

y. Let us focus on the lower half of the curves8I‘k (y) in Figure 6, that is on A 2 O and
0 S a S 325. In all cases begining at the left endpoint (A, 00 = (0,325): the curve8l‘,< (y)

descends to a single minimum value of or before returning to the right endpoint (A, 00 =

34

 

A

(A (y),%). The approach to this right endpoint from the minimum or-value may occur in

one of two ways. The first type of approach occurs, for example, when y = 3. It involves

monotonic increase in both A and oral] the way to the right endpoint so that the

curve 8 Pk (y) approaches (A (y ) ,g) from the left. The second type of approach occurs for

example when 7 = 6 and 15. It involves monotonic increase in A and or to a value of

Abeyond A (y) which is then followed by a decrease in A so that the curve81‘k(y)

approaches (A (y ) ,g) from the right.

When8Fk(y) approaches (A (y),%) from the left it follows that the range of

A that supports mechanically consistent elastostatic shocks is 0 S A S A (y). However
when 8110/) approaches (A (y),%) from the right it follows that the range of A that

supports mechanically consistent elastostatic shocks is 0 SA S A y) where

max (

A

max (7) > A (y). In what follows we determine the values of y associated with these two

different behaviors, and determine the function Amax (y) when this approach is from the
right.

Before clarifying these issues, we notice in figure 6 that the curves8I‘k (y) are
nested with respect to y. Namely if y, > y, then I), (y,) C I], (y;). This is a consequence of

the following:
Theorem 1. IfD (A, m, y) = 0 and y’ > y then D (A, m, y’) > 0.

Proof:

 

 

 

=(i_2.)( 1 _ 1 _;)_

l" 1’ A6 A‘Sm2 A2

_ (3-3][(_1__1)._1_+_1_)
Y Y' m2 A6 A2

Now [a — 37) > 0 since y >% and y’>y. The other factor is also positive since m2 S 1.
Y Y
Thus D (A, m, y’) > 0. QED.

This result allows us easily to consider the behavior of the loci 8I‘k(}/)as y ——> 00.

Taking this limit in (7.2) and requiring (7.1) yields:

4
2
D(A, n,,Y) ~-n, +F— 32 O,

(as y —) 00). (7.8)
The case n,= l (ora=%), gives A S l in (7.8), which is the y —> oo limit that is
obtainable from the definition of A (y) in (6.16). Conversely as n,—> 0it follows from

(7.8) thatA ——) \[E which locates A (y) in the limit as y -—> 00. Sincefi > 1 this
3 max 3

result opens the possibility that mechanically consistent elastostatic shocks may involve

fiber extension on IT for sufficiently large y.

To obtain the minimum'y consistent with fiber extension on IT let A = l in (7.1).

 

ThenD(1, n,,y)=l - 11% — 2 .Sincey>0 and n,)->0, the condition D(1, n,,y)20

Y“,

is equivalent to the condition - y n? + y 11:,2 — 2 .>_ 0. In particular D(1, n,, y)=0 yields

2-87

 

2_‘l’i Y
,_

2 giving 7 Z 8 as the range of ythat support mechanically
' 7

consistent elastostatic shocks involving fiber extension on IT. Note that n? S 1 is clearly

satisfied for all such y. It is interesting to note that the condition y > 8 is precisely the

36

 

condition found in [l] for simple shear at certain fiber orientations to involve negative shear

stress in the shearing direction for certain positive shears.
Additional properties of the81‘k(y) curves follow by analyzing Z—fi. Extracting

da

 

from (7.7), using 11, = - sin a and fixing y yields:

4 2 6

 

 

 

 

2
— - - cot or

d a A 3 y A 3 y A 7
= . (79)

d4 . 2

cos a srn a —l
y A 6 sin 4 0:
Since a ¢ 0, the denominator in (7.9) vanishes if either cos CL = O (or = g), or if
)6: 2 . (7m)
yfin4a

This locates the points of vertical tangency on t11e8F k(y) curves. Namely, in Figure 6,

I 1
a = g is consistent with the vertical tangent at the right endpoint (A, 00 = ( 1-2—— g).
7
Pairs (A, 00 obeying (7.10) locate vertical tangents fora :3 %. Now such vertical tangents

for 0: ¢ :2:— exist if only if 8 1“,, (y) approaches the endpoint (A 0),?) from the right. To

 

clarify this direction of approach we analyze the slope :Aa near (A, 00 = (A (y ) g). To

this end allowance must be made for the observation that this slope is infinite at (A, 00 =

A

(A (y),—72£). Note as 0L—> 12:- that (7.9) gives

da~ 41%2y-0
d1 cosay(A6——2—)
Y

 

+ 0 (cos a)

37

which, since A = (I - 31— at the right endpoint, establishes
Y

.. .—, ”ii-l at].

. d or)
SignT
d (A,a): 2y, 2

 

A direct analysis now shows that
2 l 3 > Oif y < 3.283,
[——(1--2-—) ) = Oif y = 3.283,
7 y < 0 ify > 3.283.

This establishes that the curve 8 Pk (y) approaches (A (y ) ,g) from the right if and only if y

> 3.283. In particular, a i gvertical tangents exist if and only if y > 3.283. For this case of

y > 3.283, the valueAmax (y) is subject to (7.10). Substituting from (7.10) into (7.7) to

eliminate n, yields
—163F+A4\[§(4-3)-A31+3F=0 (7.11)
Y Y Y Y Y Y
This equation gives the Amax (y) fory > 3.283.

. 1
We summarize these results as follows. For y S2 there are no values of
A consrstent With an elastostatic shock. For y > 2 an elastostatrc shock is consrstent wrth

the range of A values 0 S A S Amam (y) where Amax (y) = A (y) if %< y S 3.283

whereasAmx (y) follows from (7.11) if y > 3.283. As shown in Figure 7, the

function Amax (y) is monotonically increasing in A with Amax(-;-) = 0,Amax (8) = 1 and

4
lim Amax(y) = (3': 1.155.

r—M

38

l6r
14:
12'
10:

 

 

 

ONAO‘OO

0.2 0.4 0.6 0.8 1 1.2

Figure 7. The curve represents the maximum A—values, calledAma, (y), of each

curve81‘k(y) in Figure 6. This curve is monotone increasing, which follows from the
l

nesting property of the curves81‘k(y), with respect to 7. Note for y S - that there are
2

no A-values consistent with elastostatic shock existence. At y = 3.283 the formula
defining this curve shifts from (6.16) to (7.11). Also, Am“ (8) = l, which indicates that

y > 8 gives the range of 7 that supports mechanically consistent elastostatic shocks with
fiber extension on IT, i.e., A > 1. There is a vertical asymptote given

4
byrm,(oo) = \E =1.155.

The or-values associated with A =Amax (y) are displayed in Figure 8 for different values of

the reinforcing parameter y. For 7 < 3.283 this value is a = %. For y > 3.283, this value

1/4
6
follows from (7.10) as a = sin '1 [AA m2ax(7))

39

16 r
14 :
12
10 r

 

ONAOOO

 

. . . . - . - . . 0t
25 50 75 100 125 150 175
Figure 8. Values or associated toAmax (y). The curve is symmetric with respect

rt
too: = 3. The left side of the curve is related to the lower half of the curves8I‘k(y) in

Figure 6, while the right side of the curve is related to the upper half of the

1 7t
curves8Fk(y). For -<y S3283, the curve indicates that this angle 150: =-.
2 2

sinceAmax (y) = A (y).

The curves8Fk(y) also give the range of Oton the interval 0 < or < 7: that are

consistent with an elastostatic shock. Namely this range is 0: min (y) S a S tr — or m,,, (y)
wherea m,,,(y) is associated with points of horizontal tangency obeying 0 S or S; to

the8I‘k(y) curves. Thusam,,,(y) is associated with the vanishing of the numerator in

 

 

(7.9), namely
1
3 l-
4 sin2 a
A = (7.12)
(1- 2 7)

Substituting from (7.12) into (7.7) yields

3 ,/3 (1—sin2 or)3
(Zr-1) (277

 

 

(-sin2a-3)—isin3a+isina =0 (7.13)
Y Y

40

Roots Ct to (7.13) fory > % givea min(y) as shown in Figure 9.

16r
14*
12*
10’

N-BQOO

 

 

‘ ‘ ‘ k * Lg g (X
25 50 75 100 125 150 175

Figure 9. Values (1 vs y obeying (7.13) define two regions: the region interior to the
curve involves values a that support mechanically consistent elastostatic shocks, while
the region outside the curve involves values a that are not admissible for the particular y.

it
The curve is symmetric with respect toa = —. The left side of the curve is related to the
2

lower half of the curvesdl‘kQI) in Figure 6, therefore, giving the minimum (rt-value for
each 7, while the right side is related to the upper half of the curvesBI‘kQ'), hence
giving the maximum a-value for each 7.

Finally, we recall in Section 6 that special orthogonal shocks and weak shocks were
examined separately. The former are represented by horizontal lines a = 32: in the (A, 0t)-

diagrams given in Figure 6. Weak shocks solutions can also be represented with respect to
these diagrams. Figure 10 shows the k = 0 weak shock curves for the particular value of the

reinforcing parameter y = 15. For any value v, the weak shock curves satisfy (6.17) and

gives points (A, 0t) 6 Pk (y).

41

175 ’
150 ’
125 t
100 *
75 '
50 L
25 *

 

 

r i r L . r A.
0.2 0.4 0.6 0.8 1

Figure 10. The plot shows the k-limit points in the (A, a)-plane oraf‘kQ') curve, as
given in Figure 6, along with the weak shocks curve (k = O) for y = 15. Both curves are

It
symmetric with respect to a = 2. The curve k = 0 is inside the curvedf‘k (y), as it
corresponds to values (A, (1)6 I‘k(y). At the k-limit points, for shock angles
7: 7r
0 < a < 3, (A, 0t)-values correspond to k > 0, while for shock angles —2- < a < 7:,

(A, 0t)-values on the k-limit points correspond to k < O.

8 THE ELASTOSTATIC SHOCK MANIFOLD E(y)

The analysis of Section 7 resulting in the loci 8110/) provides a useful
characterization of values (A, 0t, k) that support mechanically consistent elastostatic shocks
for the reinforced neo-Hookean material. Namely, if (A, 006 ark (y) then there exists
exactly one k given by (7.5) such that the resulting values (A, 0t, k) support a mechanically
consistent elastostatic shock. Similarly if (A, 006 Pk (y) then there exist two values k given
by (7.3) such that (A, 0t, k) support a mechanically consistent elastostatic shock.
Consequently values (A, (1, k) that support a mechanically consistent elastostatic shocks are
envisioned as the surface of a simply connected region in (A, (1, k)-space and so can be

described as a two-dimensional closed surface manifold in (A, 0t, k)-space. We shall denote

42

1
this closed surface manifold by E(y). To be precise, for fixed y > — we say that (A, 0t,
2

k)EE(y) ifA> O, 0 <0t <1t and P(A, -sin (1, cos a, k, y) =0, where P is given by
(6.14). The projection of an E(y) manifold onto the (A, 0t)-plane then gives the associated
k-limit points. As such, the locusai‘k (y) bounds the projection of any other curve from

E(y) onto the (A, 0t)-plane.

8.1 ANALYSIS ON THE (A, k)-PLANE

The pre-image of the k-limit points on E(y) can in turn be projected into the (A, k)-
plane in a simple fashion since (7.5) gives k = % cot a for such points. In view of the well-

behaved monotone nature of the cot function on O S a S 7:, the projection of the k-limit

points onto the (A, k)-plane gives curves with the same features as the curves in the (A, (1)-

plane. This is shown in Figure 11. In particular, the point (A, CL) = (0 ,g) corresponds to the

point (A, k) = (O, 0) while the point (A, OL) = (A 0),?) corresponds to the point (A, k) =

(A (y ) , O). In addition, the points of vertical and horizontal tangency to the k-lirnit points in

the (A, 0t)-plane also correspond with the points of vertical and horizontal tangency to the

projection of the k-limit points on the (A, k)-plane. Here we just summarize this result in the

(A, k)-plane. Using (7.6) it follows from (7.10) that vertical tangents to the k-limit points in

the (A, k) plane are given by

2 ( 9 + 4 k2)2
81 y

 

A6 = (8.1)

and it similarly follows from (7. 12) that horizontal tangents are given by

43

4_ 4k2

_ 3(2y—l)'

(8.2)

Recall that the issue of vertical tangency arises if and only if y > 3.283. Finally, since k is

given by (7.5), the (A, k)-image of the k-limit points for varying yobey a similar

containment (nesting) condition as that of the k-limit points curvesaf‘k (y) in the (A, (1)-

plane as established by Theorem 1.

 

y=15

y6

 

Figure 11. Projection of the k-limit points into the (A, k)-plane for the values of the
reinforcing parameter 7 = 3, 6, 15. The curves are symmetric with respect to k = 0.

11’
Values k > 0 in each curve are associated to shock angles 0 < a < 2 , while values k < 0

7:
in each curve are associated to shock angles — < a < 7:. The curves are similar to the
2

curves in the (A, 0t)-plane shown in Figure 6. In particular, for each y, points of
horizontal and vertical tangency in each curve correspond with points of vertical and
horizontal tangency in each curve of Figure 6. Further, (A, k) = (O, O) and (A, k) =

(A (y), 0) correspond to (A, OL) = (0.1;) and (A, 0t) = (A (10,3) respectively for each

‘Y.

It is important to realize that the (A, k)-image of the k-limit points as depicted in
Figure 11 does not give the projection of the loci E(y) onto the (A, k)-plane. This follows
immediately from Figure 3 which shows the curves in the (A, k)-plane associated with
mechanically consistent elastostatic shocks orthogonal to the fiber on IT‘. Namely the y = 3
curve in Figure 3 is not contained within the y = 3 curve defined by Figure 11. Note for
example that these two y = 3 curves intersect at (A, k) = (0.638, 1.019) and (A, k) = (0.638, -

1.019) in the (A, k)-p1ane. However the value of 0t associated with these intersection points
on the Frgure 3 curve rs CL = -2—, whereas the value of (X. assocrated wrth these mtersectron

points on the Figure 11 curve by (7.5) is OL = 0.315 1t if k > 0, while CL = 0.684 at if k < 0.
We now turn to an examination of the projection of the loci E(y) onto the (A, k)-

plane. The boundary of this projection will be denoted by 3 Fa (y) and will be referred to as

the a—limit points associated with E(y). The interior of this projection will be denoted by

I}, (y). Recasting (6.13) as a polynomial in n1 gives

P(A,n1,n2,k,y)= nf2yA4(—2+k2) +n?n2 6yA4k+

 

 

 

(8.3)
+ 111202 - 11-2.» 6yA4- 2yA2) + r2.
Introducing x = tan [3, where 0 =a+—723, Z25< fl <2;- allows the
substitution n2 - 1 n2 — x2 n n — x Then (8 3) as a
1 l+x2, 2 l+x2,12 l+x2' .
polynomial in x yields
Q(A, x,k, y): x4A-2 + x2 [AZ + 14+ 2yA2(3A2 -1D+ (84)

+x 6yA4k+2yA4 (1 +18) +A7-(1 - 2y).

Now, if Q (A, x , k , y): 0, then Q (A, -x , -k , y): 0, which is consistent with the

k<—-)-k symmetry of the (A, k)-curves in Figure 11, and further establishes the k<—>-k

45

symmetry of each point in the (A, k)-plane. Namely, if (A, or, k)6 B(y) then (A, 1t-0L, -k)
E E(y). Hence the curvesaf‘a (y) will be symmetric with respect to k = 0. Denote the
coefficients of the polynomial Q as:

a4 = A‘2

a2 = 12+ 14+ 2yA2(3A2 - 1)

al = 6 M4 k

a0 = 2M4 (1 + k2) + 12(1- 2y). (8.5)

Since Q(A, x, k, y) = a 4 x4 +32 x2 +al x +30 is a polynomial in x of degree four, it has

two pair of complex conjugate roots if (A, k)9£ Fa (y). The transition from values
inconsistent with the existence of elastostatic shocks to values consistent with the existence
of elastostatic shocks involves at least one pair of complex conjugate roots remaining
complex conjugate and the other complex conjugate pair giving way to a double real root.
Here it is useful to recall some elementary facts regarding roots to quartic

4 2

polynomials of the general form a 4 x +32 x + a x + a0 [18]. Namely

l
1) two pair of double roots will exist if and only if a1 = 0, and a g - 4 a 4 a0 = 0,

ii) triple roots will exist if and only if

 

A(A, k, y)=0and B(A, k, 7) =0, (8.6)
where
2 2
E - 33 3+a2 3’- a1, , B a ——32,—a—°, (8.7)
108 a4 3a,, 8a,, 12 a4 a4

iii) the transition between two pairs of complex solutions to at least one double real root

(i.e., the breakdown of real solutions) is given by C = 0, where C = 27 A2 + 4 B3 with A
and B given by (8.7).

In view of this development the transition 3 Fa (y) between pairs (A, k) 6 Fa (y) and

46

pairs 0., 105 ram is given by C0». k, y)=0, where c = 27 A2 + 4 B3with A =
A (A, k, y) and B = B (A, k, y) given by (8.7). This permits a numerical calculation of
the (at-limit points. They are displayed in the (A, k)-plane in Figure 12 for y = 3, 6, 15, 50.
For comparison, the image of the generators of the k-limit points are also shown on this
Figure. The significance of this Figure is that:

(A, k)6£ I‘a(y)=> there does not exist any 0t associated with a mechanically consistent
elastostatic shock,

(A, k)6 3130/) =>there exists exactly one 0t, 0 < a < 7:, associated with a mechanically
consistent elastostatic shock, obtained fom (8.4)

(A, k)6 Fa(y)=> there exist multiple 0t, 0 < a < 7:, associated with a mechanically

consistent elastostatic shock, obtained from (8.4).

47

        
 
 

  

 

 

 

 

  

 

 

 

4 r 4
3 i 3 . . .
2 . (rt-limit points 7:3 2 p a—ltmrtpornts 7:6
1 1
‘ A * A
1 ‘ 1 1 . 1
-2 v __2 b
-3 . _3 .
-4 * -4
k
15 .
10 -
a—limit points
5 >
A A
-5 {
-4 ( -10
—6L —15

 

 

Figure 12. The plots give the (A, k) values at the (it-limit point curves and the
projection of the k-limit points curves into the (A, k)-plane, for y = 3, 6, 15, 50. All the
curves are symmetric with respect to k = 0. The a-limit points are (A, k)6 31}, (y) and

bound all the pairs (A, k)6 ram. The points (A, k) = (A (y), 0), related to the angle
7!
a :2, and (A, k) = (Amuw), k), related to a unique angle shown in Figure 8, satisfy

simultaneously the Ot-limit point curves and the k-limit point curves. Otherwise, the
projection of the k-limit points curves are inside the a-limit points.

In the same way, and just for comparison, Figure 13 shows the k-lirnit point curve and the
a-limit point curve in the (A, 0t)-plane for y = 15. Returning to Figure 12 we note that the
0t-limit point curve and the (A, k)-curve associated with the k-limit points appear to coincide
on their rightmost portions. The 0t-limit point curve involves C(A, k, y)=0 whereas the k-

limit point curve involves D (A, - sin or, y) = 0. In fact the k-limit point curve is interior to

48

I‘a(y) except at locations of vertical tangency. For all values y this includes the intersection
with the A—axis (A, k) = (A (y ) , 0). In addition, if y > 3.283 then the k-limit point curve and
the oc-limit point curve also intersect at two other points corresponding to A =Amax (y) as
given by (A, k) = ( Amax (y), ik) where k follows from (8.1). Conversely, with reference to

Figure 13, the (A, (JO-curve associated with the a—limit points is interior to I‘k (y)except at

these special points. For g < y < 3.283 there is only one such special point in the (A, 0t)-

plane, namely (A, OL) = (A (7),?) For 7) 3.283, there are two additional special points,

namely (A, CL) = (Amam (3!), 0t), where the two values 0t obeying 0 < 0t < 1: (both symmetric

about 0t = 325) follow from (7. 10).

175 *
150
125
100 *
75 .
50
25 '

     

(it-limit points

 

 

0.2 0.4 0.6 0.8 1
Figure 13. For the particular value of the reinforcing parameter 7 = 15, the k-limit
points and the (A, 0t)-curve associated to the a-limit points are shown. Both curves are

It
symmetric with respect to a = 3. The k-limit points bound all the (A, (Jo-points

consistent with shock existence. The curves coincide at locations of vertical tangency
where A =A (y) and for y > 3.283, A =Amax (y). Otherwise, the a—limit points are
interior to the k-limit points in the (A, 0t)-plane.

49

It remains to determine the number of real roots or associated with a given (A, k)6

I}; (y). For the particular values displayed in Figure 12 (y = 3, 6, 15, 50), it is found that
there exist at most two real roots x = tan (01+?) of (8.4) for each (A, k)6 Fa(y).
However, it shall presently be shown for certain sufficiently large y that there exist four real
roots x = tan (a + :25) for certain (A, k). (Recall for comparison that, by (7.3) any (A, (1)6

I‘ k (y) is associated at most with two values k).

8.2 CONSTANT A CROSS-SECTIONS OF E(y)

The analysis of the projections I‘ k (y) and Fa (y) of B(y) onto the respective (A, (x)
and (A, k)-planes establishes that each B(y) is a well behaved two.dimensional manifold in
(A, 0t, k)-space. Of particular interest are constant-A cross sectional slices of this manifold,

as this gives a representation of the mechanically consistent elastostatic shocks associated
with a specific reinforcing y and a specific F+ as given by (4.23). Also, these cross sections
of E(y) reveal different qualitative features as A varies. These constant-A slices can be
obtained numerically by (7.3). The procedure involves fixing y and A, and varying a,
whereupon (7.3) will give zero, one or two values k consistent with shock existence. These

then define the curve corresponding to that particular (A, y)-cross section. Figure 14

displays various slices for y = 15. For this particular y—value it follows that Amax (y) =

1.045 >A (y): 0.983. The four specific cross sections corresponding to A = 1.02, A (y),

0.8, 0.3 are displayed. ForA (y)< A < Amax (y) these cross sections consist of two distinct

symmetric components. These components coalesce at A =A (y ). For A <A (y), there is

only one component to any such slice. For each constant-A slice, in addition to the central

50

symmetry about the point (or, k) = (g, 0), it is also apparent that there exists at most two

values k for any a, and two values 01 for any k. The k-limit points give the extremal 0t-

values and are associated with vertical tangent points

 

d k .
d— : 00 on these cross sectrons. The
a .

(it-limit points give the extremal k-values and are associated with horizontal tangent points

91 = 0. The nearness of the horizontal and vertical tangent points for A near and greater

da

than A = 1 accounts for the closeness of the oc-limit point and the k-limit point curves in

Figures 12 and 13. It can be shown from (7.3) that

 

cos a [ 4

- 6 $
dk _ ,(D (sin a, A, y)
d a _ 4 sinza

[16 _ 2 — 25in2a)i2cosa\[D(sina, A, y)
y srna

 

 

Now, at the k-limit points, D (sin CL, A, y) = 0, which implies

 

d kl .
-— = 00 so long as nerther
d a

cos (1 = 0 nor A 6 = 2 . The first exception of cos 0t = 0 implies a = E which in
y sin 4 a 2
x 2 . . .
turn is associated withA (y). The second exception of A 6 = —— 1S assocrated wrth

ydn4a

(y) for y > 3.283.

max

51

A: 1.02 ..
A=A=0.983

 

 

 

 

175

 

25 50 75 100 1 150 175 25 50 75 1A A 130
-2»
-4 .
-6»
k
6.
A=0£5
A=.8 4’
N 2» m
L #_A a A A A A—A‘

 

 

25 50 75 125 0 175 2 25 50 75 100 l25 150

 

Figure 14. Different cross sections of the B(y) manifold for y = 15 in the (0t, k)-plane
shows features representative of values y > 3.283. For values Amx (y) > A > A (y) the

E(y) cross sections will involve two symmetric components as shown by the cross
section at the upper left comer (A = 1.02). In particular, at the maximal value A
=Amax (y), the “cross section” first emerges as the two separate points (a, k) = (anm,
kmax) and (0t, k) = (rt-am“, km) where km“ follows from (8.1) and am, follows from
(7.5). At the value A = A (y) these two components coalesce at the unique point (A, 0t) =

A It A
(A (y), 2). As A decreases belowA (y) the cross-sections of E(y) develop following

the patterns shown by the two lower figures. For 7 = 15, each cross section associates at
most two values k for each angle or, and at most two values (1 for each k. The k-limit

da

= 0°, while the Ot-limit

 

 

points are related in each cross section to points obeying

d
points are related in each cross section to points obeyin — = 0.
d a

52

a
175

Figure 15 shows two cross sections of E(y) when 7 = 3 at the same A-values given

in the two lower plots of Figure 14. Since 3 < 3.283, the E(y) manifold for y = 3 does not

include values A >A (y). Figure 16 shows the same cross sections of B(y) as Figure 14

when 7 = 1000. Figure 16 verifies for Y > 3.283 that B(y) retains similar features as

A

displayed in Figure 14 fory = 15. Namely, for A (y)< A < Amax (y) the cross sections of

E(y) consist of two distinct symmetric components which coalesce at A =A (y). There is
however one qualitative difference for small A. Namely, the horizontal dashed line
corresponding to k = 3 in the cross section A = 0.3 of Figure 16 (y = 1000) intersects the
B(y) cross-section four times. This shows that there exist (A, k)6 Fa (y) associated with

four real solutions of (8.4).

Define y* as the minimum 'y-value that can provide four real solutions to (8.4).
Whence, for y < 7* there exist two values 0t associated with each (A, k)6 I‘a(y) (for
instance y = 3, 6, 15, 50), while for y > y* there may exist up to four values or associated
with some (A, k)6 Fa (y) (for instance 7 = 1000). It shall be established in Section 8.3 that

* = 403. Prior to Section 8.3 we focus attention on the simpler case where y < 7*.

53

30*

20
10

_10.
_20.
—30*

30

20*
10'

-10

—20*
-30.

 

A=0.3

Y=3

 

 

25

50

75

l 125 150

Figure 15. Different cross sections of the of the B(y) manifold for y = 3 in the (or, k)-
plane, at the same A-values displayed by the two lower plots in Figure 14, i.e., A = 0.8,

0.3. Recall that for y < 3.283, the B(y) manifold does not include values A > A (y), and

. .. 71'
the cross section of B(Y) at A =A (y) includes a unique point (A, at) = (A (y), 3). In

each cross section for y = 3 there exist at most two values k for any a and two values oz

for any k.

A: 1.02

 

 

A=O.8 7:1000

 

 

25 50 75 100 125 5

_10.

-15'

175

 

 

 

 

 

 

.. y=1000
A=A=O.9997
25 50 75 100 125 75
y=1000
5 100 125 150 175

Figure 16. Different cross sections of the of the B(y) manifold for y = 1000 in the (CL,
k)-plane, at the same A-values that in Figure 14, i.e., A = 1.02, A (y), 0.8, 0.3. Note

that for y = 1000, A (y) = 0.9997. The character of the B(y) manifolds as shown in

Figure 14 for y = 15 and Figure 16 for y = 1000 are similar. Nevertheless, in the lower
right cross section of the B(Y) manifold for y = 1000, the horizontal dashed line k = 3
intersects the B(y) cross section four times. This indicates the existence of four real
solutions at for some (A, k)-pairs, as opposed to Figure 14 where pairs (A, k) are

associated with two real solutions a.

54

The main characteristics of the E(y) projection into the (A, k)—plane if y < y* are as
follows. The angles 0t vary continuously with (A, k) on I}, (y) as seen in the different cross

sections of Figures 14, 15 and 16. Those angles 0t associated with negative roots x to (8.4)

obey 0 < a < 325, while angles 0t associated with positive roots x to (8.4) obey 1::- < a < 7:.

Values (A, k, y) associated with (1 z; are given by (6.15) and the associated (A, k)-curves,

as shown in Figure 3, partition the interior of I“), (y) into three regions as shown in Figure

17 for the particular value v = 15. One finds that the cenhal region in Figure 17, which

includes the A-axis (k = 0), involves one angle 0t obeying (1 <1; and the other obeying 0t
>1. The region wholly contained in k > 0 involves both angles 0t obeying 0t <1; whereas
the region wholly contained in k < 0 involves both angles 0t obeying 0t >325. At the OL-limit
points these two angles coalesce to a common value, again obeying 0t <1; in k > 0 and
obeying 0t >% in k < 0. The a—limit point (A, k) = (A (y), 0) is associated with the

7:
common value (1 = 3.

55

0t=90°

 
    

0t- limit points

 

 

it
Figure 17. The plot shows the a-limit points and the (A, k)-values related to (1 =- for
2
y = 15, partitioning I‘a(y) in three regions: the upper region (wholly contained in k >

77:
0) involves two shock angles 0t < —; the middle region containing the A-axis involves
7t 7t ,
two shocks, one or < -—-, and the other at > —; the lower region (wholly contained 1n k <
71’
0) involves two shock angles 0t >—. Similar plots will follow for values 7 obeying
2

1
3 < y < y *. In the plot the symmetry has been considered to draw the direction of the

mechanically consistent elastostatic shocks with respect to the fiber on IT”. Recall that in
the event of elastostatic shock existence, the fiber will not keep the horizontal direction
on the 11' side of the mechanically consistent elastostatic shock, giving rise to a kinking
angle.

Further, just as the a = — curves partrtroned F a (y) into three regrons 1nvolv1ng zero, one

or two of the two angles 0t obeying 0 < a <% (Figure 17), the k = 0 curve partitions

I", (y) into three regions involving zero, one or two of the two shock strengths k obeying k

> 0 (Figure 18). Nevertheless, it should be noted that eventhough Figure 18 is representative

56

for any value of the reinforcing parameter 7, Figure 17 is only representative for values

7 supporting two real solutions to (8.4), namely 7 < 7*.

175 *
150 *
125 '
100 '
75 ’
50 "
25 ’

 

. . k- limit points‘ .
0.2 0.4 0.6 0.8 1

 

Figure 18. The plot represents the k-limit points and the (A, 0t)-values associated to k =

it
0 (weak shocks) for 7 = 15. Both curves are symmetric with respect to a = —, and
2

partition I‘k(7) into three regions. The upper region is associated to two negative values

k, the middle region to one positive and one negative value k, while the lower region to
two positive values k. These features are representative for any 7-value, since any (A,
or) E I‘k (7) is associated at most with two values k.

Figure 5 summarizes the kinking angle behavior at the k-limit points. The kinking
angle given by (6.5) can be analyzed for any value k. Namely, for 0t obeying 0 < a < 7r
(6.5) gives

k>O 2) a ~71: < 42 < 0,

k<0 => O<¢<a.

57

Fork>0, if g < a < 7t,then the kinking angletpobeys €—< 4) < O. Fork>0, if
0 < a < Zr- and the denominator in (6.5) is greater than zero, then the kinking angle (1)
obeys g— < (p < 0. For k >0, if 0 < a < % and the denominator in (6.5) is less than
zero, then the kinking angle (1) obeys a - 7: < 4) < 1;. For k < 0 there is a symmetric state
of affairs about 35. Namely, for k < 0, if 0 < a < 325, then the kinking angle 11) obeys
0< (f) < %° Similarly, for k < 0, if g < a < 7: , and the denominator in (6.5) is
greater than zero, then the kinking angle¢obeys O < (b < %,while if g < a < 71: and

the denominator in (6.5) is less than zero, then the kinking angle (1) obeys %< (p < a .
Hence, the features of the kinking angle 4) for any value k are kept as shown in Figure 5,

although the transition through the q) zig— kink angle will in general not occur at an angle

1

__ . _l .
a—+ srn — res trvel.
(75) p“ y

8.3 SHOCK ANGLE MULTIPLICITY

We now turn to clarify the issue of the number of mechanically consitent elastostatic
shocks that are associated with the projection Pa (7) of E(7) onto the (A, k)-plane. In
particular we find that 7* = 403. The results are summarized in Theorems 2, 3 and 4 which

follow:

Theorem 2: For fixed (A, k, 7) obeying :- < 7 < 403 and A > 0, there exist at most two

values 0t obeying 0 < a < 71: such that (A, or, k) E E(7).

58

Proof: It is equivalent to show that Q (A, x , k , 7) has at most two real roots x for
the given (A, k, 7). There are 4 roots x to (8.4) and the definition of the k-limit points
ensures that at least two are real. We first exclude the possibility that the 4 roots consist of 2

pair of double roots. The existence of two pair of double root in (8.4) requires that both

a1: 0, and ag-4a4ao = 0. Now a1: 0 only if k = 0, in which case
-4a4 a°|k=0 =-4(27 (AZ—l)+ 1) Upon use of the second inequality in (6.16),

2 0, with equality only if A 2 =( - 2L], but then,
1’

-4a a

4 o|k=o

27

=((1- 31;) (47- 2) + 2:71)2>0

1 . . . . . .
since 7 > 5. Therefore, the possrbrlrty that there exrsts two pair of double roots assocrated

2
ag-4a4aO)/12_[l 1]=(A2+A”2+27A2(3A2-1))=

 

with (8.4) is excluded. This also excludes quadruple roots as special case. The existence of
triple roots to (8.4) requires (8.6) where the two functions A(A, k, 7): R3 —) R and
B (A, k, 7): R3 —> R are given by (8.7). If a1=0, i.e., k = 0, then (8.6) holds if and only
if a2: 0 and a0: 0. However, the argument in which the existence of two pair of double

roots was ruled out, shows that a,, a,, and a0 can not simultaneously vanish. Thus, the

existence of a triple root requires al at 0, i.e., by (8.5),, k :5 0. To satisfy (8.6), in (8.7),,
since a 4 > 0 it is required that 30 < 0, with this, (8.7)l requires that a2 < 0. Without loss of

generality, consider k > 0. To make aO < 0, by (8.5),, it is required that k

 

< [Eli-[1 — 722-) -—1 (viz. (6.15)), with the positive sign. To make a2 < 0, by (8.5),, a2 is a
1’

59

cubic polynomial in A with one negative solution and two positive ones. By a standard
method associated with root determination for cubic polynomials, it can be shown that these
two positive roots are complex conjugate if and only if 7 < 31.825. Therefore if 7 < 31.825
then there are no triple roots to (8.4). Further, by a continuity argument there can not be

four real roots to (8.4). Now, for any 7and k > O, to get a triple root it is required
simultaneously that a2 < O and a0 < 0. Application of Descartes rule of signs to (8.4) shows

that this possibility is associated to three positive solutions and one negative solution. The

. . . 71' . . 7t
posrtrve solutrons are related to angles ~2- < a < 7:, whrle the negative one to 0 < a < 3.11:1
. . . . . . 7!
us turn to (7.3). The possrbrlrty to get a triple solution 1nvolves angles -2- < a < 71'. For each

angle 0t satisfying % < a < n, and fixed A and 7, there exist only one positive k given by

the positive sign in (7.3). Hence, for fixed A and 7, triple roots or will not be associated to
. . . . d k d k

self mtersectron pornts of the E(7) manifold or values -d—— : 00, but to values d— = 0.
a 0:

Hence, to eliminate the existence of 4 solutions x to (8.4) for a given 7-value it is enough to

guarantee that g—lS-S 0 for any (A, 006 I‘k(7), a >%. Taking the positive sign in (7.3),.
a

—d—k- yields
d a

93.
da

 

(8.8)

—6+ 2cosa ( 2

JD (A, -sin a, 7) 7A 6 sinza

_ 4 sin2

— sinza) -2 cos a JD (A, -sin a. 1’)

 

a

Now % is well defined since D (A, - sin a, 7) :6 0, A at 0, and a ¢ 7:. It is easy to see
a

that for any 7, (A, a) = (A, g), gives :—k < 0, since cos :2;- = 0. Similarly, the point (A, or)
a

60

 

satisfying JD (A, -sin a, 7): - 3 cos (X, for each 7, which implies by (7.3) k = 0, also

k
gives :— < 0. For a fixed 7, these two (A, 0t)-values will bound all the (A, 0t)-values of
a
. . . d k . . . .
interest in the analysrs of ? (srnce they bound all the pornts which can be triple roots of
a

(8.4) involving k > 0 and g<a <7t). Therefore it is enough to check that

(g—k—J S 0. A numerical search then shows that 7 = 403 is the lowest 7—value at
a (A. a. Y)

which —d—k- = 0. QED.
d a

Comment: The triplet (A, 0t, k) associated with 7* = 403 is found to be (A, 0t, k) = (.38,
0.777 n, 2.38).

Theorem 3. If (A, k, 7) is such that there exist more than two values 0t obeying 0 < 0t <

1t with (A, 0t, k)6 E(7), thenA < J; = 0.577.

Proof: For 7 > 403, there exist triple roots solutions to (8.4). By continuity, the
existence of 4 real solutions requires the existence of triple roots, which in turn has been

related, for values k > 0, to values a2 < 0. Examination of a2, yields that a2 < O

requiresA < J; = 0.577 for any 7—value. Further, the existence of triple roots involve, for k

> 0, an angle obeying %< a < 7:, and a A-value bounded between the two positive A-
solutions obtained from a2 = 0. Application of Descartes rule of signs shows that four real
solutions of (8.4) involve three angles obeying 32?- < a < 7:, and one obeying 0 < a < E if

and only if a2 < 0 and 30 < 0, while the existence of four real solutions of (8.4) involve two

61

angles obeying %< a < 7t, and two angles obeying O < 0: <25 if and only if a.2 < O and a0

> 0. QED.
Theorem 4. If 7 > 7* then there are (A, k) such that there exist four values 0t obeying 0 <
or < 1t with (A, or, k)6 E(7).

Proof: It needs to be shown that if 7 obeys 7 > 403 then there exists some (A, k)6

111(7) that give rise to 4 real solutions of (8.4). It is equivalent to show that if ad—k— is
a

positive for 7, then Ell—If— is positive for 7’ > 7. Recall (8.8) and that the existence of triple
or

2
878a

 

. . n . . . . 7r
solutrons involve angles a > 2' Our interest rs 1n . In (8.8), srnce a > —-2—, cos a <

0. Note in (8.8) that ——2——- sinza > 0, since otherwise, for a < 3:, g-l-(-< 0, a
7 6sin205 2 da

. . . . . . . 7: d k .
contradrctron, srnce the Ot-lrrnrt pornts involve values a < —2- and a—— = 0. Now, srnce
a

 

 

 

dD(A,-sina,7)__ 2 + 2 2

+ >0,
dl’ 72A6 72Aésin2a 7242

2

878a

 

a tedious but straight forward calculation shows that > 0. Whence, for 7 > 403 there

exist four real solutions of (8.4). QED.

Theorems 2-4 indicate that the type of plot displayed in Figure 17 must be modified
for 7* > 7, so as to include a region of pairs (A, k) involving four shock angles 0t that are
real solutions of (8.4). This is shown in Figure 19 and 20 for the particular values 7 = 410
and 7 = 1000 respectively. Let us focus on values k > 0. Recall that a necessary condition to
get 4 real solutions of (8.4) involves, by continuity, the existence of triple roots, which in

turn involves values az< 0 and ao< 0. Pairs (A, k) related to triple roots of (8.4) are

62

associated with one angle 0t, triple root of (8.4), obeying %< 0: <7: and one angle

0t obeying 0 < a < % . For7 = 410, it can be shown numerically that a2< 0 is satisfied for

values 0.192 < A < 0.573. Thus, triple solutions of (8.4), and therefore 4 solutions, will be

bounded by these two A-values. Now, by our argument above, the existence of triple roots

which obey (8.6) has been shown to be related to values 3115— = 0, where g—k— is given by
a a

(8.8) since k > 0. A straight forward numerical analysis of (8.8) shows that triple roots of
the E(410) manifold are given by the points (A, 0t, k) = (0.356, 0.759 1t, 2.041) and
(A, a, k) = (0.4, 0.784 1t, 1.679). Further, by the symmetry of (8.4), triple roots are also the
points (A, 0t, k) = (0.356, 0.241 1:, -2.041) and (A, or, k) = (0.4, 0.215 1:, -1.679) (if k < 0).
These points bound the A-values that support four real roots of (8.4) as seen in Figure 19.

Even more, it follows from our previous argument that the boundary of these regions
. . . d k . .
supporting four real roots are pornts (A, k) obeyrng -d—— = 0 (although thrs boundary W11]
0:

not obey (8.6)). Some of the points of these boundaries correspond in the lower right plot

of Figure 16 with pairs (A, (1) obeying 0: >325 and :11—15- = 0. Whence, each (A, k)-pair on
a

the boundary of the regions supporting four real roots of (8.4) is associated with one angle

0t that is a double root of (8.4). Figure 19 just shows how these regions begin to develop.
The unfolding of the (A, k) regions sustaining four real roots is fully appreciated in Figure
20, which represents Fa(7) for 7 =1000. By a similar procedure, it can be shown that if 7 =
1000 the triples roots of (8.4) are given by the points (A, 0t, k) = (0.22, 0.673 1:, 4.045),
(A, 0t, k) = (0.5, 0.842 11:, 0.908), (A, a, k) = (0.22, 0.327 1:, -4.045) and (A, (X, k) = (0.5,
0.157 1:, -0.908).

Points (A, k)6 Pa (7) are associated with four angles 0t as explained in theorem 2

and Theorem 3. In particular, and for k > 0, the existence of four solutions or given by three

63

angles 0t obeying 325 < a < 7: and one angle or obeying 0 < a < % requires ao< 0. Further,

 

the condition ao< 0 implies k <\/-il—2(l—E—l-—)—l (viz. (6.15)). In the same way, the
l’

. . . . 71'
exrstence of four solutrons 0t given by two angles 0t obeyrng 2 < a < 77:, and two angles 0t

obeying 0<a<12€ requires a0 > 0. Further, the condition a0> 0 implies k

 

>\/—l—[1 — ——1—) -1 (viz. (6.15)). It follows then that (A, k)-pairs on the boundary of the

A2 27

regions sustaining four real solutions of (8.4) are associated with one angle athat is

a double root of (8.4) obeying g < a < 7:, one angle a obeying 325- < a < 7: and one angle
0t obeying 0 < 0: <1; if a0< 0. Also, (A, k)-pairs on the boundary of these regions are
associated with one angle 0t that is a double root of (8.4) obeying %< a < 7: and two
angles 0t obeying 0 < a <% if ao> 0. The case k < 0 gives a symmetric state of affairs

about %. For the particular cases 7 = 410 and 7 = 1000, as seen in Figures 19 and 20

respectively, it follows that pairs (A, k) associated with four solutions of (8.4) satisfy ao< 0.

Whence, for these particular values of 7, if k > 0, (A, k)-pairs sustaining four roots of (8.4)

involve three angles 0t obeying 1;- < a < 7:, and one angle a obeying 0 < a < 125.11: k < 0,
(A, k)—pairs sustaining four roots of (8.4) involve three angles 01 obeying 0 < a < %, and

one angle a obeying g—< a < 7:. Further, since ao< 0, if k > 0, (A, k)-pairs on the

boundary of these regions involve one angle or that is a double root of (8.4) obeying

64

g<a <7r,one angleaobeying O < a<~72£, and one angle 0t obeying -72£< or <7z'. If k <
0, (A, k)-pairs on the boundary of these regions involve one angle 0t double root of (8.4)

obeying 0<a<~72£, one angle 0t obeying 0<a<g, and one angle 0t obeying

7t
—<a<m
2

 
 
  

20 f k 4 real roots

   

Ot-limit points

 

 

 

 

 

 

 

 

Figure 19. Values (A, k)6 I‘a(7) for 7 = 410 depicted here is qualitatively similar to
previous Figure 17 for 7 = 15. However, since 410 > 7* = 403 certain (A, k) are
associated with four real solutions of (8.4). The (A, k)-pairs occupy a small region that
appears as two small (symmetric) line segments. The line segment endpoints (see points
TI and T2 in the blow-up) are triple roots of (8.4). For 7 >> 7* the four solution region
more fully unfolds as clearly seen in Figure 20.

65

40

30

20 1

10 i

——10

—20

-30

-4o

 

    
 

  

T1 4 real roots
a=90°

;/::f/"i 025 . 04. 21

   
  

 

 
    

 

4- g

 

’4

0. (16 k as
I\\\\\“-‘1 .25 o.
a=90°

4 real roots

 

 
   

Figure 20. Values (A, k)6 Fa(7) for7 = 1000. The unfolding of the region given by

(A, k)-pairs sustaining four shock angles a is more fully developed compared to Figure
19. The (A, k)-points T1, T2, T3 and T4 are associated with triple roots of (8.4). Tl and T2

7r
(cases k > 0) are related to one angle (1 triple root of (8.4) obeying a > — and one angle
2
7t
0t obeying a < -2-. T3 and T4 (cases k < 0) are related to one angle (1 triple root of (8.4)

7t 7t
obeying a < -2- and one angle 0t obeying a > -2-. For k > 0, pairs (A, k) on the boundary

of these unfolding regions are associated with one angle a double root of (8.4) obeying

7t 7t it
a > —, one angle a obeying a > - and one angle 0t obeying a < —. For k < 0, pairs
2 2 2

(A, k) on the boundary of these unfolding regions are associated with one angle at double

root of (8.4) obeying a < 3, one angle at obeying a > g and one angle a obeying
a < 3, (this is appreciated in the lower right plot of Figure 16). For k > 0, pairs (A, k)
inside these regions are associated with three angles obeyinga >325 and one angle
obeying a < 3 (Theorem 3, case a, < 0 and a0 < 0). For k < 0, pairs (A, k) inside these
regions are associated with three angles 0t obeying a <32:- and one angle a obeying

a>—.
2

66

9. QUALITATIVE FEATURES OF THE ELASTOSTATIC SHOCKS
In this section we deal with two qualitative features of the deformation on both sides

of a mechanically consistent elastostatic shock, namely the issue of fiber contraction vs.
elongation and the issue of possible loss of ellipticity. Recall that the deformation on 11* is
completely determined in terms of A. The deformation on IT by (4.24) depends on A,
0t and k. As shown in section 7 a weak shock involves the loss of ordinary ellipticity of the
governing differential equations on IT. Since F" = F" for a weak shock, it follows that a
weak shock involves a loss of ordinary ellipticity on II" as well. More generally, the
necessary and sufficient condition for the loss of ordinary ellipticity of the differential
equations (2.6) under plane deformations for (3. 1) is given by equation (4.6) in [2]

HHM§+H22 Mf-zle MIMZ =0 (9.1)
where

H11 = fl C11+2 “ 7(3 C11" C11) MI

H12 = [.1 C12 +2 fl 7(3 c11 C12 - C12) M? (9.2)

_ 2 2
H22 ‘“ C22+2“7(C11C22‘ C22+2C12)M1'

Here C11, Clzand C22 are the components of the C = FT F tensor, 7is the reinforcing

parameter and p. is the shear modulus.

For a given deformation characterized by C, the differential equations are elliptic if

and only if (9.1) has no real solutions M1 and M2 with M? + M; = 1. On the other hand,

if (9.1) has real solutions M1 and M2 with M? + M; = 1 then ordinary ellipticity does not

hold and M1 and M2 are the components of the normal vector to an associated weak

elastostatic shock in the undeformed configuration. A detailed analysis of (9.1) and (9.2)

was carried out in [2]. Further, the region of (C11, C12, 7)-space associated with loss of

ordinary ellipticity was characterized by various analytical and numerical means. In

67

particular, after some manipulation of (9.1), it is shown in [2] that triplets (C11, C12, 7) are
related to incipient loss of ellipticity if and only if

where 0 =27 5124-453. The functions §1=§1(C11, C12, 7) and§2=§2(Cll, C12, 7) are

fully given in Appendix D. Further, it follows that triplets (C11, C 7) define an elliptic

12 ’
deformation if and only if 0 < 0, while triplets (C11, C12 , 7) define a non-elliptic
deformation if and only if 0 .>- 0.

The deformation on IT and on IT may separately be elliptic or non-elliptic, whence,
the ellipticity status of a mechanically consistent elastostatic shock generates four
possibilities: the deformation on IT“ is elliptic while the deformation on Il‘ may be either
elliptic or non—elliptic; or the deformation on 11* is non-elliptic while the deformation on IT
may be either elliptic or non-elliptic. In fact, direct calculation for 7 = 15 shows that all four
possibilities may occur. Namely, the following triples are all associated with mechanically

consistent elastostatic shocks for 7 = 15:

31 e = -9.4 10‘5 on w (elliptic)
(A, a, k) = (1.02, —7t, 2.15) = _ , ,
180 0 = 1.07 107 on [‘1 (non -ellrpt1c)
0 = -9.4 106 on 1'1+ elli tic
(A, 0t, k) = (1.02, -3—l7t, 2.83) =5 _ ( ,p, )
180 0 = -1.7 106 on II (elliptic)

0 = 6.71 106 on IT“ (non -elliptic)
0 = 1.54 109 on ['1' (non - elliptic)

31
A, ,k=0.8,— ,1.36
( a ) ( 1807: )=>(

31 0 = 6.71 106 on 11* (non - elliptic)
(A, 0t, k) = (0.8, —7r, 3.63) = _ , ,
180 '0 = -1.1 108 on II (ellrptrc)

68

It is worth remarking that the ellipticity status on l‘l“ is more simply determined by whether
or not A >A (7) (elliptic) or A SA (7) (non-elliptic). For the case 7 = 15 we note that

A (15) = 0.983.

9.1 FIBER CONTRACTION

The loss of ordinary ellipticity in plane deformation for the strain energy density
(3.1) requires contraction in the fiber direction [2]. A preliminary step to consider the loss
of ellipticity on any side of a mechanically consistent elastostatic shock is to analyze the
fiber stretch due to deformations (4.23) and (4.24) on both 11* and IT. On I'l+ the fiber is
contracted if A < 1 and extended if A > 1.
Theorem 5. A mechanically consistent elastostatic shock involves fiber contraction on at

least one side of the shock.

 

Proof: consider first the fiber stretch on IT at the k-limit points. Using (4.28) and
(7.5) gives

- 3 l 3
C11: 12[1 -Zn%]=12(:+z 1117'). (9.4)

Clearly ifA < 1 then C11 < 1. Further, ifA: 1, then C11 < 1, since n2 ¢ 0 implies nI < 1.

For A > 1, using (7.4), then (9.4) can be expressed

 

Therefore, the fiber is contracted on IT at the k-limit points.

69

We now consider the region interior to the k-limit points curves I‘k(7) for A Z 1.

At those interior points, the shock strength k obeys (7.3). Consider first the negative sign in

(7.3). Then the fiber stretch by (4.28) is

 

l
D n .A, —
C11: A2 1'%n% + (14 y)+-;—n2(D(nl,A, 7))2 (9.5)

now from (7.2)
1
2
0< D(nl,A, y)<n2+4(-/1—2—1) (9.6)

For A > 1, (9.6) shows that D (n1, A, 7) < 11%, so that (9.5) gives

3 n2 1 l
- 2 2 2 2 _
Cll< A (I —Zn2 +—4 —1+—A2+-2—n2]—1.

If the positive sign is taken in (7.3) then, the last term in (9.5) is preceded by a minus sign,
so repeating the same steps, we arrive at the same result. QED.
Therefore, we conclude that a mechanically consistent elastostatic shock involves

fiber contraction on at least one side of the shock.

9.2 ELLIPTICITY ON IT AND IT

The example of (A, 0t, k) = (1.02, 13810”, 2.83) for7 = 15 given at the beginning of

Section 9 indicates that the defamation can be elliptic on both sides of a mechanically
consistent elastostatic shock. Here it is instructive to recall from [5] that under such
circumstances an intermediate deformation must lose strong ellipticity. Since the loss of
strong ellipticity does not imply the loss of ordinary ellipticity, the question arises as to
whether or not a mechanically consistent elastostatic shock also necessarily involves the loss
of ordinary ellipticity in some sense. In the context of plane deformation, it has been shown

under certain conditions for isotropic compressible materials that the loss of strong

70

ellipticity implies the loss of ordinary ellipticity [4]. The result for isotropic incompressible
materials is given in [8]. In the context of three-dimensions, the analogous result is given for
isotropic compressible materials in [10] while for isotropic incompressible it is given in [7].
Here we examine similar issues for the transversely isotropic incompressible material (3.1).
In what follows ellipticity refers to ordinary ellipticity. For (3.1) the loss of strong ellipticity
is given by (9.1) with the equal sign replaced by the S sign.

We now turn to analyze whether ordinary ellipticity holds on IT and II‘. The

mechanically consistent elastostatic shocks under consideration establish C11 = A 2 and

C12 = 0 on IT, so that it follows from [2] that ordinary ellipticity is lost on IT if and only

Ill

if A S )1 —51- A (7). Further, substitution of (4.23) into (9.1) yields equation (6.18),
Y

which is the equation for weak shocks. We now examine ellipticity on IT. Consider first the

case of a strong shock perpendicular to the fiber as treated in Section 6, i.e.,a = 12!. For

these particular mechanically consistent elastostatic shocks C11 = 1- -2—1—— and C12 at 0. It
1’

is shown in [2] that C , C = l——l—, C involves ellipticity if C at 0, but that
11 12 2 Y 12 12

this ( C11, C12) is on the loss of ellipticity boundary if C12 = 0. Thus for these orthogonal

shocks there is loss of ellipticity on 11* but not on II". In general, the ellipticity status of the
deformation on IT given by the sign of 0 depends on A, Ot and k by means of (4.28).

We now turn to examine the conditions under which a mechanically consistent
elastostatic shock may appear for a deformation gradient given by (4.23) on II”. The main
result is stated as follows:

Theorem 6. If (A, 0t, k, 7) correspond to a mechanically consistent elastostatic shock, then

there exists k such that the homogeneous deformation (4.24) is non-elliptic for (A, or, k, 7).

71

Notes:
l.- k = 0 retrieves the deformation on IT and k = k retrieves the deformation on IT.

2.- In general, (A, 0t, k, 7) will not correspond to a mechanically consistent elastostatic

shock.
Proof: Let C be the Cauchy-Green strain tensor associated with (A, 0t, k) via
(4.28). We shall show that there exists values k such that (C11, C12, C22) is associated

with a non-elliptic deformation for the given (A, (1)6 I‘ k (7). In particular, we shall focus

attention on the case M = N (the weak shock coincides with the strong shock direction in
the undeformed configuration). The case M = N is thus sufficient but not necessary to

establish loss of ellipticity.

Substituting from (4.25) into (4.28), the components of C corresponding to

 

 

(A.a.k.7)me
- -2 -2 2
1‘ A2N%+A'2Nf
_ 2 _2 _
E _ k(A2N2—A Nf)+lt2N,N2
12—

12193 + A'2N12

(9.7)

 

E _ 41+ -2EN1N2+E2/t-2N§
22 A2N%+A‘2Nf

with Nland N2 given by (4.7). Introducing (9.7) into (9.1), and using M1: N1 and M2 =
N 2, after a long computation gives

£215ny+E12712N§N2+12N§+241~I¥+

9.8
+2y(3/t4-}.2)NfN%+ 27(1-A'2)Nf'=0. ( )
Substituting from (4.25) into (9.8) yields
k267A4nf + k127A4ngn2 -
(9.9)

-(27A2n12(-2A2n%- A2 + 1)—(A2 n12+ A'2n%))=0.

72

Now (9.9) is a quadratic equation in E. For values (A, 006 Pk (7) there exist real roots k to

(9.9) if and only if the discriminant of (9.9) is nonnegative, namely

 

 

4"2+ 42+ 26_4 262 22
A 7A 7A 111 1’4

2 0. (9.10)

Clearly, every (A, 006 I), (7) , satisfies (9.10), since it satisfies (7.1). QED.
Both (9.9) and (6.14) are simultaneously satisfied when either k = 0 or

k2-3n2
2nl

 

The case k = 0 in (9.8) yields (6.17), the condition for the differential equations to lose
ellipticity on IT. This is expected since k = 0 gives a continuous deformation gradient

across the shock, so that the distinction between a strong shock and a weak shock on IT is

3n

2n

2
1

 

lost. The other case k = - is precisely the definition of the k—limit points (viz. 7.5).

Whence, we can conclude that at the k-lirnit points there is a loss of ellipticity on IT and,
further, that the shock angles obtained in the analysis of (6.13) define characteristic lines.
Hence we verify that the mechanically consistent elastostatic shocks may be regarded as
evolving from characteristic lines associated with the loss of ellipticity of the differential
equations for a homogeneous deformation on IT.

It is important to realize that the (A, 0t)-pairs consistent with shock existence, i.e.,
given by (7.1), are necessarily (rt-values given by (9.1) from which weak shocks are

obtained. Further, Theorem 6 indicates that the one parameter family of elastostatic shocks
with deformation gradients given by (4.23) and (4.24) is possible since there exists a 12-

value such that (A, 0t, k) define non-elliptic (CH, C C22). It shall now be shown that, in

12 ’
fact, there exist an intermediate state defined by (A, 0t, k) associated with a loss of
ordinary ellipticity in between the homogeneous deformation gradient defined by A on IT

and the homogeneous deformation gradient defined by (A, 0t, k) on IT. Since we consider a

73

family of elastostatic shocks depending continuously on the parameter k, it is enough to

show that if k > 0, then at least one of the two k given by (9.9) also obeys k Sk. Recall that
the solution of (6.13) has been denoted as E(7), and describes a two-dimensional closed

surface manifold in (A, a, k)-space. Similarly the solution of (9.9) will be denoted as E(7),

and describes a two-dimensional closed surface manifold in (A, 0t, k)-space. Note that if
a = :2:- then k > k, except atA (7) where k = k = 0 (just by a simple application of the
quadratic formula to (9.9) and (6.13) for a = 125). Now, for the purpose of the discussion,

restrict attention to a particular A-cross section of both E(7) and E(7) obeying A <A (7).

The only intersection of the E(7)-cross section and E(7)-cross section, on k2 0 are at k = 0

3 n _.
and k = - 2 2 (the k-limit points). It follows by continuity that the E(7)-cross section on
n
l

 

k2 0 is a continuous curve connecting the two k = 0 points on the E(7)-cross section via

the k-lirnit point on k > 0. Further, since the 0: :3; point on the E(7)-cross section is

contained within the E(7)-cross section, it follows that this E(7) curve: begins at one of the

k = 0 points, proceeds into the interior of the region defined by E(7) where it includes the

a = % point, exits the E(7)-cross section at the k-limit point, and proceeds to the other k =

0 point. It follows that g—Z-atoo at k-limit points. Since k is given by the root of a

quadratic equation, it can only “double back” on itself once. This establishes the existence

of a k < k for each cross section obeying A <A (7). It follows by continuity of E(7) and

E(7) that this result holds for A Z A (7), since E(7) and E(7) only intersect at the k—limit

points.

74

Figure 21 depicts the same (or, k)-slices of both manifolds E(7) and E(7), for the

value 7 = 15 as given previously in Figure 14. This figure confirms the intersection of E(7)
and E(7) at k = 0 and at the k-limit points. Further, it can be observed that for each k > 0,
there exist a k-value such that k > k, and for each k < 0, there exist a k-value such that k
(k. Therefore, in the present context k pararnetrizes a sequence of deformations connecting
the deformation on IT (k = 0) with that on IT ( k = k) and at least one of the k ensured by
Theorem 6 obeys 0 S lkl S |k|. This establishes that a mechanically consistent elastostatic

shock involves the loss of ordinary ellipticity at some deformation on the k parametrized

path connecting F+ and F‘. It has also been shown that such ellipticity loss is associated

with a weak shock which evolves into a strong shock. Furthermore, triplets (A, or, k) E E(7)
give a non-elliptic deformation (4.24) on [1‘ if there exist values k, and k, obtained from
the quadratic formulas as applied to (9.8) such that k, < k < 15,. This follows by the simple
observation that all triplets (A, or, k)6 E(7) obeying k, <k <k2, (where k, and k, are
obtained from (9.8) for the given (A, or» generate, by means of (4.28), (Cu, C12)-pairs
surrounded by (C11, C12)-pairs as given by (A, or, k)6 E(7), non-elliptic. Whence, these
particular (C11, C12)-pairs are fully contained inside the (C11, C12)-space which gives
non—elliptic deformations [2]. Therefore, each (oak) in the interior of the region defined by

the E(7) cross section is associated with a non-elliptic deformation.

75

H‘ A=I.02 7:15

\
“s -‘5
~~~~~
--------

._.A

 

 

 

 

 

 

   

*.: ........ - or a
25 50 75 100 “i 7150‘ 175

-2 . \ ‘\‘ -2 i
_4» ‘x ‘1' -4»
-6* \i' —6’
6 6 t
4 i
2L

0. (I
_2 .
-4 )
—6 -6‘

 

 

Figure 21. Different cross sections of E(7) and E(7) manifolds for 7 =15. The
manifold E(7) is related to values k, while the manifold E(7) is related to values k.

 

_ _ 3 n
Both manifolds intersect at k = k = 0, and k =k = - 2 2 (the k-limit points). Points

n

1

(A, or, k)_E E(7_) are associated with a non-elliptic deformation on IT i_f there exist points
(A, or, k,)6 E(7) and (A, or, k2)E E(7) such that kl < k < k2. Otherwise, the
ellipticity status of (A, or, k) E E(7) is checked by means of (9.3). The complete analysis
of the ellipticity status of (A, a, k)6 E(7) is shown in Figures 22 and 23. Fr_om the
cross sections, it is easy to verify that there exists at least one k obeying 0 S lkl S Ikl.
This establishes that a mechanically consistent elastostatic shock involves the loss of
ordinary ellipticity at some deformation on the particular path connecting F” and F' that
is parametrized by k. Further, the direction of every consistent elastostatic shock is a

characteristic line, since values a as obtained fromE(7) are also values a as obtained
from E(7).

More generally, the ellipticity of (A, 0t, k)6 E(7) on IT can be checked numerically

using the condition for 0 as explained after (9.3). The ellipticity status on IT for each (A, 0t,
k) E E(7) has been checked for 7 = 15. The results are shown in Figures 22 and 23. Figure

22 gives for k > 0 the projection into the (A, 0t)-plane of triplets (A, or, k)6 E(7) associated

76

with an elliptic deformation on IT. The significance of the elliptic region in Figure 22 is that

each (A, a) in this region is associated with a unique k > 0 (via

 

3cosa+JD (A, -sin a, 7) , , ,
k: 2 . ) such that (A, or, k)EE(7) rs assocrated wrth an
srn a

 

elliptic deformation on IT. Values k < 0 give a symmetric state of affairs about a =25

(namely, if (A, or) is contained in the IT elliptic region in Figure 22, then (A, 1t—0t) is

contained in the corresponding IT elliptic region involving k < 0 (via

3cosa- JD( -sina, 7)

k = )). Similarly, if (A, 0t) is contained in the IT non-
2 sin a

 

elliptic region in Figure 22, then (A, tt—a) is contained in the corresponding IT non-elliptic
region involving k < 0. This follows from (4.28) and [2]. By (4.28), if values (A, 0t,
k)EE(7) give rise to pairs (C11, C12), then values (A, 1t-0t, -k)EE(7) give rise to
(C11; C12). In [2] it was shown that if (C11, C12) defines an elliptic (respectively, non-
elliptic) deformation, then (C11; C12) also defines an elliptic (respectively, non-elliptic)

deformation.
. . . . . 71
Notice that mechamcally consrstent elastostatic shocks assocrated to angles a = —

are elliptic on IT as it was remarked at the beginning of this section. Also, the k-limit points
give rise to non-elliptic deformations on IT. This is fully appreciated in Figure 23. Figure
23 plots shows how each cross section is divided into segments according to its ellipticity
status. In particular, we find that each cross section involves four segments: two segments
giving rise to values (or, k) that define deformations that are non-elliptic on IT, and the other
two segments giving rise to values (0t, k) that define deformations that are elliptic on IT.

Each elliptic or non-elliptic segment in each cross section follows the central symmetry

77

about (or, k) = (g, 0), i.e., if (or, k) belongs to the elliptic (respectively, non-elliptic) segment

on IT then (rt—0t, -k) belongs to the elliptic (respectively, non-elliptic) segment on IT.
Although this discussion of the IT ellipticity/non-ellipticity status on the E(7)-cross

section focused on the particular value 7 = 15, we find that similar features follow for any

 

 

   

 

 

 

 

othervalue7.
or
175 l
150 ) k=0
125 T ’,/’/non-elliptic \“x‘
. I,” “\ non
100 E " \ x elliptic
75 7» ’/
50 r -----------
25 - '
u .ls-lirtitpoims. -A- ,
0.2 0.4 0.6 0.8 1

Figure 22. For values k > 0, the plot gives the projection into the (A, Ot)-p1ane of the
triplets (A, a, k) E E(7) elliptic on IT when 7 = 15. Every point (A, or) inside the IT
elliptic region is associated with a unique point of the E(7) manifold. Recall Figure 18,
pairs (A, (1)6 T k (7) inside the curves k = 0 correspond to a unique point (A, (1, k) E E(7)
such that k > 0. Pairs (A, (1)6 I‘k(7) between the (lower) dashed line k= 0 and the k-
limit points correspond to two points (A, or, k)6 E(7) such that k > 0. In the latter
region, the (A, 0t)-pair associated with an elliptic deformation on IT is the one with the
greater value k. This is further appreciated in Figure 23. The k-limit points are associated

It
with non-elliptic deformations on IT. Points (A, (1) = (A, —) are associated with elliptic
2

deformations on IT. The closeness of some of the boundaries of the IT elliptic regions
with the k-limit points is fully appreciated in Figure 23. Values k < 0 involve a

n:
symmetric state of affairs about a = 2’ namely, if (A, or) is associated with an elliptic

(respectively, non-elliptic) deformation on IT then (A, n—a) is associated with an elliptic
(respectively, non-elliptic) deformation on IT.

78

elliptic ’
7 non
enu;§;::::> 2-

 

A: 1.02
7:15

A

 

‘ ‘ ‘ ‘ ‘ T (1
25 50 75 100 1 150 175
non _2 _
. . lliptic
elllptlc

 

elliptic

2 . no.“ .
elllptlc

 

 

 

25 50 75 100 125 150
I101)
tflhpfic

—4 - elliptic

 

Figure 23. Different cross-sections of the E(7) manifold in the (01, k)-plane, for 7 = 15,
showing the IT ellipticity status of points (A, or, k) E E(7). Each cross-section involves
four segments: two segments giving rise to (A, or, k) E E(7) non-elliptic on IT and two
segments giving rise to (A, (1, k) E E(7) elliptic on IT. The plots at the left comer depict
the closeness of the k-limit points and the transition between pairs (or, k) associate with
elliptic and non-elliptic deformations on IT. In each cross section the central symmetry

7:
about (or, k) = (3, 0) is kept, namely if pairs (0t, k) define an elliptic (respectively, non-

elliptic) deformation on IT then pairs (rt—or, -k) define an elliptic (respectively, non-
elliptic) deformation on IT. The 1-1 correspondence of the projection into the (A, or)-
plane of the E(7) manifold involving deformations elliptic on IT and either values k > 0
(Figure 22) or values k < 0 is obvious from each cross section.

791

 

175

10. ENERGETICS

The existence of an elastostatic shock modifies the mechanical energy balance of the
elastic body. Consistency with the purely mechanical version of the second law permits
certain directions of shock advance and excludes others. The argument by which some
shocks are admissible and the others are not admissible is given below. Consider a quasi-
static time dependent (time playing the role of a history parameter) family of equilibrium
states. Let u be the quasi-static particle velocity given in material coordinates, then, in a three
dimensional case, it is required that

j t-ndA-i[W(F)dv20 (10.1)
at) dt D

for every regular sub-domain D of R at each instant of the time interval. Here, t is the
nominal traction vector. The first term gives the rate at which work is done by forces that are
external toa D, while the second integral is the elastic energy storage rate. If the field
quantities have the classical smoothness, then these two terms are equal and (10.1) is
satisfied with equality. It is the presence of mechanically consistent elastostatic shocks that
allow the inequality (10.1). Physically this inequality gives the requirement that the rate at
which energy is put into the system cannot be greater than the rate at which energy is stored
into the system.

For a complete discussion of the dissipation condition that separates physically
admissible solutions among the many possible ones, see [6]. Here, we just summarize the
procedure. Consider, as for (10.1) a quasi-static time dependent family of equilibrium
states. Further, suppose that these equilibrium states involve the existence of a mechanically

consistent elastostatic shock surface 5 which advances into IT. The time-rate of change of

the energy U = dC-l—t- [W (F) dv in the system is given by [6]
D

U: [ t-udA-[sv-NdA. (10.2)
a

8D
t

80

In (10.2) 5, gives the material shock 8 at time t, N is the normal to the shock 5,, V is the

velocity of the points on 8, and S is

+

s .=.[W(F)—0aBNfi F,,, N, _. (10.3)

The fact that the shock is considered to advance into IT means that V o N > 0. If the shock
is considered to advance into IT then V 0 N < 0. Based on the discussion following (10.1)
the existence of an energetically admissible elastostatic shock requires in (10.2):

S 2 0 if the shock advances into II”,

S S 0 if the shock advances into IT. (10.4)
If S > 0, admissible shock motion involves IT being converted into IT, in which case IT is
favored. On the other hand, if S < 0 then IT is converted into IT, in which case IT is
favored. If S = 0 then neither state IT nor IT is energetically favored.

For the case of an anisotropic incompressible non-linear elastic material under plane

strain (10.3) is equivalent to [9]
= + Z
s _ [W(F)]_ + k 1'12. (10.5)

Here Z is a coordinate system located along the spatial shock so that the 1 direction is the
vector defining the shock direction in the deformed configuration land the 2 direction is the
normal vector to the shock 11.

In the literature [9], S is called shock driving traction, and is seen as the magnitude
of a fictitious nominal traction acting on the shock by the surrounding material. From a
variational point of view, and for the particular case of dead load, it has been shown that S =
0 is a necessary condition for the displacement field it to minimize the potential energy over

the class of functions considered [19].

81

10.1 FAVORABILITY OF IT VERSUS IT

To obtain a useful expression for S, we note that the strain energy on II+ and IT is

given by (6.2) and (6.6) respectively. The component 1122 by a standard procedure yields

1
7122:” nln2([A2-F) +2}’/12(Az-I)]. (10.6)
Thus

_1 4 44 3 4 3
S— -2-(-/,tk 7A nl -/.tk 47A nln2+

(10.7)
+2,u k2 7A2niz(-2A2ng- A2 +1)—[.tk2(A2 nf+ A°2 n3».
Since (6.13) is satisfied for any mechanically consistent elastostatic shock (10.7) yields

_ 1 4 3 3
s_ an M k nl (2n2 +knl). (10.8)

This is in agreement with [7] where it was shown that the energy jump across a shock for
any incompressible anisotropic material is cubic in k to leading order.
Note also that (6.12) and (10.7) yields

p— - p+ =i S. (10.9)

k2
Namely if p— - p+> 0 then S > 0, so that IT is favored, and vice versa. This is analogous to
results obtained in the context of gases, for which a gas dynamical shock moves, relative to
the gas, into the zone with less pressure. This, at least for the neo-Hookean reinforced
material, gives an additional physical meaning to a hydrostatic pressure which is a pointwise
constraint due to the incompressibility of the material.
Also, by (4.27) and (4.28) we can write

- 2
C -c+

11 11" “szznf

 

S. (10.10)

Whence, if S > 0, then CII - Ch > 0 and vice versa. Recall that './C 11 gives the fiber stretch

and, that for a mechanically consistent elastostatic, the fiber is compressed on at least one

82

side and possibly on both sides (Theorem 5 of Section 9). Thus the energetically favored
state is that which involves either elongation or else less fiber contraction. This is consistent
with experimental results of kink-band formation and broadening in fiber reinforced
composite materials, where it is found that the mechanism of kink-band broadening involves
an incremental gain in the length of the kinked fiber with respect to the unkinked fiber. The
argument follows from a simple observation of the deformation: to accommodate the
contraction of the unkinked region, the kinked region must suffer an incremental gain in
length with respect to the unkinked region [34]

Our interest now focuses on the sign of S, which according to (10.8) is given by
sign [S] "=— sign [k 111 (2 n2 +k "1)]- (10.11)
Consider first the k-limit points. Introducing (7.5) into (10.11) yields
sign [S] E sign [--3— n2 :l.
4 2
Whence S S 0 at the k-limit points. For the particular case of shocks perpendicular to the

fiber, i.e., a = %, then (10.11) yields

sign [S] 5 sign [k2],
whence, S 2 0 whena = g. In general, for a (A, (1)6 T k (7) the sign of S depends on the

associated k-value given by (7.3). By means of (7.3)1 the parenthesis in ( 10.11) yields

 

2n2 +an =%[n2i\/D(A, n1, 7)) (10.12)

We now turn to consider how the sign [S] depends upon the various possible combination
of nl and n2. Consider first values (A, a, k)6 E(7) such that k 2 0. For all these
mechanically consistent elastostatic shocks 111 < 0, but 112 may be less, equal or greater than
zero. Let 112 < 0, then, as we know, the only positive value of k is obtained if the negative

sign is taken in (7.3),, which in turn gives the negative sign in (10.12) and yields S > 0. The

83

case 112 = 0, i.e., a = %, gives 8 _>. 0. For 112 > 0, if the positive sign is taken in (10.12), so

that (7.3) gives positive values of k, then S < 0, in agreement with our discussion of the k-

limit points. On the other hand, for 112 > 0, if the negative sign is taken in (10.12), so that

 

(7.3) gives positive values of k, then s > o if n2 —\fi)( A, n1, 7)< 0, but s < 0 if

 

n2 - JD ( A, n1, 7)> 0. The values k S 0, associated to values 1t-0t (here 0t would be the

angle associated to k_>. 0) give the same result. Namely, if values (A, or, k) give S > 0 then

values (A, “It-(X, -k) give S > 0, and if values (A, 0t, k) give S < 0, then values (A, no, -k) give
S < 0. Hence the sign[S] will preserve the central symmetry about the point (a, k) = (125, 0)

on each fixed A—cross section.

The projection of the E(7) manifold satisfying S > 0 on the (A, (1)-plane is shown in
Figure 24 for 7 = 15 and k > 0. In Figure 24, the transition between S > 0 and S < 0 defines
the mechanically consistent elastostatic shocks that are stable in the sense of [19]. Further,

in Figure 24 values (A, (1)6 11(7) satisfying S > 0 are associated with a unique point (A, (1,

 

3 cos a+JD (A, -sin a, Y)

k)6 E(7) (via k = .
2 srn a

 

). The procedure for determining the S >

0 region is the following. Recall by our discussion above that the k-limit points yield S < O.

From the k-limit points, consider horizontal lines, i.e., values of (1 constant. At a particular

 

(A, a)—value, the only possibility to get 8 > 0 is given by 112 — JD (n1, A, 7)< 0, which

yields by (7.1) the condition

 

 

4 + 2 _ 4___2—._ -_2_2_> O (10.13)
A 2 7A 6 7 A 6n? 7 A
Condition (10.13) can in turn be written as:
A 6(.4)+ A 4(4-3) +[3— 22) > 0. (10.14)
1’ Y Ynl

84

Now, an analysis similar to the one in Section 7 shows that A = 1- —l— E A (7) makes

21’

( 10.14) equal zero. Whence S = 0 at A =A (7). This is expected since k = 0 at A =A (7).

For all values 7 > g the analysis of fT/(I— in (10.14) using 111 = -sin 0t shows that if

A >A (7) then S < 0. However, if A <11 (7) the cross sections, according to (10.14) is

divided into four segments: two segments in which S > 0, and two segments in which S < 0.
As previously remarked these regions follow the central symmetry about (or, k) = (325, 0). In

each cross section there are four points obeying S = 0: two of them are trivial, in the sense
that k = 0, while the other two obey (10.14) with equality. This is fully appreciated in Figure
26, which shows values S > 0, S < 0 and S = 0 related to two cross sections (A = 0.8 and A
= 0.3) of the E(7) manifold for 7 = 15 .

The mechanically consistent elastostatic shocks that satisfy the dissipative inequality
S > 0 naturally attract our attention since they can be viewed as allowing a bifurcation away
from the simple homogeneous deformation (4.23). To the extent that this is representative
of kink-band formation, we shall refer to these as energetically admissible elastostatic
shocks for kink-band formation, which shall subsequently be shortened to energetically

admissible elastostatic shocks for the ensuing discussion.

85

175 t
150 1 k=0

 

 

k — limit points;

 

0.2 0.4 0.6 0.8 1
Figure 24. Projection of the E(7) manifold satisfying S > 0 on the (A, (1)-plane for
7 = 15 and k > 0 is contained within the solid curve. This bounding curve is defined by a
k = 0 segment (above) and an S = 0 segment (below). The (A, (IQ-pairs obeying S > 0 are

inside the k-limit points. Further, and in correlation with Figure 22, there is a 1-1
correspondence between (A, (1)6 Tk(7) obeying S > 0 and points (A, (1, k) E E(7), as can

be seen in Figure 27. All the points (A, a) obeying S > 0 satisfy A <A (7), therefore,
the shocks associated to S > 0 lose ellipticity on IT. For values k < 0, there is a

It
symmetric state of affairs about a =;, namely, if (A, (1)ETk(7) obeys S > 0

(respectively S < 0), then (A, 1H1) E F k (7) obeys S > 0 (respectively S < 0).

It follows from the above discussion that A <A (7) for energetically admissible
shocks, i.e., energetically admissible bifurcation from the simple homogeneous defromation
(4.23) only occurs under compressive loading. Furthermore, the energetically admissible
elastostatic shocks lose ellipticity on IT, although the converse is not always true, i.e., not
every mechanically consistent elastostatic shock that involves a loss of ellipticity on IT is an
energetically admissible elastostatic shock. In addition, it follows from the analysis of
section 9.2 that energetically admissible elastostatic shocks may involve either elliptic on
non-elliptic deformations on IT. This is easily appreciated by comparing each

corresponding cross section of Figure 23 and Figure 26, which represent different cross

86

sections of the E(7) manifold when 7 = 15. Although comparison of the cross section A =
0.8 in Figure 26 and Figure 23 shows that the non-trivial points S = 0 (i.e., k 7: 0) are
elliptic on IT, comparison of the cross section A = 0.3 in Figure 26 and Figure 23 shows
that the non-trivial points S = 0 are non-elliptic on IT. The points of intersection between
the curves S = 0 and the boundary of pairs (A, 0t) giving the transition between elliptic and
non-elliptic deformations is (A, or) = (0.61, 0.179 1t) if k > 0, and (A, or) = (0.61, 0.821 1t) if
k < 0. Therefore, the curves S = 0 give elliptic deformations on IT for pairs (A, 01) such that
A > 0.61, while the curves S = 0 give non-elliptic deformations on IT for pairs (A, (1) such
that A < 0.61. In summary, among the four choices that an elastostatic shock generates
regarding its possible ellipticity status, the energetic analysis finds admissible only two
cases, namely, an energetically admissible elastostatic shock is related to a non-elliptic

deformation on IT and to either an elliptic or non-elliptic deformation on IT. Further, for A-

values close and less thanA (7), the incipient loss of ellipticity, the ellipticity status of an

energetically favorable mechanically consistent elastostatic shock involves a non-elliptic

deformation on IT and an elliptic deformation on IT.

10.2 MAXIMALLY DISSIPATIVE ELASTOSTATIC SHOCK MOTION

Recall that S determines the local rate of mechanical energy dissipation. In order to
be energetically admissible a mechanically consistent elastostatic shock that creates IT at the
expense of IT must satisfy S > 0. For a given A, Figure 22 shows that (1 pararnetrizes the
energetically admissible elastostatic shocks obeying k > 0. Namely, k = k((1) at fixed A and
7 (viz. (7.3),), which in turn gives S = 8(a). It is now natural to seek the particular
energetically admissible shock that maximizes the energy dissipation rate S. According to
the shock driving traction interpretation, this gives the particular shock for which the traction
exerted by the surrounding material is maximum. For a fixed A-value, such maximally

dissipative elastostatic shocks are found via

87

2s-

0. 10.15
d a ( )

Similar considerations apply to energetically admissible shocks obeying k < 0, however, in

view of symmetry we may restrict attention to values k > 0. To get a useful expression for

 

 

(10.15), and since 111 = - sin (1 and n, = cos or, we write (10.8) by means of our previous
argument,
S 4 = _ l k3 sin3a (cos (1 —JD (A, sin (1, 7)). (10.16)
u M 4

Now, (10.15) using (10.16) and (7.3),, with the positive sign, after a long computation
yields

 

cos a
7A 6 sin4

 

— cos a+JD(A, sin a, 7) — 0, (10.17)

a =
where use has been made of the fact that k > 0 and sina at 0. For 7 = 15 the root or = a(A)

obeying (10.17) can be obtained numerically. The result is shown in Figure 25 for 7 = 15

where the curve is denoted as Smax. In particular, when 7 > i it is found that (10.17) has

exactly one root, indicating that there is exactly one maximally dissipative elastostatic shock

obeying k > 0 for A <A (7). Notice that Smax involves shock angles (1 <25 when k > 0

(symmetric state of affairs for k < 0, i.e., a > g). In fact, this angle restriction is valid for

any value 7, not only 7 = 15. First, it is easy to check that :—S < 0 at or = %. Now, the
a
argument follows from a simple observation of (10.16), since to satisfy (10.16) it is

necessary that 01 < 3 (cos 0t > 0). Correlation of Figure 25 with Figure 22 shows that

maximally dissipative elastostatic shocks involve elliptic deformations on IT. In fact this

follows for any value 7 as will be shortly explained in connection with Figure 28.

88

Figure 26 shows sections of the E(7) manifold for 7 = 15 and the separate segments

obeying S > 0 and S < 0, The (01, k)-pairs associated with S = 0 and SW are also shown.
Cross sections involving both S > 0 and S < 0 must obey A <A (7) since values (A, or, k)6

E(7) with A >A (7) always give S < 0

175 *
150 * k
125 *
100 ’
75 .
50 *
25 *

0

 

 

k—Iimit points

 

0.2 0.4 0.6 0.8 1

Figure 25. In addition to the correlation with Figure 24, pairs (A, or) 6 Sum are shown
for values k > 0, for the particular value of the reinforcing parameter 7 = 15. The picture

It
shows that Smax involves elastostatic shocks obeying (1 < —. In fact, this remains for
2

7:
any value 7. Values k < 0 involve a symmetric state of affairs with respect to —.

7t
Namely, for values k < 0, Sum involves elastostatic shocks obeying or > —. This is
2

clearly appreciated in Figure 24. Even more, correlation with Figure 22 indicates that the
shocks associated to Sm involve elliptic deformations on IT.

89

 

 

 

 

 

 

Figure 26. Here we show the only two plots of Figure 15 which involve A <A (7) as

only these cross sections give values 8 > 0. For values A <A (7), each E(7) cross

section is separated into four segments according to the value of S: two segments in
which S > 0, and two segments in which 8 < 0. The symmetry of each segment in each
cross section is as follows: if pairs ((1, k)6 E(7) satisfy S > 0, then (rt-a, -k)E E(7)
and S > 0, and if pairs ((1, k)6 E(7) satisfy S < 0, then (rt-(1, -k)€ E(7) and S < 0.

71:
Therefore, each segment keeps the central symmetry about (a, k)=( -2-, 0). Correlation

with Figure 23 shows that Smu involves elliptic deformations on IT, while pairs obeying
S > 0 may involve either elliptic or non-elliptic deformations on IT.

90

Figures 27 and 28 combine the main features associated to values k > 0 of the E(7)
manifold when 7 = 15. These figures give the cross sections of E(lS) at A = 0.8 and A =
0.3. In each figure the point E is the generator of the oz-Iimit points and the point G is the
generator of the k-limit points. The point D is associated with S = 0 as given by Figure 26.
In particular, the segment AD gives (or, k)-pairs obeying S > 0. Hence the segment DH
gives points ((1, k) obeying S < 0. The points F and B are associated with the transition
between (or, k)-pairs giving rise to elliptic deformations on IT and (or, k)-pairs giving rise to
elliptic deformations on IT as given by Figure 23. In particular, the segments AB and FH
are associated with (01, k)-pairs giving rise to non-elliptic deformations on IT, while the
segment BF is associated with ((1, k)—pairs giving rise to elliptic deformations on IT. Note
that the point F does not belong to the segment AD at the A = 0.8 cross section (Figure 27)
but it does belong to the segment AD at the A = 0.3 cross section (Figure 28). This
indicates, that the energetically admissible elastostatic shocks may involve either an elliptic
or a non-elliptic deformation on IT, as has been previously remarked. Recall that the point

D and F coincide at (A, or, k) = (0.61, 0.179 1:, 3.17). The point C is associated with Smax as

given by Figure 26.

91

 

 

 

6 .'
: E A=0.8
4 . Y=15
l
2 r
l
.- - ..1 -. - <1
; 150 175
-2 i
-4 E
-6 1

Figure 27. Cross section of the E(7) manifold for 7 = 15 at A = 0.8. Points A-H have
special significance in the k 20 portion of this cross section. The point B is the
generator of the Ot-limit points and the point G is the generator of the k-limit points. The
segment AD gives (01, k)-pairs obeying S > 0 and the segment DH gives points ((1, k)
obeying S < 0. The segments AB and FH are associated with (or, k)-pairs giving rise to
non-elliptic deformations on IT, while the segment BF is associated with ((1, k)-pairs
giving rise to elliptic deformations on IT. The point C is associated with Sm“.

 

 

 

k
6 r
: A=0.3
4 T 7:15
i
2 r
i .A
-. .. - . ..... a
; 25 125 150 175
-2 E
-4 l
l
_6 C

Figure 28. Cross section of the E(7) manifold for 7 = 15 at A = 0.3. Points A-H
maintain the same significance as in Figure 22. Note however that point D has migrated
around points E and F. Once again, the point E is the generator of the a-limit points and
the point G is the generator of the k-limit points. The segments AB and FH are associated
with (a, k)-pairs giving rise to non-elliptic deformations on IT, while the segment BF is
associated with (or, k)-pairs giving rise to elliptic deformations on IT.The segment AD
gives (01, k)-pairs obeying S > 0 and the segment DH gives points ((1, k) obeying S < 0.
The point C is associated with Sm“.

92

Figure 29 gives the cross section of the E(7) manifold for 7 = 15 at A = 1.02. Since 1.02

>A (7 )= 0.983, every ((1, k) on this cross section gives S < 0. As in Figures 27 and 28, the
point B associated with the (1-limit points and the point G associated with the k-limit points
are shown. Also, the point I associated with the or-limit points and the point I associated
with the k-limit points are shown. Further, in correlation with Figures 27 and 28, the
segment BF is associated with (a, k)-pair giving rise to elliptic deformations on IT. Since S
< 0, the points A, D and H do not appear on the plot. Therefore, the point C associated with
Smax can not appear on the plot either. In particular, the points A, B, C, D, H, I and J will

coincide at the value A =A (7) which is associated with a = g— and k = 0. This is clearly

seen in Figure 30 that gives the cross section of the E(7) manifold for 7 = 15 at A = A (7 )=

0.983. In this plot the point A has been marked. Recall that this particular cross section also
obeys S < 0 except at precisely the point A which generically is representing all the above

mentioned points.

93

 

 

 

 

 

F

t A: 1.02

. F E

’ r

I
--.S--L 0t

; 25 50 75 100 125 150 175

t
Figure 29. Cross section of the E(7) manifold for 7 = 15 at A = 1.02. The points E and
I are the generators of the or-limit points and the points G and J are the generators of the
k-limit points. In correlation with Figures 27 and 28 the segment BF is associated with
((1, k)-pairs giving rise to elliptic deformations on IT. Note that the points A, H and D
do not appear on the plot because S < 0 everywhere on the cross section. It follows that
C can not appear on the plot either.

F

i

L

................ 0t

: 175

l

l

Figure 30. Cross section of the E(7) manifold for 7 = 15 at A =A (7 )= 0.983. The

points E and G are the generators of the a-limit points and the k-limit points
respectively. In correlation with Figure 27 and 28 at the point A coincide B, C, D and H.
Thus, the ((1, k)-pairs in the cross section obey S < 0 except at A where S = 0. Further,
the segment AF is associated with ((1, k)-pairs giving rise to elliptic deformations on IT.
In correlation with Figure 29, at the point A also coincide the points I and J.

94

10.3 EVOLUTION OF THE KIN K ANGLE

The criteria of maximum dissipation singles out an energetically admissible
elastostatic shock from the range of energetically admissible elastostatic shocks at fixed A
and 7. It is useful to study the kinking angle (1) and the shock angle or for these special
solutions. Figure 31 shows the kinking angle (1), and the shock angle or vs. the stretch A for
the Smax family of solutions involving values k < 0 for 7 = 3, 15, 50, 150. For comparison,
(1) and Ct for the S = 0 family of strong shock solutions involving k < 0 are also shown. The
procedure to get the kinking angle is the following: for S = 0, points (A, (1) satisfying

2 cos a
sin a

 

(10.14), with a 2 g are associated to a value k = , which in turn gives the kinking
angle (11> 0 via (6.5). The curve is denoted as 05:0. It is easy to see that if A —9 0, then

t) z —> 0 (by (6.5) since as a—) 11—) 0). For 3 , points (7., (1) satisfying (10.15), with
S 0 2 max

at 2 g are associated to a value k by (7.3), with the negative sign, so that k < 0, which in

turn gives the kinking angle (1) > 0 via (6.5). The curve is denotes as 05“,”. Now, if A —> 0,

cos a
’16

 

then cm —> 75’- (since as 0t—9 325, by (10.15) and (7.3),, then k——> , which draws by

 

(6.5) tan 95...... —) — , indicating that its“, —> g). Similarly, the curve 015:, gives the

cos a
shocks angle or associated to S = 0, and (13,,“ gives the shock angle 01 associated to SW.

Both curves approach :2:- as A —-) 0 , which is consistent with our analysis of Section 7.

95

 

 

 

 

 

175 r 175 i
150 » F3 150 >
125 l 125 »
100 i 100 .
75 l 75 , ¢Snm
50 > 50 t
25 » 25
o A o A
175 t 175 »
150 - 150
125 . 125 .
ioo . 100 .
75 » 75
50 . 50 »
25 . 25 »
o A 0 - - g - A

 

 

 

 

 

 

 

 

0.2 0.4 0.6 0.8 l 0.2 0.4 0.6 0.8 1

Figure 31. Curves am“, om“, “3:0 and ¢s=o that give the shock angle 0t and the
kinking angle (it at Sm, and S = 0 respectively for the particular values 7 = 3, 15, 50,
150. The values k associated to the curves are values k < O. The curves as”, and “3:0

7!
involving values k > 0, are symmetric about a=—, while the curves tbs”, and
2

(DH involving values k > O, are symmetric about (it = 0. Note that the curves am“ and
its”, given by Sm do not intersect nor do the curves (15:0 and (DH given by S = O. This
is in agreement with our analysis of (6.5), where it was shown that the angles a and iii
can not coincide. As 7 increases, for fixed values A the plots depict ever greater values of
4), i.e., the more anisotropic is the material the more rapid is the rotation of the kinked
fiber. Further, for each 'y-value, both $3,“, or ¢S__o show a first stage at which, small
changes of A cause a very rapid rotation of the kinked fiber. This is followed by a more
steady state, in which the kinking angle ¢ does not vary much with A. Nevertheless, the

7:
behavior of both curves departures asA —) 0, since om“ —-> 3 and (354,—) 0. Note also

that the curve (15:0 involves angles a greater than OLSW.

The deformation in the kinked region can be finally established by (4.28) and our
discussion following (10.10). Recall that the fiber stretch is given by A on IT“ and by what

we shall denote f1? = t/Cl-l on IT to avoid cumbersome notation. The curve S = 0 gives A

96

as the fiber stretch on both sides of the shock, i.e., the fiber stretch on IT and IT coincide.
The curve Sm involves S > 0, so that by (10.10) the elongation of the fiber on IT is greater

than the elongation of the fiber on IT. Figure 32 shows the fiber stretch A on IT vs. the
fiber stretch fl? on IT as given by SW for y = 3, 15, 50, 150. Notice that these curves,
apart from satisfyingJ—IE > A, may involve fiber extension on IT, i.e., K > 1. Further,
values J75 > 1 are typical at the formation of the kinked zone as 7 increases. The dashed
line in the Figure gives values A =JE , so that it corresponds to the curve S = 0. Also, note

that for the different curves Smax depending on y, as A —9 O, x/E —>A (y). This can be

shown, and in fact is true for any (A, 0t, k)-curve satisfying k—> oo asA —> 0. Taking

theA —> 0 limit in (6.4), since 0t—> :2;— the fiber stretch R on IT yields

 

JE—aAJ—zk cosa+ k2. (10.18)
Now, by (7.3)2 , (10.18) can be written

1
_1_ _
K 2 —) A(J—r % cos aJD (A,-sin a, y) +iD (A, -sin a, ”)2 (10.19)

 

Upon use of (7.1), in theA—>0 limit, (10.19) draws finally that J? —> l—-2—l— =
Y

A (y ). This result together with Figure 32 allows one to conclude that the curves Smax obey

[If > A (y). This indicates that among the two choices that an energetically favorable

elastostatic shock generates regarding its possible ellipticity status, maximally dissipative
elastostatic shocks follow one pattern, namely, the deformation on IT is non-elliptic, while

the deformation on IT is elliptic.

97

 

 

 

 

Smax

y=150
1.1L

y=50

1 7:15

09 > 7

Y:3 /
08 i ‘/
0.7 a . . / . - A

02 04 06 08 1

Figure 32. Fiber stretch A on IT vs. the fiber stretch x/E on IT as given by Smax for
y = 3, 15, 50, 150. In all cases A SA (y)< 1, but x/E may obey either J]? < l or

\[E > 1. This is consistent with Theorem 5 in Section 9.1. Further, for each curve, R
> A, indicating that the elongation on IT is greater than the elongation on IT. This is in

agreement with our discussion of (10.9). Values x/E > 1 follow immediately after initial
kinking for values 7 sufficiently large. Note that for each curve, asA —> Othen
if]; —) A (y). Further, for each curves/E > A (y), therefore, maximally dissipative
elastostatic shocks keep an elliptic deformation on IT. The dashed line represents values
A =4; , so that it corresponds to values of the fiber stretch A on IT vs. values of the
fiber stretch \[E on IT as given by S = 0.

10.4 DISCUSSION

The significance of these results is related to the fiber kinking mechanism. During
kink-band formation, experimental results show that the kinked fiber rotates from its initial
direction to a final position, which will remain almost fixed in the propagation of the kink-

band. That is the reason to denote this essentially fixed angle as a lock-up position. At this
stage the band angle, while material dependent, is observed to keep within the range $1?

[35]. The lock-up angle of the fibers determine the angle of the kink-band broadening
propagation which is considered a mechanical mechanism rather than a thermodynamical
one in a steady-state analysis, as our case. Furthermore, this process is considered to satisfy

a volume preserving deformation requirement which determines the value for the band angle

98

as or = g + %, where 11) is the fiber kinking[34]. Further, experimental results [35] show

three stages in the behavior of a fiber at the beginning of kink band formation. An early
stage of slow fiber rotation, followed by a rapid rotation due to a geometric softening
mechanism, prior to the final stage which is the so called lock-up position. Although the
band angle propagation is steeper than the band angle formation, angles close to the
direction of the unkinked fiber are not expected. Figure 25 and 31 show a and (p for (A, 0t,
k)6 E(y) obeying S = O and SW. All these curves single out shocks consistent with

experimental results of the fiber kink mechanism in the sense that for k > O, the favorable

elastostatic shocks obey O < a < 325 and for k < O, the favorable elastostatic shocks obey

7: < a < %. Even more, as it can be noticed in Figure 33 (a magnification of Figure 27

corresponding to values A in an admissible range of 5-10% of deformation) as the material
becomes more anisotropic (i.e., y increases), both curves show the three stages that the fiber
goes through, as explained above. A first stage of slow rotation from its initial position,
followed by a rapid rotation without almost modifying the displacement, and a final position

which does not vary as much as the previous stages.

99

175 » 175 l
150 » 7:3 150 l y=15
125 ~ -------- Z ............

 

 

 

 

 

   

0.8 0.85 0.9 0.95 l 0.95 0.96 0.97 0.98 0.99 1

175 . 175 .
7:50 150 r
125 -

 

 

 

 

 

 

 

0.95 0.96 0.97 0.98 0.99 l 1.01 0.95

Figure 33. These plots are just a magnification of the plots in Figure 31 in a 5-10% of
admissible deformation. In all cases, although it is further appreciated as *1 increases, the
angle (1), either for 05mm, or (195:0, as A decreases (compressive load), shows a first stage of
slow rotation, followed by a second stage of rapid rotation, and a third stage in which the
rotation is much slower than in the previous stages.

Nevertheless, the curve S = 0 involves shock angles less meaningful for the kink-band
broadening than Sm. Further, while all the energetically admissible shocks are precisely
those satisfying that the elongation of the fiber on IT is greater than the elongation of the
fiber on IT, stationary ones (S = 0) do not have this distinction, i.e., the fiber stretch on the
unkinked region and the kinked region coincide. This would give an additional requirement
to consider Smax as the supplementary condition which singles out a physically admissible
elastostatic shock, according to the fiber kinking mechanism. In addition, the shocks

obeying Smax keep elliptic deformations on IT. Since elliptic deformations are energetically

100

favored, we could conclude that the shock creating IT at the expense of IT will create an
elliptic deformation on IT, and whence, that Sm may be a reasonable condition for singling

out a physically occurring shock.

101

SUMMARY AND CONCLUSIONS

The existence of equilibrium shocks or elastostatic shocks has been analyzed for a
particular material, namely, the Reinforced neo-Hookean. The existence of these local
solutions involve a continuous displacement field but a discontinuous deformation gradient,
i.e., a discontinuity in the first derivative of the displacement field. It is precisely the surface
carrying this discontinuity the so called elastostatic shock. The deformation gradients
consistent with traction continuity requirements and displacement continuity requirements
together with the surface carrying the discontinuity give the so called mechanically
consistent elastostatic shocks. There is a distinction between weak mechanically consistent
elastostatic shocks and strong mechanically consistent elastostatic shocks. In particular,
weak mechanically consistent elastostatic shocks involve discontinuities in the second
derivative of the displacement field in a direction normal to the shock. In agreement with
similar analysis carried out for isotropic materials, the necessary condition for the existence
of weak mechanically consistent elastostatic shocks for the Reinforced neo-Hookean
material is shown to be the loss of ordinary ellipticity of the governing partial differential
equations in global plane strain on IT (section 6). Furthermore, in section 9, it is shown that
the existence of strong mechanically consistent elastostatic shocks necessarily involves the
loss of ordinary ellipticity of the governing partial differential equations in global plane
strain at some intermediate deformation in between the existing deformation gradients on IT
and on IT. This important result links the directions of the mechanically consistent
elastostatic shocks with the characteristic lines of the governing partial differential equations
in global plane strain for the Reinforced neo-Hookean. In particular it establishes that a
mechanically consistent elastostatic shock evolves from a characteristic line. The ellipticity
status of a mechanically consistent elastostatic shock is further carried out. It is shown that
the ellipticity status of a mechanically consistent elastostatic shock generates four

possibilities, namely, either side (IT or IT) can be either elliptic or non-elliptic. In section 9,

102

also, an important kinematics result is established for the mechanically consistent
elastostatic shocks, namely, the existence of a mechanically consistent elastostatic shock
involves fiber contraction at least on one side of the shock. This result rules out the
possibility to obtain such elastostatic shocks if both sides are in fiber extension.
Nevertheless, it is the energetic analysis in section 10, the one which allows to rule out some
of the mechanically consistent elastostatic shocks obtained in section 7. The dissipative
inequality used, based on the idea that the existence of a mechanically consistent elastostatic
shock can not increase the energy of the system, leads to consider as energetically
admissible elastostatic shocks those ones which simultaneously move towards the side of
the shock with a lower hydrostatic pressure and either with higher fiber contraction (if both
sides have fiber contraction) or with fiber contraction (if only one side has fiber
contraction). The former result is in analogy with gases, for which a gas dynamical shock
moves, relative to the gas, into the zone with less pressure. The latter result is in analogy
with the kink-band formation in fiber reinforced composite materials, where it is found that
the mechanism of kink-band broadening involves an incremental gain in the length of the
kinked fiber with respect to the unkinked fiber. Furthermore, the energetically admissible
elastostatic shocks are viewed as a bifurcation away from the simple homogeneous
deformation in the fiber direction. Whence, energetically admissible elastostatic shocks
necessarily involve the loss of ordinary ellipticity on the same side that the weak shocks

(on IT). Even more, it is established for the energetically admissible elastostatic shocks that

on the IT side the ellipticity status may involve either an elliptic or a non—elliptic
deformation. At last, a condition is given as a supplementary condition to get a physically
admissible shock among the energetically admissible elastostatic shocks, namely, it is
proposed to maximize the dissipative inequality. Whence, the energetically admissible
elastostatic shocks yield under this condition the so called maximally dissipative elastostatic
shocks. These maximally dissipative elastostatic shocks are shown to follow features

observed in different experiments of the kink-band formation in fiber reinforced composite

103

materials. In these experiments it is observed an early stage of slow fiber rotation, followed
by a rapid rotation due to a geometric softening mechanism, prior to the final stage which is
the so called lock-up position. The family of maximally dissipative elastostatic shocks
capture these features. Further, maximally dissipative elastostatic shocks are shown to be
related to a non-elliptic deformation on one side (the one which gives the bifurcation point)
and an elliptic deformation on the other side (the one being created). Since elliptic
deformations are energetically favored (and these are precisely the ones being created) to
maximize the dissipative inequality may be a reasonable condition for singling out a

physically occurring mechanically consistent elastostatic shock.

104

APPENDICES

105

APPENDIX A

Kinematics of plane elastostatic shocks for isochoric deformations.
Let us derive some expressions that relate the geometry of the shock in the

undeformed and deformed configuration with the deformation gradient.

 

X2 yz

 

 

 

 

I
N L
(P n on
V r X 1 fi' y 1
Undeformed Deformed

 

 

 

Figure 34. Kinematics relation between the vectors L and N defining respectively the
shock direction and its normal in the undeformed configuration and the vectors l and 11
defining respectively the shock direction and its normal in the undeformed configuration.

Simple geometry gives: in the undeformed configuration
L = (L1, L2) and N = (N1, N2) = (-LZ, L1)
and in the deformed one:
I=(ll,12) and n=(n1,n2)=(-12,]1)

where all of them are considered to be unit vectors.

Using inditial notation:
ni:_8ij1j and Lk = 8km m

106

bythedeformationF: 1.: l F..L.
1 IFLI 1] J

1 1
L =———
k IFLI

 

8.

then: n. = zjgkijkNm

———8..F.
‘ IFLI 111k

using the identity: 8ij 8k m = (aikajm — 6im5jk)

we get:

 

 

“i :—| FL [[6ik6jm —6im6jk)ijNm :| FL |(—FkiNk+FkkNi)

OI'I

 

1 1
“12'“, I(’F11N1‘F21N2+(F11+F22)N1)=|FL |(F22N1'F21N2)

1 1
n2 =| FL I(‘F12N1‘F22 N2+(Frr +F22)N2)='I"F' L""|'('F12 N1+F11Nz)

and we can express this by the incompressibility requirement for F as:

1

F'tN or N: |FL|Ftn
IFLI

 

107

APPENDIX B

Components of the tensor G in the coordinate system Y

 

Y2
it

 

 

 

 

 

 

Figure 35. Coordinate systems Y and Z used to represent the tensor G.

Consider a coordinate system located at the spatial shock: Z = (21,22). The tensor G in this

coordinate system has the form:

G2 = (1 k]
0 1
Denote the matrix that relates both coordinate systems, Y and Z, as A. It has the form:
[21] _ [cos a sin a) [Y1]
22 - sin a cos 0: Y2
by the rule of transformation of tensors:

GY=AtGZA

in component form:

108

GY_(cosa -sina)(l k](cosa sina)_
sina cosa 0 1 -sina cosa
_ l-ksinacosa kcosza

— - ksinza l+ksin 0: cos a

or since I = (cos 0t,sin or) and n = (- sin (1, cos or), in inditial notation:

BaB=(6ocB+klanfl) a,B=l,2

109

APPENDIX C

Principal directions for a plane deformation gradient F
Let a rectangular coordinate system X = (x1, x2), and a non-singular plane
deformation gradient F be given. By the polar decomposition theorem, it is possible to find
a unique positive definite symmetric second-order tensor V, called the left stretch tensor,
and a proper orthogonal second order tensor R such that:
F = V R (AC.1)
Consider the deformation gradient F to be homogeneous; the homogeneous
deformation of the material at hand (extended from — co to 00) from its reference
configuration X is given by:
y=Fx=VRx (AC.2)
By the rule of change of orthonormal basis, a coordinate system X’ = (x’l, x’2 ) can be
found such that:
x = Q x’ (AC.3)
where Q is a proper orthogonal matrix.
Substituting (AC.3) into (AC.2):
y = V R Q x’ (AC.4)

choose Q = RT, then (AC.4) yields:

y=Vx’
from where we conclude that it is possible to find a coordinate system in which F is
diagonal since V is a positive definite symmetric second order tensor.

Following [11] for a given plane deformation gradient F:

F111212 0
F: F211222 0
0 0 1

110

the angle 0 with respect to its principal directions, and the principal stretches Aland A2 can

be calculated in the following way: V and R respectively have matrix representations

A1 0 0 cos0 sin0 0
V: 0 A2 0 R: -sin6 cos6 0
0 0 l O 0 l

by the polar decomposition theorem (AC. 1):

F F 0 A 0 0 c080 sint9 0

ll 12 1
F21 F22 0 = 0 A 2 0 - sin 6 cos 6 0
0 0 l 0 0 1 0 0 1

hence, the angle 0 and the principal stretches Aland A2 can be calculated known F.

If the plane deformation is isochoric then det F = l, or equivalently det V = 1.

111

APPENDIX D

Conditions for the Loss of Ellipticity in plane strain deformation for the

reinforced Neo-Hookean.

Here we just follow [2] to give the conditions under which the reinforced Neo-
Hookean material loses ordinary ellipticity in plane strain deformation. After several

transformations (9.1) can be expressed as a quartic polynomial of the
form y4 +b2 y2 -i-b3 y+b4, where yis related to Mland M2. Recall that Mland M2 are

the components of the normal vector to an associated weak elastostatic shock in the
undeformed configuration. Following standard root determination for quartic polynomials,
the transition between two pairs of complex solutions to at least one double real root, i.e., the

breakdown of real solutions, and hence the breakdown of the loss of ordinary ellipticity, is

given by 0 = 0, where 0 = 27 £12 + 4 63,61 andtff2 given by

 

2 2
g E -fi+32_2:4_-23_ 52 E. -b_2_b_
1 108 3 8 ’ 12 4
Now the coefficients b2 , b3 and b4, are given by equation (6.8) in [2] as
3 2
b2 = -—cl +c2,
___ 3
b3 — c1+ clc2 c3,
_ 3 4 2 1
b4 — ”275—601 +T6C1C2 -chc3 +c4,
where c1, c2 , c3 and c4 are given by equation (6.2) in [2] as
c :_2C12,
1 C

11

112

2
1+ C12

2
C11

e2 =2y(3C11—1)+1+

_ 4yc12 (3C11—1)+2c1:Z
3‘ c

 

C

9

ll

_ C122{2 y(3 Cll —l)+1}+2 y(C11—1)+1
4 _ C121 '

 

Here Clland C12 are the 11 and the 12 components of the C = FT F tensor and y is the

reinforcing parameter.

113

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114

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