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THESIS

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This is to certify that the

dissertation entitled
PLASTICITY 0F RANDOM MEDIA

presented by

Horea Tiberiu Ilies

has been accepted towards fulfillment
of the requirements for

Master's degreein Mechanics

 

 

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Date May 18, 1995

 

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DATE DUE DATE DUE DATE DUE

        

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PLASTICITY OF RANDOM MEDIA

by

Horea Tiberiu Ilies

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Materials Science and Mechanics

1995.

ABSTRACT

PLASTICITY OF RANDOM MEDIA

by

Horea Tiberiu Ilies

Effects of spatial random fluctuations in the yield condition are analyzed in rigid-
perfectly plastic media governed, generally, by a Huber-Mises-Henky or Mohr—Coulomb
yield condition with cohesion. A weakly random plastic microstructure is modelled, on a
continuum mesoscale, by an isotropic yield condition with the yield limit taken as a
locally averaged random field. The solution method is based on a stochastic generalization
of the method of slip-lines, whose significant feature is that the deterministic characteris-
tics are replaced by the forward evolution cones containing random characteristics. For the
Huber-Mises-Henky medium, an application of the method is given to the limit analysis of
a cylindrical tube under internal traction. For the Mohr-Coulomb medium, the characteris-
tic boundary value problem is studied, with an emphasis on variation of the statistical
characteristics of the field variables at the extremum point considering both uniform and
Weibull type random variates. The major conclusion is that weak material randomness
always leads to a relatively stronger scatter in the position and field variables as well as to
a larger size of the domain of dependence - effects which are amplified by both, grown

noise and inhomogeneity in the boundary data.

ACKNOWLEDGEMENT

The guidance and unfailing assistance of Prof. Martin Ostoja-Starzewski is grate-

fully appreciated.

TABLE OF CONTENTS

1. Introduction
1.1 Random Continuum Plastic Medium
2. Plasticity of Metals
2.1 Continuum Field Equations
2.2 Solution of the Slip-line Net via Finite Difference Method

2.3 Limit Analysis of a Cylindrical Tube Made of a Perfectly-Plastic
Medium Under Internal Pressure

2.3.1 Tube Made of a Homogeneous Material
2.3.2 Tube Made of a Randomly Inhomogeneous Material
2.3.3 The Computer Program
2.4 Discussion of Results
3. Plasticity of Granular Media
3.1 Basic Concepts
3.2 Continuum Field Equations
3.3 Cauchy Problem of a Homogeneous Granular Medium
3.4 Inhomogeneous Continuum Model
3.4.] Nedderman’s approach
3.4.2 Sokolovskii’s approach

3.4.3 Proposed approach

13

l7

17

22

25

28

37

40

42

46

48

49

3.5 The Characteristic Boundary Value Problem

3.6 The Characteristic Boundary Value Problem with Singular Point

3.7 The Mixed Boundary Value Problem
3.8 Discussion of Results
4. Conclusions

Bibliography

56

57

59

6O

78

81

1. INTRODUCTION

Today we are experiencing a revolution in materials used in a broad variety of
engineering applications; incremental improvements in traditional materials will not do
the job! In the course of history, major shifts to new types of materials were always
accompanied by significant design changes that better used the potential of these new
materials. Mechanicians are creating new models that aim to better describe the behavior
of real materials. Among various effects that need to be accounted for is randomness,
which may, generally, be due to:

- random external loading
- random boundary conditions and data
- randomness in physical behavior of engineering materials due to fluctuations in their

parameters and properties.

In this work we investigate the effects of random spatial fluctuations in the yield
function of random rigid perfectly-plastic media. In [Olszak et al, 1962] was mentioned
the subject of plasticity of randomly inhomogeneous media. This important reference pro-
vides, among others, a very good review and discussion of the methods used to solve
boundary value problems of plasticity of inhomogeneous media described by detenninis-
tic functions which, in principle, form the starting point for stochastic problems. These
methods are: analytical, approximate, inverse and semi—inverse, respectively. Given the
power of today’s computers on one hand and the limitations of analytical and inverse solu—

tions in deterministic non-homogeneous medium problems on the other, we adopt a com-

putational method to solve the system of quasi-linear hyperbolic differential equations that
governs the problem. The solution is based on a stochastic generalization of the method of
slip-lines, whose significant feature is that it replaces the deterministic characteristics by

cones of forward evolution.

Plasticity of randomly inhomogeneous media has recently been studied by [Nor-
dgren, 1992] from a different standpoint. The focus there has been on a stochastic formu-
lation of lower—bound and upper-bound theorems and a corresponding application to the
loading of a wedge. While we postpone the discussion of relative merits of Nordgren’s and
our approaches to Chapter 4, the principal difference between them is the recognition, in
our model, of the scale-dependence of a Representative Volume Element of a random con-

tinuum approximation (see Section 1.1 below).

The method used in this work applies to media that require a stochastic continuum
formulation, i.e. when the fluctuations in constitutive laws disappear only at scales larger

than the macroscopic dimension of the body [Ostoja-Starzewski, 1992a].

A generalization of the method of characteristics extended to the flow of rigid per-
fectly plastic and spatially random media has been developed in [Ostoja-Starzewski,
1992a], where a comparison of solutions of a specific Cauchy problem in the case of a
deterministic homogeneous medium with the yield limit kdet and a random medium case
having the same average yield limit (k) = k det and a weak random noise was illus-

trated. It was found that such a weak material randomness has strong effects in the case of

inhomogeneous boundary value data. The influence of these random fluctuations upon the
Cauchy and Characteristic boundary value problems for media whose plastic behavior can

be approximated by the isotropic form:
(ox-oy)2+4rfy = 4k§(w) (1.1)

was examined in [Ostoja-Starzewski, 1992a, 1992b] and it was found that a weak material
randomness always leads to a relatively much stronger scatter in the position and field
variables and that there is practically no difference in the slip-line nets given by the inte-
gration method (i.e. explicit or implicit). Moreover, in [Ostoja-Starzewski, and Ilies, 1995]
a practical problem was solved, namely a tube made of a random rigid perfectly plastic
medium under internal load, whose yield function may be approximated by (1.1); the
paper summarizes the findings presented in detail in the following chapter. There it was
found that, even though the scatter in the slip-line nets increase with the inhomogeneity of
data on the boundary as well as with the random noise, the average solution of the stochas-

tic problem is basically the same as the solution of the homogeneous medium problem

In the case of granular materials, the medium’s response is usually approximated

by an isotropic Mohr-Coulomb yield criterion:

. 2
2 _ (511195)

1 2 2
Z(ox—oy) +1xy — 4 (ox+oy+2H5) (1.2)

in which p5 and H8 are random variables.

The Mohr-Coulomb yield criterion (1.2) is expected to give a highly nonlinear
behavior. The slip-line theory for the deterministic homogeneous granular media was
developed in [Sokolowskii, 1965], but the effect of the random spatial fluctuations upon
the medium’s behavior was not yet studied. In this case we apply the same computational
method of solving the system of governing quasi-linear hyperbolic differential equations
as in the case of media whose yield function can be approximated by (1.1). This is studied

in chapter 3.

In micromechanics of granular media, randomness is typically accounted for by
either solving a set of deterministic boundary value problems of large system of disks (one
obstacle being in this case the computer limitations on the sizes of large lattices represent-
ing discrete media), or by solving a single boundary value problem for a medium that has
average properties, case in which there arise difficulties in finding the correct average of

the random properties such that the two solutions coincide.

1. 1 Random Continuum Plastic Medium.

By a random microstructure (or medium) we understand a family:

B = {B((1)),(oe 9} (1.3)

of deterministic media B (0)) , where u) is an index for the probability space 0. A para-

digm of derivation of a stochastic continuum model is presented in [Ostoja—Starzewski,

1992a]. This relies on the concept of a window bounding a random microstructure:

where B5((0) is a single realization and 5 = Id is a non-dimensional scale parameter
that characterizes the scale L of observation relative to a typical microscale d of the mate-
rial structure. Therefore, the window may be interpreted as a Representative Volume Ele-
ment (RVE) of an approximating random continuum B5. The effective properties display a
statistical scatter which decreases to O as 8 —-> 00. While there exists a finite scale 5 at
which this scatter may be considered negligible, such an approach does not apply in situa-
tions where 5 is comparable to, or greater than, the macroscopic (relative) dimension 5M
of the body B. In this case the stochastic formulation of a given boundary value problem is
needed. In [Ostoja-Starzewski, 1992a] six steps for determination of a random rigid per-

fectly-plastic medium are outlined.

2. PLASTICITY OF METALS

2. 1 Continuum Field Equations

The state of plastic plane flow, whose generalization to materials governed by ran-
dom yield functions (i.e. (1.1) and (1.2)) is studied in this work, is defined by the funda-
mental property that the displacements of all particles of the body are parallel to a given
plane, usually chosen to be xOy of the rectangular, or Cartesian, system of coordinates

xOyz. The displacements are considered to be independent of z coordinate. Therefore,

0

each point of the continuum will be characterized by four stress components O'x, 0y, xy 1n

the xOy plane and oz parallel to Oz axis. Since under the initial assumption of plane flow,

the tangential components sz=Tyz=0a Oz is found to be a principal stress.

Another assumption that is made is that the material can be approximated by a

rigid perfectly-plastic medium, see Fig 2.1:

CA

 

 

O
8
Fig 2.1 The rigid-perfectly plastic medium

It is noted at this stage that the elastic part of deformation and the strain hardening effects

are being disregarded in the present model, although the strain hardening can be intro-
duced later. Furthermore, we will neglect the inertia terms in the field equations because at
this time, a general solution of problems of plane flow accounting for inertia terms is not
available. We can neglect these inertia terms on the consideration that in most material
forming processes and practical problems, accelerations of the material are very small,
therefore the influence of inertia forces is negligible. An estimation of the influence of

inertial forces can be found in [Szczepinski, 1979, ch. 6].

With the above assumptions, the equilibrium equations of the field reduce to the

well known form:

36 at

_" _">’ = 2.1)
ax + 8y 0 (
do a:

_y —Xy =

By +Bx 0 (2-2)

The random yield function:

135(0)) = 0 (2.3)

is approximated by an isotropic form (Von Mises):

(ox — oy) 2 + 4r)?y = 4k§ (to) (2.4)

where the yield limit k5(m), at any point x , is a random variable that can be considered as

a sum of the mean and a random fluctuation:

k5()§,(D) : (k5(2$,(0)>+k5.(2$,(0) ! (k5.().$90))> : O (25)

Clearly, k,5 (x, to) and k5' (x, to) are random fields.

At this point, in the theory of slip-lines [Chakrabarty, 1987, Szczepinski, 1979],

two new functions p and (p are introduced:

6x = p+k5cos (2(p) (2.6)
CY = p—kacos (2tp) (2.7)
Txy = kasin (2(p) (2.8)

These expressions satisfy identically the yield function (2.4). Substituting (2.6),

(2.7) and (2.8) in the field equations (2.1) and (2.2) and setting (p = —E, one obtains a

basic set of partial differential equations in two unknowns, p and (p:

8k
g+ 21(53—1’ = 5; (2.9)
8p dtp _ aka
$41.53.; .. .87 (2.10)

where the orthogonal axes are now along the local slip-line directions. Replacing _d_ and

8x

_B_ by the tangential dertivatives _8_ and _3_ respectively along the 0t and [3 character-

By 85 asB

(1
istics, the above equations will become independent of the orientation of axes [Ostoja-

Starzewski, 1992a]. Therefore:

8k

dp+2k5dtp = 5idsO (2.11)
3k

dp-2k5dtp = 5gas,3 (2.12)
(I

This stochastic system is of a quasi-linear hyperbolic type for all possible values of
p and (p; it can be thus solved by means of the method of characteristics. Solution of par-
ticular cases may be obtained by solving the appropriate Boundary Value Problems
(BVP), when either the values or a relationship between p and (p functions are given along
certain lines. These conditions are generally sufficient, but not always [Szczepinski,
1979], to define the values of p and (p uniquely in the regions adjacent to those lines, the

so-called domain of influence. In (2.11) and (2.12), the right-hand sides are random terms.

The corresponding characteristic directions are:

._,_ i -1
dx — tan tp+ (2.13)

and

( A)
3;- tan (p 4 (2.14)

Equations (2.13) and (2.14) form the basis for the determination of the Henky-

Prandtl net of slip-lines in a given Boundary Value Problem.
2.2 Solution of the Slip Line Net via Finite Difference Method

In the following, a forward finite difference approach was used, as presented in

[Ostoja-Starzewski, 1992a, 1992b]. Consider a boundary (eg. a convex one):

 

 

Fig. 2.2 Forward evolution
from [Kachanov. 1971]

Dividing the boundary into small and (not necessarily) equal segments. as seen in
Fig. 2.2, and knowing the stress distribution along the boundary, the values of p and (p are

uniquely determined at each point xi on AB, (xi 6 ATS)

11

Starting with two arbitrary and adjacent points, say X, and xi+1 from AB , we can
set up from (2.13) and (2.14) the difference equations for coordinates xN and yN for the

new point of intersection N (Fig 2.3):

yN—yi = (XN’xi) ta"(¢1+g) (2.15)

n
yN_yi+1= (XN‘X1+1)tan(‘P1+1_Z) (2-16)

as well as from (2.11) and (2.12), the difference equation for pN and (pN:

 

 

(kN+ki) dsa
pN-p,+2—i-—(¢N-<p,) = (RN—kg}? (2.17)
B
(kN+k. 1) ds
pN—pi+1+2 2H (‘Pn‘q’nd = (kN_ki+l)2_S_B' (2°18)
(1
Solving for pN and (pN, at the new point N, we get:
ds ds
(1 (1
91-pin+¢1ikN+kij+¢i+1(kN+k1+1)+(kN‘ki)dT‘(kN‘ki+1)KB
811“”) = 2kN+ki+ki+l (2°19)

ds
pNtw) = pi-(kN+k,)(<PN-<p,)Him—19);; (2.20)

12

where the dependence of all k’s on a) is implicit and:

1/2
dsa = [(xN—xi)2+ (yN—yi)2] (2.21)

1/2
(1513: [(XN_xl+1)2+ (yN_yi+l)2:I (2'22)

and the derivatives of k with respect to so, and SB are treated in the finite difference sense.
Note here that system (2.19) (2.20) may also be solved via backward differencing. In the

following, two other schemes are also considered - one recommended by [Hill, 1950]:

yN—yi = (xN—xi) tan [ (q)i +(pN)/2 +1t/4]

(2.23)
yrs-yin = (XN'xi+1)tan[(‘I’1+1+‘PN)/2‘“/4]
and another, recommended by [Szczepinski, 1979]:
l
yN - 3'1: 5(xN — xi) [tan ((pi + 1V4) + tan ((pN + n/4)]
(2.24)

1
yN—yi+l = 5(xN—xiH) [tan((pi+l—1t/4) +tan((pN-1t/4)]

System (2.19) and (2.20) is used to determine p and (p at every point of the finite
difference net. There is a random scatter in location of point N, since, according to (2.8)

and (2.11), (p, and (p2 are random variables:

l3

1"i+1

-1 - Ti _1 .
(“03(0) - iflSlfl£mJa (pi+1().$, (1)) — 2381n[k8 105,0») (2.25)

This scatter is indicated by an intersection of two cones of forward evolution
shown in Fig 2.3. These cones replace the characteristics of the deterministic medium
problem, Bdet. From (2.15) (2.16) [or (2.23) or (2.24)] we can find the mean (ensemble

average) coordinates of point N, by averaging in the dashed quadrangle, Fig. 2.3:

 

 

i i+1
Fig. 2.3 Exact stochastic method showing
two cones of forward evolution

according to:

1t 1t
yi+l-yi+xitan (91+; ‘Xi+1tan (phi-Z

TC 1:
tan((pi+Z)— tan((pi+l -Z)

1t 1t
x1+1 _xi +intg(‘Pi+ Z)‘Yi+1°tg(‘9i+ 1 ‘1)

Ctgi‘pi + :15)“°tg(<vi. 1 {15) > (2'27)

 

(xN> = < ) (2.26)

(yN) = (

 

l4

Formulas (2.26) and (2.27), derived from (2.15) and (2.16), establish the so called
average characteristics or average slip-lines. The yield limit, k5(0)), is taken to be a ran-
dom variable which, in addition, gives randomness in pN and (pN as well as in xN and yN at

the new point N which amplifies the uncertainty in the further evolution.

2.3 Limit Analysis of a Cylindrical Tube Made of a Perfectly-Plastic Material

Under Internal Pressure.

2.3.1 Tube Made of a Homogeneous Material

In the following we will study a practical problem for which an analytical solution
of the deterministic medium is known. Let us consider the slip-line field around a circular
hole of radius a, loaded on the interior surface. Let r and 9 be the polar coordinates used to
describe this state of plane stress (does not depend on the z coordinate). The most general
case under the above assumptions is when both or and Ire are non-zero on the boundary.
When the hole is uniformly loaded with a pressure p and a constant tangential load 1,9, the

problem becomes axisymmetric.

I) The case when 1:91).

Since there is no tangential stress on the edge of the hole, the equilibrium condi-
tion gives Tre=0- Therefore, at every point of the field, the principal planes have radial and

circumferential directions. The slip-line will be a curve which intersects at each of its

15

points a ray, emerging from the centre, at an angle i}: [Kachanov. 1971]. But only the

logarithmic spirals exhibit this type of property:

(p—In(-) = B (2.28)

tp+1n(a) = or (2.29)

which generate two orthogonal families. These lines have been observed in experiments

[Kachanov, 1971]:

Fig 2.4 Logarithmic Spirals [Kachanov, 1971]

16

For the so-called pressure boundary conditions in polar coordinates given by:

0' = ~p<0 Trtp = 0 at r = a (2.30)

with G¢>0, 6r<0 in the neighborhood of the boundary and the yield condition of the form:

C —o = 2k (2.31)

the stresses are determined by the formulas:

r
or = -p+2k1n(a) (2.32)

(5‘p = or+2k (2.33)

Note here that, if the yield condition has the form:

sq, — or = —2k (2.34)

the stresses are determined by the formulas:

or = (—p)—2kln(§) (2.35)

o,p = or-2k (2.36)

From (2.32) we get:

 

01':

The variation of Gr=6r(r) is shown in Fig. 2.5:

 

"‘ll

 

Fig 2.5 o=o(r)

(2.37)

(2.38)

From (2.38) above we can determine immediately the value of the radius

b(p"')=bmax at which or=0:

18

II) The case when or ¢ 0,_1:re at 0;

The yield condition now has the form:

(or—09) 2 +413, = 4k2

and the differential equations of equilibrium can be written:

 

 

99r+or_60 : 0
dr r
freight) : 0
dr r

Suppose that the boundary conditions are

G=-p trezq atr=a

(2.39)

(2.40)

(2.41)

(2.42)

(2.43)

where, of course, Iql S k. Integrating by separation of variables in (2.42) with the BC

given by (2.43), we get:

l9

2

From (2.40) and (2.44) we get an expression that gives the yield condition:

’2 2 a 4
Oe—Cr=i k --C] (T) (2.45)

Substituting (2.45) in (2.41), integrating and imposing the BC, we obtain:

 

 

Note that when Tre = q ¢ 0, the slip-lines are no longer logarithmic spirals.

2.3.2 Tube Made of a Randomly Inhomogeneous Material

We briefly saw in the former paragraphs the analytical solution for this particular

problem of a homogeneous medium (i.e. the yield limit k is constant). In case of a tube

made of an inhomogeneous material, the slip-line net will have a random scatter in posi-

tion given by the randomness exhibited by k5(a)).

Random fluctuations in the slip-line net have been observed experimentally. An

20

example is shown in [Kachanov, 1971]:

 

 

 

Fig. 2.6. Random fluctuations experimentally observed
in the slip line net distribution: from [Kachanov, 1971]

The formulas (2.32) and (2.46) no longer apply and the system (2.19) and (2.20) together
with either (2.15) and (2.16) or (2.23) or (2.24) has to be used in order to determine the
slip line net and the stress field in any particular realization of a spatially inhomogenous
medium B5((1)) of the family 35- Recall here Fig. 2.5. The extrapolation of this result for

the case of an inhomogenous material gives a dependence of or=or(r) according to:

21

0'5 1)

bmax

_—-——-—oq

 

Fig. 2.7 05:65(r5(0)))

Thus the condition Gr=0 plays the key role in the definition of an excursion set of a

random field Gr(r, (p) = {or ((1)),(0 e (2} [Adler, 1981]:

A0(or, D) = {(r,(p) e D|or(r, (p) 20} (2.47)

This leads to the definition of a so-called set of level crossing:

8A0(or, D) = {(r,(p) e D|or(r,tp) = 0} (2.48)

The set 3A0 (or, D) is a set of closed contours of plastic zone, which, in the case

of a homogeneous medium with no shear loading, is a circle of radius given by (2.39):

22
b = ae (2.49)
2.3.3 The Computer Program

A computer program was developed to implement the finite difference method

presented in cap 2.2.

First, the program transforms the stresses from the polar to cartesian coordinates,

using the well known tensorial transformation formula:

6 = ATo'A (2.50)

where 0’ and o" designate the stress components in the cartesian and polar coordinates

respectively, and A is the transformation matrix:

A = cosa sina (251)
—sin0t cosoc

at being the angle between Ox and Ox’.

Next, having the stresses, the two variables p and (p can be computed at each point

P1 on the boundary, using the formulas (2.6), (2.7) and (2.8), from which we get:

23

 

 

6 +6 0 -O
p = ‘2 3’ andtp = %acos[ x y] (2.52)

2k6

Now, having the values of p and (p on the boundary, using (2.19), (2.20), and
either(2.15) and (2.16) or (2.23) or (2.24) of the finite difference method, we can march
forward to the next row, and so on. In the end we will have all the variables p, (p, x and y,
that uniquely determine the position and the stress components, at each point of the slip-

line net. A testing follows, if or 2 0 , the program draws the net and stops.

The program was tested for the homogeneous medium, by comparing the values
obtained by running the program with those given by the analytical solution presented in
ch. 2.3.1. A table of values for the radius bmax at which or becomes zero is presented
below. Note that the data were obtained for Tr9=0, constant k (i.e. the homogeneous

medium), 60 points on the boundary, radius of hole a=1 and p=—1.9223.

Table l: The comparison of results between the Analytical and the Finite Difference

 

 

 

 

 

 

 

 

Method
11 <k(a))> Agglgtgga‘ 13.23322... 4 (as). [$33311 13.23.23...
Method Method
1 1.0 2.6147 2.6158 7 1.6 1.8236 1.8239
2 1.1 2.3959 2.3961 8 1.7 1.7601 1.7605
3 1.2 2.2277 2.2272 9 1.8 1.7057 1.7060
4 1.3 2.0946 2.0950 10 1.9 1.6584 1.6588
5 1.4 1.9868 1.9874 11 2.0 1.6170 1.6174
6 1.5 1.8979 1.8985

 

 

 

 

 

 

 

 

 

 

24

The mesh dependence in the case of a homogeneous medium with

(k,S ((0)) = 1.5, p*=-1.9223, without tangential load and with internal radius of the hole

a=1, is given below:

1‘
7

radius, r (0'50)

—————_——_—_——

on
0

3'0
number of points, 11

Fig. 2.8 The dependency of the maximum radius
upon the number of points on the boundary

 

where b0 is the radius obtained by the analytical solution at which or = O. The mesh

dependence has an exponential shape and for 30 points the accuracy of the finite differ-

ence method is below 3%.

25

2.4 Discussion of Results

In any given inhomogeneous material the set of level crossings is a random set in
plane, which, due to the spatial homogeneity of field k5, is a circle containing all the possi-
bilities. Let bmax and bmin be the maximum and minimum distance for a given realization,
respectively, from the origin to the contour. Since bmax determines the minimal amount of
material needed for a tube to withstand the internal pressure p*, the natural questions to ask

are:

Q1: “Is bmax(p*) smaller, equal to, or larger than b(p*) of the homogeneous medium prob-
lem?”
Q2: “What is the probability distribution of bmax and bmin?”

Q3: “What is the effect of shear traction in boundary condition (2.43) versus (2.30)?”

Two particular cases of noise were investigated

ks’ 6 {—0.0094, 0.0094] (2.53)

and
k 5’ 6 {-0.025, 0.025] (2.54)
Figures 2.9. a) and b) show the slip-line net of the tube made of a homogeneous
material that withstands a normal pressure p* = —3 and a normal and tangential loading

p* = —3 and q* = 1.3, respectively. The slip-line networking correspond to

26

(k8 ((0)) = 1.5 and internal radius a=l. As one expects, the amount of material needed
(i.e. the radius of the slip-line net) is greater when we have besides normal loading a tan-
gential one. For an infinite number of points on the internal boundary, the outer boundary
at which (5'Ir = 0 will be a circle of radius given by either (2.39) (i.e. no shear loading) or

a similar expression derived for the case when 0'r :4 0, ‘10 ¢ 0 on the outer boundary.

Figures 2.10 a) b) show the slip-line net given by a single realization B5(u)) of a ran-

dom medium with (k8) = 1.5 and k5’ sampled according to (2.54).

27

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b)

Fig. 2.9. Slip line patterns in a homogenous material under: a) pressure boundary
conditions (2.30) and b) hydrostatic pressure and shear boundary conditions (2.43)

28

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b)

Fig. 2.10. Slip line patterns in a randomly inhomogenous material under: a) pressure
boundary conditions (2.30) and b) hydrostatic pressure and shear boundary
conditions (2.43). In each case, a single realization of 85(to) is used.

29

In Fig. 2.11 a) we plot patterns of slip—lines under boundary condition (2.30) corre-
sponding to four hundred realizations 85((0) of a random medium with (k,S ((0)) = 1.5
and k5’ sampled according to (2.54). The set of level crossings is shown as a ring contain-
in g all four hundred piecewise-constant non-circular closed curves. Next, in Fig. 2.1 1 b) we
plot slip-line patterns for the same type of medium under boundary condition (2.43) also

for four hundred realizations.

3O

 

a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b)

Fig 2.11. Slip line patterns in a randomly inhomogenous material under: a) pressure
boundary conditions (2.30) and b) hydrostatic pressure and shear boundary
conditions (2.43). In each case, four hundred realization of B5((0) are used.

31

It is immediately seen that the tangential loading produces a sensitive increase in

the scatter of the slip-line net.

Finally, in Figs. 2.12 a) and b) we plot probability distributions of bmax and bmin for
the cases, respectively, of the pressure boundary condition (2.30), and b) of the pressure and
traction boundary condition (2.43). In order to demonstrate the sensitivity to random noise

in the yield function, in each one of these two figures we show data corresponding to (2.53)

and (2.54).

1

32

 

0.9 '-

0.0 '-

0.7 -

0.0 '-

0.5 -

0.4 r

003 P

0.2 -

0.1 -

 

P(bm1n)(2.54)

P b -
M I min)(2.53)

 

r

l

P(bmaxl(2.sa)

Pibmax)(2.54)

T

 

 

269

2.7

2.71

g 2.72

2.73

b(p”) = 2.723

2.74 2.75 2.76

 

0.9 -
0.51-

0.7 r-

0.5 '-

0.4 -

 

Pibmin)(2.54)

     

F’(bmln)(2.53)

 

. r .
Pibmax)(2.53/ .

F’(bmax)(2.s4) q

 

L

 

b)

Fig 2.12 Probability dlSIl‘lbUIIODS P(bmax)(2_53), P(bmax)(2_54), P(bmin)12.53)' and
P(bmin)(2.54) for the case a) of pressure boundary condition (2.30). and 1)) pressure
and shear boundary condition (2. 43): k5‘ is sampled according to (2. 53) and (2. 54).

In each case four hundred realizations of B((0) are used. Also. the deterministic

2.98

3

3.02

3.04

b(p', q') = 3.028

cases b(p )= 2. 723 and b(p, q )= 3. 028 are shown.

3.06 3.08

33

It follows now that the answer to the first question is “always larger”, whereby bmax
increases as the noise level in k5' increases. Thus, the principal conclusion is that the pres-
ence of material inhomogeneities requires a thicker tube than what is predicted by the
homogeneous medium theory. Let us note here that the higher the pressure p, the greater is
the external radius b, and hence, the greater is the spread of the forward evolution cones
implying an increase in the scatter of bmax and bmin- Interestingly, both random variables
bmax and bmin are symmetrically distributed about the deterministic radius b(p*) of the
homogeneous medium problem; this answers Q2. The same qualitative conclusions carry
over to the case of the pressure and shear boundary condition, but one must note here that
the addition of shear traction has a strongly amplifying effect on the scatter of dependent
field quantities, and, most notably, on the spread of slip-lines - compare Fig. 2.11 a) and b);

this addresses Q3.

We observed that the choice of the forward (as used here) versus backward differ-
encing scheme has only a small effect on the solution. The results outlined above are in
accord with those obtained by [Ostoja-Starzewski, 1992a, 1992b], namely that in the case
of inhomogeneous boundary data, the sensitivity of field quantities to the ‘magnitude’ of

the randomness of plastic limit k increases as this randomness grows.

3. PLASTICITY OF GRANULAR MEDIA

3.1 Basic Concepts

Statics of granular media studies two types of stress states: stress states in which a
small change in body or surface forces do not destroy the state of equilibrium and stress
states in which a change, no matter how small, in the applied forces causes a loss of equi-
librium. The ones that lie in the second category, the so-called limiting stress states,
depend directly on the basic mechanical constraints which characterize the resistance of a
granular material to shear deformation and form the basis of the theory of limiting equilib-

rium

In 1773, Coulomb, the originator of this theory, formulated the basic theorems of
limiting equilibrium. Later in 1857, Rankine introduced the concept of slip surfaces and

found the condition of limiting equilibrium.

Let us now take a look at a point I’ in a granular medium and consider an element

of area containing this point. This area element is loaded with a stress vector p which

forms with the normal 11 to the surface an angle 5 as can be seen in Fig. 3.1:

34

35

 

 

 

 

Fig.3.] Limiting Condition

The components of the stress vector p are on and In. The experiments show that
the resistance to shear over this area in a granular medium with some cohesion can be

expressed as

l‘tn' = ontanp + H0 (3.1)

which applies when the equilibrium is about to be destroyed. This resistance consists of a

resistance from internal friction and a resistance from cohesion H0.

The constants p and H0 are the angle of internal friction and the coefficient of
cohesion, respectively, and they can be looked upon simply as parameters which charac-
terize the total resistance of the granular medium to shear. There are two special cases: if
H0=0 the medium is cohesionless and when p=0 the medium is an ideally cohesive, one

described by (2.4).

36

It was shown [Sokolowskii, 1965] that no slip occurs if:

ltnl S Ontanp + H0 (3.2)
where
on 2 —H0cotp (3.3)
Define now:
H = Hocotp (3-4)

where H is the ultimate resistance to uniform three-dimensional tension [Sokolowskii,

1965].

Assuming that there is an equivalent stress vector p' acting on the element of area
at an angle 5' with the normal 11, its components are on + H and In. Inequality (3.2)

becomes:

I’tnl S (0'n + H) tanp (3.5)

where on 2 —H

37

The special limiting equilibrium is given when equation (3.5) has the form:

|tn| = (0'n + H) tanp (3.6)
Equation (3.6) can be expressed [Sokolowskii, 1965] as:

. 2
1 2 2 _(s1np) 2
Z(ox—oy) +th — T(Gx+oy+2H) (3.7)

which is the well known Mohr-Coulomb yield criterion.

3.2 Continuum Field Equations

The field equations of equilibrium are:

-a—-x-x+5?xy = 7511]“ (38)
Boy xy
3; +$ = ycosa (39)

where y is the density of the medium and Otis, for generality, the angle between x axis and

the horizontal.

38

At this point, two variables 0' and (p are introduced [Sokolovskii, 1965]:

0x = (5(1 + sinpcos2tp) —H (3.10)
0’), = 0(1— sinpcos2tp) —H (3.11)
txy = osinpsin2tp (3.12)

We observe that if the value of o is sufficiently large, the coefficient H will cease to
have any real influence on the stress components and in this case, it can be neglected.
Therefore, as 0' increases, the limiting equilibrium tends to the corresponding limiting

equilibrium of an ideal granular medium.

In the development of this chapter we will consider p and H as random fields,

namely:

1350590)) : (95(Kvw)>+p'5(2$1(0) a (p.5(52m)> = 0 (313)

H5()~(,(t)) = (H5()~(,0)))+1H'5(§,(D) 1 (H'5()5,(0)) = 0 (3-14)

Formulas (3.10), (3.11) and (3. 12) satisfy identically the Mohr—Coulomb yield cri-

terion (3.7).

39
3.3 Cauchy Problem of a Homogeneous Granular Medium
In the case of a homogeneous material, H and p are constants, and their values
depend on the particular material. In the following, an analytical method to solve for the

slip-line net was developed.

For a weightless homogeneous medium, the differential equations for the two

characteristics are [Sokolovskii,1965]:

dozr201anp'dtp = O (3.15)

which can be integrated by separating the variables:

2
Cc; mm”) = constant (3.16)

where equation (3.16) holds along an a (B) line (see Fig. 3.2).

Now, if the stresses (i.e. 6x, Cy, Ixy) are given along a boundary, say AB (recall
here Fig. 2.2) which is divided into n segments, then the two variables 6 and p can be
computed at every single point along that boundary. Marching forward, from every two

adjacent points, say i and i+1, on the boundary, the next point N can be found:

40

(a)
N (13)

 

i i+1
Fig 3.2 Forward evolution for a
deterministic medium.

From (3.16) we get:

 

J 2(tanp,¢,-lanpi+1‘P1+1)
0N " :F cVila-HIE

1 Ci
2tanpN [ln[(3—N] + 2tan (pi) (91]

 

‘91s:

01':

 

 

— 1 1 a” 21
(pN—Ztanpn n 01.1 + an(p,,1)<p,,,

Introducing the auxiliary angle 8:

28

II
NIP-I
l
.0

(3.17)

(3.18)

(3.19)

(3.20)

41

the angle between the two slip-lines, 01 and B, and the x axis will be respectively:

are (3.21)

Equations (3.17), (3.18) or (3.19) together with:

dy = dxtan ((pqie) (3.22)

allow us to find the slip-line net within the domain of influence.

The family of characteristics given by the upper signs are called the first family or
0t characteristics, while the family given by the lower signs, the second or the B character-
istics. The basic set of equations has two real different families of characteristics and are,

consequently, quasi-linear hyperbolic differential equations.

In the region under consideration, the xOy plane, the two sets of slip-lines intersect

at each and every point through which they pass at an angle 28 and hence the whole region

is covered by a network of slip-lines.

3.4 Inhomogeneous Continuum Model

Consider a random medium B = {B ((1)),(0 6 Q} , where (0 is an element of the

probability space 9. The random yield function:

42

F5(G(co)) = 0 (3.23)

which is generally known to have a Mohr—Coulomb form for granular media, is approxi-

mated by:

. 2
2 (smps)

l 2 2
I; (0x ‘00 + ”R, = 4 (0,. + 6,+ 2H,) (3.24)
where ()5 and H5 are two random fields:
13505.0» = (95(25,w)>+p'5(2<,m). (0'50, (0)) = 0 (3.25)
H6(57 (0) = (118(5, (D)>+H'8(§v ('0) a (H.5(Ea 03)) = 0 (326)

By assuming a scatter in the values of p 5 (x, (0) and H 8 (x, (0) , we will study the
effects of the material randomness upon the slip-line nets for materials that obey the

Mohr-Coulomb yield criterion.

The equilibrium equations (3.8) and (3.9) can be written for a weightless medium

(i.e. 7:0) as:

.__" +_._"y = 0 (3.27)

43

—y + —xy = (3.28)

Substituting equations (3.10), (3.11) and (3.12) in (3.27) and (3.28) above, we get:

[1 + sinpcosZ(p]— :4- smpsm2tpg—-—2Gs1np(sin2(p%— - costhg—cfij

+ ocosp( cos2tpg—p + sin2tp3—p) -g—:I- — 0 (3-29)

. . ac . ac . ( dtp - 3‘9)
smpsm2tp-a—; + (1— smpcosth)$+ 20'smp costh-a-i + s1n2tp-a—y-

8p 8H

8p —cosZ(p§—)—E=y0 (3.30)

+ O'cosp( sin2q)ax

Several attempts to adapt and generalize some existing methods of obtaining the
equations of characteristics for the case of a deterministic medium to our randomly inho-

mogeneous granular medium were made. They are presented below.
3.4.1 Nedderman’s approach

Multiplying (3.29) by [l — sinpcosth] , (3.30) by [—sinpsin2(p] and adding

them, one gets:

§;(cosp) — 2Gsmp|:sm2tpa+ (s1np—cos2tp)-a—-)7 (331)
_o'cosp [(cos2tp —— sinp) 3% + sin2tp3—2]

8H . 0H . .
+3; (1 — smpcos2tp) --3—y-smps1n2(p

44

Multiplying (3.29) by [—sinpsin2(p] , (3.30) by [l + sinpcos2tp] and

adding them, one obtains:

do 2 _ _ . . 3(1) - 9(1)]
5y (cosp) — 2osmp [ ( smp + cos2tp) 8—1: + Sln2tp-a? (3.32)
+ ocosp [—sin2(p% + (cos 2(1) + sin p) 3—5]

8H . . 3H .
fis1np51n2tp+5§ (l + smpcos2tp)

Now, making use of the angle E, and of the property:

3:3 = g—Ecosfi +%§sin§ (3-33)

from (3.31) and (3.32) we get:

do _ _d_<_P
a; (cosp)2 —20'sinp [sin (2(p— é) — sinpsin§]— coséds (3.34)
1 dp

4005p [cos (24>- fi) - sinpcosgl— cos—Eds

+[1— sinpcos (2(1)— §) col—s2]?

140511113 {—[sin (RP—é) —sinpsin§] tan§+ [sinpcosfi—cos (2(p—§)]}g—;lj
+Gcosp { [cos (2(p—§) — sinpcosi] tan§— [sin (2(p—§) — sinpsiné] }%

+ [sinpcos (2(p—fi) tanfi— [—sinpsin (hp—g) ] 13—1:

45

It can be now seen that if the coefficients of g—(B, g—yp and g1: respectively vanish,
the equation (3.34) becomes an ordinary differential equation. For the homogeneous case
this happens when a = (p i e, where e is given by equation (3.20), therefore, for that
value of 5,, there are two directions (i.e. the characteristics) along which equation (3.34)
reduces to an ordinary differential equation. It can, however, be easily verified that, in the

case of an inhomogeneous medium, the other two coefficients vanish at a different value

of angle é, namely when E, = (p. Therefore, equation (3.34) cannot further be utilized.

3. 4.2 Sokolowskii ’s approach

Multiplying equations (3.29) and (3.30) by sin ((1) i e) and cos ((p i 8) respec-

tively, and using the trigonometric identities:

sin ((pie) = sin ((pZFe) cos2€i sin2ecos ((pIFe) (3.35)
cos ((pie) = cos ((pqie) cosZerF sin2€sin ((ere) (3.36)
one gets:
Bo 81p 8p 3H 8H ]
[8x T 2(71'tanpa—x i oa—y— 3x _a—y tanp cos ((p 2F e) (3.37)
do 81p 8p 8H 97H]
+[ay q=2<5tanp—a—y q=oax :1:—— 3x tanp- a sin ((1); s) =

46

Equation (3.37) reduces, for the case of the homogeneous media, to equation

(3.15). However, when (p and H are random fields (i.e. inhomogeneous media), we see

that equation (3.37) contains mixed partial derivatives, and the method reaches a dead

end.

3.4.3 Proposed approach: the method used for HMH materials (ch. 2), adapted to gran-

ular media.

In this section we will present a method similar to the one developed in [Ostoja-

Starzewski, 1992a] and already used in the second chapter of this thesis. Substituting

equations (3.10), (3.11) and (3.12) in (3.27) and (3.28) above, differentiating and setting

(p = —8, we get:

. 2 _a_o_ . a_o . ( a_q) . 09)
[1+ (smp) 18x s1npcospay+2651np cospax+smpay
+ Ocosp( sinpg-E— cospg—S)—%§-I = 0

—sinpcos pg; + (cos p) 2%? + 2osinp( sin pg—i) — cospg—(E)
31>

—ocosp(cosp5;+ smp-3%) —% = O

(3.38)

(3.39)

Here, the rectangular axes are now along the local slip—line directions. Replacing

a
as 8sB

a

a; and Bay by the tangential dertivatives _d_ and _8_ along the CL and B characteristics,

the above equations will become independent of the orientation of axes. Therefore, we

47

get:

. 2 . dsa . . dsa
[1+(s1np) —s1npcosp—]d0’+2osmp cosp+smp— dtp

dSB d dSB
s
+ O'cosp[ sinp - cosp—3]dp _ dH = 0 (3-40)
dsB
_ - £13 2 . , ELSE _.
SWPCOSP + (0081)) d6+2051np smp cosp dtp (3.41)
dsa ds

ds
—(Scosp[cosp(—i—S-—[3 + siandp — dH = 0
a

The corresponding directions of the two families of characteristics, 0t and B, are:

dy = dxtan (on) (3.42)

System (3.40), (3.41) can be solved by using a finite difference approach. Denot-
ing the coefficients of do, dtp and dp in (3.40) and (3.41) as A2, Bz, C2 and A], B], C],

respectively, equations (3.40) and (3.41) can be written:

11
O

Azdo + 2 <G)B2d(p + (o)C2dp — dH (3.43)

Aldo+ 2(o)Bld(p— (o)C1dp —dH

11
O

(3.44)

where:

48

 

 

 

 

 

 

 

 

 

 

 

A1 = —sin(pN:p1)cos(pl;—p—1):—EE+[003(EP-g—Ej]2 (3.45)
B] = sin(pN:pl)[sin(E-Iigfl):—:TE—cos(&q—2:El)] (3.46)
C] = cos(pN;pl)l:cos(pN;-pl)fT:-E+ sin(E-E-;fl)] (3.47)
A2 = 1+ [sin(pN:p2):|2— sin(pN:p2)cos(pN :- p2)§:—§ (3.48)
B2 = sin(pN;p2)[cos(pN:p2)+ sin(pN:p2):—:E] (3.49)
C2 = cos(pN;p2)|:sin(EN—-2+&)—cos(fl:——pz)cai-:—:] (3.50)

System (3.43) and (3.44) can be solved by using a finite difference approach. Thus,

the two parameters, 6N and (111.; at the new point N are given by:

C
Hts-Hz +°2(A2’Bz(‘9N“P2) + “2‘2 ”11‘ 92))
ON = (3.51)

C2
A2+B2(¢N"(p2) ‘3' (pN-pzl

 

(o'N+o])
I{bl-H]_A1(6N-O'l) T——2_C1(pN—pl)

(PN = (131+ (ON+61)B] (3.52)

 

where the dependency of all p’s and H’s on (0 is implicit.

49

Using (3.51) and (3.52), one can determine both 0' and (p at every next point of the
finite difference net. Having 0’ and (p all over the net, one can find the coordinates xN and

yN of the new point N:

1: rt
yi+l_yi+xitan (91+; ”hula“ (VHF;

(KN) = < tan(¢i+§)—tan(¢i+l_§)

1t 1t
x1+1 “Xi+intg(‘Pi+Z)‘y1+1Ctg(‘Pi+1‘1)

1t 1t
C‘gi¢i+z)'“gi¢i+r2)

 

) (3.53)

(yN> = <

 

) (3.54)

as well as the stress distribution by using formulas (3. 10), (3.11) and (3.12). System (3.53)

and (3.54) establishes the average coordinates of the slip-line net.

However, it was found that in the case of granular media, whose behavior is
approximated by the Mohr—Coulomb yield criterion, the differencing scheme given by

(2.15) and (2.16):

yN-y, = (xN—x,) tan(<p,+:—:) (3.55)

it
yrs—yin = (Xn‘xi+1)ta"(¢i+1‘;{)

that give (3.53) and (3.54), does not stabilize the slip-line net with respect to the number

of points that are chosen on the boundary. Moreover, the scheme proposed by (Hill, 1950),

50

i.e. formulas (2.23):

yN—yi = (xN—xi) tan [ ((pi +(pN) /2+1t/4]

(3.56)
yN-yi+1= (XN‘XiH)ta“[(‘Pi+1+‘PN)/2‘n/4]

shows the same weakness. Interestingly, by using instead the scheme proposed by [Szc-

zepinski, 1979], i.e. formulas (2.24):

l
yN—yi = 5(xN—xi) [tan((pi+1t/4) +tan((pN+1t/4)]

1 (3.57)
yN-yi+l = §(XN’X1+1) [tan((pi+1—1t/4) +tan((pN—1t/4)]

solution of the deterministic boundary value problem converges to the analytical one as

the number of points on the boundary increases.

In order to verify the computer program as well as the accuracy of the finite differ-
ence method, a problem for which an analytical solution exists was solved. The problem

(Sokolovskii, 1965, ch 1, pp49) is presented below:

 

 

 

Fig. 3.3 A Cauchy boundary value problem from [S-Okolovskii. 1965']. ch 1. pp49

The input data for the i‘h point on the boundary were as follows:

2

(Mi) = 0 (3.59)
. _ - _L.
x(1) — (1-l)n_l (3.60)

II
C

MD (3.61)

It was observed that the finite difference method gives very accurate results for this

problem, with an error of less than 3%. This suggests that the finite difference method is

52

suitable for the study of the effects caused by the randomness in material properties upon
the slip line net distribution. The analyzed problem in case where randomness exists is
presented below. The noise was taken to be pfi’ e {-0375, 0.375] . and

115’ 6 {—0.00625, 0.00625]

 

 

Fig. 3.4 The inhomogenous solution of the Cauchy boundary value
problem from Sokolovskii, 1965. ch 1. pp49

53

3.5 The Characteristic Boundary Value Problem.

The characteristic boundary value problem was investigated for the case of a gran-
ular medium in more detail in Section 3.8, and is defined as one in which the values of the
function p and of the angle (p are given along the characteristics AB and AC belonging to
two different families of characteristics. Henceforth, ya(x) stands for AB and yB(x) stands

for AC.

 

 

 

Fig 3.5 Characteristic boundary value problem

The angle (1) is determined by the direction of the characteristic, namely:

54
d
(p AC = MIL—:3] —e (3.62)

61
(9A8 = atanidlxa] +8 (3.63)

The values of function 0' along each of the two families of characteristics are found

immediately if 0' is given at a single point, not necessarily point A. Relations (3.16):

2
d-e:F mm)“, = constant (3.64)

hold along 0t and B characteristics respectively and allow to determine 0' within the
domain of influence. The numerical procedure is the same as in the case of a Cauchy
boundary value problem. Partitioning both curves in 11 segments, one can find the values
of 0' and (p at every point on the boundary. Starting from points 1 and 2 which are in the
immediate vicinity of point A, using the same sequence of calculations as in the case of a
Cauchy problem, one can find the values of GM and (pM at the new point M as well as its
coordinates xM and yM. Having found the magnitudes of all the points M on a row, or col-
umn, we can march forward and find the solution in the whole domain of influence, there-

fore the stresses may be found.
3.6 The Characteristic Boundary Value Problem with a Singular Point.

This problem is a very important particular case of the Characteristic Boundary

55

Value Problem since it occurs in most plasticity problems of a certain difficulty. The prob-
lem can be obtained from the previous one when one shrinks one of the characteristics, i.e.
its length tends to zero whereas the increment in the angle (1) on that segment has a finite

value, Atp :

a)

 

 

Fig 3.6 Characteristic boundary value problem with singular point

from [Szczepinski, 1979]

The total increment Atp must be first divided into a number of small increments
corresponding to each of the points on the segment AB. shrunk into a single point, A:
81p = 9312. To show how the numerical procedure works, we will imagine that at A there
are a number of overlapped points, from A0 to An (Fig. 3.6 b). All of them possess the
same coordinates, but the value of the angle (p increases by 81p from one point to the
other: at A0 (pA0 = (9A and at the ith point we have (11],, = (1)}, + (i - 1) . 8(p. Using the
same procedure as described above, one can find values of all the variables within the

domain ACB.

56

3.7 The Mixed Boundary Value Problem.

In the following we will discuss just one of several boundary value problems

which exist. More details can be found in [Szczepinski, 1979. Chakrabarty, 1987].

In a mixed BVP values are given along a characteristic line and moreover along a
non-characteristic one. Let the segment AC be the characteristic. along which the values
of o and (p are known. Along AB of equation y=y(x) that is known, the angle (p is known.
These data are sufficient in order to find determine the problem in the whole region ABC.
Excepting the points lying on the segment AB. similar numerical calculations have to be

performed.

911

 

 

Fig. 3.7 Mixed boundary value problem
from [Szczepinski, 1979]

Let AC belong to the first family of characteristics. The coordinates xM and yM of
the point M, at which the characteristic of the second family through the point 1 intersects

AB, can be obtained from:

57

yM_yl = éltan ((P] -8) +1211] “I’M-8)] (XM_X1) (3.65)
and

yM = y (XM) (366)

of the line AB. The angle (pM is known, therefore GM can be found using (3.52). In the
same way one can solve the case when AC is a characteristic that belongs to the second

family.
3.8 Discussion of Results.

In the present study the initial gradients of yB(x) and ya(x) are 30° and (-30+2e)O
respectively, where e is given by formula (3.20), at the origin of an xOy-coordinate system.
In order to study the effects of homogeneous versus inhomogeneous boundary data, this
boundary value problem is studied here in four cases covering various combinations of
ya(x) and yB(x) being either straight lines or second-order polynomials with negative cur-

vature. Thus, we set up four different boundary value problems:

Problem #1 :

ya = xtan (—30°+2£) yB = xtan (30°) (3-67)

58

Problem #2:
x2
ya = xtan (—300 + 23) _ E
Problem #3:
Ya = xtan (—30° + 28)
Problem #4:

2
X

ya = xtan(—30°-1-2t~:)—15

x —x.
In all the above we takex(i) = (i—l) .M fori=0,1n.

yB = xtan (30°)

2
o x
yB — xtan(30 ) —30

2
o x
yB — xtan(30 ) —§)

n—l

(3.68)

(3.69)

(3.70)

Each one of these figures was obtained from 200 simulation runs, using exactly the

same sequence of random numbers and employing forward differencing. The angle p and

the ultimate resistance to the uniform three dimensional tension H -see figure 3.8, were

taken to be:

(135(0)) = 15°, H5010) =

Haw»

 

t411035010),

(H05((0)) = 0.6

(3.71)

59

 

Int)

Hfl

P . L.
H ° U 0"

 

 

 

Fig 3.8 The Mohr stress diagram

Numerical results of simulation of Problem #1, in three cases of randomness:

p' e {—0.015,0.015], HO' 6 [—0.0006,0.0006] (3.72)
p' e [—0.025,0.025] , HO' 6 [~0.001,0.001] (3.73)
p' e [—0.075,0.075], HO' 6 [—0.003,0.003] (3.74)

are shown in the sequence a), b) and c) of Fig. 3.9, while those of Problem #2 in the
sequence (1), e) and f) of this figure. For computational reasons, we considered H0 to be the

random variable instead of H, the relationship between these two being given by (3.71).

One notes here a gradual sensitivity of the slip-line nets to material randomness and
to the inhomogeneity in the boundary data. This latter type of sensitivity is confirmed by
Fig. 3.10, which gives solutions to Problem #3 - a), b) and c) - and to Problem #4 - d), e)

and f) - also in three cases of randomness given by (3.72), (3.73), and (3.74)

60

 

 

 

a)Prob1cm #1, noise:x (1) Problem #2, noise:
according to (3.72) according to (3.72)

 

 

 

b) Problem #1, noise x e) Problem #2, noise X
according to (3.73). according to (3.73).

 

 

 

c) Problem #1, noise )1 1‘) Problem #2, noise x
according to (3.74). according to (3.74).

Fig. 3.9 Characteristic boundary value problems

61

 

 

 

a) Problem #3, noise x (1) Problem #4, noise
according to (3.72) according to (3.72)

 

 

 

b) Problem #3, noise x e) Problem #4, noise x
according to (3.73). according to (3.73).

 

 

 

V

c) Problem #3, noise x 1‘) Problem #4, noise x
according to (3.74). according to (3.74).

Fig. 3.10 Characteristic boundary value problems

62

We investigated for the characteristic boundary value problem of an inhomoge-
neous granular medium - illustrated in fig 3.10 f) - the probability distribution of the max-
imum distance from the origin and of the three stresses, i.e. (1’x , 0'), and txy at that point.
It was observed that, even though for the maximum polar radius, the average solution of
the random problem does not differ substantially from the deterministic BVP as can be
seen in Fig. 3.11 a), the stresses have an average solution of the inhomogeneous problem
always smaller compared to the one of the homogeneous medium, see Fig. 3.11 b) and Fig
3.12 a) and b) respectively. So we do need more material, but the stresses predicted by this
theory are smaller, in average, than those predicted by the classical one. In figures 3.10

and 3.11 the variables p and H0 were taken to be:

1105(0))
tan (135(0)) ) ’

 

(9560)) = 15°, 115(6)) = 0105(6)» = 0.6

and the noise was according to (3.74)

63

 

 

   
   

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

1 .2 V 1 T I I
1 -- ‘ ------- 4
Legend:
-— ution of the
homogeneous medium
0.8 - ' - - ~ average solution of the ' n
inhomogeneous medium
0.6 _. . . . . —.
0.4 - _
0.2 - I
o 1 l l l l
12 12.1 12.2 12.3 12.4 12.5 12.6
a) of the polar radius r((0)
1 .2 t r r 1— fi 1 t t
1 -— ‘ -1
Legend:
— ution of the
homogeneous medium , :
03... _averegesolutionofthe ~
inhomogeneous medium -
0.4 o - - - ~~ 4
0.2 - -
o l 1 J J 1 J l 1
-O.26 —0.24 —O.22 -O.2 -O.1 8 -O.16 --O.1 4 -0.12 —O.1 -0.08

b) of the stress 6,,(03)

Fig. 3.11 The probability distribution at the comer point

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
 
 
  

 

 

 

 

 

  
   

 

 

 

 

 

 

1.2 l l l l l
1 ... . .1
Legend:
— solution of the
homogeneous medium
0.8 _ ' ' ‘ ' average solution of the ‘ ' j
inhomogeneous medium
0.6_...,. .' , .. ..... . ....... .r
0.4 —' . A
0.2 - a
o 1 l 4
0.76 0.78 0.8 0.82 0.84 0.86 0.88
a) of the stress 03(0))
1 .2 l r r i f i r l
1 ._ ,
Legend:
— solution of the
homogeneous medium
0.8- averagesolutionofthe
inhomogeneous medium
0.6 - - - - . -
0.2 - -
0 . t . 1 i i i 1
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

b) of the stress Ixy((n)
Fig. 3.12 The probability distribution at the corner point

65

In figure 3.13 we make a comparison of the same boundary value problem for two
different media: one governed by the HMH criterion - a), the other by the MC criterion -

b). In both cases, the B-characteristic is the same and specified by:

yB = x— _ (3.75)

whereby the Ot-characteristic is given by the yield condition used. In both cases the noise is

5% about the mean, that is:

ka'e [—-0.025,0.025] , (k6) = 0.6 (3.76)

for the first case, where k5 is a uniform random variable. For the second case (Fig. 3.13 b)):

[35' 6 {—0.375, 0.375] H05' 6 {—0.015, 0.015] (3.77)

where (p 5) = 150 , (H08) = 0.6 and p5 and H05 are both independent uniform random
variables. The relatively stronger sensitivity of the MC-type material versus the HMH-type
material, accompanied by a departure from the orthogonality of the slip-line net, is a typical

feature here.

66

 

 

 

Fig. 3.13 The characteristic boundary value problem for materials of

(a) HMH-type, and (b) MC-type.

67

Based on the assumption that the internal friction coefficient, 11 = tan (p) , is a
random variable, a probabilistic failure criterion was developed in [Alpa and Gambarotta,
1990] where the Peano-Jordan derivative of the probability of failure as a function of the
resolved tractions was assumed to have a Weibull form. There, the main conclusion was
that the stresses given by the probabilistic criterion fit better the experimental data than

those given by the deterministic MC yield criterion.

To better understand the effects of the randomness in p5 and H05 on the response of
an MC-type medium, we plot in Fig. 3.14 the mean of r5((i)), its difference with respect to
the deterministic medium Bdet = B(Pdev Hdet)’ and the standard deviation of the position
r5((u) of the extremum point with respect to the origin. Figures. 3.15, 3.16 and 3.17 present
the corresponding specifications for the stress components 0x00), O'y(0)) and Ixy((u) at the
same point. These graphs correspond to the characteristic boundary value problem shown
in Fig. 3.13 b). The sequences a), b), c) of figures 3.14-3.17 correspond to uniform random
variates while (1), e), f), of the same figures, to Weibull type random variates. For sequences
a), b) 0) each point on the surfaces was obtained from 200 simulation runs, i.e. 200 different
realizations of 8(a)), while for d), e), f) 40 simulation runs were used for each particular
point, due to computational reasons: in order to generate the Weibull type random variates
we used the von Neumann acceptance-rejection method. For a single realization there are
needed 200 random variates and for each of these numbers there are needed, in average,
approximately 41 000 do loops of the random number generator, therefore we encountered
some problems with the data storage. However, for 40 realizations the problem is stabi-

lized.

68

 

100 100

Anglo!» <H> Wt“ 41>
a) The mean polar radius, r(cu) d) The mean polar radius, r(0))

   

”‘40 1° 0 (H)
b) Its difference from the homogeneous e) Its difference from the homogeneous
medium medium

   

1° ‘0
M10. 40> ° <H> Angie. <10> 0 d4)

c) The standard deviation, STD(r(0))) f) The standard deviation STD(r(u)))

Fig. 3.14 Statistical specifications of the polar radius
a), b), c)— uniform type noise; d) e), f)- Weibull type noise

69

   

I ,<lu> 10 0 db WC,<10> '0 0 db

a) The mean of the field variable (5(0)) (1) The mean of the field variable cx(a))

   

Ange. 41» db AM, <1!» db
b) Its difference from the homogeneous e) Its difference from the homogeneous

medium medium

   

Awe, (Nb ‘° 0 at

c) The standard deviation, STD(0'x(u))) f) The standard deviation, STD(GX(0)))

Fig 3.15 Statistical specifications of the field variable Ox((.t))
a), b), c)- uniform type noise; (I) e), O- Weibull type noise

70

 

Angie, <Io> ‘0 9 <H> Angle. dc) ‘0 ° <H>

a) The mean of the field variable oy(0)) d) The mean of the field variable O'y(0.))

   

b) Its difference from the homogeneous e) Its difference from the homogeneous
medium p medium

   

Anymci» ‘0 0 db

c) The standard deviation, STD(0'y(w)) f) The standard deviation, STD(Gy(u)))

Fig. 3.16 Statistical specifications of the field variable cyan)
a), b), c)- uniform type noise; (1) e), f)- Weibull type noise

71

 

‘0 0 Angle, 40> db

 

Angle. (10> '0 0 (H) Anglo, (ID) <H>
b) Its difference from the homogeneous e) Its difference from the homogeneous
medium medium

   

Andi. <m> ‘0 ° 41> Angle, <m> <H>

c) The standard deviation, STD(1xy((n)) f) The standard deviation, STD(1:xy(w))

Fig. 3.17 Statistical specifications of the field variable txy(0))
a), b), c)- uniform type noise; d) e), f)- Weibull type noise

72

For both uniform and Weibull types of noises, standard deviations of all four depen-
dent field variables have an increasing tendency with the increase in H0. However, with
increasing angle p, these standard deviations decrease and go to a minimum in the vicinity
of 15 degrees, beyond which they start to increase. This is an interesting observation. The
limiting value of p = O that corresponds to the Huber-von Mises-Henky yield criterion
could not be obtained by running the program for the MC-type material. Note here that the
variables p and H0 have bounded domains due to the instability of the inhomogeneous
problem at small values of angle p and relatively big values of HO. On one hand, the expec-
tations of r and Txy have a very small or no dependence upon H0, as can be seen in Figs
314-3. 17; on the other, the means of the two stresses Ox and 0”, increase continuously with
the increase in H0; by increasing p the means of all four variables decrease. Moreover, the
means of r((u) and txy(o)) are approximately linear and quadratic in p. It is doubtful that the
above results could be obtained analytically, in view of the nonlinear and stochastic nature

of the problem.

A study of all the boundary value problems for a granular medium leads to the fol-

lowing principal conclusions regarding the effect of noises p 8’ and HOS':

i) The scatter in dependent field variables increases continuously with decreasing angle p
and with the increase in the ultimate resistance to uniform three dimensional tension, H.
However, p and H must be confined to certain finite ranges, dependent upon a particular

noise level, in order for the problem to remain stable.

73

ii) Parameter p has a stronger influence than H; the limiting case of p = 0 could not be
achieved with the MC-medium program, due to the instability of the inhomogeneous prob-
lem. However, as H tends to O, i.e towards the cohesionless medium H = 0, the problem

is still in the stable range.

iii) The average of the inhomogeneous stress distribution is always smaller than the deter-
ministic one, and the difference increases with the noise and inhomogeneity of the bound-

ary data.

iv) The standard deviations of the polar radius, r(0)) and stresses 6x (0)) , 0y ((0) and
txy (0)) , of an MC-type of material, present a minimum in the interval p = 15° i 3° ,

which means that the characteristic boundary value problem is most stable in that interval.

v) Although in case of Weibull type noise the position and the field variables have smoother
variations, the tendencies of variation do not change compared to those corresponding to
uniform type noise. Moreover, in the latter case, the standard deviations are greater, which
suggests that the characteristic BVP is less sensitive to Weibull type noise than to uniform

type noise.

vi) The slip-line net of a granular material is dependent upon the integration method. For-
mula (2.24) recommended by [Szczepinski, 1979] stabilizes the problem while the others,
e. g. (2.23) recommended by [Hill, 1950], do not; here, the explicit formulas were used first

as a ‘predictor’ and then, without any convergence problems, the implicit formulas were

74

employed as a ‘corrector’.

vii) For values of noise above 5% there is an increasing change in the average, and very

amplified scatter, in the random fields of stress as well as position.

vii) The changes due to the Weibull type noise are more quantitative than qualitative. How-
ever, for this type of noise, the graphs showing the difference from the homogeneous
medium and the standard deviations respectively are smoother. Moreover, in this case the
standard deviations are smaller than in the case of uniform type noise, which suggests that
the random rigid perfectly plastic medium is less sensitive to Weibull type variates than to

uniform ones.

Thus we conclude that, in case of very small noise (e.g. given by (3.72)), one may
replace the average solution of a stochastic problem by the solution of a deterministic prob-
lem with pen? = pdet = (p 5) as well as Heff = Hdat = (H 8) ; that is Beff = Bdet- Given the fact

that the governing system is a nonlinear stochastic one, this is an interesting observation.

All results in this chapter were obtained under the assumption of independence of

random variables p5, H05 (i.e. H5 as well) at all the points of the finite difference nets.

4. CONCLUSIONS

A major motivation for the research presented in this thesis has been the observa-
tion that the slip-line patterns observed in experiments on metals ([Kachanov, 1971] and
[Mellor and Johnson, 1985]) differ from those predicted by classical deterministic
medium theory. Similar discrepancies have been observed in soil mechanics ([Alpa and
Gambarotta, 1990]). Our method allows an assessment of such differences through the
determination of the statistics of slip-lines and stress fields. As mentioned in the Introduc-
tion, plasticity of random inhomogeneous media has recently been studied by Nordgren
[1992] with a focus on stochastic theorems on limit load coefficients and an application to
the loading of a wedge. Solution of this latter problem has been based on finding the mean
of the minimum energy dissipation on the multiple branches of possible zig-zag velocity
paths along the rigid-plastic boundary. This methodology differs from ours: we propose
solving a given stochastic boundary value problem directly by calculating a large number
(two hundred, say) of responses in a Monte Carlo sense. Given the power of today‘s com-
puters, this is done in a couple of minutes, unless the mesh resolution is more than about
fifty points, and yields practically the whole range of possible behaviors - that is, the prob-
ability distributions of slip-line fields and stress fields. This is in contrast to [Nordgren,

1992] which reports a need for extensive computational tasks.

Regarding the effect of random variates (k'a) (HMH-type media) or ps’ and

H08’ (MC-type media) respectively, the following major conclusions can be drawn:

75

76

i) There is practically small difference between the ensemble average net of slip-lines of
the stochastic problem (i.e. for a random medium B) and the net of a corresponding deter-
ministic problem (i.e. for a homogeneous medium Bdet) for very small noise (e. g. given by
(3.72)); however, this difference increases with the growing inhomogeneity in the boundary

data.

ii) The slip-line net of media governed by HMH yield criterion (i.e. metals) is independent
upon the integration method, while is not true with MC-type media; for these types of
media formula (2.24) recommended by [Szczepinski, 1979] stabilizes the problem while

the others, e.g. (2.23) recommended by [Hill, 1950], do not.

iii) The choice of the forward (as used here) versus backward differencing scheme has only

a small effect on the solution.

iv) More material is needed to withstand a given load than according to the classical theory.
Inhomogeneities in the boundary data have a strongly amplifying effect on the scatter of
dependent field quantities; for noise growing above 5% there is an increasing change in the

average, and very amplified scatter, in the dependent field variables.

v) The granular medium governed by a Mohr-Coulomb yield criterion is more sensitive to

noise than the Huber-Mises-Henky type medium (recall Fig. 3.13).

We conclude that in case of very small noise one may replace the average solution

77

of a stochastic problem by the solution of a deterministic problem with average preperties,

that 18 Beff =Bdet'

It should be mentioned here that this method can be applied to the determination of
the velocity fields and can be extended to the hardening behavior and anisotropic yield con—
ditions. Indeed, anisotropic yield conditions of a random character are expected from a

micromechanical derivation; the subject remains a challenge for future research

Finally, it is of interest to mention that the present study has relations to another
topic: transient stress waves in randomly linear and non-linear inhomogeneous media
[Ostoja—Starzewski, 1991]. Mathematically these both problems are governed by stochastic
quasi-linear hyperbolic systems. They both display Markov properties and share the con-
cepts of forward evolution cones in place of unique characteristics of deterministic homo-
geneous media problems. However, the major difference between them lies in that the
spacing of characteristics in a plasticity problem corresponds directly to a mesoscale, while

no such mesoscale concept appears in the wavefront studies.

BIBLIOGRAPHY

78

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