LIBRARY Mlchlgan State Unlverslty PLACE IN RETURN BOX to remove this chockout from your record. TO AVOID FINES return on or bdoro duo duo. DATE DUE DATE DUE DATE DUE MSU loAnAfflmdMAcflon/EqudOppomwylmwon mm: ABSTRACT On Directional Reinforced Incompressible Nonlinearly Elastic Solids: Simple Response and Correlation to the Linear Theory, and Failure of Ellipticity in Plane Deformation By YUE QIU A model for finitely deformable incompressible elastic materials with fiber reinforcements is established. The engineering significance of this material model is verified since it fits well with real material data. The fully nonlinear responses of this material is discussed for a number of important deformations, and the loss of monotonicity of the material response is studied. The correlation to the linear theory is provided in terms of the elastic constants. The ellipticity of this anisotropic material and the plate bifurcation problem are further studied. It is revealed that the loss of ellipticity phenomena under planar deformation involve different patterns, depending on loading path. These patterns involve simple loss of ellipticity, loss-recovery-loss of ellipticity, and primary loss-secondary loss of ellipticity. The geometry of the ellipticity boundary is made clear. The discussion on the directions of the weak discontinuity surfaces is also presented. Finally, it is concluded that there is no explicit correlation between loss of ellipticity and loss of monotonicity of the stress response, and no explicit correlation between loss of ellipticity and bifurcation. DEDICATE TO MY PARENTS iii ACKNOWLEDGMENTS I would like to express sincere appreciation to my advisor, professor Thomas J. Pence, for his very helpful guidance and support throughout this work. The financial sup- port for this work from the Research Excellence Fund for composite materials, adminis- tered by the Michigan State University Composite Materials and Structures Center, is also gratefully acknowledged. TABLE OF CONTENTS 1. Introduction 1 2. Material Model and Its Properties 10 2.1. Material Model with Fiber Reinforcements ......................................................... 11 2.2. General Stress-Deformation Relation .................................................................. 18 2.3. Homogeneous Deformations ............................................................................... 19 2.4. Material Properties (Single Fiber Family) ........................................................... 23 2.4.1. Material Response to Homogeneous Deformations ................................... 24 2.4.2. Material Properties at Infinitesimal Limit ................................................... 51 2.5. Material Properties (Two Balanced Fiber Families) ............................................ 56 3. Ordinary Ellipticity in Planar Deformation 60 3.1. The Ordinary Ellipticity Condition for Arbitrary Deformations ......................... 61 3.2. Specialization of the Ordinary Ellipticity Condition to Planar Deformation ...... 65 3.3. Loss of Ellipticity in Local Plane Strain .............................................................. 70 3.4. Basic Results for both Global and Local Plane Strain Ellipticity ........................ 71 3.5. An Alternative Form of the Global Plane Strain Ellipticity Condition ............... 73 3.6. Loss of Ellipticity in Global Plane Strain ............................................................ 75 3.7. Global Plane Strain Ellipticity Boundary: Cross-Sections at Constant C12 ............ 80 3.8. Projection of the Ellipticity Boundary onto the (C12, y)-Plane ............................ 91 3.9. Multiplicity and Orientation of the Discontinuity Surfaces for C12 = 4 ............... 95 3.10. Orientation of the Discontinuity Surface at First Loss of Ellipticity ................. 97 3.11. Cross Sections of 5 at Fixed 7 .......................................................................... 102 3.12. Asymptotic Pr0perties of the Global Plane Strain Ellipticity Boundary ......... 106 3.13. Ellipticity under Special Deformations ............................................................ 110 4. Summary and Conclusion 119 Appendix A: Algebraic Root Analysis of the Characteristic Polynomial ................ 120 Al General Algebraic Procedures upon the Quartic Polynomial ............................ 120 A2 Criteria for Double Real Roots .......................................................................... 122 A3 Criteria for Two Pairs of Double Real Roots ..................................................... 123 A.4 Criteria for a Triple Roots and a Quadruple Roots ............................................ 124 Appendix B: Bifurcation from Homogeneous Defamation under End Thrust ..... 125 B] Homogeneous Plane Deformation ..................................................................... 126 B2 Incremental Formulation .................................................................................... 128 B3 Inhomogeneous Bifurcation ............................................................................... 131 Appendix C: An Alternative Way to Obtain the Ellipticity Condition in the Buckling Problem 140 References 142 1. Introduction In nonlinear elasticity, important and closely related issues, which have obtained wide attention, are uniqueness, bifurcation and associated stability of the solution, as well as ellipticity and material stability. There are two kinds of bifurcations, namely homogeneous bifurcation and inhomogeneous bifurcation, from homogeneous equilib- rium solutions have been discussed in the literature. The homogeneous bifurcation is a phenomenon in which more than one homogeneous deformation states correspond to one single stress state, and in which the shape of the material body is immaterial, as pointed out by Ball and Schaeffer [Ball and Schaeffer 1983]. It is, therefore, reasonable that the homogeneous bifurcation analysis be placed in the scope of material stability. The material stability analysis is a different aspect from the traditional (structural) stability analysis, since in the former there is no particular structure fabricated from the material involved. In the analysis of the inhomogeneous bifurcation, the shape or the structure of the material configuration must be addressed. We shall thus refer the inhomogeneous bifurcation as structural bifurcation, and the associated stability structural stability, in contrast to material bifurcation and material stability. It should be noted that the term material stability is placed on the overall material properties and is not at the microscopic level herein. One famous example of the homogeneous bifurcation and stability is Rivlin’s cube problem [Rivlin 1948b], where the deformation states of an incompressible, isotropic, neo-Hookean material point under equi-triaxial dead load traction were studied. Further discussion on this problem for a widened class of materials is given by Ball and Schaeffer [Ball and Schaeffer 1983]. The homogeneous bifurcation and stability of incompressible, 2 isotropic elastic material subject to in-plane equi-biaxial dead load traction was studied by Ogden under plane strain condition [Ogden 1985] and under plane stress condition [Ogden 1987]. Certain restrictions for material stability were then deduced from these studies. Restrictions pertaining to material stability can be found in the literature. As Rivlin has summarized [Rivlin 1981], one condition for material stability is that the strain energy density function must be positive definite if expressed as a function of strain tensor. This condition is evident from the fact that if the strain energy is non-positive for some nonzero strain tensor, then the material will be unstable in its undeformed state. Another condition for material stability is Hadamard’s condition, it requires that the speeds of all plane waves, propagated in the material body occupying a domain in three-dimensional space, must be positive [Beatty 1987]. Even in the case of infinitesimally small deformations, the above mentioned two conditions give different restrictions on material properties. In the cases where finite deformations are involved, the conditions for material stability can be that the strain energy density function, in terms of the defamation gradient tensor, be either positive semi-definite or locally convex [Truesdell and N011 1965, Rivlin 1981]. Although the mechanisms that cause the material instability have not been made completely clear yet, we will adopt the condition that the strain energy density function must be positive semi-definite (and of value zero only for rigid body motions) in establishing our material model. Another significant issue that concerns the regularity of the solutions in the theory of finite deformation is the ellipticity analysis of the governing field equations. Loss of ordinary ellipticity happens if, in the material body, there exists a surface across which 3 weak discontinuity of the solution field is capable of taking place. That is, singular equilibrium field exists in homogeneous material body. One such phenomenon observed in real world is localized shear band. Strong ellipticity, which concerns the material stability, is often correlated with bifurcation of equilibrium solutions. Conditions for ellipticity can be found in [Knowles and Stemberg 1975, 1977] for compressible materials, and [Abeyaratne 1980, Zee and Stemberg 1983] for incompressible materials, respectively. Conditions for ordinary ellipticity are usually expressed as the nonsingularity of acoustic tensor. In comparison, conditions for strong ellipticity, derived from Hadamard’s condition, are usually expressed as the positive definiteness of the acoustic tensor. Further derivations of the ellipticity conditions to explicit inequalities for easy verification can also be found in these papers. Hill and Hutchinson have established the plane strain incremental governing equations for the tension problem of the class of time-independent piecewise linear or wholly linear materials [Hill and Hutchinson 1975, Hill 1979]. The governing equations change types among elliptic, strong elliptic, parabolic, hyperbolic types, depending on material and load parameters through the nature of the roots (real or complex) of the corresponding secular equations. [Rosakis 1990] has further discussed the derivation of ellipticity conditions and their mechanical interpretations for compressible materials. Together with a later paper, [Rosakis and Jiang 1993], they have analyzed the problem of existence of equilibrium deformations with discontinuous gradients. Recently, Horgan has analyzed the ellipticity problem for generalized Blatz-Ko material [Horgan 1996]. He has found the range of material parameter for possible loss of ellipticity, and further discussed the deformation parameter ranges which never fail ellipticity. A discussion on ordinary and strong ellipticity can also be found in [Ogden 1984]. A phenomenon that is associated with the structural instability is the structural (inhomogeneous) bifurcation (buckling) of equilibrium configuration. Among various types of inhomogeneous bifurcations, out-of—plane bifurcation is of special interest. Examples of this type of structural bifurcations can be found in [Sawyers and Rivlin 1974] for homogeneous material, and in [Pence and Song 1991, Qiu et. a1. 1994] for composite material. The so called Euler’s stability criterion [Beatty 1987] states, an equilibrium configuration of a structure is unstable for dead loading if there exists, for the same loading, another equilibrium configuration situated in the neighborhood of the assigned one. In the utilization of Euler’s criterion, and in the scope of finite elasticity, the structural stability analysis is often carried out in the context of the theory of small deformation superposed on finite deformation [Biot 1965]. It has shown that the structural stability of equilibrium configuration can be affected by the material anisotropy [Polignone and Horgan 1993], and by the composite structure [Horgan and Pence 1989]. It has been shown that the out-of-plane inhomogeneous bifurcations under in-plane loading are possible to take place when the corresponding incremental governing equation is either elliptic or non-elliptic [Hill and Hutchinson 1975, Hill 1979, Kurashige 1981]. It is also found, however, that loss of ellipticity of the incremental governing equation and in-plane inhomogeneous bifurcation take place simultaneously, for incompressible isotropic elastic membranes under in-plane loading [Haughton 1987]. With the anisotropic material modeled by adding fiber stiffness to the Blatz-Ko material, Kurashige [1981] studied the bifurcation of a transversely isotropic slab under axial loads. It is usual that the difficulty of solving the nonlinear partial differential equations 5 of finite elasticity largely confines the number of exact solutions to problems with a high degree of symmetry, or to problems with fewer dimensions. It is also usual in solving these problems that certain kinematic constraints and other simplification assumptions are introduced to restrict the variety of deformations. By the study of Ericksen and Rivlin, the total number of kinematic constraints can be as many as six [Ericksen and Rivlin 1954]. Among them, the incompressibility and inextensibility in assigned direction are frequently included in the research in finite elasticity. The incompressibility constraint is a good approximation for various real materials such as natural rubber, synthetic elastomers and biological tissues [Beatty 1987]. The inextensibility constraint is usually used to describe the effect of strong continuous fiber reinforcement [Adkins and Rivlin 1955]. Material models (constitutive relations) play an important role in the research in finite elasticity. These models enable one to analyze in deep the mechanical phenomena of a given system, and to predict the response of the material to specified traction and displacement boundary conditions, and thus provide convenient tools for engineering design. The material models can also provide guidelines for the design of material experiment and for the interpretation of experimental data. Two simple but widely-used isotropic materials models are the neo-Hookean type [Rivlin 1948a] for incompressible materials and Blatz-Ko type [Blatz and Ko 1962] for compressible materials. In finite elasticity, it is mathematically convenient to express the material model as a strain energy density function, while in the classical linear elasticity the material properties are usually described by a constant elasticity tensor. The strain energy density is usually expressed as a function of a set of deformation parameters. These parameters are regularly independent, complete and irreducible strain 6 invariants under the symmetry transformation appropriate for the material. For compressible isotropic materials, the number of such strain invariants is three. For incompressible isotropic materials, this number becomes two. According to Green and Adkins [Green and Adkins 1960], the number of such strain invariants describing compressible transversely isotropic material is exactly five, while this number is four for the incompressible counterpart. In engineering practice, material anisotropy is often introduced by fiber reinforcement. This provides unprecedented flexibility to economically meet performance setpoints by tailoring the directional response of the individual substructural constituents. For example, a material reinforced with one family of straight parallel fibers belongs to the class of transversely isotropic materials, and the corresponding strain energy density function involves five strain invariants or four strain invariants for compressible or incompressible materials, respectively. In linear elasticity this material is modeled by five elastic constants in the elasticity tensor [Christensen 1979]. For materials reinforced with multiple families of fibers, the number of such strain invariants depends on the number of families of the reinforcements and depends on the direction of each family of reinforcement. Besides, the anisotropic effects due to several fiber reinforcements will be coupled, since reinforcement in one direction may affect the material properties in other directions. The complete modeling of fiber reinforcements may proceed along the following line. Given a system of fiber reinforcements, first, derive the complete and irreducible set of strain invariants under the symmetry transformation group that is associated with the given system of fiber reinforcements. The material model is usually a polynomial of these strain invariants. Second, identify each term in this polynomial by its 7 nature in deformation and attach each term with appropriate material parameter. The first step is within the theory of representations for tensor functions, which has obtained the efforts of a number of researchers and is presented pretty completely in [Zheng 1994]. In literature, there appear two straightforward ways of modeling the effect of fiber reinforcement. One way is to model the fiber reinforcement by the kinematical constraint of inextensibility in fiber direction, as can be seen in [Adkins and Rivlin 1955, Pipkin and Rogers 1971]. This modeling procedure may fail to yield sufficient deformability if the reinforcement consist of several families of fibers in different directions. Another way of modeling the fiber reinforcement, that has been used for the analysis of materials reinforced in single direction, is by adding extra stiffness in fiber direction. Referring to the works presented in [Kurashige 1981, Triantafyllidis and Abeyaratne 1983], the modeling of compressible material reinforced with a single family of fibers is gained by adding fiber stiffness to the Blatz-Ko material in the direction of reinforcement. In a recent paper [Polignone and Horgan 1993], a similar modeling is obtained in extending the incompressible neo-Hookean material by fetching in additional stiffness in the radial direction for the spherical problem considered. In the early 1970s, Pipkin, Rogers and Spencer developed what is known as the ideal continuum theory of fiber-reinforced material (ideal theory, for abbreviation) [Pipkin and Rogers 1971, Rogers and Pipkin 1971b, Spencer 1972], its origin may date back to [Adkins and Rivlin 1955]. Here the term ideal has three aspects: (1) the fibers are continuously distributed so that they can be represented by a field of unit vectors and the material can be treated as continuum, (2) the material is inextensible in the fiber direction, and (3) the material is incompressible. A number of problems have been solved employing 8 the ideal theory [Rogers and Pipkin 1971a, 1971b, England et. al. 1992 and Bradford et. al. 1992]. Several interesting phenomena have been predicted by the ideal theory, such as: that stress is channeled, mainly due to the inextensibility, both parallel and normal to the fiber direction without attenuation; that there exist stress concentration layers; and that shear deformation can only take place along the fiber direction (see [Pipkin and Rogers 1971] as well as [Rogers 1975]). These phenomena can be expected for highly anisotropic materials (such as fiber-reinforced composites with stronger fibers and weaker matrices). For example, as we will see later, the shear modulus for shear normal to fiber direction will be significantly larger than that along fiber direction for anisotropic materials if the deformation is beyond infinitesimally small. The standard mathematical approach for solving a boundary value problem is that one determines a physical field, often the deformation field, from the governing equation, the constitutive relations and the boundary conditions. In view of the nonlinearity of the differential equations of finite elasticity, the standard mathematical approach is usually not applicable. Instead, one adopts the semi-inverse approach. The semi-inverse approach can be summarized [Beatty 1987] as follows. First, a suitable class of deformations characterized by a number of parameters is chosen for study. Then the constitutive relations is used to determine the stress distribution that satisfies the differential equations of equilibrium. Finally, the surface loading necessary to maintain the deformation in its equilibrium configuration is determined. The deformation that can be produced in a material body by the application of surface traction alone is called the controllable deformation. In the studies that follows, we will, in chapter 2, establish a material model for 9 materials reinforced, in general, with multiple families of fibers. The material reinforced with single family of fibers will be studied in detail. The nonlinear response of this material to certain important deformations will be addressed. The loss of monotonicity of the material response will be discussed. The elastic constants corresponding to the linear theory are then derived. In chapter 3, we shall discuss the ordinary ellipticity of this material reinforced with single family of fibers. The phenomena of loss of ellipticity under planar deformation will be clarified. The generation of discontinuity surface in the material body, and the orientation of the discontinuity surface will be discussed. The loss of ellipticity is further studied in the context of certain special deformations introduced in chapter 2. Two substantial appendices are also given. The first, Appendix A, develops the algebra associated with the polynomial root type changes. Criteria for double real roots, for two pairs of double real roots, as well as criteria for triple roots and quadruple roots are discussed. The second, Appendix B, correlates the results of this thesis with the out-of-plane inhomogeneous bifurcations of a thick plate consisting of the material reinforced with single family of fibers under end thrust. A short appendix, Appendix C, provides an alternative way to obtain the ellipticity condition in the buckling problem discussed in Appendix B. 2. Material Model and Its Properties The description of nonlinear mechanical behaviors of anisotropic materials is still in its developing stage. In the present studies we do not attempt to model the fiber reinforcement in its complete nature. Instead, we take into account the effect of each family of fibers by including individually additional stiffness in specified directions given birth by each family of fiber reinforcements and omit any explicit coupling effect. It is evident that the most significant and direct effect introduced in material body by fiber reinforcement is the increased stiffness and strength in fiber direction. This modeling procedure allow us to isolate and investigate the most important effect of the fiber reinforcement and neglect any other effects at least at this stage. It is assumed that the material is incompressible, but we are not going to include the constraint of inextensibility in fiber direction. The fiber inextensibility can be simulated as the additional stiffnesses in the fiber directions become very large. It is assumed that fibers are continuously distributed throughout the material body and thus each family of fibers can be modeled by a field of unit vectors of which the trajectories are fibers. This makes it possible to employ continuum theory. In this chapter, the material model is introduced and discussed, including the connection to material symmetry. Then the fully nonlinear response is discussed for a number of important deformations. The loss of monotonicity of the material response is studied, especially for simple shear deformation. Finally the correlation to the linear theory is provided in terms of the elastic constants that follow from the specialization to infinitesimal deformations. 10 11 2.1. Material Model with Fiber Reinforcements The mechanical responses of a nonlinearly elastic material that can sustain finite deformation are determined by two configurations, namely, original (or reference) configuration and current (or deformed) configuration, but do not depend on the time history from one configuration to the other. The deformation from original configuration to current configuration is then an invertible mapping between these two configurations and can be, in general, written as x = x(x), (2.1.1) or in component form as x1 = x1(X1,X2,X3), x2 = x2(Xl,X2,X3), x3 = x3(X1,X2,X3), (2.1.2) which is not written as a function of time here since we are to deal with static processes only. The corresponding deformation gradient tensor that describes the configurational relation in between is defined as _ ax F — 87‘. (2.1.3) The strain energy density function W, defined over the material body with respect to the original configuration, is important and convenient for determining the mechanical responses, such as the deformation and stress field, of an elastic material. This strain energy density function W, in general, is a function of deformation gradient tensor W = W(F, X) or W = W(F) (2.1.4) for an elastic material that is initially inhomogeneous or initially homogeneous, respectively, in its original configuration. 12 Among the various strain energy density functions for finite elastic materials, the neo-Hookean type is the simplest for the purpose of analysis and gives reasonable description for incompressible isotropic finite elastic materials. Here, we consider an incompressible anisotropic finite elastic material model that is extended from the neo-Hookean material by taking into account the effects of reinforcements of M families of fibers by adding extra stiffnesses in the fiber directions. This material model given by the strain energy density function is expressed as M (m) W = gal -3)+ z B—z-(KWL l)°“"", (2.1.5) m = 1 where r, = me), Km) = A(m)-CA("‘), c = FTF. (2.1.6) Here I, is the first invariant of the right Cauchy-Green strain tensor C given by (2.16),. The incompressibility constraint requires detF = 1. (2.1.7) For the material reinforced with M families of fibers, quantities with the superscript (m) (m = 1,2,...,M) are associated with the mm family of fibers. For the case where materials are reinforced with only one family of fibers, we will omit the superscript (1). The fiber reinforcements are assumed to be continuously distributed. Thus, vector Am) is the m‘h unit vector field modeling the mth family of fibers in the undeformed configuration and is determined by the orientation of the fiber of the mth family at a specified point in a given coordinate system. Now, it is clear that K‘m-l is the elongation in the direction of the mth family of fibers. Note here that the unit vector field Am”) could in general be a function of 13 X. If so, the strain energy density W, in view of (2.1.5), takes the form of (2.1.4)]. If the material is uniformly reinforced with only straight parallel fibers, then Am) is independent of the position vector X and the strain energy density function W takes the form of (2.1.4)2. We shall say that the fiber reinforcement is in-plane fiber reinforcement if all of the fibers lie in parallel planes. This is the common case for fiber-reinforced composite materials. For in-plane fiber reinforcement, we shall set the rectangular Cartesian coordinate system in the way that the planes determined by the in-plane fibers are normal to the Xz-axis. We can, therefore, determine the fiber orientation of the mlh family of fibers at a specified point by an angle 0“") in planes parallel to the (X1,X3)-plane. This angle 0"“) is measured from the positive direction of the XI coordinate to the tangent direction of the fiber at that point by a rotation about the Xz-axis, and 0"") is positive for rotations that obey right hand rule, and vice versa (refer to Figure 1). Thus, for straight parallel in-plane fiber reinforcement, the unit vector Am) = {c0s9(m), 0, sin9(m)}. (2.1.8) IZTXZ o o . o 0(2) X1 i1 Figure l. The graphical description of a composite thick plate reinforced with multiple families of fibers. The strain energy density function of this type of composite material is modeled in expressions (2.1.5). 14 The fibers in the deformed configuration will in general change their orientation. The counterpart of Au") in the deformed configuration is am) (m = 1,2,...M), given by F A(m) ah“) = , ,/K(m) (2.1.9) In the material model (2.1.5) u>0 is the shear modulus of the neo-Hookean material. The parameter [30“) 2 0 characterizes the increased stiffness contributed by the mth family of fibers in the fiber direction. This in turn will depend on the stiffness and the fiber volume fraction of the mth family of fibers. The neo-Hookean behavior is retrieved if 13"“) = 0 for all m. It is evident that W is symmetric in C. In its undeforrned state, I1 = 3 and K“) = 1, in which case the strain energy density function W given by (2.1.5) is zero. Anisotropy introduced by the fiber reinforcement is characterized analytically by the symmetry transformations that do not affect mechanical response. For M = 1 the material is locally transversely isotropic, and we may let 0") = 0, so that A“) = i1. Under transverse isotropy, arbitrary rotations about the il-axis, and reflections about the plane with normal i1, do not affect the local mechanical response. For M = 2 if either A“) . Am = 0, or [3(1) = [3(2) and or“) = (la), then the material is locally orthotropic. If A(1)-A(2) = 0, let 9‘” = 0 and 0‘” = -1t/2, then A“) = i1 and A‘” = i,. This situation describes perpendicular reinforcing fibers which need not provide identical reinforcing. If, on the other hand, [3(1) = [3(2) and a“) = (2), let 0“) = -0(2) = 0, then i! and i3 are the perpendicular bisectors of the fiber axes in this plane. That is, i1 = (A(1)+A(2))/(2cos0), i3 = (A(2)-A(1))/(2sin0). This situation describes biassed reinforcing fibers with balanced reinforcing. Note in both cases that reflections with respect to planes that are normal to i,, i2 or i3, respectively, do not affect mechanical 15 response. For M = 2 with AW-AW¢0 and either [Swatbm or amateur”), the material is said to be locally clinotropic (see [Zheng 1994]). This situation describes biassed reinforcing fibers with different reinforcing. Under clinotropy, n—rotation about i2 and reflection with respect to a plane normal to i2 have no effect on mechanical response. In the linear theory, transversely isotropic materials are characterized by 5 elastic constants, orthotropic materials are characterized by 9 elastic constants, and clinotropic materials are characterized by 13 elastic constants. In sections 2.4 and 2.5 we give these elastic constants derived from material (2.1.5) for the transversely isotropic and biassed orthotropic cases respectively. In this regard it is necessary to take of”) = 2 (m = 1,2,...,M) for the material (2.1.5) to have finite additional stiffness in the fiber directions under infinitesimal deformations, since 0t(m)>2 involves no detectable strengthening in the an) infinitesimal limit and a‘ <2 gives no extension along the A‘m-axis in the infinitesimal limit. In other words, the exponent a0") = 2 gives a nontrivial consistency with the familiar linear elasticity theory. It is worthwhile noting that, in the similar material models [Kurashige 1981, Triantafyllidis and Abeyaratne 1983, Polignone and Horgan 1993], this exponent of the additional stiffness, though not mentioned, is taken to be 2. Henceforth we take or“) = 2 (m = 1,2,...,M) so that (2.1.5) is rewritten as M (m) w = gal—3).» 2 [SE—(K0104)? (2.1.10) m = 1 It is clear that the specific augmentation applied to the neo-Hookean form is only one particular model for fiber reinforcement. In the present study we consider only materials reinforced with families of straight parallel fibers, hence from now on we will say fibers instead of straight parallel fibers for abbreviation. 16 The generality of the material model (2.1.10) within the symmetry classes corresponding to transverse isotropy and to orthotropy is clarified by referring to the work of Zheng [1994] specialized to incompressible materials, and hence isochoric deformations. For transversely isotropic incompressible materials, it follows that the most general form for the strain energy density function is given by ([Zheng 1994]) W = W(tr(C), tr(C2), A ~ CA, A - CZA). (2.1.11) In our case A = {1,0,0} and so gives a dependence on the four scalar arguments: tr(C) = C11+C22+C33 , tr(CZ) = C121+C§2 +C§3 +2(C%2 +C123 +C§3)2, ( .1.12) In particular, (2.1.10) with M = 1 gives and so represents only one particular model for the finite elastic response of a transversely isotropic incompressible material. Nevertheless, just as the utility of the neo—Hookean model for isotropic incompressible materials is widely acknowledged, the form (2.1.13) provides an idealized form for investigating the effects of transverse isotropy in finite deformation. In a similar fashion, the general form of the strain energy density function in an orthotropic incompressible material is given by ([Zheng 1994]) W = W(tr(C), tr(Cz), tr(MC), tr(MCM), tr(CMC), tr(MCZM)) , (2.1.14) where 17 M = il®il—i3®i3. (2.1.15) It can be shown that the dependence upon the six scalar trace variables appearing in the argument list of (2. 1.14) is equivalent to dependence upon the six quantities: C11, C22, C33, C122, C132, C232. This is because, in addition to the first two equations in (2.1.12), tr(MC) = ell—€33, tr(CMC) = C121+C122—C%3 —c§3 , 6 (2.1.1 ) tr(MCM) = C11+C33, tr(MCZM) = c121+c122+2c123 +c:§3 +C§3, which, in turn, gives (211 = %(tr(MC)+tr(MCM)), C22 = tr(C)-tr(MCM), (233 = %(—tr(MC) +tr(MCM)), cf, = %(—tr(C)2+tr(C2)+2tr(C)tr(MCM)—tr(MC)tr(MCM) (2.1.17) —tr(MCM)2+tr(CMC)-tr(MC2M)), ct, = :11-(2tr(C)2-2tr(C2)-tr(MC)2—4tr(C)tr(MCM) + tr(MCM)2 + 4tr(MC2M)), (:3, = %(-tr(C)2+tr(C2)+2tr(C)tr(MCM)+tr(MC)tr(MCM) — tr(MCM)2 — tr(CMC) — tr(MCzM)). For the orthotropic case arising from perpendicular reinforcing fibers, (2. 1 . 10) with K“) = C ,1, Km = C33 provides the strain energy density representation (1) (2) W = %(C11+ C22 + C33 — 3) + £3E—(Cll — 1)2 + %(C33 —1)2. (2.1.18) For the orthotropic case arising from biassed, balanced reinforcing fibers, (2. 1.10) with 18 K“) = (cos0)2C11+2cos0sin0C13 + (sin0)2C33 . (2.1.19) Km = (cos0)2C11— 2cos0sin0C13 + (sin0)2C33 , provides the strain energy density representation = Ll(c: + C + C — 3) 11 22 33 (2.120) + [3(((cos9)2C11+(sin0)2C33 — 1)2 + 4(cos95in0)2Cf 3). As required, C” enters into this last strain energy density expression only through its square. 2.2. General Stress-Deformation Relation The Cauchy stress tensor for the incompressible anisotropic finite elastic material (2.1.10) is aw M _a__w aK-FA—1)FA0, i = 1,2,3. (2.3.4) In the literature, this deformation is often referred to as homogeneous pure deformation. We shall call it homogeneous triaxial deformation to reflect the directional feature shared with the anisotropic material that will be studied. The constraint of incompressibility (2.1.7) now becomes Secondly, a general plane deformation taking place in the (X1, X2)-plane can be written, in the deformation gradient tensor, as 21 1:11 1:12 O F = 1:21 F22 o , (2.3.6) 0 O 1 and F12 at F 21 in general. For this plane deformation, the left Cauchy-Green strain tensor is C11 C12 0 C = CT = C12 C22 0 , (2.3.7) 0 0 1 With Cij = 1:“ij SO that Cll > 0, C22 > 0. (2.3.8) The corresponding incompressibility constraint is An important specialization of (2.3.6) is generated by requiring F12 = F2. = 0. This gives the biaxial deformation described by the deformation gradient tensor 7).] o 6; it, 0 o F = 0 x2 0 = 0 All 0 , (2.3.10) 0 O l 0 O 1 wherek,=1,and 2.2 = A,“ (2.3.11) is the incompressibility constraint directly obtained from (2.3.5). Here the deformation is described by the principal stretch 1, alone. Biaxial deformations taking place in other 22 coordinate planes can be similarly defined. Consider now the simple shear deformation with respect to coordinate planes. There are in total six different canonical cases of simple shear deformations regarding the direction of shear and the coordinate plane, in which the deformation take place. For a simple shear of amount k in XP-direction taking place in the (Xp, Xq)-plane, the deformation gradient tensor has the components F”. = 6ij+k8ip8jq. (2.3.12) The incompressibility constraint (2.1.7) is satisfied by the simple shear deformation. Finally we set here more general simple shear deformation in the (X ,, X2)-plane. Consider a unit vector e, lie in the (X1, X2)-plane and pass through the origin of the coordinate system set forth. This unit vector e, is apart from il an angle \[I by right hand rotation about i3. Let e2 = i3 x el and e, = i3. We define simple shear in the direction of e1, upon which line elements in the direction of e, in the planes normal to i3 do not change their length and orientation, but have slides along el-direction. Furthermore, the slide of a specific line element is proportional, by factor k, to the distance measured along e2 from e, to the line element. By this definition, the deformation gradient tensor with respect to the {i,,i2,i3} coordinate system is given by 1 —kcos‘\|lsin\y kcoszw 0 —ksin2\y 1 +kcoswsint|l 0 ° (23°13) 0 0 1 — Simple shear defined in (2.3. 12) with indices p = 1 and q = 2 is obtained by letting \[t = 0 in (2.3.13). Simple shear in (2.3.12) with indices p = 2 and q = 1 is obtained from (2.3.13) by 23 setting \|I = M2 and replacing k with -k. The stress state of a uniaxial load T in the Xp-direction is given, in components, by T.. = T5. 5- . (2.3.14) 2.4. Material Properties (Single Fiber Family) If the material expressed in (2.1.10) is reinforced with only one family of fibers (M = 1 in (2.1.10)), then it has three mutually orthogonal material principal directions and three mutually orthogonal planes of symmetry. Furthermore, one of these planes of symmetry is the plane of isotropy. This type of materials is the transversely isotropic material and is characterized by five independent elastic constants in linear elasticity. We now investigate the mechanical responses of this material model through prescribed simple homogeneous deformations in accordance with simple loading conditions, such as uniaxial load in fiber direction, uniaxial load transverse to fiber direction and simple shear. For this purpose, we arrange the rectangular Cartesian coordinate system so that the Xl-axis is in the direction of the family of fibers (0 = 0) and, therefore, the X2 and X3-axes are in the material isotropic plane. Thus, the strain energy density function (2.1.10) becomes W = 2(Il 3)+2(K l) . (2.4.1) The components of the Cauchy stress tensor, following from (2.2.4), are given by The unit vector field A for the single family of fibers is simply given by 24 A = {1,0,0}T or A- = on (2.4.3) in the coordinate system set previously. This, for M = 1, reduces the Cauchy stress tensor (2.4.2) to as In the case that the material is reinforced with single family of fibers (or families of fibers that only differ from orientation), we define a dimensionless material parameter (i.e. stiflfness ratio) as Y = 13/11 2 0. (2.4.5) 2.4.1. Material Response to Homogeneous Deformations Here, we examine the mechanical responses and investigate the properties of the material (2.4.1) reinforced with one family of fibers through simple homogeneous deformations corresponding to some simple loading conditions. (i) Uniaxial load in the X ,-direction (fiber direction) We seek the stress state of uniaxial traction in Xl-direction, given by (2.3.14) with index p = 1, corresponding to a homogeneous triaxial deformation (2.3.3). In view of the equivalence of the X2 and X3 material directions in this case, we seek solutions with the additional lateral symmetry A; = M. This, in connection with equation (2.3.5), yields The corresponding left Cauchy-Green strain tensor is diagonal and given by 25 2.,2 0 o O O 1.71 We now show that there is a single family of such solutions and that they can be parametrized by 7t,>0. That is, we obtain functions T11 = 11104). it, = X201), 2., = 513(21), (2.4.8) for each 21>0. Substituting equations (2.3.3) with (2.4.6) into equation (2.4.4), the only possible nonzero components of the Cauchy stress tensor are The hydrostatic pressure p which makes T22 = T33 = 0 is p = 113.1“ . (2.4.11) Therefore, the uniaxial load required to produce the deformation corresponding to deformation gradient (2.3.3) with (2.4.6) is T11 = uOLf—ltfl)+2|3(hlz— 1)).12. (2.4.12) Equation (2.4.6) and (2.4.12) yield the following asymptotes: 12—900, 23 —>oo, T11 ~-u7tf1 —>-oo, as 7.1—>0, (2.4.13) and 26 22—90, 7&3 —>0, T11~ZB7tf—>oo, as 7.1—ioo. (2.4.14) It can be seen from (2.4.13)3 that to compress the material body to a flattened state such as a deformation (2.4.13)”, the dominant portion of load T“ is required to balance the hydrostatic pressure p which is generated for preserving volume. From (2.4.14)3 we see that if the material body is forced to extend very large, such as a deformation (2.4.14)”, the dominant portion of load Tu is contributed to overcome the resistance generated by the additional stiffness in the fiber direction. Figure 2 and Figure 3 are the responses of material (2.4.1) in uniaxial load in fiber direction. In Figure 2, A.) and he, given by equation (2.4.6), is independent of the material properties and is determined by the incompressibility constraint only. Noting (2.4.5) and (2.4.7), we derive from (2.4.1) the normalized strain energy density function for this deformation, expressed in terms of it, and y, as well as its partial derivative with respect to 3.1: Wavy) 5&W(C(h1), [1, [3) = %().12+ 2).;1— 3) + £70.; — 1)2, (2.4.15) %-W(k, 111,7) = 1.1- 172 + 270.12 - 1)).1, (2.4.16) 1 respectively. Comparing (2.4.16) with (2.4.12) and noting (2.3.5), one verifies the well-known result 1 _ A1 3 ~ fiT]1(l]: Y) "' k112A3-a—Av—1W(xl’ Y)! (2:417) in the anisotropic problem studied here. 27 0 of2 0.4 0.6 ate 1 1:2 1:4 1:6 1.8 2 11 Figure 2. Lateral deformation vs. extension/contraction in the loading direction for the case of uniaxial load in the fiber direction. Here L2 or 33 is independent of the material properties and is determined by the incompressibility constraint only (2.4.6). 2., = 1 represents to the undeforrned configuration. T1 141 _60 l l l I P L 1 J 0 0.2 0.4 0.6 0.8 1 1.2 1 .4 1.6 1 .8 2 11 Figure 3. Dimensionless loading Til/u vs. extension/contraction in the loading direc- tion for the case of uniaxial load in the fiber direction (2.4.12). The stiffness ratio 7 takes the value: 7 = 0, 1, 10 and 100. For p4.960745, Tn behaves non-monotonically for Md. Loss of monotonicity first take place at 7: 4.960745, 3., = 0.5194. This gives rise to the possibility that, in compression, three deformed configurations could corre- spond to one load level. 2.1 = 1 represents the undeforrned configuration. In Figure 3, for larger stiffness ratio 7, the responses (T1,) behave non—monotoni- 28 cally in compression (A, < 1) and give rise to the possibility that three deformed configurations correspond to one load level. The minimum 7 at which T,, becomes non-monotonical is 7 = 4.960745. The cases of ‘y = 0, 1, 10 and 100, are plotted in Figure 3. For the case that y = 0, the response is simply that of the neo-Hookean material. Denoting the value of it, at which T,,/u takes its local maximum by min“) , and the value of it, at which T,,/u takes its local minimum by A'lIFrinin) , we now investigate the relation between these special values of it, and 7. For this purpose, we take the partial derivative of (2.4.17) with respect to it, and obtain the following equation a l which, for every given 7, determines M311“) and Halli"). Since A, > 0, the second equality in (2.4.18) is equivalent to 872.15 — (4y — 2)7t? + 1 = 0. (2.4.19) This is a fifth order polynomial equation and there is no standard solution procedure for it. However, the number of positive real roots may be determined by the Descarte’s rule of sign, which states the number of positive real roots of a polynomial equals the number of sign variation of the polynomial, or less by an even number. The number of sign variation of the left hand side of (2.4.19) is 0 for y S 1/ 2, and is 2 for y > 1/ 2. By the Descarte’s rule of sign, we know that equation (2.4.19) has no positive real root for y S 1/ 2, and equation (2.4.19) may have either 0 or 2 positive real roots for y > 1/ 2. We have found numerically that local maximum and local minimum exist for y> 4.960745. We now solve equation (2.4.19) numerically to obtain km“ (7) , Am“ (7) , and plot it in Figure 4. 1(max) 1(min) 29 200 . . . . 180~ - 160 - Region of decreasing T, ,/u “ 140- 120 l- .1 y 100- ~ uni , °°~ 1.1mm) “mm ‘ 60h - 40- . 2° ’ (0.5194. 4.9607) ‘ 84 a} 05 0) ask, a} a} as 74 Figure 4. Curves of Magma) and with”) l,(m,,,)(y) corresponding to the non- monotonic uniaxial response behavior of T,,/|.1 which occurs in Figure 3 for Y > 14£Mfll745. (ii) Biaxial load associated with biaxial deformation If the deformation is biaxial as given by (2.3. 10), then 1,2 0 0 C = 0 11.2 01- (2.4.20) 0 0 1 For these deformations the Cauchy stress tensor is diagonal. If, in addition, the hydrostatic pressure p is chosen so that T22 = 0, corresponding to a traction free boundary with normal i2, then the only nonzero components of the Cauchy stress tensor are T11 = 1101.12 — hfz) + 213(7tf—1)).2, (2.4.21) T33 = u(1 —>.,-2). Here the hydrostatic pressure p which makes T2, = 0 is p = 11).;2. (2.4.22) 30 The nonzero component T33 (2.4.21), is required to maintain biaxial deformation condition 7),, = 1. Here T,, (2.4.21), is plotted against 3., for “y: 0 (neo-Hookean), 1, 10 and 100 in Figure 5. As anticipated, ITIIO‘lllbiaxial load & biaxial deformation > ITIIO‘I )luniaxial load & triaxial deformation ’ if 7‘1 T 1 ' Making use of (2.4.20) in (2.4.1), it is found that the normalized strain energy density function, expressed in terms of it, and y, as well as its partial derivative with respect to A, are given by Wm. v) a fiwrcrtn, it. B) = £042 + >42 — 2) + $70? - 02. (2.4.23) g-Mk’ (V, y) = 2.14.73 + 270.?- 1)).,, (2.4.24) 1 for this biaxial deformation. Comparing (2.4.24) with (2.4.21), and noting (2.3.5), one verifies, again, the well-known result 1 _ A1 3 ~ l1'1‘11()"19 Y) "' xllzx3mW(Al,Y)’ (24°25) in the anisotropic problem studied here. Figure 5 indicates, for larger stiffness ratio 7, that the load responses T,,Ot,) behaves non-monotonically in compression (7t, < 1) and so also give rise to the possibility that three deformed configurations can correspond to one load level. The minimum 7 at which T,, becomes non-monotonic for biaxial deformation is y = 14.950393 as shown below. Denoting the value of it, at which T,,/u takes its local maximum by M’lmax) , and the value of it, at which T,,/p. takes its local minimum by M’lmin) , we now turn to locate M’Emax) and M’émin) for changing 7. For this purpose, set the partial derivative of (2.4.25) 31 with respect to it, equal to zero: i(1T,,(}t,,'y)) = 2}», +Af3+27(4}t,3—27L,) = 0, (2.4.26) 37‘1 11 which, for every given 7, determines possible values Ailmax) and M’lmin) . Since it, > 0, the second equality in (2.4.18) is equivalent to 473.16 — (27— 1)}t‘,‘ + 1 = 0. (2.4.27) Expressing equation (2.4.27) in terms of C,, by noting C,, = 2,2, yields the following cubic equation 47C?1 — (27— 1)C,21 + 1 = 0. (2.4.28) This cubic equation can be solved by following standard procedure. It is to be noted that equation (2.4.27) has a positive real root for it, if and only if equation (2.4.28) has a positive real root for C ,,. It is to be further noted that the transition from monotonicity to nonmonotonicity of T,,/u corresponds to a positive real root of (2.4.28). Following the theory of cubic equations [Kurosh 1980] indicates that equation (2.4.28) has a double real root if and only if 873 — 12072 + 67 — 1 = 0. (2.4.29) Following, again, the procedure as mentioned in [Kurosh 1980] indicates that (2.4.29) has only one real root 7 = 14.950393. This is the lower bound for nonmonotonic behavior. For ‘Y > 14.950393, one may then solve equation (2.4.28) ((2.4.27)) to obtain ”Emanw) and Nahum”) , as shown in Figure 6. Figure 7 compares the nonmonotonicity regions for this response with that of the uniaxial response as given previously by Figure 4. 32 4o» 7:100 F10 - 20- Til/l1 OF' Figure 5. Dimensionless loading T, ,/u vs. extension/contraction in the loading direc- tion for the case of biaxial load (2.4.21) associated with biaxial deformation (2.3.10). The stiffness ratio 7 takes the value: 1: 0, l, 10 and 100. For y>14.950393,T,, behaves non-monotonically for 3.,<1. Loss of non-monotonicity first takes place at 7 = 14.950393, 1, = 0.5676. This gives rise to the possibility that, in compression, three deformed configurations could correspond to one load level. A, = 1 represents the undefonned configuration. 200 180 ' r 160 ' Region of decreasing T1 ,lu 140- 20 . (0.5676, 14.9504) Figure 6. Curves of M’fmx)(y) and Miriam”) corresponding to the nonmonotonic biaxial response behavior of T, ,Ip. which occurs in Figure 5 for ‘y> 14.950393. 33 A‘llmaio (Y) 80 .- 60 _ 40- A'lllltinax)(7) A'llfltitin)(7) 20 _ 1 8.1 0f2 0:3 0:4 0L5 0:6 0:7 0.8 74 Figure 7. Comparison of the nonmonotonicity regions between the uniaxial response and the biaxial response. (iii) Uniaxial load in Xz-direction (transverse to the fiber direction) Here we seek the stress state of uniaxial traction in Xz-direction, given by (2.3.14) with i = 2, corresponding to the homogeneous deformation (2.3.3). We now show that there is a single family of such solutions and that they can be parametrized by 1,. That is, we show that given 2», we can obtain it, = 1,0,2), T22 = "122(12), 2.3 = 5.30.2). (2.4.30) These relations are, however, not given explicitly. The nonzero components of the Cauchy stress tensor, by substituting equations (2.3.3) and the incompressibility constraint (2.3.5) into equation (2.4.4), are T11 = uk,2+2[i(}tf- 1)}t,2—-p, (2.4.31) T22 = mtg — p, (2.4.32) 34 Here T22 is the load required to produce such a deformation as described in equation (2.3.3). The uniaxial stress state (2.3.14) with i = 2 requires that T,, = T,, = 0. This permits the elimination of p between equations (2.4.31) and (2.4.33), and so gives the following relation between it, and 71,, p(hlz-kg) +2B(}t.12— 1)?\.,2 = . (2.4.34) Then eliminating h, from equation (2.4.34) by using equation (2.3.5), yields f,2(}t,, 7‘2) sZBhf+ (u—2B)X‘,‘—u}t§2 = 0. (2.4.35) Alternatively eliminating h, from equation (2.4.34) by using equation (2.3.5), yields f320t3, 12) E — uh? + (p. — 2B)}e§21§+ 2133.54 = 0. (2.4.36) From equations (2.4.35) and (2.4.36), the corresponding 3., and L, at every given 21., can be determined. In fact, let x = 2,2 or x = 70,2 respectively in (2.4.35) or (2.4.36). We then get either 213x3 + (u—2[3)x2—u7t§2 = 0, (2.4.37) or — ux3 + (p. - 28)}tgzx + 282.54 = 0. (2.4.38) According to the Descarte’s rule of sign, both equation (2.4.37) and equation (2.4.38) have exactly one positive real root for x. It follows that equation (2.4.35) or (2.4.36) each gives one positive real root to it, or h, respectively for every given AQ>0 and so define two functions it] and 51.3 in (2.4.30)”. The load deflection relation (2.4.30), is then 35 T22( 1,2) = 11.0»; — 13112)} . Using a numerical procedure, we can graph the responses of the material defined in (2.4. 1) in uniaxial load transverse to the fiber direction according to equations (2.3.5), (2.4.31)-(2.4.36). Figure 8-Figure 10 are examples of these responses plotted for y = 0 (neo-Hookean), 1, 10 and 100. This numerical procedure indicates that the dimensionless loading T,,/u monotonically increases with 2.2, so that for each value of load T22, there is one corresponding deformed configuration of type (2.3.3). Figure 8. Lateral deformation in the fiber direction vs. extension/contraction in the loading direction, for the case of uniaxial load transverse to the fiber direction. The stiffness ratio 7 takes the value: 7: 0, 1, 10 and 100. A12 = 1 represents the undeformed configuration. 36 Figure 9. Lateral deformation orthogonal to the fiber direction vs. extension/contrac- tion in the loading direction, for the case of uniaxial load transverse to the fiber direc- tion. The stiffness ratio 7 takes the value: 7: 0, l, 10 and 100. A12 = 1 represents the undeformed configuration. -h (D .1 ., Tzz/li -5 — an 5v Figure 10. Dimensionless loading T2241 vs. extension/contraction in the loading direc- tion, for the case of uniaxial load transverse to fiber direction. The stiffness ratio 7 takes the value: 7: 0, 1, 10 and 100. it, = 1 represents the undeformed configuration. 37 Equations (2.3.5), (2.4.31)-(2.4.33) with T,, = T,, = 0, as well as equation (2.4.35) and (2.4.36) also yields the following asymptotes in terms of 20,: -1/6 ‘ x3 .. (23)”352/3 _) 0°, * as 12 —> 0 . (2.4.39) [1 '1‘ ._, _(2uZB)1/3A§4/3 _) _°°, —1/4 ‘ x3 .. (1 _ -2—B)1/4h§“2 _, 0’ t as A2 —> °° - (2.4.40) u T ~ ”Ag -> 00, , (iv) Simple shear deformation with respect to coordinate plane We now turn to simple shear deformation in the (X,,X2)-plane. First, if the material is subjected to a simple shear in the X,-direction (in the fiber direction) then the deformation gradient tensor is given by (2.3.12) with indices p = 1 and q = 2. The Cauchy stress tensor associated with this deformation is obtained from (2.4.4) as u(1+k2)—p 11k 0 T: ”k VP 0 . (2.4.41) 0 0 u-p We note that the associated shear stress, T12 = T21 = uk (2.4.42) is not affected by the additional stiffness in the X,-direction, since this deformation (2.3.12) with i = 1 and j = 2 involves no stretching of the fiber. 38 If instead the simple shear is prescribed in the Xz-direction (transverse to the fiber direction) in the (X,,X2)-plane, then the deformation gradient tensor is given by (2.3.12) with indices p = 2 and q = 1. It then also follows from (2.4.4) that the Cauchy stress tensor is u-p+2[3k2 uk+213k3 0 T = uk+2Bk3 11(1+1t2)—p+2[ik4 0 - (1443) O 0 l-l-P Now, we find that T,2 = T2, = (u+2[3k2)k. (2.4.44) The additional shear stiffness 213k2 to the total shear stiffness is due to the additional stiffness, introduced by the fiber reinforcement, in the X,-direction. For infinitesimally small shear (Ikl<<1), this additional shear stiffness is negligible, but this term has significant effect if the shear deformation is not infinitesimally small. For strongly anisotropic fiber-reinforced materials, the stiffness in the fiber direction can be greater, by orders of magnitude, than the stiffness in other direction. If [3 tends to infinity, as is the case that simulates an inextensible fiber, then T,, is required to be infinitely large to maintain nonzero amount of shear k in the Xz-direction. This is the same result as obtained by the ideal theory where the only direction for shear to take place is the fiber direction, due to the constraint of fiber inextensibility [Pipkin and Rogers 1971]. By symmetry of the X2 and X3-directions, the results obtained above hold for simple shear in the (X,,X3)-plane if subscript 2 is everywhere replaced by subscript 3. On the other hand, for simple shear in the (X2,X3)-plane, the fiber stiffness [3 plays no role, and one easily finds that 39 regardless of whether the simple shear is prescribed in the Xz-direction or in the X3-direction. The nonzero normal components of stress in (2.4.41) and (2.4.43) is due to the in-plane coupling effect (extension/shear), that, in finite elasticity, normal stresses are required to support simple shear deformation. The simple shearing states (2.4.41) and (2.4.43) are supported by in-plane loads if T,, = 0. This corresponds to p = u, when utilized in (2.4.41) and (2.4.43) leads to the response diagrams given in Figure 11 and Figure 12. In Figure 11, the stress components T,,, T,, and T,, in (2.4.41) are plotted, and in Figure 12, the stress components T,,, T,, and T,, in (2.4.43) are plotted. 0.6 '- Til/l1 T1241 0-4 T1241 T2241 T1 1/11 I 0.2 " .1 Figure 11. Simple shear in the fiber direction as given by equation (2.4.41) for y = 10. This choice of ycorresponding to u = 4.9 and B = 48.9, is motivated by the linear prop- erties of the boron/epoxy material as exemplified later in Table l. 15 I I I T I I T j T1 1’1;l 10 r- -. Trill»l T1241 5’ 12, ‘ Tat/u 0 T2241 . -5 l .l l 1 l l l 1 l 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 it Figure 12. Simple shear transverse to the fiber direction as given by equation (2.4.43) for y: 10. (v) General simple shear deformation in the (X ,,Xz)-plane For the more general simple shear in the (X,,X2)-plane with shear direction rotated from the fiber direction by an angle 11!, the Cauchy stress tensor is obtained, by substituting (2.3.13) into (2.4.4), in component form as T,, = p(kzcoszw — ksin2w) + 2B(k2sin 2W — ksin2tll)(l — ksin2\|t + kzcoszwsinzty), T12 = p(kcos2w + ékzsinZZV) + 23(kzsin 2W — ksin2\|l)(— ksinzut + kzcoswxlhavfi) T = p(kzsinzw + ksin2ty) + 2B(k2sin2w — ksin2\|l)(k2sin4\|1). 22 Here once again the hydrostatic pressure p is eliminated by equating p = u so as to give T,, = 0. Equation (2.4.46) reduces to (2.4.41) with p = u if \[I = 0, and reduces to (2.4.43) with p = [.1 if \V = 1t/2 and k is replaced by —k. The responses under this simple shear (2.4.46) for single fiber family material with a number of shear directions are shown in Figure 41 13-Figure 15. These responses further reveal the in-plane coupling (extension/shear) effects, which present themselves not only in finite elasticity but also in linear anisotropic theory. 10 -2 .- \V=31t/8 - -10 l 1 l 1 1 1 1 l 1 0 0. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 k Figure 13. Normal stress T, ,/p. in (2.4.46) varies with simple shear k for material with one family of fibers, where Y = 10 (u = 4.9 and B = 48.9). The material response to the simple shear (2.3.13) changes with the orientation ‘11- The existence of normal stress in simple shear is characteristic of anisotropic materials. 42 5 r 0 T1241 _5 - -1o 1 l l 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 14. Shear stress T1241 in (2.4.46) varies with simple shear k for material with one family of fibers, where y = 10 (it = 4.9 and B = 48.9). The material response to the simple shear (2.3.13) changes with the orientation \tI. 0 0. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 15. Normal stress T2201 in (2.4.46) varies with simple shear k for material with one family of fibers, where y = 10 (p. = 4.9 and B = 48.9). The material response to the simple shear (2.3.13) changes with the orientation ‘1'- The in-plane shear stress resolved on surfaces parallel and perpendicular to the shearing 43 direction is then given by Telez = uk + B(kzsin 21y — ksin 2W)(2ksin 21p — sin 2y) . (2.4.47) This resolved shear stress is plotted in Figure 16 for various w at the fixed value 7 = 10. The most immediate feature is the loss of monotonicity. 1 = 1:72 ‘ w = N8 2.5 - X1 _ Telez/u 2* 111:11/4 ~ 1.5 — — 1 l- ‘ 0.5 r- 4 _ . = 3111/8 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.3 0.9 1 k Figure 16. Shear stress Tent/u in (2.4.47) varies with simple shear k for material with one family of fibers. As before 7 = 10. The material response to the simple shear (2.3.13) changes with the orientation \y. This loss of monotonicity is correlated to a progressive contraction and subsequent relaxation/elongation of a material line element in the reinforcing direction. It is Convenient to describe the associated line element deformation in terms of “fiber” Shortening and lengthening. Then ,lCll gives the fiber stretch, where, for this deformation, (2.3.13) yields 44 I- q 1— 2kcosulsinw + kzsinzw k(cos2\|t - sinztll) — kzcoswsinw 0 C = k(cos2\|l - sinzw) - kzcostysinw l + 2kcos1|1sintv + kzcoszw 0 ' (24°48) _ 0 0 1, As shown in Figure 17, if 0 < 11! < 1t/ 2 this then gives rise to a regime of fiber contraction (C,, < 1) for moderate shear k which then gives way to fiber extension (C,, > 1) for large k. 5 t 4 e - 3 r .. C” 2 " "‘ ‘ fiber extended 1 lfiber contracted o _ \[I = 0.028 _1 l l l 1 4L l_ l l L 0 1 2 3 4 5 6 7 8 9 10 k Figure 17. Fiber length changes with k, the amount of shear in general simple shear deformation. It is shown that fiber experiences first contraction (C,, < 0), then elonga- tion(C,,>O) for O<1y<1r/2. This effect is due to the rotation that the fiber experiences under the deformation. Denoting this rotation by 0, Figure 18 illustrates this phenomena for original fiber orientation w = M4. For ‘1’ = 71:14 the fiber is contracted for k: 0 —> 2 corresponding to q): 0 —> —1t/2 , after which the fiber is extended for k: 2 —) 00 corresponding to 11): -7t/ 2 —> -31t/ 4. More generally Figure 19 shows for any 0 < 1|; < 1t/ 2 that the fiber is 45 contracted for k: k(o) a 0 —> k(b) E 2cott|1 corresponding to fiber rotation 0: 0 -) —11: + 2w. The minimum deformed fiber length occurs at k(a) E catty , corresponding to C11 = sin 211! and 0 = —(1t/2 — 1|!) . The fiber is extended for k: k(b) -—> 00 corresponding to fiber rotation 0: -1t + 211! —) w — 11:. In particular, the fibers align with the simple shear direction \l’ as k -—> 00. Xz‘ %’ e, Fiber is contracted from its original length / / Fiber is extended for O < k < 2 from its original length 2 for k > 2 _. : fiber and its orientation p X1 Figure 18. Simple shear deformation of a unit block with increasing k for W = 1t/4. The angle 4) is the fiber orientation with respect to the X,-direction. Here, as k: 0 —) oo , the angle 91 0 —) -31t/ 4 . The fiber is first contracted and then extended. The maximum contraction occurs at q) = -1tl4. 46 Fiber extended / / I Fiber / relaxing / / er ' /(b) F 1ber . / contracting / Ma“ / / // (0) y 1 ¢==—(1t/2-W) / ¢=-1t+2\|l X2 ‘ Fiber contracted I —> : fiber and its orientation > X1 Figure 19. Deformation of a fiber under simple shear in arbitrary \y-direction. Phases of fiber contraction, relaxation and extension are shown. Here point (0) represents the original fiber length and orientation. At point (a) the fiber is mostly contracted and at point (b) the fiber retrieves its original length. Angle a is the deformed fiber orientation with respect to X,-direction. To correlate this fiber contraction/relaxation/extension with the loss of monotonicity in the stress response (2.4.47), it is instructive to consider changes in the elastic strain energy density as a function of simple shear k. By making use of (2.4.48), the strain energy density function (2.1.13) gives that W(k, w, y) a &W(C(k, w), 11, B) = $18 + %y(kzsin2w — ksin2w)2. (2.4.49) The partial derivative of W with respect to k is 331mm 1,7,7) = k + 7(kzsin2w — ksin2\|t)(2ksin2\y — sinzty), (2.4.50) so that (2.4.47) shows that 1 _ a ingot. \v. 7) - fill/(k. 111.7). (2.4.51) In Figure 20, Telez/tt for a material with y = 10 is plotted against k over a larger extent of 47 k than that plotted in Figure 16 for fiber orientations \V = 0, 7:18, 1:14, 31:18 and 1:12. It is to be noted that Tel 12,/u for \v = 0 is the normalized shear stress without the effect of fiber reinforcement. For other values of w, deviation of shear stress Telez/u from that for 1|! = 0 exhibits the effect of fiber reinforcement. Note that the curves for 11! = 1:18, 1:14, 31:18 each cross the 111 = 0 (unreinforced) curve twice. This is due to the effect of fiber contraction-relaxation-elongation. On these curves the points corresponding to minimum fiber length are marked by ‘(a)’ (when k = k(a) ) and points corresponding to the retrieval of the original fiber length are marked by ‘(b)’ (when k = k(b))' The fiber is continuously contracting for 0 < k < k(a) , it relaxes back to its original length for k(a) < k < k(b) , and it elongates for k > k(b)' Points (a) and (b) are exactly the crossing points of these curves with the straight line for ‘l’ = 0. For each of these curves, it is seen from Figure 20 that T.,.,(k. \II, 10) > Te,e2(k’ 0» 10), for 0 < k < krar (2.4.52) Telezflt, 11!. 10) < Twat, 0, 10), for it“) < k < km. For fixed reinforcing value 7 and fiber orientation w one finds that local maxima and minima to these nonmonotonic response curves occur at kmax = cotw- chotzw—éycscfl'w, kmin = cotw-t- N/gcotzw—écsc‘tw, (2.4.53) respectively. The quantity in the radical in (2.4.53) must be nonnegative for km and km,n to exist. This requires that 1 . 2 1: 1 . 2 “y _ 2, 2arcstn(J;) _ \y - 2 2arcsm(J;), (2.4.54) 48 for nonmonotonic response. For 7 = 10 this gives 0.2318 < 1|! < 1.3390, so that all 3 curves ‘1’ = 1:18, 1:14, 31:18 in Figure 20 are nonmonotonic. In general, the corresponding shear Stl'CSSCS are Tgpgnw. 1) = T,,.zrkmax, 111.7) magnum) = T,,.zrkmin, v.1) (2.4.55) These km and km,n are mapped to the (k, C,,)-plane and plotted in Figure 21 for ‘y = 10, together with the response previously given in Figure 17. 6 T I I I T I T l I W=1V2 w=31r18 w=1r14 5 - (b) 4 .. _. W: 3 *- .. 9 l 2 2 _ (b) _ =1r18 (a) 1 L (b) - 0 (a) 1 0: point (a), minimum fiber length x: Point (b), original fiber length _1 l l l l l 1 4L # l 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 20. Shear stress Tc! e2/1.1 in (2.4.51) ((2.4.47)) varies with simple shear k for a material with y = 10 and various fiber orientations v with respect to the direction of shearing. For each curve, the fibers contract from the origin to the point marked (a). The fibers then relax back to their original length between points (a) and (b). From point (b) onwards the fibers continue their elongation. The energy relation (2.4.51) then shows that the decreasing portions of these response curves are associated with energetically unstable behavior in the standard sense of giving 49 gig-W“, 1|!, 7) < 0. This unstable regime includes the point of minimum fiber length k = k(a)° It is also to be noted that the area between any 0 < 1|! < 1:/ 2 response curve and the 1|! = 0 baseline curve for 0 < k < km is the same as the area between the two curves for k,,, < k < k(,,,. This shows, with one exception, that the work required to obtain any of these k > 0 deformations in the fiber reinforced materials with 0 < 1|! < 1:/ 2 is always greater than that required in the unreinforced material. The exception is the k = k(b) deformation, associated with regain of original fiber length, which is work neutral with respect to the unreinforced response. It is also apparent from Figure 20, for the 'y = 10 material with fiber orientation 1|! = 1:14, that Te] .32 < 0 near the local minimum of the response curve. In particular, for relative fiber strength 7: 10 and original orientation 1|! = 1:14, one finds that Tellez = 0 is associated with shearing k = 0, k = 1.7236 and k = 1.2764. For the y = 10 material, one also finds that fiber orientation in the range 0.5536 < 1|! < 1.0172 gives an interval of negative Te,e, near the minimum of the response curve. The effect of the relative fiber stiffness parameter 7 is, according to (2.4.50), merely associated with a simple linear scaling of any 1|! ¢ 0 response away from the unreinforced response (which is formally provided by the 1|! = 0 curve). Although nonmonotone response will occur in materials obeying 'y 2 2 and arcsin(../2—/_'y) S 21|! < 1: — arcsin(x/27{) , this does not necessarily ensure Te,e, < 0. Instead, T0192 will only attain negative values for k near k,,,," if the y—scaling away from the unreinforced response is sufficiently large. We find that the lower bound value 'ycapable of providing such negative Te, (,2 response is y: 8. At this value 7: 8 one finds that Te,e, = 0 for 1|! = 1:14 at k = kmin = 1.5. As 7 increases from y = 8, one finds that an increasing range 50 of original fiber orientations 1|! are capable of supporting Te,e, < 0 for k near km. This negative Tel,2 response phenomenon is actually energy releasing under the general simple shear deformation. The region of (1|!, 7) supporting this energy releasing is plotted in Figure 22. 1.5 I l l l I I =7” =31t/8 1|!=7:14 v=0 Local minimum to Te,e, for 7:10 C11 =0.028 0.5- ‘V Minimum Local maximum/ / fiber length toTemfory=10 =1:/8 V0 0.5 1 1 5 2 5 3 3 5 4 FN- Figure 21. Changes of fiber length with k (Figure 17) are plotted again here together with the curve of minimum fiber length. Fibers are continuously contracted up to min- imum length, then relaxed to their original length. and then are elongated, with increasing k for 0< 1|! < 1:/ 2. The corresponding local maximums and local mini- mums are also plotted for the material with y: 10. 51 Possible energy releasing region 60% 40* 20- (W. Y) = (TI/4. 8) l L l l 1 I l 0.2 0.4 0.6 0.8 1 1.2 1.4 \V Figure 22. Region of (1|!, 7) supporting energy releasing behavior (Telez < 0) under general simple shear deformation. Such behavior will occur if and only if y 2 8 . 2.4.2. Material Properties at Infinitasimal Limit Here we consider the behaviors of the material given by (2.4.1) in the case that deformations are restricted in the neighborhood of the undeformed configuration. The corresponding elastic constants in linear elasticity are then derived. In the linear theory [Christensen 1979], a transversely isotropic material is described by five independent elastic constants, namely, the Young’s moduli E,, and 13,2, Poisson ratios v,2 and v2, and shear modulus C,,. All other elastic constants can be obtained from E,,, En, v,2, v22 and G,2 by supplementing relations, such as E22 2___(1 + v23) (2.4.56) G23 = G32 = in the plane of isotropy, and 52 which is required by the symmetry of the elasticity tensor. In the case of uniaxial load in the fiber direction, the relation between the load and the stretch in the load direction was found to be given by equation (2.4.12) and the lateral deformation was found to be given by equation (2.4.6). It follows from equation (2.4.12) that the corresponding Young’s modulus E,, in the fiber direction is dT Ell = CIT] = 3fl+4B. (2.458) Al=l From equation (2.4.6) it follows that the Poisson’s ratio corresponding to uniaxial load in the fiber direction is V21 = -J-l' (2.459) ‘11:] This is clearly due to the constraint of incompressibility. Here, the first subscript of v indicates the direction of lateral contraction/extension, and the second subscript of v indicates the direction of loading. Similarly, the corresponding material properties in linear elasticity related to uniaxial load transverse to the fiber direction, derived from equations (2.3.5), (2.4.31)-(2.4.33) with T,, = T,, = 0, and equation (2.4.35) and (2.4.36), are given by (”22 3|l+4B E22 — FX; 11+ B , (2.4.60) 12:1 AI = A2 = 53 and V =—dA3 : £+—ZB 32 dlz A 1 2|l+2B' k3: 2= (2.4.62) Note also that, by symmetry, E,, = E,,, v,2 = 11,, and v3, = v23. Here again, the subscripts of v are defined as before. It is clear from (2.4.42) and (2.4.44), for infinitesimally small deformation of simple shear, that the shear modulus is = 11. (2.4.63) By symmetry, the same results are obtained for 6,3 = G3, = u. Finally, it is seen from k = 0 Thus we have obtained the elastic constants corresponding to the linear elasticity for material (2.4.1) under the symmetry of transversely isotropy, and given by equations (2.4.58)-(2.4.63). It can be readily verified that relations (2.4.56) and (2.4.57) are satisfied. In comparison, since |.t 2 0 and B 2 0 , (2.4.64) it follows that and 54 If B = 0, then the equality holds everywhere in equation (2.4.65) and in the first of (2.4.66). In this case the material defined in expression (2.4.1) reduces to the incompressible isotropic neo-Hookean material, and we have 1 E11 = E22 = 311’ 612 = G21 = ”1 V21 = V31 = V12 = V32 = 5- (24-67) In the extreme case that |.t = 0, material (2.4.1) has only stiffness in extension/contraction in the X,—direction, and in such a case we have E11 = 451 E22 = 01 612 = G21 = 01 1 (2.4.68) V21 = V31 = 2’ V12 = 01 v32 = 1° It should be noted in all cases that which are required by incompressibility, and which, again, is required by the symmetry of the elasticity tensor [Christensen 1979]. For the five independent elastic constants EL 2 E11 , ET 2 E22 , VLT a 02, , 1).”- E 1132 and On a 62, , four of them, namely E,, E,, Ga and v”, are significant under the plane stress condition as regards the in-plane stress-strain relations. These four elastic constants, E,, E,, G,,- and V”, can be obtained from [Zweben et. al. 1989] for various fiber-reinforced composite materials as listed in Table 1. Here, the subscript L denotes the fiber direction and T denotes the transverse direction. For the elastic constants using L and T notation, we follow Zweben's convention in which the first subscript of v indicates the direction of loading, and the second subscript of v indicates the direction of lateral 55 contraction/extension. Although the strain density function of material (2.1.5) has only two parameters, it can still be used to approximate the elastic constants of these composite materials. Some results are listed in Table 1. We have obtained these results by using least square method to determine the values of the two parameters u and B, which make the elastic constants E,’, E,’ and G,,’ derived the best fit to those of measured values of the various fiber-reinforced composite materials. Note that VLT’ = 11,, = 0.5 derived from (2.4.6) does not depend on u and B. In certain cases (e.g. boron/epoxy) the match is very good as regards E,, EI and GL, Elastic Constants from P Mafgi 'n Derived Elastic Composite [Zweben et. al. 1989] (2 l 5) 1 Constants Fiber/Matrix ' ' 131. 15“Ir GLT VB: 11 B EL, El" GLT’ E Glass] Epoxy 45.0 12.0 5.5 0.28 3.2 8.8 45.0 12.0 ' 3.2 s Glass! Epoxy 55.0 16.0 7.6 0.28 4.3 10.5 55.0 16.0 4.3 KcharEii‘xuya’md’ 76.0 5.5 2.1 0.34 1.4 17.9 76.0 5.5 1.4 Gfggggfgg'éfiy 145.0 10.0 4.8 0.25 2.5 34.3 145.0 10.0 2.5 gigxgfijy 220.0 6.9 4.8 0.25 1.7 53.7 220.0 6.9 1.7 . Elsuahfi't‘jéfii‘y‘ 290.0 6.2 4.8 0.25 1.6 71.3 290.0 6.2 1.6 Boron/Epoxy 210.0 19.0 4.8 0.25 4.9 48.9 210.0 19.0 4.9 Alumina! Epoxy 230.0 21.0 7.0 0.28 5.4 53.5 230.0 21.0 5.4 Table 1. Here EL, E,, G; and V1.1" are obtained from Zweben et. al. (1989) with volume fraction of fiber = 0.6. EL’, Er’ and GL1” are given by equations (2.4.58), (2.4.60)-(2.4.63) for parameters 11 and B listed, which are determined by least square method to best fit to BL, E, and G”. All the quantities except for Vu- have unit GPa. 56 2.5. Material Properties (Two Balanced Fiber Families) The material modeled by expression (2.1.10) is reinforced with two balanced families of fibers if M = 2 with B“) = B”). Then, as discussed in section 2.1, it will, again, have three mutually orthogonal material principal directions and three mutually orthogonal planes of symmetry. This type of materials is classified as orthotropic material and is described by nine independent elastic constants in the scape of linear elasticity. This set of nine elastic constants can be chosen as E11. E22, E33. V,2, V13. V23. G12. 0,3 and G23. Let us set up the rectangular Cartesian coordinate system in the way discussed in section 2.1 that the planes determined by the fibers from these two fiber families are normal to the Xz-direction and both the X,-direction and the X,,-direction bisects the angle between fibers from these two families. Thus, the coordinates X,, X, and X, are in the three material principal directions, respectively. We thus have 9“) = _9(2) and B“) = [3(2) .1 B, (2,.51) since the material considered here is reinforced with two balanced families of fibers. Now, the strain energy density function (2.1.10) becomes W = I§l(I,—3)+-g(K(”—1)2+-g—(K(2)-l)2. (2.5.2) The Cauchy stress tensor is given by T,,. = prikrj, — p5,]. + 2B[(FranA§1)A§1) — 1)Fiijqugl)Agl) 2 2 2 2 (2.5.3) +(FranAg )Af )—1)FiijqA‘() )Aé l], where A“) = {c, 0, —s}T and Am = {c, 0, s}T, (2.5.4) 57 and c = case“) and s = sine“) (refer to Figure 1). With the coordinates so set, we shall call the reinforcement discussed here the reinforcement of two families of symmetrically balanced fibers. For the triaxial deformation (2.3.3), together with (2.5.4), the potentially nonzero components of the Cauchy stress tensor, given by (2.5.3), are Tll = 1111.12 — p + 4B(02}t,2 + 521% — l)c2}tz, T22 = 113.,2 —p, (2.5.5) T33 = pkg — p + 4B(c2}\.,2 + $271.;- - 1)c27t§. By considering uniaxial loads in the X,, X, and X3-direction in turn, as given by (2.3.14), the equations (2.5.5) yield a set of three equations for X,, 26,, X,, p and the uniaxial load. This set of equations, in connection with the incompressibility constraint (2.3.5), can be used to derive the corresponding elastic constants: Young’s moduli and Poisson ratios, as proceeded in subsection 2.4.2 by considering the limit 1., -) 1.2 —> 3.3 —> 1 . The results are written here 58 = 31:2 + 8|.1Bc4 + 8|.tBs4 - 8|.thzs2 u+2Bs4 ’ = 3112 + 8111304 + 8|lBs4 — 8|chzs2 |.t + 2B04 + 2Bs4 - 4Bc2s2 _ 3u2 + 8uBc4 + 8|.1Bs4 — 8|.1Bc2s2 |.:+2Bc4 ’ = 2|1+ 16Bs4-8Bs2 2‘ 4|.l+8Bs4 = 211-1-8Bc2s2 31 4|:+8Bs4 ’ v = 2|.J.+16Bs4—8Bs2 ‘2 411+ 16Bc4+ 16Bs4-8B’ v = 211+ 16Bc4—8Bc2 32 4|1+16Bc4+16Bs4-8B’ = 211-1-8B02s2 13 4|.t+8Bc4 ’ v = 211+ 16Bc4—8Bc2 23 4u+8Bc4 ° {1'1 8 I (2.5.6) It is readily to verify that the following relations are satisfied, E11V12 = Ezzvzrt E11V13 = E33V31’ (2.5-7) E22V23 = E33V329 as required by the symmetry of the elasticity tensor. If the material is subjected to a simple shear as described by (2.3.12), then the corresponding shear stress can be calculated from (2.5.3), and the corresponding shear moduli obtained, as proceeded in the limit k —) 0. The results are listed here G12 = Gzr = G23 = G32 = ll, (2.5.8) It can be seen that the shear moduli in the plane of fiber reinforcement are enhanced, if 59 0 it 0 and 0 at 1:/ 2. As one would expect, these B-dependent elastic constants in (2.5.6) and (2.5.8) reduce to those corresponding to transverse isotropy as discussed in subsection 2.4.2, if 0“) = 0(2) = 0 and [30) = 8(2) = 13/2. 3. Ordinary Ellipticity in Planar Deformation The ordinary ellipticity concerns the smoothness of the solution field. The loss of ordinary ellipticity corresponds to the possible appearance of discontinuity in the derivatives of the solution field. That is, if loss of ordinary ellipticity takes place, then, in the material body, there may exist certain surfaces across which the highest order directional derivative of the solution fields in the direction of the normal of these surfaces has finite jump. This kind of discontinuity is usually called weak discontinuity. The analysis of ellipticity usually yields restrictions on deformation parameters, if loss of ellipticity is to be avoided. The deformation parameters are usually the principal stretches of the deformation gradient X,, lo, and A, or the left Cauchy-Green strain tensor C,,. The restrictions are imposed on deformation parameters depending on material parameters, such as u and B in (2.4.1). It is convenient to choose principal stretches X,, 71.2 and 2.3 as deformation parameters for isotropic materials, but not for anisotropic materials. The reason is that the principal directions of the deformation, i.e. the principal directions of the left Cauchy-Green strain tensor, will not coincide, in general, with the material symmetry axes, during deformation. For this reason and for the problem to be considered being more tractable, we shall choose the components of the left Cauchy-Green strain tensor as the deformation parameters. In this chapter, we shall analyze the ordinary ellipticity of material (2.4.1) reinforced with one single family of fibers. We set, again, the rectangular Cartesian coordinate system as that set in section 2.4, with X, in the fiber direction. In next two sections, we discuss, first the ordinary ellipticity condition given by [Zee and Stemberg 1983], and then the specification of this condition to planar deformation. The distinction 60 61 between local and global plane strain ellipticity is given. Basic results for both global and local plane strain ellipticity are presented in section 3.3 and 3.4. In section 3.5, we relate the global plane strain ellipticity condition to a root type problem for a polynomial of degree four. A parameter space representation for loss of global plane strain ellipticity is developed. This involves the clarification of the complex phenomena of loss of ellipticity. The orientation of discontinuity surface is also clarified, especially with respect to first loss of ellipticity. Then the loss of ellipticity is discussed in the context of the special deformations introduced in chapter 2. 3.1. The Ordinary Ellipticity Condition for Arbitrary Deformations The necessary and sufficient condition for the ordinary ellipticity of the displacement field for an incompressible material was given by [Zee and Stemberg 1983]. Here, we outline their work in deriving this necessary and sufficient condition. The Piola-Kirchhoff stress tensor for incompressible hyperelastic materials is given by _8W_ 4‘ S’fif pF . (3.1.1) Denoting by u the displacement field, and noting Fij = 8n- +ui’ j, the equilibrium equation divS = 0 gives rise to the set of governing equations -1 _ _ Ciqu“p,qj"P,iji — 0 detF — 1 (3.1.2) for the displacement u and hydrostatic pressure p, where 82 ———W(F) (3.1.3) arfiarpq Cqu is the elasticity tensor. The result F11} j = 0 for volume preserving deformation has been 62 used in arriving at (3.1.2). Now consider a surface S lying within the reference configuration and described by x = 510;, :2), (3.1.4) where (C,, £2) is orthogonal curvilinear coordinate system on S. Further points near S in the reference configuration can be described in terms of a local orthogonal curvilinear coordinate system (C,, C2, C3), such that X = m, Cato = x(§,,§,)+g,n(g,,g,), (3.1.5) in which n is the unit normal vector of S. Thus, fields u and p on and near S can be expressed in terms of (C,,, C2, C3). By the definition of ordinary ellipticity, the possible jumps in the derivatives of these fields across S are [lu,,q,11= [[g—Zé‘flta ,Ca ,. [[pjll = [[32 ——]]t3,- (3.1.6) Here [[h]] denotes the jump of a function h across S. Now, the governing equation (3.1.2) yield, on S, 11113—221144 1211211th -1 3211p qu 3T; C3c1";3i=0 Note that VC3/|VC3| on S coincides with the unit normal 11 of S. Defining . {[331] q . .v.,.-r||g_v§||, (3.1.8) (3.1.7) 63 equation (3.1.7) is now written as Qv-qF‘Tn = 0, v - (F'Tn) = 0, (3.1.9) on S. Here Qip = Ciqunan’ 0,, = Qpi. (3.1.10) is the acoustic tensor. Equation (3.1.9) constitutes four linear homogeneous algebraic equations in the jumps v and q, which do not have nontrivial solution if and only if _ -T det Q] F n :20, v us (13, (3.1.11) nTF— 0 where 113 = {n l n,,,nm = l, m = 1,2,3}. In other words, if condition (3.1.11) is satisfied, there can exist only zero jumps v and q. Therefore, (3.1.11) is the necessary and sufficient condition for the ordinary ellipticity. It is to be noted that the problem here is formulated in the reference configuration, so that the normal n determines a surface of weak discontinuity in the reference configuration and Fn/IFnI determines the counterpart of the surface of weak discontinuity in the deformed configuration. If (3.1.11) is not true at some point P, so that the inequality is replaced by equality for some n, say n*, then a surface through P with normal n* in the reference configuration is capable of supporting a weak discontinuity in the displacement field. Specifically, recalling that |§3| denotes perpendicular distance in azu, 3? associated discontinuity values in the vector [v, q]T are given by nontrivial solutions to the direction of n*, then and 32% may suffer finite discontinuities at P. The 3 It _ gr :1: 0“” F n [v] = 0. (3.1.12) n*TF“ o ‘1 Since _ -T —T T -1 Q F n = F 0 FQF—n F 0, (3.1.13) nTF-l 0 0T 1 M 0 0T 1 we now have the relation _ -T T det Q F n = (detF‘1)2det F QF '“ . (3.1.14) nTF—l O “T 0 Thus the necessary and sufficient condition for ordinary ellipticity can equivalently be expressed as det H ‘ ¢o, v ne (1, (3.1.15) nT 0 where H = FTQF , HT = H (3.1.16) is a 3 x 3 symmetric matrix. With the coordinate system set-forth, it follows from (2.4.1) and (3.1.3) that the elements of c are given by ciqu = ufiipojq +2[3{2Fi1Fp1 +(F51Fsl - l)8ip}8j15ql , (3.1.17) so that (3.1.10) and (3.1.16) then give Hpq = uCpq + 213(C11 - Unfcpq + 4Bn12CplC1q. (3.1.18) For a given deformation, and hence given C, and given material parameters [.1 and B, 65 condition (3.1.15) with (3.1.18) suffices to determine whether or not surfaces of weak discontinuity may exist in the three-dimensional displacement field. 3.2. Specialization of the Ordinary Ellipticity Condition to Planar Deformation We henceforth restrict attention to deformations occurring in the (X1, X2)-p1ane either locally or globally. In turn, we restrict attention to possible surfaces of weak discontinuity with unit normals that have no component in the X3-direction at a material point under consideration, so that the tangential planes of the surfaces of weak discontinuity at the material point are perpendicular to the (X1, X2)-plane. Now, the elements of HI can be calculated, for the plane deformation described by (2.3.7), as H11 = ”(311+ 23(3Ci1‘C11N‘12’ H12 = H21 = PC12+25(3C11C12‘C12)“12’ H22 = uC22+2B(C11C22-C22+2C122)n12 , (3.2.1) H33 = 11+ZI3(C11" ”“12 H13 = H31= H23 = H32 = 0- The restriction on possible surfaces of weak discontinuity, n3 = 0, when combined with (3.1.15) and (3.2.1) gives H33(Hnn§+H22n12-2H12n1n2) $0 , V n e ‘112. (3.2.2) Here, if = {n | nan“ = l, a = 1,2, n3 = O}. The satisfaction of (3.2.2) is then the simultaneous satisfaction of both of the following two conditions H33 ¢0, v n e 212, (3.2.3) and H11n%+H22n12—2H12n1n2¢0 , V ne ‘112. (3.2.4) Note that conditions (3.2.3) and (3.2.4) are derived from condition (3.1.11) for general 3-dimensional incompressible materials. Following a standard classification [Knowles and Stemberg 1975], a local plane strain state involves a displacement field that has no components in the X3-direction at the material point under consideration, and the global plane strain state is a deformation in which the displacement components in X3-direction vanish identically. Thus, the normal of the surface of weak discontinuity for local plane strain happens to be in the (X,, X2)-plane, and the surface of weak discontinuity for global plane strain is a cylindrical surface with generators parallel to the X3-axis. Conditions (3.2.3) and (3.2.4) must be satisfied simultaneously for ellipticity in local plane strain deformation. For ellipticity of materials under global plane strain deformation the only condition required is (3.2.4). Actually, condition (3.2.4), which can be obtained here by eliminating the third row and the third column in the left hand side of condition (3.1.11), corresponds to the counterpart of condition (3.1.11) in the global plane strain deformation analysis given in [Abeyaratne 1980]. The ellipticity problem is now in terms of the deformation parameters C1,, C12 and Cu, and the material parameters 11 and B, in seeking nonzero unit normal n = [n,, n2, 0] to the surface of weak discontinuity. These five parameters are not independent. By making use of the stiffness ratio 7 defined in (2.4.5) and the incompressibility constraint (2.3.9), the number of parameters is reduced to three, namely, C”, C12 and 7. Their respective domains are given by 67 Note, by replacing every appearance of C22 with (1+C122)/C11, that C22 has been completely eliminated by using the incompressibility constraint (2.3.9). Hence, the restriction imposed by (2.3.9) on the remaining deformation parameters C“ and C12 has been released. Triplets (CI 1, C12, 7) obeying (3.2.5) that satisfy (3.2.4) will be said to constitute the global plane strain ellipticity set (GPSE). Triplets (C1,, C12, 7) obeying (3.2.5) that satisfy both (3.2.4) and (3.2.3) will be said to constitute the local plane strain ellipticity set (LPSE). Thus, membership in LPSE implies membership in GPSE, but not vice versa. It is to be noted for any pair (C11, C12) obeying (3.2.5)” that there exists an infinite number of associated plane strain homogeneous deformations. The infinite set of deformation gradients F can be parametrized by its F2, component on the range F11 = :l:,/Cll —F%l , (either sign), (3.2.6) F12 = (F11C12‘F21)/(F121+F%1) = (‘le 1C12JC11"F%1)/C11 ’ F22 = (1+F12F21VF11 = (i (C11-F%I)+C12F21«/C11”F21)/(C11'JC11’F21)° In particular, choosing F21 = 0 gives 0 0 1 68 The homogeneous deformation associated with the positive signed deformation gradient in (3.2.7) can be obtained as a sequence of two homogeneous deformations. First an axial expansion/contraction in the fiber direction with stretch JCT] accompanied by a volume preserving in-plane stretch 1/ ( «[51—l ). This is followed by an in-plane simple shear of amount C12 in the fiber direction. For —./C“ < F21 < [C11 the positive and negative roots in (3.2.6) represent two distinct branches of deformation gradients associated with the given C. These branches join together at both P2] = -,/C11 and F21 = JC“ , where o 1/ C11 0 o —1/ c110 F = , F = ,— , 3.2.8 _ C11 -C12/ C11 0 C11 C12/ C11 0 ( ) _ 0 o 1_ o o 1_ respectively. This forms a “loop” of deformation gradients associated with the given C. Any F on this loop will generate the full loop upon being post multiplied by the full range of an in-plane rigid body rotations, i.e. rotations about the X3-axis. In particular, the member of this loop given by the positive roots in (3.2.6) for F _ cnctz 21 — (3.2.9) (1+ C”)2 +C122 gives the special symmetric, positive definite deformation gradient ' 1 JC11(1+C11) JC11C12 j(1+c,,)2+c§2 j(1+c11)2+c122 U = , (3.2.10) ECU O 1+Cn+Cfi J(1+C11)2+ C122 JC11«/(1+C11)2+Ci2 O l 69 corresponding to the pure deformation U in the polar decomposition F = RU of any other member of this loop. Since it can be noted from (3.2.1) that the ellipticity is affected by deformation only through the right Cauchy-Green strain tensor C, it will not be influenced by the rigid body rotation R. Actually, in view of (2.3.2), the mechanical effect of a deformation described by C to a material body is equivalent to the mechanical effect of the pure deformation given by U. The post rotation R is then viewed as post determined by requiring compatibility with the actual deformation field. One need not, however, consider this post rotation R to extract the pointwise material properties. That is, the post rotation R can be set to the identity for the consideration of mechanical response. 80, the “putative normal” is Un/IUnI that determines the possible surface of weak discontinuity and is produced by the pure deformation part U of the overall deformation F as determined from the polar decomposition F = RU. The “actual normal” Fn/ anI in the actual deformation field satisfying the compatibility condition will, in general, be rotated rigidly with the purely deformed material body, but we will not consider the rigid body rotation here. Similarly, regardless of the post rotation R, the deformed fiber putative direction given by (2.1.9) is now rewritten as UA/ JR. Unless stated otherwise, all subsequent development will be with respect to quantities associated with the pure deformation part U of the overall deformation. For the plane deformation in the (X,, X2)-plane and the original fiber direction given by (2.4.3), we denote the angle from the Xl-axis to the deformed fiber direction by (I), which is measured by a right hand rotation about the X3-axis. In view of the above discussion, it is taken as 70 (3.2.11) 3.3. Loss of Ellipticity in Local Plane Strain A sufficient condition for loss of ellipticity in local plane strain is for the existence of n e ‘u2 such that H33 = 0. Now from (3.2.1).,, it follows that H33 = 0 if and only if “12 = (27(1 — C“))‘1 . Since 0 S nf S 1 it follows that the sufficient condition H33 = 0 for loss of ellipticity in local plane strain is met if and only if 1 0 S —— S 1 , 3.3.1 21((1 - c“) ( ’ which, since 1 — 1/2'y < 1 , is equivalent to c s 1 - i. (3.3.2) 11 27 In view of (3.2.5), this condition cannot be met if 'y S 1/ 2, which includes the neo-Hookean case 7 = O. For Cll obeying (3.3.2), it is sufficient to find an n e ‘112 that violates (3.2.3). On the other hand, the satisfaction of the opposite inequality 1 cu>1— 5? (3.3.3) is equivalent to the satisfaction of (3.2.3). Thus, (3.3.3) is necessary for ellipticity in local plane strain, however, sufficiency requires the satisfaction of both (3.2.4)and (3.3.3). Since (3.2.4) alone characterizes the ellipticity in global plane strain, it follows that triplets (C11, C12, 1) in GPSE are also in LPSE if and only if Cll obey (3.3.3). In particular, triplets (C,,, C129 7) Obeying 71 1 CI] = 1— 27 (3.3.4) constitute a cylindrical surface in the (C1,, C12, y)-space involving transitions from local plane strain ellipticity to the loss of this same property. As an illustration of loss of ellipticity in local plane strain, consider the biaxial deformation (2.3.10) for which 2.12 O O (212—11—2)+2'y(112—1)kf O O C = 0 A? 0 , T = u 0 0 0 , (3.3.5) _o o 1_ _ 0 0(1—lfz) where the hydrostatic pressure p has been chosen so that T22 = 0. Then, as A, is decreased from the undeformed value A, = 1, the loss of ellipticity sufficiency condition (3.3.2) is never met if y S 1/ 2, that is, if the additional stiffening due to the fibers is sufficiently weak. On the other hand, if y > 1/ 2 , that is, if the fibers are sufficiently stiff, then loss of ellipticity is ensured under decreasing 71., starting with A] = m. At this value of 71., a surface of weak discontinuity with normal direction 11 = [1, O, O] in the reference configuration can be sustained. Note that this corresponds to the surface with normal in the fiber direction. Under continued decrease of 9», this potential surface of weak discontinuity rotates away from this initially perpendicular fiber intersection and such a rotation can occur either clockwise or counterclockwise. In particular, as it] -—> 0 , the normal to the potential surface of weak discontinuity tends to the value n = [1/2'y, 1J1 — 1/472, 0] . 3.4. Basic Results for both Global and Local Plane Strain Ellipticity The loss of ellipticity results obtained in the previous section were based solely on 72 the consideration of (3.2.3). The associated analysis was elementary. Now we turn to the consideration of (3.2.4) which, we recall, is the sole defining requirement for global plane strain ellipticity. In contrast to the analysis of (3.2.3), the analysis of (3.2.4) presents some challenge. Condition (3.2.4) can be expressed as [n1 n2]{(1—2n127)D+2nny}|:n1:|¢0 , V us 112, (3.4.1) n2 where we have introduced 1+Cfi 2 -3C C 3C2 -C12 C11 11 12 11 We now examine the properties of matrices D, E and (ED). It is verified directly from (3.4.2) using Cu>0 and detD = 1, that matrix D is positive-definite. In addition since E,, = 1+3C122>0 and detE = 3C112>0, matrix E is also positive-definite. It can also be obtained from (3.4.2) that matrix (ED) is positive-definite if and only if C,,>1. This is because (E-D)22 = C,,(3Cu-1) and det(E-D) = (3CH-1)(C”-1). If Cll = 1, then (ED) is positive-semidefinite. According to the above discussion and following from the sufficiency conditions (3.3.3) and (3.4.1) for local plane strain ellipticity, we arrive at the following conclusions: (EI) If C11 2 1, then local plane strain ellipticity is guaranteed. Here condition (3.4.1) is satisfied since D is positive-definite and (E-D) is positive-semidefinite. Condition (3.3.3) is also trivially satisfied. Hence, it is made clear that material (2.4.1) is 73 elliptic for planar deformations when fibers are in extension, so that loss of ellipticity can only take place when fibers are compressed. (EII) If Y = 0 (B = O, the neo-Hookean material), local plane strain ellipticity is guaranteed. Here condition (3.4.1) is satisfied and, further, condition (3.3.3) is trivially satisfied. This result can be extended to materials obeying y S l/ 2. Since E is positive-definite, (3.4.1) gives that loss of global plane strain ellipticity is possible only if 1-2nlzy<0. This guarantees global plane strain ellipticity for y < 1/ 2. Furthermore, when 7 = 1/ 2 , the first term in the left hand side of (3.4.1) may have its minimum value (zero) only if r11 = l , but now the second term in the left hand side of (3.4.1) is strictly greater than zero. Thus, global plane strain ellipticity is assured for materials obeying ‘y S 1/ 2. Note, simultaneously, condition (3.3.3) is again trivially satisfied for y _<_ 1/ 2 . We can thus conclude that materials obeying ‘y S 1/ 2 guarantee local plane strain ellipticity. (EIII) For all those unit normals n e ‘le with 111 = 0, both condition (3.2.3) with (3.2.1)4 and condition (3.4.1) are satisfied. Thus, any possible unit normal describing a surface of weak discontinuity must have nonzero component n,. 3.5. An Alternative Form of the Global Plane Strain Ellipticity Condition We now turn to examine condition (3.2.4) in more detail. This corresponds to the global plane strain deformation problem. Since that n = [n,, n2, O]T is the unit normal of the surface of weak discontinuity of the displacement field taking place in the (X,, X2)-plane, this suggests the substitution 111 = cosa and 112 = sina. In seeking the existence of this unit normal, and equivalently the existence of the surface of weak discontinuity, n and -n play the same role. Thus, noting (EIII), we restrict that -1t/2 < a < n/ 2 . Here, on is the angle 74 in the (X,, X2)-plane measured from the Xl-axis to n. It is positive if it is a right hand rotation about the X3-axis, and vice versa. By further making the substitution x = tana, —oo < x < co, and expanding the left hand side of condition (3.2.4), using 111 = 1 2, + x 2 2 __ X _ . . . n — and n n — —, one obtarns the followrn 01 normal of x, P x, C , 2 1 +x2 l 2 1 +X2 g P y ( 11 C12, 7), of degree four: P(x, c”, C12, 7) a {x2 +27(3C11—1)+1}{(C11x—C12)2+1}—47C“. (3.5.1) The polynomial P(x, C“, C12, 7) expressed in the form of a sum of powers of x is P(x, C11, C12, 7) a Cf-l (x4 + a1x3 + a2x2 + a3x + a4) , (3.5.2) where 2C12 a1 531(C11’C12) = “T, 11 1+Cfi 4 C (3c 1)+2cl:1 (3'53) 7 2 11' 12 a3sa3(C11,C12,y) = l C r 11 Cf2{27(3C”-1)+1}+27(C11—1)+1 a4Ea4(C11’C12’ 'Y) = Cf] The condition (3.2.4) for global plane strain ellipticity is now equivalent to the requirement that the polynomial P(X, Cu, C12, 7) has no real roots. That is, if for a given triplet (Cu, C12, 7), there exists no real x such that P(x, C11,C12, Y) = 0, (3.5.4) then the triplet is a member of the global plane strain ellipticity (GPSE) set. It is evident from (3.5.1) that, if x is a root of equation (3.5.4) at (C11, C12, 7), then 75 -x is a root of equation (3.5.4) at (C,,, -C,2, y), and vice versa. Further, the unit normal determined by -x is the reflection about the (X1, X3)-plane of the normal determined by x. The deformation described by (C1,, -C,2) is a similar reflection to that described by (C,,, C12). One may, thus, restrict the discussion to the case C12 2 0. 3.6. Loss of Ellipticity in Global Plane Strain We begin analysis on (3.5.1) by giving an example of triplets (CH, C12, '1) that are associated with loss of global plane strain ellipticity. We consider here the triplets (C, 1, C12, 7) which support the existence of surfaces of weak discontinuity that are normal to the Xl-axis (x = 0). That is, the normal of the surface of weak discontinuity is in the fiber direction. Let x = O in (3.5.1), equation (3.5.4) becomes P(O, C“, C12, 7) = O, and yields (27-1)(C,22+1) .. 11= 27(3C1224-1) E(311(C312.Y)- (3.6.1) Equation (3.6.1) defines a surface in the (C,,, C12, 7) parameter space as shown in Figure 23. In view of (3.2.5), and the restriction C12 2 O , it is seen that this surface is associated with pairs (C12, 7) e 2, where For every triplet (C11 , C12, 7) on this surface, the polynomial P(x, C“ , C12, 7) has, at least, two real roots, one of which is x = O. In addition to the root x = 0, the other root is, in general, nonzero and corresponds to a second possible surface of weak discontinuity whose normal is rotated away from the Xl-axis by an angle ii. The contours of this angle at for triplet (C11, C12, 7) obeying (3.6.1) is plotted in Figure 24. These angles are found by substituting from (3.6.1) into (3.5.1) and determining the other real root x of the 76 resulting polynomial. The angle a is then given by or = tan'lx. This other real root is seen to coincide with x = 0, indicating a double root, on the boundary of 2, that is on 32 E{(C12, y)| either C12 = O, or y = 1/2}. (3.6.3) 77 / Q i I 'I‘ "‘ ’o m O l“ : .43 \ la 9\ 9/; Q §\ 12 “ 9 \ ° " E \ .. g \ P(x C12 “Cll Q v—‘fi c Figure 23. Surface consisting of triplets (CH,C12,y) obeying (3.6.1). The asymptotes of this surface are also shown. Points associated with this surface are not members of GPSE and can support weak discontinuities with normals in the fiber-direction. (3) 78 -A C) I l I l l l I I I o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 C12 Figure 24. These contours give the angle 6: , in degrees, for the direction of the normal to the second possible surface of weak discontinuity for triplets (Cu ,C12,y) obeying (3.6.1). According to (El), plane strain ellipticity is guaranteed for C11 2 1 ; on the other hand, for C“ given by (3.6.1), loss of plane strain ellipticity takes place. Consequently, it follows for each (C12, 7) e )3 that there exists at least one value C11 obeying C11 S C11 S 1 associated with the transition from ellipticity to non-ellipticity. Defined at (C12, 7) e 2 , these C“ '5 associated with the transition between ellipticity and non-ellipticity furnish the global plane strain ellipticity boundary fl: C11 = C11(C12, y). We shall find that there exist more than one such C11 for certain (C12, 7) e 2 so that C11 = C11(C12, 7) may be multiple—valued. In general Bis a two-dimensional manifold in the (C11, C129 Y) Sme. Since P(x, C“, C12, 7) is a polynomial in x of degree four, it has two pair of 79 complex conjugate roots for triplets (C1,, C12, 7) in the ellipticity region. The transition from ellipticity to non-ellipticity then corresponds to the first transition of a pair of complex conjugate roots of P(x, C,,, C12, 7) to a pair of real roots. It is convenient to investigate this root type transition phenomenon in a “well-drilling” fashion. That is, for fixed (C12, 7) e 2, C11 is decreased from C11 = 1 to C11 near zero. In the well-drilling process, a complex-real transition occurs if a pair of complex conjugate roots of P(x, C,,, C12, 7) changes to a pair of real roots. Accordingly, a real-complex transition occurs if a pair of real roots changes to a pair of complex roots. The Cll values at which a complex-real transition takes place will be denoted by C“, and the C11 values at which a real-complex transition takes place will be denoted by Q”. It will be shown that more than one complex-real transition and more than one real-complex transition may take place in a single well-drilling, depending on the value of (C12, 7). Thus, the manifold in the (C11, C12, 7)-space associated with changes in global plane strain ellipticity is much more complicated than the smooth manifold (3.6.1). In general, C“ and Q“ are multiple-valued functions of (C12, 7) e 2, and each branch will be written in the form C” = C“(C,2, 7) and Q“ = 911(C12. 7), respectively. In the following discussion, a superscript in parentheses will be used to mark each single-valued branch of C,,(Cn, 7) and each single-valued branch of C,,(Cn, 7). The set of all triplets (C,,, C12, 7) and (C,,, C”, 7) will, in general, form a two-dimensional transition manifold 5. In general, {B is a submanifold of 5 since real-complex transitions and complex-real transitions may or may not involve the root pair associated with loss or gain of ellipticity. We now seek to determine the properties of this manifold 5. Consequently, triplets (611(C12, 7), C12, 7) and (211((212’ 7), C12, Y) on 5 in the (C11, C12, 7) parameter Space. are 80 associated with values x' that are double real roots of (3.5.4). That is they simultaneously satisfy the following equations P(x,c1,,c,2, 7) = 0 and §;P(X,C11,C12, 7) = 0. (3.6.4) 3.7. Global Plane Strain Ellipticity Boundary: Cross-Sections at Constant Cu It is convenient to explore the properties of 5 by considering its cross section at various fixed values of C12. In general one find that this cross section then evolves in a fairly complicated way as C12 increases from zero. In particular we find that there exist 6 special values of C12: 0 u—v“ N" v II 1.4444, C53) = 2, c9) = 2.026, (3.7.1) 2.0945, C13) = 2.13, egg) = 2.2635, n u—v" N& V II at which the cross section of 5 changes its qualitative form. The cross section of 5 for C12 = O is given by those (Cu, 7) pairs that support real roots x satisfying both equations in (3.6.4). For C ,2 = 0, one obtains from (3.5.3) that a1 = a3 = 0, so that P(x, C,,, O, 7) is quadratic in x2. Accordingly x = O is a root of %P(x,C11,O,Y) = 0. Then one finds that P(O, C“, O, 7) = 0 gives C” = 1—217 ((3.3.4)) as an exact expression for the cross section of 5 at C12 = 0 (Figure 25). Note this expression coincides both with (3.3.4) and with (3.6.1) for C12 = O. This completes the analysis of 5 for C12 = 0 since it can be shown that the other two roots x of %P(x, C“, O, 7) = 0 do not satisfy P(x, Cu, O, 7) = O for triplets (C,,, O, 7) obeying (3.2.5). 81 0'9 _ Emmmizi ’Y) C12 = _ 0.8 - 4 0.7 h a 0.6 - _ cll 0.5 - A 0.4 ’- ~ 0.3 - _ 0.1 '- .. O 20 4O 60 80 1 OO 1 20 1 40 1 6O 1 80 200 Y Figure 25. The cross section of 5 for C12 = O is given by C11 = 1-1/27 and so coincides with both (3.3.4) and with 6,.(0, 7) . Here 5coincides with a. We now turn to the consideration of cross-sections of 5 for C12 > 0. Some results are plotted in Figure 26-Figure 33. The procedure employed here and in what follows for generating these cross sections is based on the discussion in Appendix A. There it is shown that a point (C,,, C12, 7) is a member of 5 if and only if it satisfies the equation 19(C11,C,2, 7) = 0, where 1‘) = 27§f+4§%. The two functions §1: 119—)3, and 4 5 l() l() «:2: K3 -) .‘K are given by (A.l). It is also shown that cusp points (see and in Figure 29-Figure 33) to 5 are associated with the simultaneous solution of the equations §1(C“, C12, 7) = 0 and §2(C“, C12, 7) = 0. Such cusp solutions exist only if C12 2 2. This simultaneous set of equations can be treated numerically to locate potential cusp points. In addition, it is shown that self intersection points (see 1‘” in Figure 29-Figure 33) of the manifold 5 are associated with the simultaneous solution of b3(C11, C12, 7) = 0 and b2(C11,C12,7)2-4b4(C“,C12,7) = 0, where b1: xii—MK, b2: K3—>x and 82 b3: 1&5 -—) x are given by (A. 1). These self intersection points also occur only if C12 2 2 and can again be located numerically. Consequently, the cross section of 5 at fixed C12 is found from a numerical root search of the equation 13(C11, C12, 7) = O , where the cusp points and the self intersection points are given by the separate analysis of (A.l), and (A.l)u, respectively. These special values of (C,,, 7) associated with the limit points 1"), 1(2) and 1“” in Figure 28-Figure 33 are located by refining the numerical root search of the equation MC“, C12, 7) = O. The special values C12“), i = 1,3,4,5,6 are also determined numerically. On the other hand, C120) = 2 is an exact result associated with the unique location on 5 corresponding to a quadruple root of (3.5.1). For Cu in the interval 0 < C12 < Cum = 1.4444, two additional cross sections of 5 at fixed C12 = 0.5 and C12 = 1.4 are plotted in Figure 26, Figure 27, respectively. Well-drilling at (C12, 7) obeying C12 < C 12(1) intersect 5 only once, and this intersection is associated with the first and only complex-real transition. This root transition is hence a transition associated with loss of global plane strain ellipticity. We denote this single-valued part of 5 by C,,") = C,,“)(Cu, 7), which is part of the ellipticity boundary 3: C11 = C11(C12, 7) , for C12-points of Z, and the corresponding subregion of 2 possessing this property will be denoted by 21. 83 0.8 '- o.7 - ........................................................................................ - 0.6 - _ C” 0.5 - .. 0.3 l- - 0.2 - _ l l l l O 20 4O 60 80 1 OO 1 20 1 4O 1 60 1 80 200 Figure 26. Cross section of 5 for C12 = 0.5, showing the global plane strain ellipticity boundary 3. Pairs (C12, 7) above 5 are in the GPSE, and pairs below 5 involve loss of global plane strain ellipticity. The dotted line shows the cross section of the surface previously given in Figure 23, which is now interior to the loss of ellipticity region. 0.9 ,_ C12=1.4 —l (1) 0.8 P C11 (C12, Y) 0.7 - 0.6 - Cno.5 _ C11(C12, Y) . .. ... . ... . . ....--..-.........- ,...... -..-...... ,............... .-. ... ... .. .. -.- u o 4 g... 0.4 L 0.3 - 0.2» E - O l L I 1 L l l 1 l O 20 40 60 80 1 00 1 20 1 4O 1 60 1 80 200 Figure 27. Cross section of 5 for C12 = 1.4, showing the global plane strain ellipticity boundary 9. As in Figure 26, the cross section of the surface associated with (3.6.1) is again shown by the dotted line. It can be noted from Figure 26 and Figure 27 that, with the increase of C12, 84 manifold 5 is beginning to develop an inflection point. At C“ = 0.5181, C12 = C12“) = l 2 l( ) l( ). 1.4444 and 7 = 7.4009, this inflection point begins to yield two limit points and These two limit points I“) and 1‘”, with the further increase of C12, generate two space curves in the (C11. C12, 7) parameter space. The associated (C11, 7) values may be parametrized in terms of C12 as 1(1) a (611’, ”7“”) = (o<1>(c,2), r<1)(C,2)), for C12 2 cg), (3.7.2) 1015(611’, 7“”) = (om(c,2), F‘2)(C12)), for c12<1>sc12scggx (3.7.3) These vertical tangency points are shown in Figure 28 for C12 = 1.8. This is a typical cross . l 2 section for C12” < C12 < C12” = 2. For C12 in this interval, manifold 5 is no longer single-valued with respect to 2. It is triple-valued for F(1)(C12) <7< F(2)(C12). Well-drilling at (C,,, 7) obeying c,,"’(c,2) < 7 < r(2>(c,2) intersect 5 three times. These intersections are, in turn, associated with complex-real transition, real—complex transition and complex-real transition. Manifold 5 is thus split into three single-valued branches by l") and 1(2) . We denote these three single-valued branches of 5 by C1,“) = C1,“)(C12, 7), C,,") = C,,“)(Clz, 7) and Cum = Cum(C12, 7), as illustrated in Figure 28. The second intersection at C,,") corresponds to the recovery of ellipticity with increasing compression (Cll decreasing) of a material particle. Values (C12, 7) e 2 obeying C,,“’(c,2) < 7 < r<2)(c,2) are classified as <0, 2, 0, 2>-points of 2, and the corresponding subregion will be denoted by 2‘9. Well-drilling at (C,,, 7) obeying c,,“’(c,2) or 7 > P(2)(C12) , intersects the manifold 5 only once. This intersection is associated with complex-real transition, and so these (C12, 7) pairs are <0, 2>-points of E, i.e. 85 (C12, 7) e 2"1 - All Err“) = 611(“(C129 Y), Q11“) = g11(1)(C12a 'Y) and 511(2) = EII(Z)(C12’ 'Y) for C12“) < c,2 < C,,“) = 2 are part of the ellipticity boundary a (“:11 = (“211(c12, 7). 0.9 _ C12: 1.8 (l) 0'8 " E11 (C12, 7) ‘ 0.7 i- (l l) (1) £11 (C12, 7) 2 2 l() _, 0.6 L Clp.5 - 0.4 - 0.3 - 0‘2 _ a11(2)(Cizt Y) A 0.1 b 4 o 20 4o 60 so 1 oo 1 20 1 4o 1 so 1 80 200 'Y Figure 28. Cross section of 5 for C12 = 1.8, showing the global plane strain ellipticity boundary fl. Here 1(1) and 1(2) are limit points associated with vertical tangency. In this and future figures for 5 at constant C12, the cross section associated with (3.6.1) is no longer shown. Note in Figure 28 that there is a sharp bend in the Cum branch of 5 just below 1‘”. Starting at C,, = 0.3883, C12 = Cum = 2 and 7 = 17.0815, this bend unfolds into three branches. TWO of these, denoted by Cum) = CHOWC”, 7) and Cum) = C,,(22)(C12, 7), continue to be associated with complex-real transition. In between them, the third unfolded branch, denoted by Qua) = CHOKCH, 7), is associated with real-complex transition, as illustrated in Figure 29-Figure 32. This generates 3 special points on each cross-section, which are denoted by 1‘”, 1(4) and 1(5). As C12 increases from C12 = 2, this generates 3 new space curves. All 3 curves originate at T: (C1,, C12, 7) = (0.3883, 2, 17.0815) which is the sole quadruple root to (3.6.4) ,. The associated (Cu, 7)-values on 86 these curves can also be parametrized in terms of C12 as 1(3)=(Cii), 7‘”) = (o<3>(c,2), r<3>(c,2)), for Clzzcggx (3.7.4) 1<4>=(c1‘1’,’y‘(4’) = (6(4)(C12),I‘(4)(C12)). for Cuzcgg), (3.7.5) 1(5)=(Cii), “(5’): (o<5>(C,2),r<5>(c12)), for clzzcg). (3.7.6) The space curve 10) is the intersection of CumKCn, 7) with C“(22)(C,2, 7) for Cum = 2Cj4): — 2. 0945. Curve 1‘” corresponds to two distinct root transitions, and triplets (6(3)(C12), C12, F(3)(C12)) thus provide two double real roots for equation (3.5.4). Triplets (6(4)(C12) , C12, P(4)(C12)) and (9(5)(C12) , C12, P(5)(C12) ), which are the cusp points in Figure 29-Figure 32, on the space curves and respectively provide triple real roots for equation (3.5.4). Now, branch Cu(2”(C,2, 7) is bounded by 7 = 1/2 and 1(5), branch C,,(22)(C12, 7) is bounded by l“) and 1(2), and branch Q, 1mm”, 7) is bounded by 1(4) and 1‘”. 87 1 T I r C = 2.01 0-9 ” 11 (C12, 7) '2 0.8 *- _ 0.7 _ 0.3377 . . f 4 _ (l) - (22) 0.6 e C11 (C12, 7) . (1) 0.3377 - (5) 11 (C12. 7) - 1 C1165 — - 1(2) 0.3377 - 0.4 [- - (2) 0.3 - o.am- 911 (C12, 7) - 0.2 e _ 0.3377 > 1(4) .. (2|) . ‘ . q . . _ 0'1 611 (C129 7) 17.33 17.331 17.332 17.333 17.334 17.335 17.333 0 l L 1 I I l I I l O 20 4O 60 80 1 OO 1 20 1 4O 1 60 1 80 200 Y Figure 29. Cross section of 5 for C12 = 2.01 is typical of C12 values obeying C120) < 3 . . 2 . . C12 < C]; ). Unlrke the cross sections for C12 < C]; )= 2, the cross section now inter- sects itself (inset). Points on the internal branches created by this intersection are mem- bers of 5 but not members of 3. Triplets (C1,, C12, 7) within the triangular region enclosed by these internal branches have 4 real roots to (3.6.4)1. In the well-drilling process, branches C,,(21)(C12, 7) and C,,(22)(C,2, 7) are associated with complex-real transition and branch Q,,(2)(C,2, 7) is associated with real-complex transition. For C l2(2) for 1/2 <7< F(1)(C12) , <0, 2, 0, 2> for 1‘“)(C12)<7< P(4)(C12)1<092101 2,4,2> for F<4)(C12) < 7< r<5>(c12), <0, 2, 0, 2> f0r F(5)(C12) < y < P(2)(C12) , <0, 2) for 7 > F(2)(C,2). The subregion of 2 containing <0, 2, 0, 2, 4, 2>-points will be denoted by 23. 88 At C13) 5 2.026, it is found that FW(C12) = FW(C12) . This gives F(1)(C12) < P(4)(C12) < P(3)(C12) < P(2)(C12) < P(5)(C12), o<3>(clz) < 907C”), (3.7.8) for C8) < C12 < C53) 5 2.0945. The values of (C12, 7) e 2 , for C,2(3) for 1/2 < 7 < F(1)(C12) , <0, 2, 0, 2> for F(')(C12) < 7 < F(4)(C12) , <0, 2, 0, 2, 4, 2> for r<4)(c,2) < 7 < when), <0, 2, 4, 2> for F(2)(C12) < 7 < 1"(5)(C12), <0, 2> for 7 > F(5)(C12). The subregion of 2 containing <0, 2, 4, 2>-points will be denoted by 24. 0.9 _ C12=2.08 _ ) C11 (C12: Y) 0.8 b 0.7 b (5) 7 _. 22 (21) 1(1) C11( )(C121 C11 (C12,?) (1) °'°°" C11 (C12,?) 0.6 b C110.5 '- 0.3335 - (2) 0'4 " Cu (C12, 7) 0.3 ‘- O.383 '- O.2 - 0.1 - 10.15 10.2 10.25 "' O 20 4O 60 80 1 00 1 20 1 4O 1 6O 1 80 200 Figure 30. Cross section of 5 for C12 = 2.08 is typical of C12 values obeying C120) < C12 < C12“). Note that the vertical tangency point 1(2) is approaching the self intersec- tion point 1(3). At C14) 5 2.0945 , it is found that F(2)(C12) = 1"(3)(C12) a 19.52 and 6(2)(C12) = 9(3)(C12) 5 0.3828 . This gives 89 F(1)(C12) < F(4)(C12) < P(3)(C12) < P(2)(C12) < F(5)(C12), 9(3)(C12) > 8(2)(C12), (3.7.9) for C111,) < C12 < CB) 5 2.13. The values of (C12, 7) e 2‘. , for C,2(4)-points, <0, 2, 0,~ 2>-points, <0, 2, 0, 2, 4, 2>—points, <0, 2, 4, 2, 4, 2>-points, <0, 2, 4, 2>-points and <0, 2>-points, in turn, for 7 intervals divided by F“), 1"“), 1"”), I'm and 1'“), as can be seen in Figure 31. The subregion of 2‘. containing <0, 2, 4, 2, 4, 2>-points will be denoted by 25. C12 = 2.11 0-9 ' C11 (C12, Y) 0.8 r a 0.7 - -~ ~ (5 (1) l ) 1 0.33215 . 1 0.6 - l) (l) 7 ( 11 (C12, Y) 0332- 911 (C12, 7) C11 0'5 ” (2) - (C121 7) 0.4 - 0.301s - _ 0.3 - 0.301 _ 0.2 *- 0.3005 [(4) .1 - ‘6 (21) n - 0'1 611 (C129 7) 01305 100 10.05 20 2005 201 2015 O O 20 4O 60 80 1 00 1 20 1 4O 1 60 1 80 200 Figure 31. Cross section of 5 for C12 = 2.11 is typical of C12 values obeying (212(4) < C12 < C126). Here the vertical tangency point 1(2) has passed through the self intersec- tion point 1(3). These two points coincide only at the special value C12 = C12“), which is intermediate between that of Figure 30 and the present figure. At C13) 5 2.13 , it is found that F(3)(C12) = F(4)(C12) . This gives r<1>(c,2) < r<3>(c,2) < r<4>(C12) < F‘2’(Ctz) < F(5)(Cl2)’ (3.7.10) for CE) < C12 < C13) .2. 2.2635. 90 The values of (C12, 7) e Z , for C 12(5)-p0ints, <0, 2, 0, 2>—p0ints, <0, 2, 4, 2>-points, <0, 2, 4, 2, 4, 2>-points, <0, 2, 4, 2>-points, and <0, 2>-points, in turn, for 7 intervals divided by 1'“), F”), I“), I“) and 1'“), as can be seen in Figure 32. 1 f _. ) C = 2.16 _ 0'9 C11 (C12, 7) 12 0.8 - ~ 0 7 __ 0.3315 2 1(1) 0.301 - (5) 0.6 7 (1) 0.3000 ~ 1 - C 911 (C12. 7) cm- (1) . ll 0 5 - 0.3700 C11 (C129 Y) . T 0.4 _ 0.370 e (2) _ 0.3735 r 911 (C12! 7) 0-3 ' 0.373 >- “ (2) (22) 0.3773 - 1 E1 1 (C 121 Y) 0.2 ~ .. 0.377» 1(4) — 2l . . . . r 0-1 ' C“( )(C12’ 7) 037.31.: 21.3 21.4 21.5 21.0 21.7 1 0 0 20 40 60 80 1 00 1 20 1 40 1 60 1 80 200 Y Figure 32. Cross section of 5 for C12 = 2.16 is typical of C12 values obeying Cum < C12 < C12“). Note that the vertical tangency point 1(2) is approaching the cusp point l“). At C56)§2.2635, it is found that P(2)(C12) = P(4)(C12) 524.4047 and 9(2)(C12) = o<4>(c12) 5 0.3694. Hence the branch associated with c, ,‘22’(c,,, 7) and the limit point 1‘” no longer exist for C12 > C12“). However, a new branch Cum = 011mm”, 7) is generated along with the new limit point 1“”, parametrized by the space curve 1(6)a(€1i’, 7“”) = (6(6)(C12),F(6)(C12)), for clzzcggx (3.7.11) Now, branch E,,‘3’(c,,, 7) is bounded by 1“” and 1“”, and branch 0,310,, 7) is bounded by 1“” and 1‘”. This new branch CHOKCU, 7) is associated with complex-real transition. Figure 91 33 is a typical cross section of 5 for C12 > C12“). In this C12 range: I‘(1)(C12)< F(3)(C12)< I“(6)“:12) < I“(4)“:12) < I‘(S)(CIZ)’ 6 (3.7.12) c12 > 012); 2.2635. The values of (C12, 7) e E , for C12>C12(6), are classified according to the parameter 7 as: <0, 2>—points, <0, 2, 0, 2>-points, <0, 2, 4, 2>-points, <0, 2, 4, 2, 4, 2>-points, <0, 2, 4, 2>~p0ints, and <0, 2>-points, for 7 intervals divided by I‘m, 1"”), I“), I“) and 1'“),in turn, as can be seen in Figure 33. C11 1 T T I E (l) C I I I IV 0.9— C12=4,0 I I 11 ( 12,?) I I _ I I I | | 0'8 ' I I I | I I | I I J 0.7 r (1) I 1 I I I I 0.6 ~ 3 I I | I I 0.5 - I I I I -I (l) 0 4 _ _ (21) I 911 (C12, Y) | | l(5) _I ' C11 (C12, Y)! 1 1 . 1 , I .. (2) 0.3 I | | .. 1(4) 5211 (C12, 'Yll I I | I I C11 (C12, YlI 0.1 ~ 1 1 1 1 1 1 0‘ 1 l 1 l l 1 J J 1 I 1 L 4 0 20 40 60 80 100 120 140 1 60 1 80 200 7:30 7:70 7:110 7:130 Figure 33. Cross section of 5 for C12 = 4.0 is typical of C12 values obeying C12 > C12“). The dashed lines at fixed 7 = 30, 70, 110, 130, 160 and 200 provide correlation with Figure 35. 3.8. Projection of the Ellipticity Boundary onto the (C12, 7)-Plane The cross-section analysis of the previous section reveals how the manifold 5 is l 6 11 )-l( 1 divided into branches by the space curves . The penetration of each branch provides a root transition, only some of which are associated with loss or gain of global plane strain 92 ellipticity. Changes in C” at fixed (C12, 7) e 2 involve different root transition sequences depending on the value of (C12, 7) and motivated the classifications 23,-25 as an exhaustive subregion decomposition of Z. The projections of the space curves 10)-]‘6’ onto the (C12, 7)-plane give plane curves I'm-1"“), which provide the boundaries of these subregions. The curve projections and subregions are plotted in Figure 34. We find that 7-values on the F—curves are monotonically increasing with C12. The curves 1"” and 1"(3)-l"6) involve C12 —) ea. In contrast, curve I‘m terminates at (Cum, 6)) at which point 1“” begins. The curves I'm-I‘m) intersect at the 6 points: c9) = 1.4444, 7(1) = 7.4009, ch> = 2, 7(2) = 17.0815, ch> = 2.026, 7(3) = 17.782, (3.8.1) ch) = 2.0945, 7(4) = 19.521, ch> = 2.13, 7(5) = 20.47, ch> = 2.2635, 7“” = 24.3891. The interpretation of these 6 points are as follows: (Cum, 7‘”) is the origin of the curves 1"“) and I“) associated with the development of C11 tangency (vertical tangency) on 5. (C120), 2)) is the (C12, 7) value of {E the unique quadruple root of (3.64),. This point is the origin of curves I“), I“), 1.15) associated with the creation of internal branches of 5that are not on Q. (C120), 7‘”) is associated with the intersection of the projection curves Pm and 1"“) even though the parent space curves 1(2) and 1‘” do not intersect. 93 (Cum, 4)) is associated with the intersection of the vertical tangency space 2 l() . . . 3 curve wrth branch intersectron space curve 1( ). Hence the projection curves rm and I“) also intersect. (C126), 7‘”) is associated with the intersection of the projection curves 1“” 3 4 l() l() and 1"“) even though the parent space curves and do not intersect. (Cum), 7“”) is associated with the transition of vertical tangency between branches of 5. Specifically, vertical tangency space curve 1‘” is converted to vertical tangency space curve 1(6)due to the (2)/1(6) intersection of the combined space curve 1 with cusp curve 4 1‘ ’. It is useful to note that the 4 points: (Cum, l)), (C120), 2)), (Cum, 4)), (Cum), 6)) are associated with the intersection of actual space curves 111). In contrast, the 2 points (C120), 3)) and (C126), 5)) are only associated with the intersection of the space curve projections 1'“). 94 I | | | I l | | l 2 rr‘(‘; 1 I‘m 1 | r“ I | r"1 I l l I 1 CI? CI? 152’ CI? Magnification is not plotted in scale in order to reveal the order of the F-curves. (1)_ (6) l . r11) l..(6) . . . . . give plane curves - . These, in turn, div1de 2 into regions 21, 2}, £3, 24 and 25 associated with different root transition sequences as C“ is decreased from C“ = 1. The root transition sequences are E, :1 <0, 2>, 22 :3 <0, 2, 0, 2> . Z3©<0,2,0,2,4,2>, 24m<0,2,4,2>, 25w<0,2,4,2,4,2>. Figure 34. The projections of the special space curves 1 onto the (C12, 7)—plane 95 3.9. Multiplicity and Orientation of the Discontinuity Surfaces for C12 = 4 In this section we examine the change in the normal direction of the surfaces of weak discontinuity associated with the well-drilling process at the fixed cross-section of 5 corresponding to C12 = 4. The cross-section of 5 was given in Figure 33, and is chosen here in view of its irregular character. We consider well drillings at 7 = 30, 70, 110, 130, 160 and 200 respectively. These well-drillings are marked by vertical dash lines in Figure 33. Figure 35(a)-(f) present the corresponding normal directions in terms of the angles 01, which are measured in degree in the (X,, X2)-plane from the Xl-axis to the normal direction. This measurement is positive if it is the result of a right hand rotation about X3-axis, and vice versa. Figure 35(a) corresponds to 7 = 30. Here, for C11 S Cfillm, 30) two families of normals 11 form a half loop, which is initiated at C11 = Cfillm, 30) where the normals from these two families have the same direction (double real root of P(x, C 1,, C12, 7)). This behavior persists for all 7 < 54.6295. At 7 = I‘“)(4) = 54.6295 , two new families of normals 11 make appearance and begin to form a separate upper loop near C11 = 0(1)z0.6566. A well-developed (and very thin) upper loop of normals for 9111(4, 70) s (211 s CHM, 70) is shown in Figure 35(b), together with the lower half loop of normals for C11 S Cfinm, 70). Each tip of the upper loop, at Cll = Cfl)(4, 70) and C11 = (ZIP(4, 70) respectively, corresponds to a double real root of P(x, C11, C12, 7). The upper full loop is separate from the lower half loop for 54.6295 < 7 < 127.31. At 7 = l"(3)(4) = 96.5469 , the lower tip of the upper loop reaches the top level (at C1] = Cfillm, 965469)) of the lower half loop, and this situation corresponds to two double real roots of P(x, C 11, C12, 7). For further increased 7 > F(3)(4) , there is an overlap 96 in the C,,-values of the upper loop with the lower half loop for CH)“, 7) S Cll S CHI)“, 7) as shown in Figure 35(c). In this situation P(x, C”, C12, 7) has four real roots. At 7 = P(6)(4) z 127.31 , the further developed upper loop touches the lower half loop at C11 = 0(5)(4) = 0.3025. This contact point is also a double real root of P(x, C,,, C12, 7). For 7 > FI6)(4) , this contact point splits both the upper loop and the lower half loop, and forms two extremum points at Cll = CH)“, 7) and C11 = Cfi)(4, 7) , as can been seen in Figure 35(d). Now in two C,,-intervals: cII)(4, 7) s 011 s (25%)(4, 7) and 0&4, 7) 3 C11 s ("21%”(4, 7) , it follows that P(x, C,,, C,,, 7) again has four real roots. With the further increase of 7, the extremum points at Cll = CH)“, 7) and C11 = CI})(4, 7) approach each other; these two points finally merge at (C1 1, 7) = (9(4)(4) , F(4)(4) ) = (0.2660, 138.57), so that the interval of four real roots QI})(4, 7) < Cll < CWM, 7) vanishes and P(x, C11, C12, 7) has a triple real root. Figure 35(e) displays the normal directions after this merge. At (C,,, 7) = (0(5)(4), F(5)(4)) = (0.3488, 197.87), the remaining two extremum points at C11 = Cfilw, 7) and C11 = Cfi‘)(4, 7) also merge and ends the four real root interval 95%)(4, 7) < Cll < Cfil)(4, 7). This merge point corresponds to another triple root of P(x, C,,, C12, 7). For 7 > F(5)(4) there are, once again, at most 2 discontinuity surfaces. A typical picture of the normal directions for 7 > F(5)(4) is plotted in Figure 35(1). Note in all six subfigures of Figure 35 that the angle 01 is typically close to 190°. This corresponds to fibers which intersect the discontinuity surface at a shallow angle. Recall however from (EIII) that 01 = i90° is not allowed so that the discontinuity surface can not actually be a fiber containing plane. 97 (a) (b) (c) C 1 r , r v v . C 1 r v . fi . a v '. . fl 1 r_ r 1_ r ”,,, C,2=4.0,7=3O, ”“5”” C,2=4.0,7=7Q C11,, E m. C12_-.,4.-021:110, OT 0.?)- 07 0.61 as» 1 M 0.11 01’ .1. (d) (e) (0 Cll'._C_tt‘_'..’f.......,C12'._=...4:QtYT-,‘:._I.3Q Cue-1.; ‘ ' ' ' ' 011' «'3‘... :_, V ‘, b m E (3) 1 11: 11 M’ n :‘3 “‘ ‘ CZ,2=4.0,7=160 : ::C,2=4.0,7=200 ‘ WEQHU as Mr “" 05> 02’ 01» 01> I Figure 35. The evolution of normal direction a of the surfaces of the weak discontinu- ity along well-drillings at fixed (C,2,7). The six separate subfigures all correspond to C,2 = 4.0, and to six separate values of 7. These subfigures correspond to the six dashed lines in Figure 33. 3.10. Orientation of the Discontinuity Surface at First Loss of Ellipticity According to the results of the previous section, the only significant deviation from 01 near 190° will typically only occur near the first loss of ellipticity. Accordingly, in this section, we examine the normal direction of the surface of weak discontinuity when it is first initiated. The triplets (C ,,, C,,, 7) that are associated with this situation are those that form part of the global plane strain ellipticity boundary, across which the first occurrence of loss of GPSE takes place in the deformation process. For the purposes of this section we 98 consider deformation corresponding to the well-drilling process: decrease of C,, at fixed C,,. The set of these triplets, denoted by (8“): C”) = CH)(C12, y) for (C12, 7) e 2, is given by C,, = EH)(C,2,7) for (osc,,) or (c,2>qg> andy> r(c,2)), 62,, = Efi>(c,2,y) for C§5> and 1/2(c,2), (3.10.1) C,, = mummy) for c,,zcg§> and 1/2<'y< 1"“)(C12). It is to be noted that C“) = éii)(C12. y) is discontinuous across l“)(C,2,y). Projecting to the (Cum-plane, this discontinuity occurs across 1'“)(C,2). The double real roots x. to the equation P(x, CH), C12, 7) = O can be obtained by solving the simultaneous equations (3.6.4) using a numerical root searcher at the values CH) as determined by the methodology outlined in section 3.7. The normal directions associated with the triplets given by (3.10.1) are then measured by angles a in the (X,, X2)-plane from the X,-axis to the normal direction, and the angle Otis given by at = tan"(x‘). The angle 0t is positive if it is the result of a right hand rotation about X3-axis, and vice versa. Figure 36, plotted in the (Cum-plane, shows the angle a in degrees for (C12, 7) e 2. All these angles 0t are positive except those for triplets (C,,, C,2, 7) along either C,, = O or 'y = 1/2, where a = 0 as discussed previously. We find that at = 0 as C12 —> co and CL = tan"(C,2) as y -> oo , these results will be verified later in section 3.12. Regardless the rigid body rotation R, the normal direction at can be mapped to the deformed material configuration by using U given in (3.2.10) to obtain a’s deformed counterpart fit. The corresponding angles (it are plotted in Figure 37. Recall that the deformed fiber direction 4) is given by (3.2.11). For triplets (C,,, C,,, 7) on E“) given by (3.10.1), one obtains directly from (3.2.11) that ¢ is always positive. 99 The positive direction of (p is similarly defined as that for 0t. Figure 38 plots the angles «1 and Figure 39 plots the angles (fit-(p), that is, the normal direction of the surface of weak discontinuity with respect to deformed fiber. It is to be noted that 6: -¢, being a difference of directions in the deformed configuration is, in fact, independent of the choice of F=RU used for obtaining the deformed counterpart. An interesting question concerns the maximum value of (it-4) on the global plane strain ellipticity boundary fl. Figure 39, for example, shows a 27° contour. Here it is important to realize that the values given on Figure 39 are associated with only the portion of 3 corresponding to the largest C,, at fixed (C,,,y). For comparison, 2P=(0.3883,2,17 .0815) which locates the unique quadruple root of (3.6.4), is on flbut is not on 3, and so is not represented on Figure 36-Figure 39. We find at T=(O.3883,2,17.0815) that x = 2.5754, so that ét-d) = 24.9°. Ann ‘W I l 180- 140— -23 120~_1o 80- 60- <40 40- 20~ H“_" _ C12 Figure 36. Constant value contours of angle at (in degree) associated with first contact of $by decreasing C,, from 1 at fixed (Cu, 7). This gives the orientation of the discon- tinuity surface in the reference configuration. The contours are discontinuous across 1"“), the projection of the 1“) vertical tangency curve associated with the termination of the upper branch of $. 101 Ann ‘uu l __50 120— so 7100— so- 60" “t 40 - 2°11 A V O 1 C12 Figure 37. Constant value contours of angle (it (in degree) associated with first contact of Qby decreasing C 1, from 1 at fixed (Cu, 7). This gives the orientation of the discon- tinuity surface in the deformed configuration, where the deformation is determined on the basis of the pure part of the polar decomposition. Ann ‘W I l 180- __40 80- [ 60* 40— 80 201 J JU“ '1 2 v0 3 7 Figure 38. Constant value contours of angle :1) (in degree) associated with first contact of 3 by decreasing C,, from 1 at fixed (Cu, 7). This gives the orientation of the deformed fibers in the deformed configuration, where the deformation is determined on the basis of the pure part of the polar decomposition. 102 nnn ‘W 7 I l I I I I Figure 39. Constant value contours of angle 6t - 4) (in degree) associated with first con- tact of $ by decreasing C,, from 1 at fixed (C ,2, y). This gives the angle between the deformed fiber direction and the normal to the discontinuity surface in the deformed configuration. This difference angle is independent of the choice of F used for describ- ing the deformed configuration. 3.11. Cross Sections of 5 at Fixed 7 Six cross sections of the manifold 5 at fixed 7 are plotted here to illustrate the evolution of 5 with increasing 7. The six values of 'x 5, 10, 20, 50, 100, 200 were chosen for this purpose. Since 7 is the sole parameter defining the material, each cross-section gives the loss of ellipticity boundary for a fixed material. It is to be noted that S 7“” 5 24.3891 . The unfolding of Shas further developed. In particular the former ver- tical tangency point 1(2) has transferred to a different branch in the guise of point 1“” I I I I C, 1(1)(C12’ Y) 0.9 '- O.8 '- O.7 " (l) 0.6 P (6) 110) C11 0-5 ' (C12, Y) 0.4 - 1(5) 1(3) 11(2) 0.3 r (C12, Y) W 1(4) ll(3) 0.2 ~ (C12. Y)/(1 E11(2l)((-212v Y) Figure 44. Another typical cross section of 5 for 7 values obeying y > 7“”. The unfold- ing of 5 is now well developed. A value of y at this order of magnitude is not uncom- mon in fiber reinforced materials. 106 0.9 b 0.8 — 0.7 - 0.6 - C,, 0.5 — 0.4 )- o.3 — 0.2 - I — l I I (311‘ )(C12, Y) F200 6 l() ](5) l(3 (2) W -1 Cl] (C12, 7) / 1(4) (3) C,, (C12, Y) 4 C12 Figure 45. Cross section of 5 for even larger ‘7. For this 7 value and practical values of C,2, loss of ellipticity takes place at C,, very close to 1. Although the manifold 5 as given by the values C,,(Clz, 'y) for (C12, 7) e E are 3.12. Asymptotic Properties of the Global Plane Strain Ellipticity Boundary not determined explicitly by the numerical procedure that generated Figure 25-Figure 33 and Figure 40-Figure 45, the asymptotic form of 5 can be found by an explicit analysis of (3.6.4) using (3.5.1)-(3.5.4) as we now show. With respect to the octant C1] > 0 , C12 2 0 , y 20 we have in section 3.7 shown that 5 is given by C11 = 1 — 1/27 when C,,, = 0, whereas in EII it was shown that 5 does not intersect y = 0. Here we complete the analysis of the boundaries of this octant by considering the separate special cases Cll —-> 0 , C12 —-> co and 'y —) 00 (since El ensures ellipticity for C11 2 0 there is no need to consider (i) C,, = 0. In this case, equation (3.5.4) in conjunction with (3.5. 1) becomes 107 (x2-2'Y+ l)(C122 +1) = 0. (3.12.1) The global plane strain ellipticity requirement that (3.5.4) has no real root is met if and only if «é, when C,1 = 0. (3.12.2) This result is not in contradiction to the previous result discussed in (EU), since there we consider only C,,>0. This confirms that the global plane strain ellipticity boundary encounters the plane C ,, = 0 along the line 7: 1/2, where equation (3.12.1) has double real root x = 0. (ii) C12 —) 00 at fixed 7. To analyze the asymptotic form of 5 as C12 —) co, we rewrite equations (3.6.4) in full as P(x,C,,,C,2,y) = C122{x2+27(3C11 - 1)+ 1} — C,2{2C1,x[x2+2y(3C11 — 1) + 1]} (3.12.3) +(C§,x2+1){x2+2y(3c,,—1)+1}—4yc,, = o, and a —P(x,C ,c ,y)=2CZx—c {2c [x2+2y(3c —1)+1]+4c x2} ax ll 12 12 12 11 11 11 (3.12.4) 2C12,x[x2+2'y(3C,, — 1) + 1] +2x(c§,x2+ 1) = o. Neglecting O(C,2) in (3.12.3) and (3.12.4) gives that the dominant balance equations as C12-)oo are x2+27(3C”— 1)+1 = 0 and x = 0. (3.12.5) Thus one obtains that 108 For the next order correction, let 1 1 C11 = 3(1— 27) + C13,. (3.12.7) Expanding the left hand sides of equations P(x, %(1 --2-i-Y)+C{51,C12,7) = 0 and $1309 3(1 - 51;) + Ci), C1217) = 0 in the Taylor’s series to the first order near x = 0 and Cf), = 0, gives 27—1—9C727Cf, = 0, (3.12.8) Equation (3.12.8), yields c9, = 53.5- , which, in conjunction with (3.12.7) and (3.12.8)2 12 gives 1 1 27—1 _ 27-1)2 - C =-(1-—)+ +oC2 and x=——+oC3 3.12.9 11 3 2’Y 9YC122 ( 12) 9YC?2 ( 12) ( ) as Cl2 —> 00. This also verifies the consistency of the procedure and delivers the horizontal asymptotes seen in Figure 40-Figure 45. Notice that x —> 0 as C12 —) 00 implies at —> 0 which is consistent with Figure 36. The leading order behavior of C,, as C12 —> 00 yields the same asymptote as that given by (3.6.1) which, not surprisingly, followed from the direct consideration of x=0. (iii) 7 -) 00 at fixed C,,. For this case, we rewrite equation (3.6.4) in full as (3.12.10) +(x2+1)[(C11x-C,2)2+1] = 0 , 109 and 8 a_xP(X, C11, C12, 7) = Y{4(3C11' 1)C11(C11X ‘C12)} (3.12.11) +2x[(C,,x - C12)2 + l] + 2(x2 +1)C1,(C,,x — C12) = 0. Neglecting C(70), one obtains the dominant balance equations (3C11‘ l)(C,,x-C,2)2 +C11-1 = 0 , (3.12.12) (3C11- 1)(C1,x-C,2) = 0. Equation (3. 12.12) indicates that C11~1 and x~C12 as 7—)oo. (3.12.13) For the next order correction, let C11 = 1+Ci31 and x = C12+xA. (3.12.14) Expanding the left hand sides of equations P(C,2+xA, 1 +Cf,,C,2, 7) = 0 and 8 ENC” + xA, 1 + C9,, C12, 7) = 0 in a Taylor’s series to the first order near xA = 0 and C9, = 0, gives 1+C122 +2C12xA+27Cf1 = 0 , (3.12.15) Solving equations (3.12.15)” one obtains 1 + C2 C 1 + 2C2 C13] = _ 2712 + 0(7'1) and xA = ‘2( 4y ‘2) + o(7‘1), (3.12.16) as 7 —> 00. This verifies the consistency of the procedure, which, in summary, gives C12(1+ 2C122) 1+Cfi 1 27 +o(7‘1) (3.12.17) 110 as 7 —> 00. Hence 01 —> tan ‘(C,2) as 7 —> 00, corresponding to the eventual straight vertical nature of the constant (1 curves in Figure 36. The asymptote Cll ~ 1 as 7 —> oo is consistent with Figure 25-Figure 33. Since x does not approach 0, the asymptote C11 ~ 1 does not match the 7 —) oo asymptotic value for C ,, in Figure 23. 3.13. Ellipticity under Special Deformations We now turn to examine the possible loss of global plane strain ellipticity under certain plane deformations. Simple Shear For simple shear in the fiber direction, the deformation gradient tensor is given by (2.3.12) with i = 1 and j = 2. The right Cauchy-Green strain tensor is then given by l k 0 C = k1+k2 0 , (3.13.1) 0 0 1 where k is the amount of shear. We have, under this deformation, that C,, = l and C,2 = k. Parametrized in terms of k, the loading path of this deformation can be plotted in the (C ,,, C,2)-plane, as shown in Figure 46. Hence, the material (2.4.1) is elliptic with regard to both the global and the local plane strain criteria. For simple shear perpendicular to the fiber direction, the deformation gradient tensor is given by (2.3.12) with i = 2 and j = 1, and the right Cauchy-Green strain tensor is given by lll 1+k2k0 C: k ,0, (3.13.2) 0 01 where, again, k is the amount of shear. Since under this deformation we have Cll = 1+k221, the material (2.4.1) is, again, elliptic, for both types of plane deformations, i.e., global plane strain deformation or local plane strain deformation. Parametrized, again, in terms of k, the loading path of this simple shear perpendicular to the fiber direction is plotted also in Figure 46. C11‘ Simple shear perpendicular to fiber direction Simple shear in fiber direction ’ Biaxial deformation for decreasing C ,, 1 Figure 46. Loading paths for some plane deformations. Biaxial Defamation For incompressible materials, a plane deformation (2.3.7) in the (X,, X2)-plane is a biaxial deformation if C,, = 0. Following directly from (2.3.10) with the incompressibility constraint (2.3.11), and noting that 71., = CH2 , the right Cauchy-Green strain tensor for this biaxial deformation is rewritten as -C,, 0 11 C = 0 Ch] 0 . (3.13.3) _0 O l As plotted in Figure 46, the loading path for this deformation goes vertically down with decreasing C,,. According to the discussion in section 3.7, loss of ellipticity (both local and global) takes place, if 7 > 1/2, as soon as C11 5 1- 1/27. Since Cll = A? this corresponds to 2., S m. It is to be noted from Figure 47 that there is no obvious correlation of loss of ellipticity with the loss of monotonicity phenomena associated with materials obeying 7> 14.95 as discussed in (ii) of section 2.4.1. flaw, 0.9831) 0.8 r A, 0.6 - 3.," (14.9504, 0.5676) 0.4 *- 0 l a l 1 1 L l 1 l l —1 O O 1 O 20 30 4O 50 60 7O 80 90 0.5 14.9504 7 Figure 47. The loss of ellipticity region in biaxial deformation with C,, < 1 (where C,, = if and 2c, is the principal stretch in the fiber direction) is given by k, < m. The principal stretches associated with this loss of ellipticity phe- nomena do not correlate in any obvious way with nonmonotonic stress response for this deformation. The nonmonotonic stress response, in terms of 172mm and 172mm was used previously in Figure 6. 113 General simple shear deformation in the (X ,,X2)-plane We now turn to examine the ellipticity under the more general simple shear deformation discussed in section 2.3. It was shown in that section, loss of monotonicity in the shear response curves could be addressed in terms of fiber contraction/relaxation/elon- gation, as for example shown in Figure 17, which diagramed C,, vs. k, where C,, gives the fiber stretch and k is the amount of simple shear. To address any relation that this may have with loss of ellipticity it is useful to recast Figure 17 as a diagram of C,, vs. C,,, since this is the deformation space in which we have determined loss of ellipticity phenomena (in terms of 7). One finds that Figure 17 correlates with Figure 48. In Figure 48, the \V = 0 curve corresponds to simple shear in the fiber direction and the t1! = 1tl2 curve is the k —> —k image of the curve for simple shear perpendicular to the fiber direction as shown previously in Figure 46. As discussed above, neither of these curves involves C,, < 1. Note from Figure 48 that if 0 < w < 1t/ 4 , then C,, first increases and then decreases with k, whereas Cu is monotonically decreasing with k if 1t/ 4 < 1|! < 1t/ 2. One finds that C,, is again zero for k = 2cot2tv , corresponding to fiber rotation (1) = —(1t/2 — 2w) . 114 fiber length I Fiber rotates by -1U2 k Fiber rotates 0-5 by -1r/4 Minimum fiber length ‘1’ = 0.02861: —3 —2 —1 o 1 2 3 C12 Figure 48. Loading paths for simple shear in various directions with respect to fiber direction (\v is in radian). The fiber is rotated by -1r/4 as k = l/ ( sin 2w + coswsinw) , and the fiber is rotated by -1t/2 as k = 1/( coswsinw) . The minimum deformed fiber length occurs at k = cotw, corresponding to Cll = sinztv and d) = —(1c/2-\|I). Hence the fiber relaxes back to its original length for cotw < k < 2mm: correspond- ingto -(1t/2—w)<¢<—1t+2\y. Since the general simple shear curves for 0 < \y < 1t/ 2 involve a regime of fiber contraction for moderate k, it follows that loss of ellipticity may occur. Figure 49 superposes cross sections of global plane strain ellipticity boundary for various 7 in preparation for comparisons with Figure 48. Note in this figure that the cross sections of global plane strain ellipticity boundary are extended into negative values of C,,. Figure 50 then combines general simple shear curves of Figure 48 with the cross section of global plane strain ellipticity boundary for the special value 7 = 10. As illustrated in Figure 50, loss of global ellipticity will not occur for 0.42611: < 111 S 1t/ 2 as well as \V = 0. Loss of 115 ellipticity and recovery of ellipticity, in turn, will take place with increasing k for 0< \y S 0.42611t. Furthermore, loss of ellipticity-recovery of ellipticity will take place twice for 0.01681t S w < 0.04231t. 1.5 1 l 1 l o l -3 -2 —1 0 1 2 3 Figure 49. The cross sections of the global plane strain ellipticity boundary for 7: l, 5, 10, 20, 50 and 100. Here, unlike Figure 40-Figure 45, the global plane strain ellipticity boundary is shown for both positive and negative C,2 in order to aid comparison with Figure 48. 116 1.5 . ' ' I I . Rt *1 *1 \V : C11=0-947 C,2=-O.212 1 - \y = 0 ; 111k\\ C11=0-906 C12=0.864 I w = 0.01681: C11 W ‘7: 10 0.5 — \ - c,,=0.41 : C,2=1.562 n, = 0.04231: k “ n / _3 —2 ‘1 0 1 2 3 Figure 50. Loading paths for simple shear in various directions with respect to fiber direction (\V is in radian). The cross section of 5 for 7: 10 is also plotted here. For 7 = 10 material, simple shear with 0< W 5 1.3385 involves loss of global plane strain ellipticity. For this 7 = 10 material, Figure 51 gives the shear stress response function for some of the original fiber orientation ‘1’ shown in Figure 50. In this respect the figure is similar to Figure 20 of section 2.4. However, now the loss of ellipticity portions of the response curves are shown in dot lines. It is clear that loss of ellipticity has no explicit correlation with loss of monotonicity of the stress response. Finally, Figure 52 is the counterpart to Figure 21 in that it displays the locations in the (C,,, C,2)-plane corresponding to km, and km,n for the 7 = 10 material, together with the deformation paths and the loss of ellipticity region (in shading). 117 6 I I I l I l T I 1 17:31:18 5 1- 4 — .— 3 - _ l p, ere: '- 2 - ' (b - ...... “Hr/8 1 ‘ _.(b (a) ‘ 0 (a) ' ‘ " _ 0: point (a), minimum fiber length x: Point (b), original fiber length _1 l 1 l l l l l l l 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 51. Shear stress Te,e2/” in (2.4.51) ((2.4.47)) varies with simple shear k for a material with 7 = 10 and various fiber orientations ‘1’ with respect to the direction of shearing. As in Figure 20, the fibers contract from the origin to the point marked (a), and then relax back to their original length between points (a) and (b). From point (b) onwards the fibers continue their elongation. In this figure the region of the curve cor- responding to loss of ellipticity is shown in dotted line. 118 ‘1’=0 —>k Local maximum to T for 7:10 Local minimum e,e, to T for 7:10 e,e2 0.5 -3 —2 —1 o 1 i 2 3 C12 Figure 52. Loading paths for simple shear in various directions with respect to fiber direction (v is in radian). Points on the loading paths associated with 1:1,,“ and km,“ are also plotted (see Figure 21), as is the loss of global plane strain ellipticity region (in shadow). 4. Summary and Conclusion A model for incompressible finitely deformable materials with fiber reinforce- ments is established. The engineering significance of this material model is verified since it fits well with real material data. The fully nonlinear responses of this material is discussed for a number of important deformations, and the loss of monotonicity of the material response is studied. The correlation to the linear theory is provided in terms of the elastic constants. The ellipticity of this anisotropic material and the plate bifurcation problem are further studied. It is revealed that the loss of ellipticity phenomena under planar deformation involve different patterns, depending on loading path. These patterns involve simple loss of ellipticity, loss-recover-loss of ellipticity, and primary loss-secondary loss of ellipticity. The geometry of the ellipticity boundary is made clear. The discussion on the directions of the weak discontinuity surfaces is also presented. Finally, it is concluded that there is no explicit correlation between loss of ellipticity and loss of monotonicity of the stress response, and no explicit correlation between 1088 of ellipticity and bifurcation. The buckling problem presented in Appendix B is, by its own right, an interesting research topic, including the research aspects as: complete buckled solutions for bifurcation conditions (B.l), (B.l) and (B.l), the number of the buckled solutions, the asymptotic behaviors of the buckled solutions for large values of parameters 7 and n, as well as the ordering of the buckling loads. 119 Appendix A: Algebraic Root Analysis of the Characteristic Polynomial Since the characteristic polynomial (3.5.1) is of degree 4 in x, it follows that its roots can be expressed in terms of radicals of the polynomial coefficients. In this appendix we present certain aspects behind this process that enable us to determine properties of 5, which, we recall, is the locus in the (C ,,, C,,, 7)-space associated with double real roots to the characteristic polynomial. A.1 General Algebraic Procedures upon the Quartic Polynomial By applying the linear transformation of the independent variables _ 1 _ C12 A1) X'Y‘Zal-Y+2C11, (. and eliminating the common factor C, ,2 (C,,>0 (3.2.5),), the polynomial P(x, C,,, C,,, 7) given in (3.5.2), (3.5.3) is written as 130', C11, C1217) E3'4'1'l)2y2+b3y‘l'baa (A-I) where _ 3 2 b2=b2(C11’C12) = ‘g 1"”32’ 1 b3ab3(C,,,C,2, 7) = -é~a?+§ala2—a3, (A.1) _ 3 4 1 2 1 This linear transformation will not change the nature of the roots of P(x, C,,, C,2, 7). That is, if y is a complex (real) root of equation P(Y’ C119 C12, 7) = 0 (A1) 120 121 then x given by (A.1) is a complex (real) root of P(x, C,,, C12, 7), and if y is a double root, then x given by (A.1) is also a double root, etc., and vice versa. If b, = 0, polynomial (Al) is factored as - 1 1 1 1 for b3 = 0. Its four roots are given by l ’1 1 ’1 yl = J-ib2+ zbg-b4 , Y2 —J—§b2+ 21312—134 , 1 1 1 1 Y3 = J—5b2-,/Zb§-b . Y4 = "J-ibz-szi-b4- If b3 at 0, then, following the standard solution procedure (refer to [Kurosh 1980]), the (A.l) four roots of polynomial (A.1) are given by - 1 .. - 1 - y15y1(z) = Edit/A7). y22y2(z) = Edi-E). (A.1) a (z) = —(-Ji+./A ). a (z) = —(-fi-./4 ). Y3 Y3 fl 2 Y4 Y4 J5 2 where A (“') " b b3 A (“) “ b b3 (A 1) 1 2 22 2 2 .52 Here 2 is any one root obtained from equation 3 b 2 1b2 b 1b2 - 0 A1 The three roots of equation (A.l), denoted by z,, z, and z,, can be obtained again by 122 following a standard procedure for cubic polynomials as discussed in [Kurosh 1980], for example. Equation (A.1) has a root 2 = 0 if and only if b, = 0, and in this case {y,, y2, y3, y4} are given by (A. l). A.2 Criteria for Double Real Roots It follows from the theory of cubic equations [Kurosh 1980 p228] that equation (A. 1) has a double (real) root if and only if 0(C,,,C12, 7)E27§,2+4§§ = 0, (A.1) where b3 bzb4 bi bi §](C119C129 Y) = -F)-8-+—§—--8_ 9 §2(C11,C12,Y) = -'1—2'-b4 . (A.1) It is found that equation (A.1) has double real root if and only if equation (A. 1) has double root. In fact, if z, and 22 are distinct roots of (A. 1), then examination of the expressions for the roots of (A.l) as given by say (A.l), reveals that y,(z,) ¢ y,(zz) , i = 1,2,3,4. Since {y,(zt). ya(zi). ya(zt). y4(zi)} and {y,(za). yam). yam). y4(ze)} give the same set of four roots of (A.l), there must be an 1-1 correspondence between these two sets. Without loss of generality, assume that y,(z,) = y3(zz). Now, consider 2, = 22 being double roots of (A. 1), then y,(z,) = y3(z,) are double roots of (A. 1). Furthermore, if the double roots y,(z,) = y,(z,) are complex, then it is required that y,(z,) = y,(z,) and are the complex conjugate. According to the previous discussion, equation (A.1) can only has two pairs of imaginary double roots, and this case is prohibited by (A.l). Inversely, if y,(z,) = y3(z,) are double real roots, then by the 1-1 correspondence we have y,(z,) = y,(zz). It is then follows from (A.1) that y3(z,) = y3(74) requires z, = 22 be double roots of (A.l). this procedure can be 123 exhausted for all the possibilities of the 1-1 correspondence to prove that equation (A.1) has double real root if and only if equation (A! ) has double root. Consequently, 5 is completely defined by the triplets that satisfy (A.1) which determines a two-dimensional manifold in the (C u, C,2, 7)-space. A.3 Criteria for Two Pairs of Double Real Roots We now examine the situation at which (A.l) has two pairs of double roots. By expanding (y-Y,)2(y-Y2)2 and comparing the coefficients of the powers of y in the expansion with that in (A.l), one finds that the following simultaneous equations (A.1) must be satisfied. Equation (A.l), shows that Y, and Y, are either both real or both imaginary. Examination of b2, which is real, according to (A.l)z under the condition that b, = 0 reveals that b21b3=o>° if0$C12<2, b <0 'fC >2 (A'l) 2|b3=0— 1 12- ° This result, in conjunction with (A.l), shows that if C12 2 2 , then a (Y,, Y2) double root pair involves real Y, and real Y2 whereas if C ,2<2 then a (Y,, Y2) double root pair involves imaginary Y, and imaginary Y2 and so need not concern us further. Hence for C12 2 2 , equation (A.l),”, gives the subset of 5 associated with two pairs of real double roots. This one-dimensional manifold, denoted by 1‘”, locates intersection points of 5 with itself (Figure 29-Figure 33). 124 A.4 Criteria for a Triple Roots and a Quadruple Roots A similar procedure can be execute to prove that equation (11.1) has triple (real) root if and only if equation (11.1) has triple root. Equation (A.1) has triple (real) root if and only if The subset of 5 associated with triple (real) root of (A.l) are given by the simultaneous satisfaction of both of (A. 1). Space curves along which (A. 1) has triple roots are 1‘” and 1(5) and are the cusp points in Figure 29-Figure 32. All 1‘”, 1(4) and 1(5) initiate at the unique point where (A.1) has quadruple roots. This requires that b2 = b, = b, = 0. This set of algebra equations can be solved to give C,, = C1212) = 2, and give the associated C,, and 7 by lengthy exact expressions. Together this gives the quadruple root x = 2.57539, at (C,,, C12, 7) = (0.3883, 2, 17.0815) E {P as the 0—dimensional intersection of the three l-dimensional manifolds 1(3), 1(4) and 1‘”. Following the discussion here we know that to find those triplet (C,,, C,,, 7) at which (3.5.4) has double real roots and root type transition takes place is equivalently to find the triplet (C,,, C,,, 7) that satisfies equation (A. l). Appendix B: Bifurcation from Homogeneous Deformation under End Thrust Buckling is a major failure mode for load—carrying engineering structure. Mathematically, buckling corresponds to bifurcation from some equilibrium solution. In [Haughton 1987], the co-occurrence of loss of ellipticity and in-plane inhomogeneous bifurcation was observed. This motivate us to seek the correlation between loss of ellipticity and out-of plane bifurcation here. The bifurcation analysis will be carried out in the context of the theory of small deformation superposed onto finite homogeneous deformation [Biot 1965]. The equivalent problem for the neo-Hookean plate (which therefore does not admit loss of ellipticity) was studied by Sawyers and Rivlin [1974]. They found a family of flexural buckled solutions at relatively low end-thrust and a family of barrelling buckled solutions at relatively high end-thrust. The extension to a layered neo-Hookean construction was given by Pence and Song [1991] and Qiu, et. al. [1994]. In particular, the latter work established the extension of additional families of buckled solutions, and the conjecture was put firth that an N-ply construction would admit N + 1 families of plane strain buckled solutions. Here, the composite construction involves directional reinforcing, rather than multi-layers. For the purpose of comparison, plane deformation inhomogeneous bifurcation onset from the global plane deformation (2.3.10) (or (3.13.3)) of a thick rectangular plate under end thrust will be investigated here. This plate is constructed from the material (2.4.1) reinforced with one single family of fibers oriented in X,-direction. The associated ellipticity problem was investigated in section 3.7 for C,2 = 0 and section 3.13. First addressed will be the finite homogeneous deformations, figured out by using 125 126 semi-inverse method, of this plate under end thrust. Next, the inhomogeneous bifurcation phenomena studied. B.l Homogeneous Plane Deformation The thick plate in its original configuration, placed in the 3-dimensional rectangular Cartesian coordinate system, occupies the region 11((X,, X2, X3). The region 1«Xi. X2, X3) is confined by R: '115X1511’ 42529512. -l3SX3513, (13,1) with 12 significantly smaller than either 1, or 13. Illustrated earlier in Figure l (M = 1) is the original configuration of the plate. The following traction boundary conditions will be applied on the plate and furnish the so called end thrust condition. We define the lateral surfaces as the surfaces originally at X, = i1, and X3 = :13, so that the top and bottom surfaces of the thick plate are the surfaces originally at X2 = :12. By thick we mean that the deformation along the plate thickness need not relate to each other (or to a central neutral surface) by some simple prescribed relation through thickness, so that a fully three dimensional continuum analysis is required through the plate thickness. By end thrust we mean a total normal force T in the X,-direction will be applied on the pair of lateral surfaces at X1 = 3:11 , and no shear tractions will be applied on this pair of surfaces. The top and bottom surfaces of the plate are kept traction free during deformation. In solving the problem, normal displacements will first be imposed on the pair of lateral surfaces at X1 = ill. The traction required to maintain these displacements will then be calculated following the semi-inverse method. Deformations are assumed to take place in planes 127 parallel to the (X,, X2)-plane globally. Hence, the surfaces at X3 = :13 are kept frictionlessly in their original planes. Thus, and to be more specific, we have the boundary conditions, as x, = ih,l,, on X, = i1, , 8,2 = S,3 = 0, on X, = 21:1,, S21 = S22 = 823 = 0, 0“ X2 = 31:12 . (31) S3, = S32 = 0, on X3 = :l:l3 , x3 =il3, onX3 =il3. Here S is the Piola-Kirchhoff stress tensor. As follows from (2.4.21) and (2.4.22), that the particular biaxial deformation (2.3.10) can be sustained with in-plane tractions on lateral surfaces. Since in the constitutive equation (2.2.1), for the material (2.1.10) considered in the context, the hydrostatic pressure p is arbitrary, is assumed to be constant in position X for homogeneous deformations. It is then clear from equation (2.2.1) that the Cauchy stress tensor T, and the corresponding Piola-Kirchhoff stress tensor S, is independent of position X if the deformation is homogeneous. In this case, the equilibrium equation divST = 0 (BI) is satisfied. Hence, in all the discussions on homogeneous deformations, equilibrium is ensured. For the material (2.4.1) reinforced with single fiber family, the Cauchy stress tensor is given by (2.4.2). The in-plane fiber reinforcement is represented by the unit vector field A(= Am) given by (2.43),. The potentially nonzero components of the Cauchy stress tensor T are given by (2.4.21). The hydrostatic pressure p in equation (2.4.2) has 128 been eliminated by equating T22 to zero in accordance with the traction free condition (B.l), on the surfaces X2 = :12. The potentially nonzero components of the Piola-kirchhoff stress tensor, calculated from S = F‘1T , are given by S,, = 1101., 4,3) +213().,2— 1)).,, (Bl) S33 = u(l-)t,‘2). As required, 8,, = 7122.3T,, = lf‘T“ and S33 = ).,71.2T33 = T33, where T,, and T3, are given in (2.4.21) and )0, =1, 71., = 1,1. It is then evident that the boundary conditions (B.1) can be satisfied by the homogeneous deformation (2.3.10) and the corresponding Piola-kirchhoff stress tensor (B.1). Thus, deformation (2.3.10) with 2,2 = 71.," is the homogeneous solution to the problem posed by the equilibrium equation (BI) and the boundary conditions (B.l) for the plate construction with fiber reinforcement oriented in the X,-direction. The end thrust T applied on surfaces originally at X, = i], can now be figured out as T = 41112130., — M3) + 81312131th 1,). (B.1) To obtain equation (B.l), the current area, A = 42,3213, of the surfaces on which the end thrust is exerted is applied to (B.l),. B.2 Incremental Formulation We now turn to consider the out-of-plane inhomogeneous bifurcation problem. The homogeneous deformation solution that takes place in planes parallel to the (X,, X2)-plane was given in the previous section. The small incremental deformation to be 129 superposed onto finite homogeneous deformation is assumed to be a plane deformation that takes place in planes parallel to the (X,, X2)-plane. Thus, the full deformation, by superposing a small incremental deformation 11 onto the homogeneous deformation described by (2.3.10), can be written as x, = 11X, +cu,(X,, X2), 122 = MIX, + eu2(X,, X2), (8.1) 23 = X3 , where e is a small parameter that will be used to obtain the linearized formulation that governs bifurcation onset. In the foregoing discussion, quantities associated with the full deformation (B.1) are indicated by a superposed “A”, and linearized quantities associated with the incremental deformation u,(X,, X2) and u,(X,, X2) are indicated by a superposed “-”. It is only necessary to take into account the linearized quantities, since our interest is bifurcation onset and not the post bifurcation behaviors. Thus, the corresponding incremental quantities, such as deformation gradient tensor F , hydrostatic pressure p , the Cauchy stress tensor '1‘ and the Piola-Kirchhoff stress tensor S can be calculated as 2., + cu,,, 811,,2 O f“ = sum 71.,‘1+8u2’2 O . 03-1) 0 0 1 f)(X,8) = p+8p(X)+O(£2), (B.1) T(X,e) = T+8T(X)+O(82), (B.1) S(X,£) = S+cS(X)+O(£2), (13.1) respectively. Now, we have the determinant of the deformation gradient tensor F 130 detF = l+e(}t,u2’2+}t,‘1u,’,)+O(82). (B.1) Hence, for the linearized problem, incompressibility requires that 1,112,2+A,'lu,,, = 0. (B.1) According to equation (B.l), the components of linearized incremental Piola—kirchhoff stress tensor are given by §21 = ”(111,2'1'112‘12, 1) 9 - _ (B-l) S22 = — MP + 211%, 2 , §33 = -I3 . Substituting the linearized incremental Piola-Kirchhoff stress tensor into the equilibrium equation divS = 0 , a set of three second order partial differential equations, -53 = O . for the unknown functions u,, u, and E, can be obtained. Equation (B. 1), yields 13(X,.x2.x3) = 13(X,.x2). (13.1) That is, the hydrostatic pressure p is a function of X, and X, only. Equation (B.l),z, together with the constraint of incompressibility (B.l), are a set of differential equations governing the bifurcation onset for the plate with reinforcement oriented in the direction of the end thrust. The boundary conditions are posed in (B. 1). 131 B.3 Inhomogeneous Bifurcation In the foregoing analysis, we seek nontrivial solutions (corresponding to bifurcation) for the incremental deformation u,(X,, X2) and u2(X,, X2). By using the method of separation of variables, the set of partial governing differential equations (B.l),; and (BI) can be turned into M‘QHXZ) + [u + 215(32t,2 — 1)]92U,(X2) — uU,”(X2) = 0 , 1,P'(X,) + 111 + 2130»?— 1)192U2(x2)- uU2”(X2) = 0. (13.1) Here the solution form proposed for the set of partial differential equations (B.l),; and (BI) is —sinQX, cosQX, cosQX, u, = . stX, cosQX, U1(X2)’ u2 " ._ U X , -= sinQX,} 2( 2) p }P(X2), (B.1) where Q=j1t/l, for upper case of (B.l), Q=(j—l/2)1tll, for lower case of (B.l), and j = 1,2,... gives the number of half wavelengths (mode number) in the buckled solution. Eliminating P(x,) and U,(X2) from equation (B.l), gives the fourth order ordinary differential equation Ugilez) —§22h,(7t, 7)U2”(X2) +Q4h2(k, 7)U2(X2) = 0, (Bl) where l=7tfz>0, 7c>0 (B.1) is introduced as a load parameter. Here, the load parameter 2,. describes the amount of homogeneous deformation through (2.3.10) and (BI) under given load T, and the stiffness ratio 7 characterizes the increased stiffness in the fiber direction. The two functions, h,()u, 132 Y) and h2(7~. Y). given by h,(7t, 7) = 1+7cz+27(%— I) and h2(7u, 7) = t2[1+27(%—1)], (3.1) are by virtue h,0~.7) = 32(%.0.Y) and 1120.7) = a,(,{.o.v) (13.1) in comparing with (3.5.3)“, by noting that 71. = 1/C,,. It is worthwhile noting that the differential system (B.l) (or (B.1) virtually) is a two parameter (7., 7) system. The nature of this system varies with the parameter pair (7., 7) in the domain 7». > 0 , 7 2 0. That is, equation (B.1) will change its type, such as elliptic and nonelliptic, depending on the coefficients h,(7t., 7) and h2(7., 7). The characteristic equation of (B.1) for the proposed solution form U2(X2) = erxz for unknown r is r4-(22h,(7t, 7)r2+§24h2(7., 7) = 0. (Bl) Equation is exactly that given by C,,'ZQT(d2‘li, 1/7», 0, 7) = 0 from (3.5.2), where i = J: . We shall, here, denote the four roots of the characteristic equation (B. 1) by ir, and irz. Thus, we obtain rf = (2fo , r% = szg. (B.1) Here no.7) = izih,+(h,2-4h2)“21“2. 50.7) = ith,-(h,2-4h,)“21“2. (13.1) I J5 It can be proved that 133 r,2>0, r§>0 if h2(7.,7)>0, r2>0, r§<0 if h2(7.,7)<0, (B.1) r,2>0, r§=0 if h2(7.,7)=0. Figure 53. The solution of the fourth order ordinary differential equation (B.l) changes its type with (7.,7) according to the value of h2(h., ). This deformation is biaxial with A. = llC,,. Ellipticity holds in the region h; > 0 and is lost in the region hz < 0. It is to be noted that h2(7t, 7) = 0 yields the same relation as that given by (3.3.4), by noting that 7» = 1/C,,. The graphs of h,(7., 7) and h2(7t, 7) are plotted in Figure 53. The curve of h2(7,., 7) = 0 becomes the same curve as plotted in Figure 25, if we exchange the ordinate with abscissa and make use of the substitution 7c = 1/C,,. Hence, for h2(7., 7)>0, the differential equation (B. 1) is elliptic type, and for h2(7t, 7)<0 or h2(71., 7) = 0, we shall say that equation (BI) is non-elliptic type (a parallel discussion is given in Appendix C). The general solution of the homogeneous equation (BI) is the linear combination of four linear independent solutions 134 u,(x2) = L,U§1)(X2)+ L2U§2)(X2) + M,U§3>(x,) + M2U§4)(X2), (B. 1) where U2(i)(X2) (i = 1, 2, 3, 4) are determined by the roots of the corresponding characteristic equation (B. l) and their forms change accordingly. The solution (B.l) of the fourth order ordinary differential equation (B.1) can be written, according to the value of h2(7c, 7), as U2(X2) = L, cosh(r,X2) + Lzsinh(r,X2) + M, cosh(r2X2) + Mzsinh(r2X2) , if h2(7., 7) > 0, (3‘1) U2(X2) = L, cosh(r,X2) + Lzsinh(r,X2) + M, cos(Im(r2)X2) + Mzsin(Im(r2)X,%3 .1 u,(xz) = L, cosh(r,X2) + Lzsinh(r,X2) + M, + szz , if h,(>., 7) = 0. (3'1) Thus, the existence of the incremental solution u,, u, and 5, and therefore the existence of buckled solutions, leads to the determination of the set of nonzero coefficients {L,, L2, M,, M2} of the linear combination. To obtain this set of {L,, Ll, M,, M2}, the Piola-Kirchhoff stress tensor are then calculated and subjected, with the full deformation (B.l), to the boundary conditions (8.1). This manipulation will yield a linear system, controlled by parameters 71., 7, and j (mode number), of algebraic equations. The condition for bifurcation to take place is therefore the condition of nontrivial solutions of {L,, L,, M ,, M2} for this linear system of algebraic equations. This virtually is a generalized eigenvalue problem for the triplet (7., 7, 11). Here 135 is defined as the mode parameter that is the continuous version of the discrete mode number j, scaled by the length of the plate 1,. The involvement of 12 in the definition of n is for the convenience of applying the boundary conditions on top and bottom surfaces. Note here that equation (B.l) can be written in the form of equation (B.1) provided thath is taken as an imaginary number. It can be readily shown that the boundary conditions (B.l), except (B.l)45, are automatically satisfied by the proposed solution form (BI) and (BI). The satisfaction of (B. 1),: gives four relations among these four constants L,, L,, M, and M2, therefore, gives the following 4 by 4 systems of equations for the four unknown constants L,, L,, M, and M2, Jl = 0. (B.1) Here 1 = (L,,M,,L2,M2)T , (3.1) and 'A,C, A2C2 0 0 - Jch = Bf‘ 3:82 A?S, A332 , for h2(7c,7)¢0, (13.1) - 0 0 Blcl B2C2_ -DS, 0 0 O ' J=Jd _ E33, 7:: D221 2 , for h20t, 7) = 0, (3.1) _ o 0 ES, 7.212_ where 136 A, = “247.292, A2 = r§+7uzflz, B, = —r,(7tz£22+r§), B2 = —r2(7t2!22+r,2), C, = cosh(f,n), C2 = cosh(f2n), (B.l) S, = sinh(f,n), S2 = sinh(f2n), Note that the matrices .1c and Jd are functions of load parameter 7., stiffness ratio 7 and deformation mode parameter 1]. It is clear that the system (B.1) with the matrix .1‘3 given by (B.l) can be divided into two subsystems as L JcF(M‘)E Alcl AZCZ (:11) = (g). (3.1) 1 B,S, B282 1 L JCB(M2)E A151 A232 (11:12) = (0) (B.l) 2 B,C, B2C2 2 0 The bifurcation takes place, for (7., 7) obeying h2(7., 7) at 0, if either of the subsystem (B.l) or (B.1) has nontrivial solution. If the nontrivial solution of subsystem (B.1) exists, then U2(X2) is an even function of X2. In this case, the thick plate considered here undergoes flexural deformation as given by equation (B.l) in connection with equation (B.1). For the subsystem (B.1) to have a nontrivial solution, it is required that the determinant of the 2 by 2 matrix must vanish. We, thus, have the flexural bifurcation condition ‘l’°F(7\., 7,n)EdetJ°F = 0, for h2(7., 7) #0. (3.1) Similarly, if the subsystem (B.1) has nontrivial solutions, then U2(X2) is an odd function of 137 X,, and U,(X2) is then an even function of X2. This type of symmetry corresponds to barrelling deformation. The barrelling bifurcation condition can then be expressed as ‘I’CBOt, 7,11)EdetJCB = 0, for h2(7c,7)¢0. (B.l) Now, we turn to consider the system (B.l) with the matrix Jd given by (B. 1). This system can also be divided into two subsystems as L2 81 0 L2 0 dF = = J (MZJ—IEC, 7.2][MZJ (0) (B'l) L2 C1 E L2 0 dB = = J (M2]_| ES, 2,2121%) (0) (3'1) Similarly, the bifurcation takes place, for (7c, 7) obeying h2(7u, 7) = 0, if either of the subsystem (B. 1) or (B.1) has nontrivial solution. It follows directly from (B.l) that detJ‘1F = r,7.4sinh(f,n) :6 0, for h2(7c, 7) = 0. (Bl) Thus, the subsystem (B.l) has only the trivial solution, so that flexural buckling is not a possibility for (7., 7) obeying h2(7\., 7) = 0. For subsystem (B.1) to have nontrivial solutions it is required that ‘I’dBQ, 7, n) EdetJdB = 0, for h2(7c, 7) = 0. (Bl) This is the barrelling bifurcation condition corresponding to subsystem (B.l), since the solution to subsystem (B.1) will yield U2(X2) as an odd function of X2, so that U,(X2) is then an even function of X2. Here, we turn to examine the correlation between 1088 of ellipticity and bifurcation. A preliminary numerical analysis on the fiexural bifurcation condition (B.l) 138 shows families of buckled solutions for the material with 7 = 10. The result is plotted in Figure 54 for (7., 11) pairs satisfying (B.l). It can be easily figure out from Figure 54 that h2(7, 7) < 0 if it is evaluated with these large 7's on the solution curves in connection with 7: 10. Thus, bifurcations take place with (C,,, 7) located in the nonelliptic region. We now examine whether bifurcation will take place for (C,,, 7) located in the elliptic region. Solving h2(7., 7) = 1 > 0 we obtain 7 = (1+ 7.)/27.. Pairs (C,,, 7) = (1/7., (1 + 7.)/27.) are in the elliptic region. Substituting this particular 7 values in the flexural bifurcation condition (B.l), we obtain one family of buckled solution, as plotted in Figure 55. It is thus clear that bifurcation can take place for in both elliptic and non-elliptic regions. 12 10*- O l l L l o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 11 Figure 54. Solution families of the flexural condition (B.1) for the material with 7: 10. These large 7’s in connection with 7: 10 are located in the non-elliptic region. 139 2.5 - 1.5*- 0.5 P l 1 1 l l l 0 0.5 1 1.5 2 2.5 n 3 3.5 4 Figure 55. Flexural bifurcation solution for (C1 1, 7) in the elliptic region. 4.5 Appendix C: An Alternative Way to Obtain the Ellipticity Condition in the Buckling Problem The set of governing second order partial differential equations (B.1)l 3 and the incompressibility constraint (B.1) are rewritten here as — film 4» 11011.11 + “1,22) + 213(31t12—1)u1,11 = 0, 4113.2 + Muz, 11 +u2,22) + 2130.12— l)u2’ 1, = 0, (C1) and kluz,2+kf1u1’l = 0. (Cl) Let u,,; = v1, u1.2 = v2, u2,1= w, and “2.2 = w, noting that (Cl) yields v1 = 43w] , we can obtain from the second order equations (CD a set of first order partial differential equations as —Xf1f5,n+|.L(-K12W2,1+V2’2)—2BK12(37\.12-1)w2’1 = 0, ~11p’2+u(wl,l+w2,2)+2[3(}t]2- 1)w1,1 = O, 2 (Cl) Alw2,2+v2,l = O, w1,2"W2,1= 0’ which has the characteristic equation [Whitham 1974] det(A -va) = 0, (Cl) where 4.;1 o o _ m _ 2:32.;(3112 - 1) O 1 O 0 L 0 0 0 -1 _ 140 141 and '0 no 0‘ —k100 u a=0 . (Cl) 0 001% _o 010_ Let A = M2 and 7 = B/u, equation (C. 1) gives rise to the following equation 1.2[1+27(%—1)]+[1+AZ+27(%—1)]v2+v4 = o, ((2.1) which, virtually, is the equation P(v, Ill, 0, 7) = 0 from (3.5.2). Thus, the types of the roots of (Cl) and the type of equations (C. 1) (and therefore the equations (C.1)) associated with different values of (A, 7) are as discussed in chapter 3. References 1 Abeyaratne, R. 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