, t _ ‘ .. .2... km”? :85. I 11... Imu , ,3 13.2»? y 1. it. .Vfius a: :x “no.1”. at. iii}. 1 1“]..VII‘ L. a .,- . ‘ . , . 3,. bun...” , ‘ ‘ . H.» R. THESE 4) C) '7 " 9404/23/4— This is to certify that the dissertation entitled AXISYMMETRIC PROBLEMS IN NONLINEAR ELASTICITY: EXISTENCE AND GLOBAL INJECTIVITY OF ENERGY MINIMIZERS AND NEW CLASSES OF EXACT SOLUTIONS presented by Lydia S. Novozhilova has been accepted towards fulfillment of the requirements for the PH.D. degree in APPLIED MATHEMATICS AND MECHANICS Major PMesBor's Signature (9W 8; 200‘} Date MSU is an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State University I I PLACE IN RETURN Box to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIRC/DatoDuo.p65-p.15 AXISYMMETRIC PROBLEMS IN NONLINEAR ELASTICITY: EXISTENCE AND GLOBAL IN J ECTIVITY OF ENERGY MINIMIZERS AND NEW CLASSES OF EXACT SOLUTIONS By Lydia S. Novozhilova A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics and Department of Mechanical Engineering 2004 u t ‘abl 1" n l. I'll Dr F I»? ll ABSTRACT AXISYMMETRIC PROBLEMS IN NONLINEAR ELASTICITY: EXISTENCE AND GLOBAL INJECTIVITY OF ENERGY MINIMIZERS AND NEW CLASSES OF EXACT SOLUTIONS By Lydia S. N ovozhilova Axisymmetric problems in nonlinear elasticity are investigated from two different perspectives. In the first part the existence theory for axisymmetric minimizers in Sobolev spaces, based on the approach suggested in the seminal paper by J. Ball (1977) and more recent results, is developed, and new classes of hyperelastic materials are included into the existence analysis. Under suitable assumptions, higher regularity properties and topological properties of openness and discreteness of the radial and axial components of the mappings are established. Global injectivity of axisymmetric minimizers is investigated, and stronger injectivity results are obtained compared with those known for full three-dimensional case. In the second part some classes of specialized three-dimensional axisymmetric motions in a neo-Hookean material under an internal constraint of incompressibility are examined. The original governing system of equations is found to reduce to a simpler unconstrained system of PDEs allowing for finding analytical solutions corresponding to various specialized motion classes. In certain particular cases these closed form solutions reduce to previously known results. A formal action functional whose Euler-Lagrange equations are given by the reduced system is also found. Acknowledgments I would like to thank my son Sergei, my husband Vladimir, and my daughter Elena for their support and encouragement that made this work possible. I also wish to thank the people who collaborated in the project or gave invaluable advice: my advisor Professor T. J. Pence, Department of Mechanical Engineering, MSU, Professor Z. Zhou, Department of Mathematics, MSU, Professor M. Miklavcic, Department of Mathematics, MSU, Professor D. Mason, Department of Mathemat- ics, Albion College, Professor M. Tang, Department of Mathematics, MSU, Professor H. Tsai, Department of Mechanical Engineering, MSU, Professor N. Ivanov, Depart- ment of Mathematics, MSU, and Professor A. Volberg, Department of Mathematics, MSU. Besides sharing their insight in mathematics and mechanics, they gave me a lot of encouragement and moral support and made my graduate studies a very personal experience. I am especially grateful to Professor J. Kurtz, a Graduate Director in the De- partment of Mathematics in 1999, for admitting me to the graduate program in Mathematics department at Michigan State University, which gave me the oppor- iii tunity to do things I enjoy. I wish to acknowledge former Graduate Coordinator of Mechanics program, Professor T.J. Pence, for his great and time consuming effort in resolving coordination issues when my joint Ph.D program in Applied Mathemat- ics and Mechanics was set up. Professor W. Brown, a former Graduate Director in Mathematics Department, helped me to overcome many problems in administering my joined program, and I want to express my deep gratitude for his help. I would like to acknowledge the Department of Mathematics at Michigan State University for providing me with financial support in the form of Teaching Assistantship, which at the same time gave me an exiting teaching experience. I also want to thank Barbara Miller for navigating me through the rules and regulations governing the process of my graduate study at Michigan State University. iv Contents Acknowledgments 0.1 Notation .................................. 1 Introduction 2 Setting axisymmetric variational problem 2.1 Overview of Ball’s existence theory ................... 2.2 Description of the axisymmetric problem ................ 3 Existence theorems 4 Global injectivity of axisymmetric minimizers 4.1 Some properties of mappings of finite distortion ................................. 4.2 Global injectivity theorems ........................ 5 Governing equations for TIE motion 6 Closed form solutions for TIE motion iii 20 2O 25 35 43 44 51 59 66 6.1 Controllable deformations ........................ 66 6.2 Traveling waves .............................. 72 6.3 Simple twist motion ............................ 74 6.4 Motion with a Riemann type similarity variable .................................. 77 7 Cartesian description of TIE and TIES motions 81 7.1 TIE motion in Cartesian description .................. 82 7.2 TIES motion in Cartesian description .................. 83 7.3 Cartesian description of general axisymmetric motion of neo-Hookean body .................................... 89 8 Conclusions and discussion 91 vi 0.1 Notation Q C R3 : open and bounded domain occupied by a continuous material body in its reference (material, undeformed) configuration 80. 89 : boundary of D that is assumed to be strongly Lipschz'tz ([39], Definition 3.4.1). |G| : m-dimensional Lebesgue measure of m—dimensional set G C R", m S n. B(a, R) C R" : ball of radius R centered at the point a. Function (deformation) u : D ——> R3, 11 = (119,112, u3), maps a material point X E 9 into corresponding point x = u(X) E u(§2) in the deformed configuration 3 = um). MSX3 : set of all 3 x 3 real matrices endowed with the usual Euclidian norm |A| = (A: A)“. Mi” : subset of matrices A E M3X3 such that detA > O. F(u) := Vu : Q —-+ Mi” : differential (deformation gradient) of u E WI'P(Q,1R3). In Cartesian coordinates it is represented by the matrix of partial derivatives of the components of u F(u) = (E,) = (Bui/BXj). cof F : Q —v M3"3 : matrix of cofactors of the deformation gradient. The adjugate matrix is the transpose of the matrix of cofactors, adj F = cof FT. C(u) = FT(u)F(u) : right Cauchy-Green deformation tensor. Positive square roots of its eigenvalues are called singular values (principal stretches) of the deformation gradient F(u). 11, 12, 13 : principal invariants of C, 11 = trC = |F|2, 12 = tr cof C, 13 = det C. Div D : divergence operator on (2. 0(9) : space of continuous functions in Q. 73(9) : space of C°° functions having compact support in D with the standard topol- ogy defined by uniform convergence on compact subsets. D’ (Q) : space of Schwarz distributions (the dual space to 17(9)). Di, 2' = 1,2,3 : distributional derivative with respect to i-th coordinate, i.e., for f e D’, ¢ 6 D =-. WI’P(Q,R3) (more generally, WI’P(Q,IR'"), Q C IR"): a triple (m—ple) of functions from Sobolev space WIND). For p = 3 (p = n) the latter is called Sobolev space with natural exponent. Cof u, Det u : matrix of distributional cofactors and the distributional determinant, respectively, defined by (COf U)¢‘j = i+2(uj+2uj+l,,-+1 )— Di+1(uj+2uj+1,,+2), Detu = Dj [111(C0f F)j1] , where i, j = 1, 2, 3. In the first equation the indices are to be taken modulo 3. A function f : U —-> R, where U is a subset of a Banach space V, is said to be weakly lower semicontinuous (w.1.s.c.) if for any sequence uk 6 U converging weakly to u, uk —* u, the inequality f(u) 3 lim f(uk) k—ooo holds. N(f, ) : Y —+ N U {0,00} : multiplicity function for a map f : X —) Y. For y E Y the value N (f, y) is defined as the number of elements in the set {:13 E X : f (1:) = y}. Henceforth in this work, the conventions of Ogden [44] for tensor calculus are used. In particular, the divergence of a tensor S in Cartesian coordinates (X 1, X 2, X 3) reads DIV S = USU/aXi. Cartesian coordinates of a tensor 8W/6F, W = W(F) being a scalar function of F, are written in the component form as (6W/6F)a, = BW/OFm. Summation over repeating indices is assumed. Chapter 1 Introduction This thesis is concerned with the mathematical theory of nonlinear elasticity [44], [16], [37]. Specifically, the hyperelastic version is regarded as a useful model for solids undergoing large deformations without energy dissipation. This endows correspond- ing mathematical problems with a strong variational structure that makes it possible to use modern powerful machinery of the calculus of variations. In general, a static variational problem in continuum mechanics, when the ther- modynamic variables, such as temperature or entropy, are not under consideration, is to find deformation(s) u‘:BO-—>1R3 that render absolute minimum to the potential energy E(-) of the medium under consideration inf E(u) = E(u’) = min E(u). (1.0.1) 116A uEA Here the set A of admissible deformations is usually a subset of an appropriate Banach space (e. g., Sobolev space WI'P(Q,1R3)) faithful to physical restrictions of the problem. Rigorous mathematical approach to variational theory for general three-dimensional problems in nonlinear elastostatics was started in 1977 with a seminal paper by J.M. Ball [4]. He employed the direct method in the calculus of variations to state and prove his theorems on existence of absolute minimizers for equilibrium problems in nonlinear elasticity. More generally, a motion is a time parametrized family of deformations described by a function x = 30 X [0,00) -+ R3, X(-,t) = x(wt) E 3(t), where B (t) is the current (deformed) configuration. The corresponding general varia- tional problem in elastodynamics is to find motion(s) that render absolute minimum to the action functional T Lab/{Bf éprlde-E(x(-,t)) dt over curves in a set of admissible deformations. Here p = p(X) is the inertial mass density in the reference configuration, V := x(-, t), where dot stands for time deriva- tive, and the first term in the integrand represents the kinetic energy. It is usually as- sumed that the initial deformation x(-, 0) and the velocity field V(-, 0) are prescribed, and deformations x(-, t) belong to an admissible set satisfying appropriate physical requirements. When a problem admits a variational formulation, the equations of motion, which represent in a differential form the fundamental Balance of Linear Momentum Principle in continuum mechanics, can be obtained as Euler-Lagrange equations of the action functional. In the static theory, when the inertia effect is not an issue, the equations of motion become the equilibrium equations, and they can be interpreted as necessary condition for minimizers of the potential energy. To specify the potential energy for a material under consideration a constitutive relation describing a mechanical response of the material should be included into the macroscopic model. Mathematical formulation of the constitutive laws must be con- sistent with available experimental data and satisfy certain physical restrictions such as frame indifference and (possibly) material symmetry requirements. The formula- tion must also satisfy mathematical restrictions related to such issues as existence and uniqueness of solutions to the balance equations. Other simplifying restrictions are introduced to make rigorous mathematical approach tractable. A hyperelastic material is assumed to support a strain (stored) energy density W : Q x Mi” —i IR so that W(X, F) represents the stored energy per unit volume at a material point X when the elastic body is subjected to deformation u with deformation gradient F = Vu at this point. The total stored energy in the deformed volume um) is then 1 E(u) = [0 Wm, Vu)dV. (1.0.2) Additional physically meaningful assumptions as outlined next simplify the functional 1If tractions (external surface forces) are exerted on (part of) the boundary in the reference configuration, an appropriate surface integral is added to the right hand side. form of constitutive function. Frame indifference is the assumption that physical laws are invariant with re- spect to observer orientation in space. In terms of the stored energy functions frame indifference translates into the requirement W(X.QF) = W(X. F) for all F 6 M1” and all proper orthogonal matrices Q. Material symmetry refers to a linear isometry P : R3 ——> R3 such that a material response is unaffected if the material orientation changes from B to P8. In terms of the stored energy functions this translates into the requirement W(X,FP) = W(x,F). A set of all material symmetries GB of the body is called the material symmetry group. Here we consider only isotropic materials with GB = 50(3) (i.e., all orientations equivalent). It can be shown (see, e.g., [16]) that a hyperelastic material is frame indifferent, isotropic, and homogeneous 2 if and only if W is a function only of the principal invariants of the right Cauchy deformation tensor C = FTF, A W = @(Il,12,13). (1.03) Some materials exhibit volume preservation property; they are termed incompressible. 2i.e., W is independent on X The deformations possible for such materials must satisfy the constraint detF = 1 (1.0.4) and are called isochoric. In particular, many isotropic rubber-like materials are con- sidered to be incompressible, and they are often modeled by Mooney-Rivlin stored energy A w = (Il,12) = (1(11 — 3) + 3(12 — 3), (1.0.5) where a, fi 2 0 are material constants. In the limiting case B = 0 the material is called neo-Hookean and its stored density is usually written as “ _ H W — 2(11 —- 3). (1.0.6) The material response function, the nominal stress tensor S, in the incompressible case is given via the strain energy function by [44] aw 3:535— F", (1.0.7) p = p(X) being the hydrostatic pressure (Lagrange multiplier) associated with the constraint of incompressibility (1.0.4). For compressible materials there is no restriction (1.0.4), and the term pF‘1 does not appear in (1.0.7). In all cases the Balance of Linear Momentum in the absence of body forces requires Div S = px, (1.0.8) which is the Euler-Lagrange equation associated with minimization of the action functional. Ball’s theory was based on the new notion of polyconvexity that ensures weak lower semicontinuity of the energy functional in an appropriate space of functions. Ball also showed that polyconvexity implies quasiconvexity. The latter concept was introduced by GE. Morrey [36] who showed that, modulo some technical assumptions, quasi- convexity is a necessary and sufficient condition for w.l.s.c. of a multiple integral. Policonvexity can be effectively characterized in terms of the integrand (i.e., strain energy function in the setting of nonlinear elasticity), as opposed to quasiconvexity whose characterization is still an Open question. Ball has illustrated his theory by applying it to a wide class of isotropic stored energy functions commonly used in nonlinear elasticity and referred to as Ogden materials. These functions can be written in the form M 3 N 3 W(F) = 2a.- (2 Al‘) + ij ( Z (AkAJ’J') + h(det F), (1.0.9) = k—l ' 1' 1 3:1 k,l=1 kaél where: AI: = ,\,.(F) are the singular values of F 6 Mi”; ai>0i7i>12 lsstibj>036j2171SjS N; h : (0, +00) —i IR is a convex function. Emotions (1.0.9) satisfy the hypotheses of Ball’s theory under appropriate choices of the growth exponents p = max 7,, q = mjax 6,. Ball’s contribution stimulated investigations into existence theory through a vari- ety of theoretical lenses. Materials with strain energt density with growth exponents below the values allowed by Ball’s theory received significant attention in connection with cavitation [6], [29] and other phenomena involving singularities [7]. Regularity issues for function classes of mappings u that involve information on both the gradi- ent F = Du and its adjugate matrix were studied by Sverak [51]. The degree formula used in [51] was generalized by Miiller, Qi, and Yan [42] and used for weakening Ball’s constraints on the growth exponents needed for existence of a minimizer in Sobolev space. Ball’s conjecture about identicity of pointwise and distributional cofactors and determinants was proved by Miiller [41] under the assumption that those null La- grangians (quasiaffine functions) defined in the sense of distributions are functions. This development allowed one to reformulate the existence theorems by Ball in terms of pointwise null Lagrangians rather than the distributional ones. Variational prob- lems with non-convex (non-quasiconvex, non-polyconvex) integrands are currently intensively studied. The relation between quasiconvexity and relaxation was discov- ered by Dacorogna (see [18] and references therein). Explicit formulations of relaxed problems are not in abundance, but when they are available and represented by a multiple integral, Ball’s existence theory applies if the integrand meets appropriate requirements. More recent references can be found, e.g., in [8]. Classes of admissible functions introduced by Ball stimulated new developments in geometric function theory. A class of functions having finite dilatation (distortion), which includes Ball’s admissible functions, has been defined and investigated from various points of view (see [31] and references therein). Theory of mappings of finite 10 distortion is a natural generalization of the theory of quasiregular mappings [48]. Well known topological properties of openness and discreetness of nonconstant quasiregular mappings were recently carried over to two-dimensional mappings of finite distortion by Iwaniec and Sverak [30], provided that the dilatation quotient K (see Definition 4.1.1) is an integrable function. For n > 2 the openness and discreetness of n- dimensional mappings of finite distortion was investigated by Heinionen and Koskela [26], and Kauhanen, Koskela, Maly [32]. Results in geometric function theory and degree theory allowed one to state condi- tions ensuring more realistic properties of admissible functions , e. g., global injectivity. This issue was investigated in different settings by Ball [5], Ciarlet and Necas [15], and Tang [52], while Fonseca and Gangbo [22] studied local invertibility properties of Sobolev classes with natural exponent. Despite the important contributions discussed above, genuine three-dimensional deformations of some commonly used material models are not covered by Ball’s the- ory. In particular, as was noted by Ball himself, restrictions imposed by his theory rule out three-dimensional deformations of neo—Hookean materials (1.0.6). These materials were the object of numerous investigations (see, e.g., [43] and references therein). They can also serve as a good source for testing numerical methods. The neo-Hookean strain energy density has also been suggested as a useful form for mod- eling the base matrix material response in composite materials, subject to additional reinforcing, and used for analysis of different aspects of the theory of composites by 11 a number of authors (see, e.g., [46] and references therein). On the other hand, more complicated expressions for the strain energy density function provide more flexibility for correlation with experimentally observed deformation behavior, and development of more sophisticated hyperelastic constitutive models is an active subject (see, e.g., [10], where fairly general constitutive relations for the shape memory materials are developed). Recently, Holzapfel, Gasser and Ogden [28] presented the analysis of the biomechanics of blood vessels, employing the neo-Hookean strain energy function for modeling the behavior of the matrix material. This example from biomechanics, along with other applications using nonlinearly elastic models for bodies of tubular geometries, motivated investigation of axisymmetric problems in nonlinear elasticity in this thesis. In Chapters 2-4 the existence issues for variational formulation of axisymmetric problem are studied. It seems natural to expect that restrictions on growth exponents in the framework of Ball’s theory will be milder if one confines analysis to a subclass of three-dimensional deformations. This is true for plane deformations that can be viewed as a subclass of three-dimensional deformations. We introduce a general class of axisymmetric deformations of the form r=r(R,Z), 0=w+r(R,Z), and z=z(R,Z), (1.0.10) where (R,w, Z), (r,9, z) are cylindrical coordinates in the reference and deformed configurations, respectively. Reduced restrictions on the growth exponents for the strain energy densities of Ogden materials (1.0.9) subjected to deformations (1.0.10) 12 are then obtained in the spirit of Ball’s existence theory. Although not necessary for essential conclusions, for simplicity attention is restricted to isochoric deformations with boundary condition of place on a subset I‘ of the boundary 80 in the absence of external surface forces on the remainder of the boundary. To ensure the coerciveness inequality in axisymmetric setting, the body in the reference configuration is assumed to be cylindrically hollow. Under a natural assumption that the radial component is nonnegative almost ev- erywhere, improved regularity of two-dimensional mapping determined by the radial and axial components of isochoric deformation, v = (r, z) e WI’P(D, 1R2) (1.0.11) with p = 2, is established. Here D is half of the axial cross-section of the undeformed body. It is also found that the two-dimensional mapping v is Open and discrete. This is one of the most novel and original result in this development, and it does not have an analogue in three-dimensional existence theory. Based on improved regularity and the topological property of openness of the mapping v, injectivity of minimizers is established in a stronger form than that stated in [5] and [15]. For technical reasons, if the angular deformation function r is present, a stronger restriction on the radial deformation is imposed, namely, it is assumed that originally hollow cylindrical body remains hollow after deformation. Beginning with Chapters 5, we turn from existence issues for static axisymmetric problems in variational formulation to analysis of axisymmetric motions in differ- I3 ential form. Specifically, a time dependent version of (1.0.10) subject to additional specialization is considered. Attention is restricted to neo-Hookean material response. Less is known about existence and uniqueness for elastodynamics than for elas- tostatics (see, e.g., [37] on some aspects of what is currently known). Closed form three-dimensional solutions to the equations of motion are rare, and most such solu- tions involve both specialized material response and a priori symmetry assumptions that impose severe structural restrictions on the unknown functions. Known explicit dynamical solutions for incompressible materials include the radial oscillatory solu- tions due to Knowles [33] and the circularly polarized finite amplitude wave motions studied by Carroll [13, 14]. For a Mooney—Rivlin material, detailed analysis of fi- nite amplitude plane wave motions is given by Boulanger and Hayes [11, 12]. More references on exact solutions in finite elastodynamics can be found in [43]. The focus in chapters 5-7 is on deriving the governing differential equations for the specialized forms of three-dimensional motions in neo—Hookean material and ob- taining new physically meaningful explicit solutions. The motions presented here give both space and time variation in all three principal stretches and naturally describe various wave forms in tubular geometries. In certain particular cases they reduce to previously known results. Here is an outline of the content of the thesis. In Section 2.1 an outline of Ball’s theory is given providing a framework for fol- lowing existence analysis. In Section 2.2 the axisymmetric variational problem is 14 described. By a straightforward computation it is shown that the Euler-Lagrange equations for the reduced variational problem are equivalent to the equilibrium equa- tions for the physical problem under consideration. New dependent variables that simplify the description and allow one to apply the direct method of the calculus of variations in the spirit of Ball’s theory are introduced. Two existence theorems for isotropic strain energy densities with and without dependence on the cofactor matrix are stated and proved in Chapter 3. Here cases with and without assumption that the distributional cofactor matrix and determi- nant are functions are examined, and we employ the result of [41]. Although, as expected, the restrictions on the growth parameters are significantly reduced due to the axial symmetry, materials with neo—Hookean rate of growth (p = 2) represent a marginal case for the existence theorem in the admissible set without restriction on the distributional determinant. The goal of Chapter 4 is twofold: to extend the existence results to integrands with rate of growth p = 2 and without conditions on the cofactor matrix, and to examine injectivity of admissible mappings. The cylindrical description of admissible mappings is used. Under a natural assumption that the radial component of defor- mation is nonnegative, some remarkable properties of two-dimensional mapping v, defined in (1.0.11) for a mapping u E W1'2(D, R3) from a set of admissible functions, are presented in Section 4.1. Firstly, it has been proven that v has finite dilatation. Furthermore, it is also shown that for functions of finite distortion in Sobolev space 15 with natural exponent the mapping L : Wl'"(fl,lR") —+ C(G,R"), L(f) 2 fig, (1.0.12) where G C Q C IR" is a relatively compact domain in D, is compact. Consequently, for any relatively compact G CC D weak convergence of a sequence of admissible functions 11), = (rk, rk, 2k) in W1’2(D,IR3) implies uniform convergence of the corre- sponding sequence (up to a subsequence) vk = (rk, 2k) in G. It is worth noting that the general fact of compactness of the embedding (1.0.12) is of interest in its own right. The most remarkable properties of the two-dimensional mapping v, established in this section, are openness and discreteness that follow from the result in [30] on Stoilow type factorization. In Section 4.2 two additional conditions are introduced for admissible sets. The values of the radial component are assumed to be separated away from zero, that is, r(R, Z) 2 a > 0 (1.0.13) for a fixed positive number a and almost all (R, Z) 6 D, and an axisymmetric counterpart of the well-known injectivity condition of [15] is imposed. For this smaller set of admissible mappings, existence of minimizers with p = 2 is proved, and the injectivity of minimizers almost everywhere is established for p 2 2 along the lines of [15]. In the border case p = 2 the argument of [15] needs to be modified, and we use some results from geometric function theory. Furthermore, making use of the openness of the two—dimensional mappings (1.0.11), one concludes that under 16 the injectivity condition these mappings are in fact homeomorphic. For p > 2 this implies that corresponding axisymmetric deformation is a homeomorphism too, which represents a substantial improvement compared with relevant results known for the three-dimensional case. For a two-dimensional isochoric deformation from appropriate Sobolev class, the Stoilow type factorization is also readily available. This observation allows one to sharpen previously known results on injectivity a.e. when they apply to this class of mappings, although two-dimensional deformations are not a focus in this work. Global injectivity for Dirichlet problem presented in this section relies on the result in [5]. For the two-dimensional case, the condition on the adjugate matrix introduced in [5] is found to be automatic for the mapping (1.0.11). Therefore, except for that condition, the statement about global injectivity for the Dirichlet problem in this section is otherwise identical to that in [5]. In Chapter 5 a specialized class of motions is considered. In cylindrical coordinates it is given in terms of axially varying twist function T(Z, t), radial inflation/deflation function s(Z,t), and axial contraction/glongation function z(Z, t) by the following ansatz r = Rs(Z,t), 0 = w + r(Z, t), z = 2(Z,t). (10-14) (These motions are referred to as TIE motions.) The general governing equations for axisymmetric motion of neo—Hookean mate- rial, derived in Section 2.2, are reduced here for the specialized ansatz (1.0.14) to a 17 second order system of two coupled nonlinear partial differential equations for func- tions 3 and r. This system contains two material constants, the inertial mass density p and the neo—Hookean shear modulus p, as well as an arbitrary function of time C (t) that results from a general integration. The neo—Hookean shear wave speed is given by c. = \/p/—p, a parameter that has special significance with respect to the various motions described herein. Chapter 6 presents four various classes of specialized solutions to the governing system for TIE motions. e Equilibrium deformation solutions of three different forms depending on the neo-Hookean shear wave speed c... One of the well known universal deformations for incompressible hyperelasticity emerges as a special case. 0 Travelling wave solutions of the same three forms as above at arbitrary wave speed. Further, at the neo—Hookean shear wave speed 0. additional travelling wave solutions are also available. 0 Motions with specialized forms of the twist function. For the special case of zero twist the governing equations reduce to a single linear partial differential equation which can be treated by standard means. 0 Motions for which both the twist function r and the inflation / deflation function 3 are constant on rays Z / t = constant. Although we are unable to obtain explicit solution in this general case, an analytic expression is given for a special 18 case when one of the parameters in the governing system of ODE vanishes. In Chapter 7 Cartesian descriptions of TIE and TIES motions are derived. It is shown that the reduced governing system of PDE for the radial and angular compo- nents of TIE motion, found in Chapter 5, admits a variational formulation. Formal change of dependent variables transforms the Lagrangian of this variational problem ‘ into quadratic expression with respect to new dependent variables therefore leading to a linear system of Euler-Lagrange equations. For a particular case, when the function C(t) involved in these equations is a constant, the system reduces to two identical telegraphy equations, which can be treated by standard means. In Section 7.2 more general class of motions, termed TIES, are considered. The motions include transverse ghear in addition to twist, inflation / deflation, and contraction/ elongation, describing TIE motion. Although two unknown functions accounting for the in-plane shear are introduced, the governing system for TIES motion is shown to decompose into four identical decoupled linear equations of the same type as for TIE motion. In Section 7.3 the governing system for general axisymmetric motions in neo-Hookean solid in Cartesian coordinates is derived, and it seems to be more convenient for further analysis than the original one, derived in terms of cylindrical coordinates. To the best of my knowledge, presented in this thesis results on existence, in- jectivity, and regularity for axisymmetric minimizers, as well as the development on specialized elastodynamic equations of motion and their explicit analytic solutions, are new and have not been discussed in the literature up to now. 19 Chapter 2 Setting axisymmetric variational problem 2.1 Overview of Ball’s existence theory Let a material body 80 in its reference configuration occupy an open and bounded domain 9 C R3 with strongly Lipschitz boundary BS2. Given a material point X E Q, a mapping u : O —> R3 describes the material deformation with x = u(X) E u(Q) the corresponding point in the deformed configuration, and F := Vu the deformation gradient. The material of the body 80 is assumed to be hyperelastic with the stored energy function W satisfying the requirements of frame indifference and, unless stated oth- erwise, isotropy. Thus the total stored energy in the deformed volume u(Q) is defined by (1.0.2), and corresponding minimization problem is then given by (1.0.1), where 20 a set of admissible deformations .A is a subset of Sobolev space WIND, R3) satisfy- ing appropriate physical restrictions of the problem, e. g., boundary condition of place u = no on 1" C 69, [PI > 0, (uo E WIND, R3) being a given function), specified trac- tion values on the remainder of the boundary, and the incompressibility constraint (1.0.4). For the successful application of the direct method of the calculus of variations to problems in nonlinear elasticity, one needs to formulate physically realistic hypotheses on both the stored energy density W and the admissible set .A so that the following major argument can be realized: Step—1. Ensure finiteness of the infimum of the total energy functional E(-) over the admissible set A and show existence of a minimizing sequence nn 6 A that converges weakly to some a for a suitable choice of p. _S_tep_2. Show that weak limits of minimizing sequences belong to the admissible set A. _St_ep__?: Verify that the total energy functional E() is w.l.s.c. Then the inequality E(fi) 3 lim E(un)=&161f1E(u) S E(fi) (2.1.1) n-ooo implies that a is a minimizer. To ensure the application of the above three steps to three-dimensional problems in nonlinear elasticity, J. Ball [4] assembled the following hypotheses (with appropriate 21 modifications of W and A in different settings) on the stored energy function : (H 1) Polyconvexity: For almost all X 6 fl there exists a continuous convex function W(X’ ‘i ‘1 l 3 M3” X M3” x IR —+ IR such that WlxiF) = W(X,F,C0f F,det F) fOl' all F E Mai-X3, and W(-,F,H,6) is measurable over 9 for every (F,H,6) E M3X3 x M3X3 x IR. (H2) Coercivity: There exist real numbers a > 0, B, p > 1, q > 1, s > 1 such that for almost all X E Q W(X,F,cof F,det F) 2 a (IFIP + Icof Fl" + (det F)’) + )6. (2.1.2) (H3) Finiteness: There exists an admissible deformation u E A such that E (u) < 00. It was also shown that Ogden materials (1.0.9) satisfy the hypotheses (H1) and (H2) with p = max 7, and q = max 63-. A typical existence result obtained by employ- : .7 ing Ball’s theory within the context of the above hypotheses is given by the following Theorem 2.1.1 Let a stored energy function W satisfy (H1)-(H3) with 1 p22 and 42—3— p-l' Let lim W = 00. det F—90+ 1In [42] it was shown that the right hand side of the inequality for parameter q can be replaced by 3/2. 22 Then the total energy (1.0.2) assumes its minimum in the admissible set A given by A := {u E Wl’p(Q,IR3) 3 = ‘10 a.e. in F, COI F E Lq(QaM3X3)v detF E L’(§l), detF > 0 a.e. in Q}, where F C 39, [F] > 0, and no is a specified function in WI'P(Q,IR3). Proof (Sketch). Existence of a minimizing sequence 11,. and boundedness of the cor- responding gradients in LP(Q,M3"3) follow from the finiteness and the coercivity hypotheses, respectively. Boundedness of the sequence 11,. in LP(I2, R3) is proven via the generalized Poincaré inequality (see Theorem 6.1.8 (b) in [16]) P /|f|Pda:gc /|Vf|de+ ffda . (2.1.3) 9 Q I‘ applied to the components of uk. The displacement boundary condition imposed on admissible functions is needed here to ensure that the second term on the right hand side is bounded (in fact, it is a constant). Thus 11), is bounded in WIND, R3), and the existence of a weakly convergent subsequence follows from the reflexivity of Sobolev space Wl'l’ with p > 1. The most technical part of the proof is establishing that the weak limit resides in the admissible set. It includes proof of weak continuity of the minors of the deformation gradient, and this dictates the restrictions on the growth exponents p, q. Satisfaction of the boundary condition for the weak limit relies on the compactness of the trace operator tr 6 [I (WI’P(D); LP(F)) and is proved in a standard manner by extracting a subsequence converging almost everywhere on F. 23 The polyconvexity of the integrand implies its quasiconvexity [4], which is essen- tially equivalent to the weak lower semi—continuity of the energy functional [36]. This completes the proof. I As was pointed out in [4], for three-dimensional deformation of an incompressible hyperelastic material with stored energy function independent on cof F, the bound on the growth exponent needed for the weak continuity of the determinant (distri- butional determinant) is p 2 3 (p > 9/4). The optimality of these bounds has been demonstrated in [19]. Consequently, any of these restrictions rules out neo-Hookean materials. In Chapters 3 and 4 new existence theorems for variational formulation for iso— choric, axially symmetric deformations of hyperelastic materials will be presented, and some regularity and injectivity properties of minimizers of may will be estab- lished. It is always assumed that certain boundary conditions of place are prescribed on the part of the boundary and the rest of the boundary is traction free. It is also assumed that the following conditions are satisfied. 0 W(X, F) is frame indifferent and, unless stated otherwise, isotropic; e W(X, F) satisfies the hypotheses (H1), (H2); 0 Function uo prescribing boundary condition of place for an admissible set A belongs to this set, and I (no) < 00. The existence theorems cover some classes of hyperelastic incompressible materials 24 with stored energy functions that do not satisfy growth conditions of Ball’s existence theory in genuine three-dimensional case, in particular, the class of neo-Hookean materials. The detail will be provided only for proving the fact that weak limits belong to appropriate admissible sets, since the rest of the argument sketched above is standard (see [4], [16], [18], and references therein). Parameters p, q always denote the growth exponents in the coercivity hypothesis (2.1.2) for the stored energy function under consideration. 2.2 Description of the axisymmetric problem In this section we describe the axisymmetric setting in both cylindrical and Cartesian coordinates, adjust the total energy functional to this setting, and prove the equiva- lency of the equilibrium equations for the physical problem under consideration and the Euler - Lagrange equations of the reduced minimization problem. Let a hyperelastic body in its reference configuration occupy a domain (2 6 1R3 given in cylindrical coordinates X = (R,w, Z) by Q := {(R,w, Z) : X' := (R, Z) 6 0,0.) 6 [0, 27r)}, (2.2.1) where D C R2 is an open domain with strongly Lipschitz boundary 0D such that min R = R.- > 0. (2.2.2) X’ED Introduce a class Ari(Q) of almost everywhere isochoric, axisymmetric deformations f1 : Q —» IR3 with components (r, 9, z) of the form given by (1.0.10). 25 The deformation gradient of u E Axi(Q) takes the form r _ T,1 0 T,3 F: rr,1 r/R rr,3 (2'2'3) 2,1 0 2,3 J with the corresponding right Cauchy-Green deformation tensor given by I . 731 2 + (T731 )2 + 2,12 T2731 /R 731 733 +7‘2T,1 7,3 +2.11% C = ... (r/R)2 131,3 /R , (224) L r,32+(rr,3)2+z,32 _] [— where the ellipses stand for appropriate symmetric expressions. Here and throughout this work we adopt the notation f,12=Gf/GR £22: Bf/Bw and f’32= Bf/BZ run: for any scalar function f (R,w, Z). The incompressibility condition (1.0.4) takes the form %(r,1 2,3 —r,;; 2,1) = 1. (22-5) In the three-dimensional Cartesian setting, the first invariant 11 = trC = [F]2 of the Cauchy-Green strain tensor C is a sum of the squares of all partial derivatives of the Cartesian components of deformation. Consequently, if the total energy is finite, those partial derivatives belong to LP due to the coercivity inequality (2.1.2). However, as follows from (2.2.3), in cylindrical coordinates the first invariant has the form 1. = IFI2 = n12 +(r'r,1)2 + 2.12 + war + r32 + (m3 )2 + m 2, (22.6) 26 so that the derivatives of r in this expression are directly coupled with r. Hence, for a minimizing sequence of deformations f1" 6 1111(9), one can only conclude from the coercivity inequality that the functions r,f , 2,? , r“r,[c , i = 1, 3, converge weakly in the space [f(D) thereby preventing determination of appropriate Sobolev space for the limiting angular deformation function r. One way of resolving this problem is to assume that the radial component r in the deformed configuration is uniformly bounded below away from zero, r(R, Z) Z a > 0, for almost all (R, Z) 6 D. This possibility will be explored in Chapter 4. The problem of decoupling functions 7' and r can be also eliminated through the introduction of the new dependent variables £=§(R,Z) :=rcosr and n=n(R,Z) :=rsinr. (2.2.7) In fact, then corresponding right Cauchy-Green strain tensor and its first invariant 11(C) in terms of u = (5, n, 2) take the forms I . 6’1 2 + 77,1 2 + 271 2 (7771 E _ "€11)/R 6)] 6:3 +7711 ")3 +zal 273 C = --- (£2 + 720/122 (72,3 6 - 176.3 )/R (2'28) {,3 +0.3 +2.3 and 1, = 2 (6.3.. +03. +23. ) + (£2 + 05/122. (22-9) m=l,3 Remark. It should be noted at this point that nothing prevents the radial component r from taking negative values.2 Therefore, the unique determination of the cylindrical 2The existence theorems of Ball [4] also assert only that under certain assumptions a minimizer 27 coordinates r, 0, z of the image of a point (R, w, Z) in terms of (E, n, z) is not possible. However, using (1.0.10), (2.2.7), and the standard relations zr=rcosd, y=rsin0, z=z(R,Z) with 0 = w + r, the corresponding image can be described in Cartesian coordinates (1:, y, 2) by the formulae X Y Y X 33 — R6 — Eli, y — R6 + E77, 2 — 2(R, Z) (2210) with X = Rcosw, Y = Rsinw. These equations will be referred to as Cartesian de- scription of deformation. It follows from the equations (2.2.10) that the new depen- dent variables € and n have clear physical meaning: these are the first two Cartesian coordinates of the image of the axial cross-section w = 0 in the deformed configura- tion, €= $(R,0,Z), 7) = y(R,0,Z)- If one defines a matrix valued function 5.1 “77/3 5,3 Fo= 71,1 é/R 71.3 (22-11) 211 0 213 corresponding to an axisymmetric deformation 1‘1 6 Axim), then a direct computa- tion shows that C = FTF .—. Fg‘Fo, u exists. Injectivity of a minimizer is another problem that was later stated and investigated in different settings, cf. . [5], [15], [52]. 28 so that F and F0 have the same singular values. Using the chain rule, it is easy to show that the deformation gradient F(fi) in Cartesian coordinates satisfies the equation F = Q F0 QT) where F - cos a) — sin a) 0 Q = sin a) cos a) 0 0 0 I I This representation implies that the polyconvexity hypothesis (H1) and the coercivity hypothesis (H2) hold for axisymmetric deformations with F replaced by F0 due to the frame indifference and the isotropy assumptions. It also follows from the above representation that F0 is the deformation gradient in Cartesian coordinates restricted to the section a) = 0 of the cylindrical body (2, F0 = F(R, 0, Z). The incompressibility condition (2.2.5) in terms of §, 17, 2 reads (£2 + 772))1z13 _(62 + 772),:3 z31: 2R- (2.212) For it E Axi(Q), the three-dimensional minimization problem (1.0.1) formally reduces to inf RW(X,F(u))dede, “EA 0 where X = X(R, 0.), Z). To reduce the dimensionality of the underlying space we have to assume that W depends on X, Y only through the variable R = (X 2 + Y2)1/2. 29 Then the energy functional of the problem (1.0.1) takes the form I(u) :2 3.23 RW(X’, F(u)) da (2.2.13) D with u = (r, r, 2), F given by (2.2.3), da the area element in D, and X = (R, Z). For Cartesian description of deformation, ueA I(u) :2 inf / RW(X', Fo(u)) da. (2.2.14) D with u = (5,17,21) and F0 given by (2.2.11). In each case A represents a set of admissible ordered triplets of functions that is assumed to be a subset of appropriate Sobolev space Wl'p(D, IR3) faithful to physical restrictions of the problem including the incompressibility constraint and boundary conditions to be specified later. Note that by virtue of (2.2.2) the coercivity hypothesis (2.1.2) holds for the integrand in (2.2.14), provided it is true for the strain energy density W(X, F, cof F). Before proceeding with the existence analysis, the reduced variational formulation needs to be justified from mechanical point of view. Specifically, one must show that the equilibrium equations for the problem under consideration coincide with (more exactly, are equivalent to) the Euler-Lagrange equations for the reduced functional to which an appropriate term accounting for the incompressibility constraint must be added. In the absence of body forces, the equation of motion (1.0.8) transforms into the equilibrium equation Dw3=a mam) where S is the nominal stress tensor (1.0.7). 30 In the next lemma the equivalency between the Euler-Lagrange equations for the reduced functional (2.2.13) and equilibrium equation (2.2.15) is shown. For simplicity, it is assumed that the strain energy density function W does not depend explicitly on the spacial variables. Lemma 2.2.1 Let 1. u = (r, 9, z) : (I -> R3 be the triplet of functions corresponding to a deformation u E Axi(§2), where Q is defined by (2.2.1), (2.2.2); 2. The strain energy density W = W(F) is frame invariant and isotropic; 3. The Lagrange multiplier does not depend on the angular variable, i.e., p = p(R, Z)- Then the Euler-Lagrange equations for the functional in (2.2.13) differ from the equi- librium equations by a factor R and therefore are equivalent to the equilibrium equa- tions. Proof. For a material satisfying the requirements of isotropy, frame indifference, homogeneity, and incompressibility the strain energy density can be written in the form W = W(Il,12), where 11, [2 are the principle invariants of the right Cauchy- Green deformation tensor C = FTF, and the formula (1.0.7) becomes [45] _ 8W 8W T 6W T _, S_2(611+11612)F 26,1ch pF. (2.2.16) Assume temporarily that W = W(Il), and let the prime denote differentiation with respect to 11. It follows from (2.2.15), (2.2.16) that the equilibrium equation under 31 the assumptions of lemma becomes 2W’ Div FT + 2W”F V1l — F‘T Vp = 0, (2.2.17) where we used the gradient operator V in cylindrical coordinates a 1 a a V — ERGR + EwREU—J + Ez-a—Z, and Piola identity [16], Pg. 39, which for isochoric deformation takes the form Div F‘1 = 0. To compute the first term in (2.2.17) note that the transpose deformation gradient can be written in the form FT = T,1ER®8,-+T,3Ez®€r+TT,1ER®€9+T/REw®eg + TTisEz®€9+ 2,1 Ea®ez +ZisEz®€r Then direct calculation gives Div FT = e,.(Ar — i~(vr)2 + (1712.),1 ) + e9(rAr + 2% - vi— + (T/R)T,1) +ez (A2 + z,1/R). The calculation uses the fact that the operator Div is the gradient operator followed by contraction [44] and the following elementary formulae for the derivatives of the basic vectors (ER)72 : Eu): (E(u))? = ’ER) (er)i2 : 897 (99))? : —err (er)1i: 806,51 (99))i : —e7‘0,i 2 = 1r 3 32 It is easy to verify that the second term in (2.2.17) is given by the expression eFVr -V11 + engr - V11 + ezV2 - V11. To compute the last term note that the inverse of F reads r2,3 /R 0 -7‘7‘,3 /R F—l _ — T(Ta3 zil _T31Z73) r/R T(T,3 Til —T,1 T13) —r2,1/R O rr,1/R implying that r F_T VP = R (er(Pi1 2,3 —Pi3 2,1) - ez(19i1 Tia ‘pis 2,1 )) Combining the above computations one obtains the following equilibrium equations W’ [Ar — TNT)? + (r/R),1] + W”vr1 .vr —(r/2R)(pal 213 —pi3 Zal) = 0) (2”218) W’ [rAr + 2Vr - VT + (r/R)r,1] + W”rVIl - Vr = 0, (2.2.19) W' [A2 + 2,1/R] + W"VII - V2 + (r/2R)(p,1r,3 —p,3 r,1) = 0. (2.2.20) Derivation of the Euler-Lagrange equations is standard. Under the assumptions of the lemma the energy functional in (2.2.14) modified to incorporate the incompressibility constraint takes the form I(u) = f R (W(Il) — p [%(r,1 2,3 —r,3 2,1) — 1]) deZ. D 33 Then the first Euler-Lagrange equation is I 611 I 611 [R (W 6(T,1) pTZ,3 /R)] 11 + R (W 8(T,3) +przil /R) 13 I _ R (WI-8871 - p(rrl 233 -r13 z)1)/R) : 0 Using (2.2.6) one arrives after elementary computations at the first equilibrium equa- tion (2.2.18) multiplied by 2R. Derivation of the other two equations is similar. The same argument applies when the strain energy density depends on the second invariant [2, but it requires more technically involved computations. I Remark. For neo-Hookean stored energy density (1.0.6) the equilibrium equations stated in the lemma become ”(Ar — r (voz + (r/R). ) - g0. —p.. z.) = 0, (2.2.21) rAr + 2Vr - Vr + (r/R)r,1 = 0, (2.2.22) 7' p.(AZ + 2,1/R) + E(pd T,3 '—p,3 7,1) = 0. (2.223) Relative simplicity of the system suggests that for a priori simplified forms of func- tions r, r, 2 finding exact solutions could be possible. Some such possibilities will be explored in Chapters 5-7. 34 Chapter 3 Existence theorems In this chapter admissible sets appropriate for the Cartesian description of the ax- isymmetric deformations defined in the previous section will be introduced, and the main existence results for stored energy densities with and without dependence on the cofactor matrix will be stated and proved. To handle the incompressibility constraint (2.2.12), it is convenient to introduce the following expressions that are similar to the pointwise and the distributional determinants in genuine three-dimensional setting del (11) = (£2 + 772),1Z,3 —(€2 + 772)a3 3:1: (301) Del (11) = D1 ((g2 + 172)z,3) —- 193 ((§2 + 772)z,1) . (3.0.2) Now (2.2.12) takes the form del(u) = 2R a.e. in D. 35 Next we define four classes of admissible ordered triplets of functions 11 = (E, n, z): A“ 2: {u E Wl’p(D, 1R3) : u = no a.e. in I‘, cof F0 6 L"(D,M3X3), del (u) = 2R a.e. in D} A3” := {u e WW0, 1R3) : u = no a.e. in r, Cof F0 6 L"(D,M3"3), del(u) 2 2R a.e. in D} A" := {u E Wl’P(D,lR3) : u = no a.e. in I‘, del(u) = 2R a.e. in D} A5 := {u E Wl’p(D,lR3) : u = no a.e. in I‘, Del (u) = 2R a.e. in D} where I‘ C 0D, II‘I > 0, and no is a specified function in W1’1’(D, 1R3). Although the deformation (1.0.10) is in general three—dimensional, the integrand in the energy functional depends only on two variables. It is this reduction of the space dimension that allows for the relaxation of Ball’s a priori restrictions on the growth exponents p and q in the coercivity hypothesis. In the lemma below relations between certain pointwise and distributional null La- grangians are established. The lemma relies on the following theorem from [41]. Theorem 3.0.1 Let Q C IR” be open, 1 S p < n, v E WI'P(Q), and a' E L‘1(Q; R") for l/p + l/q — l/n S 1. If the distributional divergences Diva' and Div(v a') belong to L1(Q), then Divva’ = Vv(x) ~a'(x) + v (x) Div a’(x) ' a.e. in 52. Lemma 3.0.1 Let u E W1’3(D,IR3). 1. Ifs 2 4/3, then Cof Fo(u) E L1 => Cof Fo(u) = cof Fo(u). 36 2. Ifs 2 3/2, then Del (11) E L1(Q) => Del (11) = del (11). The equalities hold a.e. in D. Proof. Part 1 is proven in [41]. To prove Part 2, we set U = {2 + 172, a’ = (2,3, —z,1), s = 3/2 and check the hypotheses of Theorem 3.0.1. Clearly, q = 3/2, Diva = 0, and Div(v a) 2 Del (11) E L1(D). By Sobolev embedding theorem (continuous em- beddings) E, 17 E L”, where s‘ := 23/ (2 — s) = 6 is the critical Sobolev exponent. Using Holder’s inequality, it is easy to verify that v e Wl’6/5. In fact, 1/5 4/5 / |€€,. IG/Sda s f l€|6 f It. Imda < oo, i: 1.3. D D D The identical argument applies to 77m. N ow the statement of Part 2 follows from Theorem 3.0.1 since 19 = 6 / 5 is a borderline case for the inequality relating parameters in his theorem. I Remark. Note from Lemma 3.0.1 that AZ” C A” and A3 C .A”. The following theorem gives sufficient conditions for the existence of a minimizer of the reduced stored energy functional (2.2.14) when the stored energy density W depends explicitly on both F and cof F. The first statement ensures the existence of a solution to the minimization problem under weaker restrictions on the growth exponents p and q, but in a smaller set of mappings for which the entries of the distributional cofactor matrices are functions. 37 Theorem 3.0.2 Let W(F) = W(F,c0f F) and q > 1. Then the energy functional (2.2.14) assumes its minimum in admissible set A in each of the following cases: 1. A = A3”, p > 4/3, and p-1+ q'1 g 3/2. 2. .A = A“, p 2 2. The only fact that needs to be proven is that the admissible sets are closed with respect to weak convergence of mappings u and corresponding cofactors. The rest of the argument is standard (cf. discussion in Section 2.2). To begin, we need the following lemma concerning weak continuity properties of Cof (), Del (-), and del (-). Lemma 3.0.2 1. Let p > 4/3. Then the mapping Cof o : Wl’P(D,R3) —» ’D’ defined by Cof 0(u) = Cof Fo(u) is weakly continuous, i.e., uk —* u in Wl’P(D,R3) => [Cof Fo(uk)],j —> [Cof F0(u)],j in D’(D). 2. Let p > 3/ 2. Then the mapping Del : W” ——> D’ is weakly continuous, i. e., 11,, —\ u in Wl’p(D,IR3) => Del (uk) —> Del (u) in D’(D). ,3. Letp > 4/3, q >1, and p’1 + q‘1 S 3/2. Then (u,c —\ u in Wl'p(D,IR3) and cof Fo(uk) —* cof F0(u) in L"(D,M3"3)} => del (uk) —-» del (11) in D’(D). Proof. 1. This fact is well known (cf. . [4], [16]) and is stated here for completeness. 38 2. From Relich-Kondrakov theorem (compact embeddings) we have that weak convergence of the sequence uk in WI'P ensures strong convergence of some subse- quence (not relabelled) 11,, in L9 for any q such that 1 < q < p‘, where p‘ is the critical Sobolev exponent. In particular, for p > 3 / 2 this yields (6")? —> 52 and (12")? —> 172 in L3. Since l/p + 1/3 < 1, it follows from Holder’s inequality that products of the form (8)2235" and (0")2zfin for m = 1,3, are integrable in D. Consequently, Del (uk) 6 D’ and therefore for any fixed 45 E D(D) < Del (11ka >:= — f ((r")2 + (77")2) (z,§¢,1 —z,'; ¢,, ) da —» < Del (u),¢> >, D thereby proving the second part of the lemma. 3. First note that del (uk) = {k (cof F0(uk))22 — '0" (cof F0(uk))12. As in Part 2, it can be inferred that there exist subsequences 6", 17" such that 5" --» g and r)“ —> 17 in L" for 1 < s < p“. Combining this observation with the assumed weak convergence of the cofactor matrix and bounds on the growth exponents concludes the proof. I Proof of Theorem 3.0.2 . 1. By Lemma 3.0.1, Part 1, Cof Fo(u) = cof Fo(u) for u 6 AZ”, and existence of a weakly convergent in Wl'P(D, 1R3) minimizing sequence uk = (5",17",z"), as well as boundedness of cof Fo(uk) in L” are established in a 39 standard way. (cf. discussion in Section 2.2). Hence, for some subsequence 11;, (not relabelled) we have cof Fo(uk) —\ H in L". Now by Lemma 3.0.2, Part 1, one concludes that H = Cof F0(u), thus proving that Cof Fo(u) 6 L9. Consequently, the assumptions of Lemma 3.0.2, Part 3, hold, thereby implying that del(uk) —) del (11) in ’D’. Weak convergence of del (uk) to 2R in L' for any r > 1 follows from the incompressibility constraint. Hence del (u) = 2R a.e. and therefore 11 E 5’”, proving Part 1 of the theorem. 2. If u 6 WM”, p 2 2, the pointwise and distributional cofactor matrices of Fo(u) coincide, since for any fixed function 4’) E D(D) and for any fixed pair of indices i,j, 1 S i,j S 3, the functionals 9.01) = / (cof Few»... cbda and Mu) =< (Cof Fo(u)).,-, 9’) > D coincide on the dense set 02(D,1R3) C Wl’P(D, R3) and are continuous in Wl'P(D, 1R3) norm. (cf. [16], Theorem 7.5.1.) Therefore Part 2 is a particular case of Part 1 of the theorem. I Remark. To compare the assertions of Theorem 3.0.2 with analogous results in the genuine three-dimensional case, recall that the restrictions on the growth exponents for the analog of the second statement of the theorem are p Z 2, q _>_ 3/ 2 [42]. If one seeks a minimizer in a set {11 6 WW”: Cof F e L", DetF 6 L1}, 40 the restrictions are p 2 3/2, p”1 + q"1 S 4/3 [41]. The next theorem provides conditions for the existence of a minimizer when the stored energy density does not depend on cof F. As in Theorem 3.0.2, the first statement ensures the existence of a solution to the minimization problem under weaker restrictions on the growth exponent p, but in a smaller set of mappings u for which the distributional counterpart (3.0.2) of the expression del (11) is a function. Theorem 3.0.3 Let W = W(F). Then the energy functional (2.2.14) assumes its minimum in admissible set A in each of the following cases: 1. A: A5, p>3/2. 2. A = A”, p > 2. Furthermore, for p > 2 any minimizer u belongs to Hb'lder space Co'“(D) with 0 S a S 2/p, and there exists a minimizing sequence uk converging to u in C°'°‘(D)—norm for O S a < 2/p. Proof 1. It follows from Lemma 3.0.2, Part 2, that for any minimizing sequence uk converging weakly in W14" to a function u one has Del(uk) —> Del (11) in D'. On the other hand, the definition of the admissible set AZ implies Del (uk) —> 2R a.e. in D for some subsequence (not relabelled). Therefore Del (11) = 2R a.e. in D. The rest of the proof is standard. 41 2. For p > 2 there exists a minimizing sequence 11,, and some q > 1 such that cof F0(uk) —\ cof F0(u) in Lq. Then the weak closedness of the admissible set follows immediately from Lemma 3.0.2, Part 3. The statement about regularityand convergence in Holder spaces follows from Sobolev- Relich-Kondrakov theorems. I Remark. Note that the first statement of the Theorem 3.0.3 ensures the existence of a minimizer for neo—Hookean materials in the admissible set A3. It is tempting to obtain existence result for the case p = 2 in the larger set A2. This would be possible if W1'2(D) C Lfifc(D), but it is well known that in general a function f E Wm'Pm), Q C R" with mp = n, and n > 1, does not belong to L°°(Q). An example is provided by a function f = Iloglxlll‘z/("'1), n 2 2, defined in the ball B(0,r), r < 1, [55]. However, as will be shown in the next chapter, the existence theorem can be extended to the marginal case p = 2 under an additional constraint on the radial component in cylindrical description of the deformation. 42 Chapter 4 Global injectivity of axisymmetric minimizers The mere existence of a minimizer for a problem in nonlinear elasticity is not quite satisfactory. It is desirable to ensure some realistic properties of solutions, e.g., in- jectivity, which physically means that interpenetration of matter does not occur. For smooth mappings u 6 01(0) local invertibility follows from positivity of the deter- minant. However, this does not prevent overlapping of parts of the image u(Q). In this chapter the global injectivity of minimizers for cylindrical description of axisym- metric deformations is investigated. We make use of some properties of mappings of finite distortion that are collected in Section 4.1. The results on global injectivity of admissible functions as well as the extension of Theorem 3.0.3 to the case p = 2 are stated and proved in Section 4.2. 43 If a uniform positivity assumption (1.0.13) is imposed on the radial component, the direct method of the calculus of variations applies to axisymmetric deformations in cylindrical description, 11 = (r, r, 2). Then existence of a minimizer for the problem (2.2.14) in the admissible set A? for p > 2, stated in the second part of Theorem 3.0.3, can be obtained in exactly the same manner for the minimization problem (2.2.13) in the admissible set of triplets of functions 11 = (r, T, 2) defined by A2 := {u E Wl’P(D,lR3) : u = uo a.e. in I‘, (2.2.5), (1.0.13) hold a.e. in D} with a > 0. 4.1 Some properties of mappings of finite distortion Ball’s existence theory in nonlinear elasticity motivated introduction of a class of mappings of finite distortion since the admissible functions he introduced belong to this class. A class of mappings of finite distortion includes well known mappings of bounded distortion (or, equivalently, quasiregular mappings) [48]. The latter is a generalization of classical quasiconformal mappings [1], [25]. In this section some properties of functions of finite distortion needed in the sequel are stated. These prop- erties will allow one to obtain essentially sharper injectivity results in axisymmetric setting compared with those presented in [5], [15], and [52]. Definition 4.1.1 Let Q be a bounded connected, open subset in R". 44 A mapping f : 9 —-> R" is said to be a mapping of finite distortion (MFD) if 1. f e W“(Q,1R") Ice 2. The Jacobian J (2:, f) of f is locally integrable and does not change sign in Q. 3. There is a measurable function K : 9 —> IR such that K (x) Z 1, finite almost everywhere, and f satisfies the dilatation inequality |Df(:r)|" S K(x)|J(:r,f)| a.e. in 9. (4.1.1) The smallest of such functions K (), K (:17, f), is called the dilatation, or dis- tortion, quotient. In the theorem below some of the properties of functions of finite distortion are listed. The theorem is similar to Theorem 1.3 in [26] (parts 1, 3, 5 in both theorems are identical.) Theorem 4.1.1 Let f E Wl'"(Q,1R") be a MFD. Then 1. f has a continuous representative. 2. The following estimate of the modulus of continuity holds Irv - yl ’1” 2R |f($) - f(y)| S C(n,B)IIVfIILn(m 108 (4-1-2) with arbitrary :r, y E B(a, R) C B(a,2R) C Q. 3. f is differentiable a.e. 45 4. For every measurable set C C (2 the inequality man s f |J(a:.f)|dx 0 holds. 5. f satisfies condition (N), i.e., |f(E)| = 0 whenever E C Q and [E] = 0. Proof. See Theorem 1.3 in [26] for references to proofs of assertions in Parts 1, 3, 5. Part 2 is a particular case of Theorem 7.5.1 from [31]. The theorem is stated there for weakly monotone functions in Orlich-Sobolev spaces WP(Q) with Orlich function P satisfying the following conditions: °° dt / P(t)2;:f = oo, 1 the function t H P (t(2"+1)/(2"2)) is convex. Clearly, function P(t) = t", corresponding to Sobolev space WIND), satisfies those conditions. Without introducing the notion of weak monotonicity, we refer to The- orem 7.3.1 in [31], which states that the coordinate functions of a mapping f E Wl'"(§2, IR") with finite dilatation has this property. Part 4 follows from Theorem 1.4 in [25], Pg.274. The assumptions of the theorem are ensured by Part 3 and Part 5 of Theorem 4.1.1, and by local integrability of the Jacobian J(:r,f). I In the following lemma it is shown that for any relatively compact set G E Q a bounded set of MFD from Sobolev space with natural exponent is pre—compact in C(o). 46 Lemma 4.1.3 Let a sequence gk 6 Wl'“(f2, IR") of mappings of finite distortion con- verge weakly in Wl’"(Q,lR") to some mapping g. Then for any relatively compact domain G C 9 there exists a subsequence converging uniformly in C. 1 Proof: We show that a sequence of ith components 9;: of mappings 9;, is equicon- tinuous and uniformly bounded. To simplify notation introduce the scalar functions fl, = gfc, f = 9‘. Clearly, it follows from (4.1.2) that given 6 > 0 one can find 6, 0 < 6 < dist(C', BSD/2, such that for any pair :13, y, E G, [x — y] < 6, the left hand side in (4.1.2) will be less than e. This proves equicontinuity of the sequence fk. For a fixed 6 as above, there exists a finite covering 81- = B(zcg, 6), j = 1,... ,N, of C. By Relich-Kondrakov compact embedding theorem the sequence fk is pre—compact in any space L’, s _>_ 1. Since for any sequence of functions converging strongly in L" there exists a subsequence that converges almost everywhere, one can assume that for some subsequence, not relabeled, fk(1~3) —) f (x3) for every j. Uniform boundedness of fk in C then follows from the inequality Ifk($)| S lfk(rv) - fk(1'3)l+ lfk(1‘3)- f(1‘3)|+ max lf(r3)|, 15j_<_N where :I: E 83-. Note that the first term on the right is uniformly bounded due to equicontinuity. The statement of the lemma then follows from Arzela-Ascoli theorem. 1It was known to Lebesgue [34] that a family of continuous and monotone functions with bounded Dirichlet integral is equicontinuous. 47 If a nonconstant mapping f has finite distortion with integrable dilatation quotient (by definition f is called quasiregular in this case), then, by a fundamental result of Reshetnyak [48], the mapping is open and discrete. Here ’discrete’ means that the preimage of a point y e f (9) is a discrete subset of Q, i.e., it does not have cluster points. These properties were recently carried over to two-dimensional MFD with integrable dilatation quotient [30] . The result is stated in the theorem below. Theorem 4.1.2 Let Q be a bounded domain in the complex plane (C,o(z)), 0(z) being the area element, and f E W1’2(Q,C) with J(z,f) Z 0 and K(-,f) E L1(Q). Then there exists a homeomorphism h : (2’ -> (2, with Q’ = h‘1(Q) and a holomorphic function (f) : Q’ —r R2 such that f=¢°h”l- Remark. If the conclusion of the Theorem 4.1.2 holds, function f is said to admit Stiolow’s type factorization. Then, obviously, f is open and discrete. We will need also a change of variables formula for functions of finite distortion from Sobolev space Wl'"(fl, IR"). The following fragment of Theorem 2.2 from [48], Pg. 99, will be sufficient for our purposes. Theorem 4.1.3 Let Q be an open set in R", and f : Q —+ IR" a continuous mapping. Assume that 1. f has property N; 2. f is differentiable almost everywhere in Q; 48 3. Function :1: --+ J (x, f) is locally integrable in 0. Then for every nonnegative function g : Q -—) IR the function y->N(y.f,g),1\’(=y.f,g)=z 9(rr f‘(y) is measurable in IR" and / N(y,f.g)dy = fg 0}, we can find a sequence of open relatively compact strongly Lipschitz subdomains G,- C {(R, Z) : R > a,- > 0} such that GICGICch..., D=UG,-. (4.1.5) It suffices to show that for any relatively compact subset G in D the dilatation quotient of the mapping f = VIG is integrable. In fact, due to the incompressibility constraint the dilatation quotient satisfies the inequality r r 2 (T/R) r 2_ I 2 K R- > 0. Therefore there exists a positive constant C so that K(X’. f) S CIDV(X’)I2, and the integrability of the dilatation quotient in G follows. Since any open set E C D can be represented as a union of open relatively compact sets, E = U321 G,- H E, this completes the proof. I Remark. In the compressible case, if u = (r, r, z) E W1'2(D,1R3) and 7' . detF = —(r,1z,3 —r,;; 2,1) > O a.e. in D, R 50 then the two-dimensional mapping v still has finite distortion provided r _>_ 0 a.e., and therefore Theorem 4.1.1 remains valid for the mapping. But openness and discrete- ness are not available for v in this case without additional assumption, for example, detF 2 B > 0 a.e. in D. 4.2 Global injectivity theorems Now we are in a position to examine the injectivity of axisymmetric minimizers. For Dirichlet boundary conditions, the main result in [5] on global invertibility can be applied to the two-dimensional mappings v corresponding to u from an appropriate admissible set without imposing any condition on the adjugate matrix. This is shown in the next theorem. We recall that a domain U E R" is said to satisfy the cone condition if for all x E U a set {x + E (e(x))} is a subset of U, where E (e(x)) is the right circular cone of fixed radius and height with vertex at the origin, and a vector e(x) specifies the direction of the axis of the cone. Theorem 4.2.4 Let p > 2, A = Afyl with I‘ = (9D. If no, defining the boundary condition of place, is such that the corresponding v0 is continuous in D, one-to-one in D, and v0(D) satisfies the cone condition, then any 11 E A (in particular, any minimizer) is a homeomorphism of D onto v0(D), and the inverse function X’ () belongs to WI'P(v0(D)). If v0(D) is strongly Lipschitz, then v : D —r vo(D) is a homeomorphism. 51 Further, in the former (latter) case corresponding three-dimensional deformation (R,w, Z) —-> (r,w + r, z) is also a homeomorphism of (2 { S2) onto its image. Proof. All assertions of the theorem, except for the last one, are identical to those in Theorem 2 of [5] with the only missing condition / |(Vv)"l(X')|p det Vv(X') dX' < 00. D This condition can be established in exactly the same manner as the integrability of the dilatation quotient in Lemma 4.1.4. The last statement follows from the well-known general fact that a one-to-one, continuous, and Open mapping f : U —> V of topological space U into topological space V is a homeomorphism of U onto f (U) I To examine the injectivity of minimizers when boundary condition is prescribed only on the part of the boundary, we need an analogue of the famous injectivity condition by Ciarlet and Neéas [15], viz. [detF(u)dX S |u(f2)|. (4.2.6) 9 For isochoric axisymmetric deformation this condition simplifies to fRdX’ S / rdx’, x’ = (r,z), (4.2.7) D v(D) which can be recast into fdet Vw dX’ S |w(D)|, (4.2.8) D 52 where the mapping w is defined as w : D —> R2 W(X') = (p, z) := (r2, 2), (4.2.9) where the incompressibility constraint (2.2.5) was employed. Inequality (4.2.8) is the injectivity condition for axisymmetric setting. Remark. Although for isochoric deformations (4.2.7) looks simpler, the injectivity condition in the form (4.2.8) is similar to commonly used three-dimensional version (4.2.6) and, more importantly, can be used for the compressible case as well. Now we are ready to carry over the injectivity results obtained in [15] and [52] to the axisymmetric problem under consideration with essential sharpening due to higher regularity and the openness of the two-dimensional mapping v. For a fixed a > 0 introduce an admissible set A? := {u 6 AZ such that (4.2.8) holds}. The next theorem is the main result of this section. Theorem 4.2.5 Let W = W(F) and p Z 2. Then I. The energy functional (2.2.14) assumes its minimum in .4”). 2. For any 11 E A’,’ (in particular, any minimizer) the corresponding two - dimen- sional mappingv : D —> v(D) is a homeomorphism, and v‘1 6 my: (v(D), 1R2) . 3. If p > 2 then for any admissible function (in particular, any minimizer) u = (r, r, z) E A? a mapping (1 := (r, w + r, z) is a homeomorphism off? onto 13(9). 53 Proof. 1. It suffices to show that the incompressibility condition and the injectivity condition are preserved by weak limits of the elements from A’;. Let uk 6 Al}, 11,, —\ u in W1*P(D,IR3). Incompressibility condition for u will follows from weak continuity of the mapping del : Wl’p(D, 1R3) ——> ’D’(D). Note that condition r(R, Z) Z a > 0 combined with the incompressibility constraint implies that det Vv,c = R/r S C a.e. in D with a constant 0 independent on k, C 2 maxb R/a. Therefore for an arbitrary fixed 3, t > 1, there exists a subsequence 11,, (not relabelled) such that r)c -> r in L’ and det Vv,c —* det Vv in L‘. Since del (uk) = 2r det Vvk, the weak continuity of the mapping del follows. To prove that the injectivity condition (4.2.8) is preserved by weak limits, it suffices to show that the injectivity condition holds for any G CC D, i.e., / deti dX’ 5' |w(G)| (4.2.10) G for any w corresponding to a weak limit of a sequence of admissible functions. Indeed, then writing this condition with G = Gk, where G), is a subdomain from (4.1.5), and passing to the limit as k —> 00, we obtain the injectivity condition in D. Firstly, we prove (4.2.10) for any w corresponding to a mapping u 6 A’}. By 54 Theorem 4.1.1, Part 4, for any measurable set E C D the inequality |w(E)| g / detwdR dZ (4.2.11) E holds. If we assume that for some G C D (4.2.10) does not hold, then, by virtue of injectivity condition (4.2.8), there must be a set E C D of positive measure such that the inequality opposite to (4.2.11) must hold. Note that this contradiction implies even more than was claimed. In fact, it has been proven that for any G C D (4.2.10) holds with strict equality. Secondly, we prove that (4.2.10) is preserved by weak limits. Note that Theorem 7.9.1 from [16] does not apply directly to two-dimensional mappings wk in case p = 2 because these mappings belong to the space W1'2(G, R2) while wk 6 W111” with p > 2 is needed. However, the argument, used in the theorem, applies since it relies on the following facts: 0 Functions wk, w have (N) property. 0 Functions wk are continuous, and the sequence converges to w uniformly in C. o det Vw)c —\ det Vw in L9 for some q > 1. The last property is just the weak continuity of the function del, which is proved above, the other two follow from Theorem 4.1.1, Parts 1, 5, and Lemma 4.1.3. Using those facts we reproduce the argument of Theorem 7.9.1 [16] below with appropriate modifications. 55 Since the set W(C) is compact, whence measurable, there exists, by a classical property of the Lebesgue measure, an open set 0(6) such that W(G) C 0(6), |0(6) \W(C7)| < 6- Then it is easy to show [16] that there exists a number 6(6) > 0 such that U3 (3136(6)) C 0(6). where the union is taken over all y’ E W(G). Hence there exists an integer K = K (6) such that wk(C—}') C 0(6), for all k 2 K, since wk converges to w uniformly in C. Employing (4.2.10) one obtains / dethkdx' s Iw.(G)I = Iw.(é)l s l0(e)l, for all k .2 K, G where we used the fact that functions wk have (N) property. Passing to the limit as k —> 00 one obtains / detidX' s 10(6)) = Iw<é>| + l0(e) \w 1 for some x’ E v(D). By Lemma 4.1.4 the mapping v is open. Therefore there exists a neighborhood U C D of x’ such that N (v, x’) > 1, V x’ E U. This contradiction implies that N (v, x’) = 1 for all x’ E v(D). Noting that a continuous, open, and bijective mapping is a homeomorphism, the proof of the first statement in Part 2 is complete. The second statement follows from Theorem 3.1 in [22]. 3. If p > 2 the tree-dimensional mapping 1‘: is continuous, open and bijective. I Remark 1. A version of Theorem 4.2.5 can be proved in the compressible case with appropriate modifications. The argument used for proving the weak continuity of the determinant in Part 1 of the theorem fails. However, this property can be established as follows. Let u), -—* u in W1'2(D, 1R3), del (11) > O a.e., and r 2 a > 0 a.e. Then corresponding two-dimensional mappings vk are continuous, and for any relatively compact G CC D we have (up to a subsequence, not relabeled) rk rkm, —\ rr,m in L2(G). Now the weak continuity of del (u) = (r2,1, 72,3) - (2,3, “(3,1) 57 follows from the compensated compactness theorem [53]. Injectivity almost everywhere for weak limits in compressible case can be estab— lished via the injectivity condition (4.2.8) in the same manner as in Part 2 of Theorem 4.2.5, but without the openness of the mapping v stronger assertions of the theorem are not in general true for the compressible case. Remark 2. Clearly, by Theorem 4.1.2, two-dimensional isochoric deformations from Sobolev space WI’P(Q, R2), 9 C R2, p 2 2, are open and discrete. This observation can be used in exactly the same manner as in the proof of Theorem 4.2.5, Part 2, for sharpening the injectivity results of [15], [52] for this class of deformations, but we do not pursue this issue here. 58 Chapter 5 Governing equations for TIE motion Beginning with this chapter, more specialized axisymmetric deformations than those examined in the previous chapters are considered, but they are assumed to depend on the time variable. Here the governing elastodynamic equations for motions involv- ing axially varying twist, radial inflation/deflation, and axial contraction/ elongation (TIE) and introduced by the equations (1.0.14), r = Rs(Z, t), 6 = w + r(Z, t), z = z(Z, t), are derived. Attention is henceforth restricted to neo-Hookean materials whose strain energy density is given by (1.0.6). The nominal stress tensor S, given in the incompressible case by (1.0.7), reduces 59 for neo-Hookean material to S = ”FT — pF-l. (5.0.1) Direct computation shows that for TIE motion the isochoric constraint (1.0.4) be- comes (92 1 — = —. 5.0.2 BZ 32 ( ) Utilizing the standard expression of the acceleration vector in cylindrical coordinates and the equilibrium equations for neo-Hookean material (2.2.21)-(2.2.23) enables the linear momentum balance ( 1.0.8) to be written as M [AT ‘ 7' (VT)2 + (r/R),1] — (r/R)(p.1 2,3 ‘R3 2,1) = P0" - ”2), (50-3) u [rAr + 2Vr - Vr + (r/R)r,1] = p(ri‘ + 2H), (5.0.4) A2 + 2,1/R + (r/pR)(p,1r,3 —p,3 r,1) = p2. (5.0.5) For TIE motion these equations transform into moss—sat) —p..z,3s = pRs—sr"). (5.0.6) ”(3T133+23r37-a3) = p(ST+2ST)7 (5"07) #2,33+Rss,3p.1-szp,3 = pi (5-0-8) We seek to investigate controllable motions associated with this system. Here the term ‘controllable’ is used in the sense that the motion (or deformation) is sustainable by surface tractions alone. For our purposes, a suitably smooth set of functions s(Z,t) > 0, r(Z, t), 2(Z, t) defined on [Zl,Z2] x [t1,t2] (where Z] and Z2 may take 60 infinite values) is said to define a controllable axially varying TIE motion via (1.0.14) for a neo-Hookean material if: (a) the mapping in cylindrical coordinates defined by (1.0.14) at fixed t is one-to- one, (b) the constraint (5.0.2) is satisfied, (c) there exists p(R, Z, t) such that (5.0.6)—(5.0.8) are satisfied. Recall [44] that the surface traction per unit area on the vector area element da in B (t) is denoted by t and is given by t da = STN dA = STdA, (5.0.9) where dA is the corresponding vector area element in 30 with associated unit normal N. Note that F = F(R, Z, t) and x = )2(R, Z, t) implying that p = p(R, Z, t). (5.0.10) Next we simplify the system (5.0.6)-(5.0.8) by eliminating the pressure. To this end equations (5.0.6) and (5.0.8) can be solved for p] and p3,: 10,1 = #R8(s.33 -sr,§) — pRS(§ - 3+2), (5.0.11) 19,3 = ,U (R23,3 (8,33 —ST,§) + 2,3 2,33) — p (R28,3 (S — 37.2) + 2,3 2) ,(5H012) where (5.0.2) gives certain simplifications. Equating the cross-derivatives of p gives an equation 3 (u(s,33 —sr,§) — p(§ — 3522)), — s,3 (u(s,33 —sr,§) — p(§ — 379)) = 0. 61 Integration then provides #(3,33 _STé) — P(5 — 3+2) ‘ C(t)s = 0, where C(t) is, in general, an arbitrary function of time. Thus the system (5.0.6) - (5.0.8) reduces to the following system of two coupled nonlinear PDE for r, r: u(s” — sr'z) — p(.§ — si'z) — G(t)s = 0, (5.0.13) p(sr" + 2s’r’) — p(si‘ + 25+) = 0, (5.0.14) where prime stands for differentiation with respect to Z. Note from (5.0.11) and (5.0.13) that p3 = RszC(t) whereupon p(R, Z, t) = §C(t)st2 + p(z, t), (5.0.15) with (5.0.12)—(5.0.15) providing 133 = —2,us_5s,3 —ps‘22 4:) p = g54 — p/s‘zi dZ + po(t). (5.0.16) The determination of controllable motions now reduces to the determination of suit- able functions s(Z, t) and r(Z, t) that satisfy the governing system given by the two second order nonlinear partial differential equations (5.0.13) and (5.0.14). The func- tion C(t) may be arbitrarily chosen. The axial contraction/elongation 2(Z , t) then follows from integration of (5.0.2) and so is determined up to an arbitrary function of time that represents an axial rigid body displacement. The pressure p(R, Z, t) fol- lows from (5.0.15) and (5.0.16), and so is also only determined to within an arbitrary 62 function of time. One finds that the principal stretches A1, A2, A3 are given in terms of functions 3, r by 1 A? = %(’n + '72)/32, A3 = §t('Yl - 72)/sz, and A3 = 32, (5.0.17) with 71 2 1+ 5, 72 = {-4s6 + (1 + fl)2}1/2, ,6 = R234s,§ +36(1 + R273 ). Thus, in general, all three principal stretches vary in both space and time. For the case r(Z, t) = r(t) it is seen that eg is a principal direction on B(t). For the case s(Z, t) = s(t) it is seen that e, is a principal direction on B(t). A case of some physical interest is that for which [30 is a cylinder whose cross- section is an annulus with inner radius R1 2 0 and outer radius R2 > R1. Notice that cross-sections of constant Z map into cross-sections of constant 2. On the lateral surfaces R = R1 and R = R2, N = :hER giving sTN = :l:(/is(Z, t) — p(R, Z,t)/s(Z, t))e. i Rp(R, z, t)s(Z, t)s’(Z, t) e, (5.0.18) and thus determining sustaining surface tractions via (5.0.9). In addition, tractions associated with ends at any fixed Z = Z require a resultant axial force N and a resultant twisting moment M given by N(Z,t) := 27r R232.(R,Z,t)RdR R] 7m ((192))2-(0’31)2 7r 4 4 4 ——2—- 32m” — ZG(t)s (Z, t) ((122) — (R1) ) — «s2(z,t) ((152)? — (R1)”) (poo) — p / :((ZZ”,’)dZ) , 63 (R?) M(Z,t) := 2n / r(R,Z,t)Szg(R,Z,t)RdR R1 = gusz(Z,t)r'(Z, t) ((122)4 — (air). As a premliminary and very simple example consider the case s(Z, t) = constant. In order to restrict rigid body motion, take 2(0, t) = 0 so that (5.0.2) gives 2(Z, t) = Z/sz. Then (5.0.14) requires r(Z, t) = h+(Z+ Mt) +h_(Z— \/;_1/_pt) for arbitrary functions h+ and h_, whereupon (5.0.13) gives that either r(Z, t) = h+(Z + Mt) or r(Z, t) = h- (Z — Mt). The motion is thus a single travelling wave. In addition, C (t) = 0 and the pressure p = po(t). According to (5.0.18), this solution is supported by uniform normal traction on the lateral surfaces R = R1 and R = R2. This includes the case of traction free lateral surfaces obtained by taking po(t) = #32. The resultant axial force N (Z, t) associated with this traction free solution is N (Z, t) = 7r,u/2((R2)2 — (R1)"’)(s‘2 — 34), also a constant. In particular, N < 0 for axial contraction (s > 1) and N > 0 for axial elongation (s < 1). The twisting moment is given by M (Z ,t) = rru/232((R2)4 — (R1)")szh§t(Z :l: \fu/—pt), where prime denotes derivative with respect to the argument. Thus M, unlike N, varies with the passage of the travelling wave. More general travelling wave motions wherein s is not necessarily constant are discussed in the next chapter, where we obtain closed form solutions for the following four classes of controllable motion: 0 controllable deformation, which is the special case for which 3 = s(Z), r = r(Z), 2: 2(2); 64 o controllable travelling waves, which is the special case for which 5 = s(Z — ct), r = r(Z _ ct), 2 = 2(Z — ct) where c is a constant; c controllable simple twist motion, which is the special case for which 7' = a2 + r0(t) where a is a constant; 0 controllable motion with a Riemann type similarity variable, which is the special case for which 3 = s(Z/t), r = r(Z/t). 65 Chapter 6 Closed form solutions for TIE motion 6.1 Controllable deformations Controllable deformations s = 5(2), 1' = r(Z), 2 = 2(Z) provide equilibrium so- lutions to the equations of motion (5.0.6)-(5.0.8). Then the equations (5.0.14) and (5.0.13) (under the replacement C —-2 [10) give sr" + 2s’r' = 0, s” — sr’2 — Gs = 0. (6.1.1) The first equation in (6.1.1) gives r”/r’ = —2s’/s, which upon integration provides 1" = cl/sz, (6.1.2) where c; is a constant of integration. Substitution from (6.1.2) into (6.1.1)2 gives 3" — (cf/s3) - Gs = 0. (6.1.3) 66 Introducing q(s) := s’ and using the relation 3” = q(dq/ds) permits (6.1.3) to be rewritten as q(dq/ds) = Cs + cf/s3, so that integration provides q2 = 032 — (cg/s2) + c2, where c2 is another integration constant. Since q = ds/dZ, one obtains := :I:Jc. Z = :l: / sds \/Gs4 + 0232 — cf Evaluation of Jc is sensitive to the sign of G giving results as follows from subsequent elementary calculations: (i) If G < 0 then 2 , s -B arcsm —— —c3, J0: A 1 2v-C where c3 is an integration constant, and the new constants A > 0, B > A take the place of c1, c2. Since Cl is necessary for the determination of r(Z) from (6.1.2), it is noted that c1 = :t,/G(A2 — 32). Hence in this case 1/2 s(Z) = (B + Asin 2a(:l:Z + 5.)) (5.1.4) with a = \/—G and c1 = :l:a\/B2 — A2. (ii) If G > 0 then a corresponding calculation yields 1 NC? Jc= lnlsz—B+\/(s2—B)2—A2[—c3, where A, B with A2 > 82 replace c1, c2. In this case 2 s(Z) = (B + éexp (2a(iZ + c3)) + 52— exp ( — 2a(:l:Z + c3))) 1/2 (6.1.5) 67 with a = \/G and c1 = :l:a\/A2 — B2. (iii) If C = 0 then s(Z) = (B2 + A2(Z + c3)2)1/2 (6.1.6) with c1 = :tAB. The functions s(Z) given in (6.1.4)—(6.1.6) provide the framework for a family of controllable deformations. Given any such s(Z) the associated 7(2) and 2(Z) follow respectively from (6.1.2) and (5.0.2) and so differ from the integral only by (distinct) multiplicative factors and (distinct) constants of integration. If s(2) is given by (6.1.4) then, to within an integration constant, HZ _ 1 t Btana(Z+c3):l:A <>-;——./B—2_—Aza‘”“ ./——32_.42 - If s(2) is given by (6.1.6) then, to within an integration constant, H(Z) = Zl—B—arctan—é—(Eg—(i) If s(2) is given by (6.1.5) then H (Z) can again be expressed in terms of elementary functions, but the expression is rather cumbersome; the particular case B = 0 gives, to within an integration constant, H(Z) = fiarctan exp2a(4Z + 63). The following theorem summarizes this development. 68 Theorem 6.1.6 For arbitrary constants a, c3, c4 and c5, and constants A, B sub- ject to the restrictions listed below, each of the following sets of functions represent controllable TIE defamation for a neo-Hookian solid on an appropriate interval in Z. (i) For [B] > [A] and any Z—interval of a length less than 1r/a: 1/2 ’ r(R, Z) = B(B + Asin 2a(Z + 03)) Btana(Z + c3) + A «E27712 ”4’ 6(a), Z) = w :t arctan 1 Btana(Z+c3)+A+ 2(Z) = —————arctan .../m m (ii) For [A] > [B] and any Z-interval: r(R, Z) = Rs(Z), dZ 6(w,Z) = wiaVAz—sz— +c4, 32(2) dZ 2(Z) = [mi—C5, where s(Z) is given by (6.1.5). (iii) For arbitrary A, B and any Z—interval: r(R, Z) = R(B2 + A2(Z + c3)2)’/2, 0(w,Z) = wimctanflBigl+c4, 2(Z) = Elgarctanw-l-cg C5. In the above formulae, constants c4, c5 represent rigid body motion, whereas constant c3 is a simple offset distance for the dependence on axial coordinate Z. Symmetry 69 with respect to clockwise and counterclockwise twist is provided by the :l: in the formulae for 6(a), Z). The associated pressure is determined directly from (5.0.11) and (5.0.12) to be H R 4 l‘ 2 p(RZ) = Elm) — 5mm» +p. where p0 is an arbitrary constant and a = 0 for case (iii) above. Formally one may take p0 = po(t) if desired. It is worth noting that Theorem 6.1.6(i) with A = 0 gives the relatively simple deformation r=\/BR, Ozwia(Z+c3)+c4, 2=(Z+c3)/B+c5 in which twist (characterized by the parameter a) decouples from the radial inflation / de— flation and axial contraction/elongation (characterized by parameter B). This spe- cial case represents one of the universal deformations for an arbitrary homogeneous, isotropic, incompressible, hyperelastic material [21]. Other than this special case, the deformations described in this section do not correspond to a universal deformation for an arbitrary homogeneous, isotropic, incompressible hyperelastic material. This deformation also represents the only solution that is physically meaningful for the infinite Z-interval. For the periodic solution in Theorem 6.1.6(i), the Z-interval is restricted by the period of the solution; for solutions in Theorem 6.1.6(ii) and (iii), the radial deformation grows without bound as Z —1 ice. Figures 1 and 2 depict the deformation of the coordinate plane 1.) = 0 for conditions representative of Theorem 6.1.6(i) and (ii), respectively. The left part of each figure 70 E \ Figure 1: Deformation of the coordinate plane or = 0 for a case in Theorem 6.1.6 with B > A. depicts the plane in the reference configuration for Z1 S Z S Z2, 0 S R S R0. In both cases the values of the parameters a, A, B were chosen a priori, and the other three parameters, c3, c4, c5, found by precluding deformation of the section Z = Z via s(Z1)= 1, r(Zl)= O and 2(Zl) = Z1. Figure 1 depicts the case A = 0.5, B = 1, a = 0.3 with Z1 = —4, Z2 = 4, R0 = 2. It illustrates radial inflation combined with axial contraction (when —4 < Z < 7r / (2a) — c3) and radial deflation coupled with axial elongation (when 1r / (2a) — c3 < Z < 4). Figure 2 depicts the case A =1, B = 0, a = 1 with Z1: 0, Z2 =1, R, = 1. It illustrates radial inflation combined with axial contraction. Notice that assigning the resultant force and moment for the section Z = 22 71 Figure 2: Deformation of the coordinate plane or = 0 for a case in Theorem 6.1.6 with B < A. generates two additional conditions that can in principle be associated with the de- termination of A and B (or equivalently c1 and oz.) The constants p0 and a remain unrestricted in this assignment. 6.2 Traveling waves Traveling waves in the axial direction are motions in which the dependence on Z and t is via the similarity variable 17 := Z — ct. The constant c is the traveling wave speed. Consequently, s = s(Z — ct), r = r(Z — ct), 2 = 2(Z — ct). The special case c = 0 retrieves the static deformations discussed in the previous section. For the purpose of the present section, the prime notation will denote differentiation with respect to the similarity variable argument 1) = Z — ct. Introducing s = s(n),r = r(n) into 72 (5.0.14) and (5.0.13) gives (u - pc2)(sr” + 2s’r') = O, (u — pc2)(s” - sr'2) — Gs = 0. (6.2.7) For any suitably smooth functions 3 and 7‘, this system is satisfied with G = 0 for traveling wave motion at the neo-Hookean shear wave velocity, c. := im- Alternatively, if s and r satisfy (6.1.1), then they satisfy (6.2.7) under the re- placements Z —+ r] and C -r (u — pc2) G. As in (6.1.1) it is necessary to take G independent of t in order to obtain solutions for this second alternative. In summary, TIE traveling waves are supported in a neo-Hookean material. For propagation at the shear wave velocities :hc.., any appropriately invertible functions s(n) > 0, r(n) define such a controllable traveling wave via r(R, Z, t) = Rs(n), 0(Z,t) = w + r(n), 2(Z, t) = /;%, 17 = Z :l: c..t. At all other traveling wave speeds c aé :l:C,.., TIE traveling waves are supported in the forms defined in Theorem 6.1.6 provided that the independent variable Z is replaced by Z :1: ct and the constant C is replaced by (u — pc’) C. Notice for fixed G of the static deformation, that, under these replacements, the associated traveling waves change their form in transitioning from the subsonic case c < c. to the supersonic case C > C... 73 6.3 Simple twist motion Controllable simple twist motion is here defined as r = aZ + r0(t), where a is a constant, so that (5.0.14) and (5.0.13) become II P 2uas' -— p(srb + 2571,) (6.3.8) II _o u(s" — azs) — p(si — 37‘02) — C(t)s (6.3.9) For the purpose of the present section, the prime notation will denote differentiation with respect to Z. We consider three special cases. (i) Suppose To is constant and a ,6 0. Then (6.3.8) is satisfied if and only if s = s(t) whereupon (6.3.9) gives .6 + p-1(C(t) + [102)3 = 0. (6.3.10) Hence any sufficiently smooth s(t) > 0 is consistent with such motion by taking C(t) = —ua2 — ps'/s. Note that (6.3.10) is of the general and standard form ,7) + h(t)y == 0. (6.3.11) This equation has been intensively studied from a variety of perspectives. (See, for example, [9].) When h(t) is periodic it is called Hill’s equation, whereupon a formal series solution is readily obtained [40]. In our framework it is possible to reduce the order of the equation (6.3.11) using the positiveness of s(t). The new dependent variable v = ln s transforms the equation into u” + (u')2 + h(t) = 0. 74 One more change of the dependent variable v = u’ results in the first-order ordinary differential equation v' = —v2 — h(t), which can be treated by appropriate standard methods. (ii) More generally, suppose s = s(t). Then (6.3.8) integrates so as to give 1'0 = c./s2 (6.3.12) with Cl a constant. Substitution from (6.3.12) into (6.3.9) gives 3' - (cfi/si‘) + p"1(C(t) + uaz)s = 0. (6.3.13) Once again, any sufficiently smooth s(t) > 0 gives rise to controllable simple twist motion, now by taking C(t) = —ua2 — p.§/s + pcf/s". The choice c1 = 0 retrieves the results for case (i). Alternatively, if s is not assigned and if C(t) = C, a con- stant, then, somewhat surprisingly, (6.3.13) is of the same form as (6.1.3) although the independent variable is now t instead of Z. Accordingly, the various solution forms (6.1.4)—(6.1.6) can be appropriated with simple modification. For an arbitrary G = G (t), (6.3.13) can be examined by appropriate numerical or qualitative methods for ordinary differential equations, although we do not pursue this issue here. (iii) Suppose To is constant and a = 0. Then (6.3.8) is satisfied identically whereas (6.3.9) becomes a single linear partial differential equation, us” — p5 — C(t)s = 0. (6.3.14) 75 Solving (6.3.14) for C(t) and requiring G’ = 0 gives III I II u(ss — s s ) — p(ssi' - s'S') = 0 (6.3.15) as a necessary and sufficient condition for s(Z, t) to satisfy (6.3.14) for some C(t). The particular solution of (6.3.15) given by s = s(t) retrieves a motion corresponding to (ii). Separation of variables on (6.3.14) formally gives two sets of solutions. The first set involves either 3 = b(t) sinh/fl/nZ) or s = b(t) cosh/B/uZ), (6.3.16) where B > 0 is an arbitrary constant and b(t) satisfies 5+ p‘1(C(t) + 6):; = 0. (6.3.17) The second set involves either 8 = b(t)exp(\/fl/MZ) or s = b(t) exp(-\/B/#Z). (6.3-18) where B > 0 is an arbitrary constant and b(t) satisfies 25+ p‘1(C(t) — 6)!) = 0. (6.3.19) Both equations (6.3.17) and (6.3.19) are of the form (6.3.11). In view of the linearity of (6.3.17) and the arbitrariness of G (t), any superposition of solutions (6.3.16) and (6.3.18) with )6 = Bk, b(t) = bk(t), k = 1,2,... will also provide a formal solution. At issue then is the positivity requirement on s. This requirement would provide restrictions on {6h bk(t)} that incorporate, for example, the length of the Z -interval on which the motion holds. 76 Note also for C (t) = C, a constant, that (6.3.14) is the classical telegraphy equa- tion for which there are standard treatments (see, for example, [17]). In particular, integral representations for solutions of various initial/boundary value problems in terms of corresponding Green’s functions are given in [54]. 6.4 Motion with a Riemann type similarity variable For wave propagation problems in one spatial dimension, solutions in terms of the similarity variable 5 := Z /t are central to the analysis of initial value problems characterized by step function initial data [50]. Riemann’s problem in gas dynamics, and shock tube problems in general, provide standard examples. In the present case (5.0.2) is inconsistent with non-trivial solutions such that both 3 = 3(5) and 2 = 2(5). However, solutions with s = 3(5) and r = r(5) can be considered, whereupon, as discussed previously, 2 = 2(Z, t) and p = p(R, Z, t) follow from (5.0.2), (5.0.11) and (5.0.12). For the purpose of the present section, the prime notation will denote differentiation with respect to 5 = Z / t. Introducing s = 3(5), r = r(5) into (5.0.14) and into (5.0.13) followed with multiplication by t2/p gives, respectively, (c. — 52)(sr" + 2s'r') — 2537" = 0, (6.4.20) (c. — 62x3” — s—r’z) — 253' — gig—’93 = 0. (6.4.21) 77 Note from (6.4.21) that solutions consistent with this framework can be constructed only if C(t)/p = kot’z, (6.4.22) where k0 is a constant. Equation (6.4.20) can be written as [(c. - 52)sr’]’ + (c. — 52)s’r' = 0. Dividing by the expression in brackets gives [In ((c. - 52)sr')]’ + (ln s)’ = O, whereupon integration and solving for r’ gives I 61 ’ z (c. —e>s2’ (6.4.23) where Cl is a constant of integration. Entering (6.4.21) with both (6.4.22) and (6.4.23) then leads to (of - 52)?! — 2(c3 - {2)63’ — 53-3 — ko(c3 — 52)s = 0. (6.4.24) We now show that (6.4.24) can be recast so as to eliminate first derivatives. This recasting is somewhat different for subsonic waves ([5] < c.) and supersonic waves (It! > 0*): (i) For —c. < 5 < c..., introduce the new independent variable c..+5 C = 1n (6.4.25) «m 5‘ 78 so that 2c, _4__c..5 2 4cfeC r_ n_ _2_ (Hz—52’ 4 (c2— é—Z—V’ 6* “(M1)? After standard manipulations equation (6.4.24) for s = 3(5) becomes d23 eC _ —3 _ _ = kls k°(eC + 1)23 0, (6.4.26) where k1 = cf/(4cf) 2 0. (ii) For both 5 < —c. and 5 > c.., introduce the new independent variable {+0. ( =1n é _ c. (6.4.27) so that (I = —&__, C” = ——4—-——C*€ , C3 — £2 : —._‘.1£3£(__ c3 — 6 ("ca—_- 52)? (e< - 1)? After standard manipulations equation (6.4.24) for s = 3(5) becomes dzs _3 6C d—C2 - kls + [Cows = 0, (6.428) where again k1 = cf/(4cf). Note that (6.4.26) and (6.4.28) are of the same form as (6.3.13) (with an appro- priate choice of C(t)), although the independent variable is now 5 instead of t. Those equations are potentially more convenient than (6.4.24) for numerical computation. They are also convenient for further analysis as described next. Here we only consider the separate special cases of k1 = 0 and k0 = 0. Case k1 = 0 implies c1 = 0 and it follows from (6.4.23) that the motion is twist- free in the sense of Section 5. The case k1 = 0 can therefore be developed directly 79 from (6.3.14) by requiring the additional specializations (6.4.22) and s(Z, t) = s(Z / t). Both (6.4.26) and (6.4.28) are then of the general form (6.3.11), and the comments following that equation apply. In the case k0 = 0 both (6.4.26) and (6.4.28) reduce to a common form. This form is integrable after multiplication by ds/dC giving (ds/d5)2 + k1 /s2 = k2, where kg 2 0 is the integration constant. Yet another integration yields 1 / C + In 1133 = 76; [$382 — ’61, (6.429) where the integration constant is written as In k3 with k3 > 0. For subsonic waves, solving (6.4.29) for s > O and invoking (6.4.25) gives +5 2 1/2 _ C. 3(5) _ (12+ (In 1:36, _ 6) ) , (6.4.30) where k4 = k1 /k§ _>_ 0. It is readily verified that (6.4.30) applies also to the supersonic waves (where 5 is given by (64.27)) provided that k3 < 0. It is to be noted that s as given by (6.4.30) is unbounded as Z/t —* :lzc... In summary, TIE motions such that s = s(Z / t), r(Z / t) are supported in a neo- Hookean material. On the subsonic characteristic curves, the function 3 must satisfy the second-order ordinary differential equation (6.4.26) where k1 Z 0 and k0 are otherwise arbitrary. On the supersonic characteristic curves, the function 3 must satisfy (6.4.28) where, again, k; 2 0 and k0 are otherwise arbitrary. In both cases, 1' then follows from (6.4.23) using c1 = i2c.\/k_1, in general giving rise to a five parameter family of solutions for s and r. 80 Chapter 7 Cartesian description of TIE and TIES motions In this chapter we obtain Cartesian descriptions of TIE and TIES motions. It is found by trial that the reduced nonlinear system of PDE for the radial and angular components of TIE motion, derived in Chapter 5, admits a variational formulation. Formal change of dependent variables transforms the Lagrangian of the correspond- ing variational problem into quadratic expression with respect to new dependent variables therefore leading to a linear decoupled system of Euler-Lagrange equations. Governing equations for a more general class, called TIES motions, are also derived. Although in addition to twist, infiation/ deflation, and contraction/elongation func- tions, describing TIE motion, two unknown functions accounting for in-plane shear are introduced into the ansatz for TIES motions, the governing system is shown to 81 decompose into four identical decoupled linear equations of the same type as for TIE motions. The governing system for general axisymmetric motions of neo-Hookean body is also transformed in the same manner as TIE motion into a system that seems to be more convenient for further investigations than the original one. 7.1 TIE motion in Cartesian description Consider the system of nonlinear PDE (5.0.13), (5.0.14) for functions 3, 1' which determine the radial and the angular components of TIE motion. The system can be recast under replacement G —+ pG into 3' -— cfs” — s(r'r2 — c3712) + C(t)s = O, s(i‘ — GET”) + 2(5+ — cfs'r') = 0. Direct computation shows that these are Euler-Lagrange equations of variational problem with Lagrange density given by c = (5'2 — 533” + (3+)2 —— c3(sr')2 — C(t)52). (7.1.1) NIH Introducing new dependent variables by formulae 5=scosr, n=ssinr the Lagrangian (7.1.1) becomes 5 = g (£2 + 7'72 — 6305” + r) — C(t)(€2 + 722)) . 82 Indeed, 3’2 + (37’)2 = ——(€:;:’:72’)2 + —————(”I€€,;’:§)2 = (5’)2 + (17’)? 'Ifansformation of the terms involving time derivatives is identical. Euler-Lagrange equations corresponding to the transformed Lagrangian take the form 5" — c325" + C(t)5 = 0, 0' — c377” + on)" = 0. (7.1.2) These equations are of the form (6.3.14), and the comments after that equation apply, except for the one concerning positivity of solutions, since neither of functions 5, 7) needs to be positive. Axisymmetric description (1.0.14) of TIE motion translates in terms of functions 5, r), 2 into the following Cartesian description dZ x: X5—Yn, y=Y5+Xr7, 2 = /—§2+Tl2 +2o(t), (7.1.3) where functions 5, n satisfy equations (7.1.2), and 20 is an arbitrary function of t. 7 .2 TIES motion in Cartesian description Here we consider more general than TIE class of motions for neo-Hookean body given by the ansatz .23 = X€(Z’t) — YTKZat) + f(Zrt)i y = Y6 + X77 + 9(ZiT)i 2 = Z(Zat)' (7"24) For f = g = 0 (7.2.4) reduces to a Cartesian description of the previous TIE motion. The addition of nonzero f and 9 can be interpreted in terms of transverse ghear. 83 Hence (7.2.4) will be referred to as TIES motion. The motions investigated in [47], [2] are particular cases of TIES. In [47] it is assumed that r) = 0 (twist free motion) and 5 = 5 (t), n = q(t) (no dependence on the axial coordinate in 5, 1)). The model in [2] does not include inflation/ deflation, and therefore, due to incompressibility, elongation/contraction. To simplify derivation of the governing equations for TIES motion the following simple technical lemma, aimed at eliminating the pressure terms in the governing equations, will be helpful. Lemma 7.2.5 Let a motion of a neo-Hookean solid be determined by functions xi = $(X1,X2,X3,t), 2:11213) that are three times continuously differentiable. Then If the body is simply connected, corresponding pressure can be found from the equations pik : $i,kai7 2" k = 1,213 (7.2.6) Here and below notation E] := 11A — p82/8t is used for d’Alambertian operator. Proof. Since the nominal stress of neo-Hookean material is given by (5.0.1), the general equation of isochoric motion for hyperelastic material (1.0.8) translates for this material into Div(uFT — pF‘l) = pv. 84 Using N anson’s formula this can be rewritten as )1 Div(FT) — F’TVp = pv. (7.2.7) Pre-multiplying (7.2.7) by FT one obtains W = [.iFT Div(FT) — pFTv, (7.2.8) which is a vector form of (7.2.6). Applying the cross product operation with the Operator V to both sides of (7.2.8) gives W x (FTDiv(FT)) = W x (FTv). In tensorial notation this reads Gmnk (Iwajjl‘m - Piixu) m = 0, 01' Gmnk ((D $i)$i,k) m = 0- Since 6mnkx,,kn = 0 this implies 6mule“:l $i)m xiJc : 0: which is (7.2.5) in tensorial notation. I Next we apply the vector equation (7.2.5) to TIES motion (7.2.4). The equation will be shown to split into a system of five equations for unknown functions 5, n, f, g, and G = G (t), where C is an auxiliary function simplifying the structure of the system just as in the case of TIE motion. To this end we need to compute the left hand side of the equation V(D x) x Va: + V(C| y) x Vy + V([:l 2) X V2 = 0. (7.2.9) 85 Clearly, the last term on the left hand side is zero for the ansatz (7.2.4). From (7.2.4) we derive Vx=[s. —n, X€’-Yn’+f’]’DV$=[D£. —Dn. XD€’-YDn’+Df’]- Then -Dn(X€' - Yn’ + f’) + u(XClé’ - YD 71’ + Elf’), p DVx x Vx = 5(XD5’ — mn'+ Df’) — maxg' — Yn’ + f’). EDn-nflé Similarly, D€(Y€’ +Xn’ +9’) - {(Yflé’ + X00’ + 09’), DVyxVy= nuIMH5MJq+Uy)—DMY6+XW+62 600-006 Now equation (7.2.9) can be written in the following scalar form A(€,n)X+B(€,n)Y+an’-€Dg’-f’Un+g'C|€ = 0, (7-2-10) -B(€,n)X+A(€,n)Y+€Df’+an’-f’D€-g’Dn = 0. (72-11) 5077 — nCI5 = 0, (7.2.12) where A = -€Un+nD€+nTM-£Ud=4d36“-@Dnfl U213 B = rfCln — 77C] 77' + 5'05 - 5D 5' = —(97;71)’172 — (~E%)I52. (7.2.14) Equation (7.2.12) implies D_’Z_% 7) €’ 86 which can be written as D5—C5=0 Eln—Cr)=0 (7.2.15) with an arbitrary function G = C(Z, t). These equations are identical to (7.1.2). Equations (7.2.10), (7.2.11) imply 2405.77) = 0. B(€.n) = 0, (7.2.16) 77C] f’ — 5E] g' — f’Clr] + g’Cl5 = 0, (7.2.17) 5D f’ + 77!] g' — f’CI5 -— g'Dr] = 0, (7.2.18) where A, B are defined by (7.2.13), (7.2.14). The first condition in (7.2.16) is true identically by virtue of (7.2.15), while the second implies C ’(é2 + 772) E 0- Assuming 52 + 172 56 0, this means that function C does not depend on the axial coor- dinate, i.e., C = C(t). From (7.2.17), (7.2.18) one obtains after some manipulations le’ — Cf’ = 0, Elg’ — Cg’ = O, or, equivalently, Elf -— Cf 2: D1(t) Clg — Cg = D2(t) (7.2.19) with arbitrary functions D1, D2. Next we compute the pressure for TIES motion. Equations (7.2.6) become 19.1 = €(XD€ - YUn+ l31f) + 7I(XCln + YES + C19). 10.2 = -n(XU€ - YDn + Elf) + E(XUn + YD€ + By). 19,3 = x'(XD5-—YDn+Df)+y'(XClr]+YD5+Dg)+2'C12. 87 Making use of (7.2.4, (7.2.15), and (7.2.19) these equations simplify to P11 = 0X32 + C(ff + 977) + 016 + D277, (72-20) p.2 = CYS2 + C(fn + 96) + D17] + D26. (7.2.21) 1 , , m = 5006 + W32) + X(C(f€ + my) + DIE + D277) + Y(C(g5 — fn + f2/2 + 92/2) — D17; + D25), + 2'C12, (7.2.22) where the notation 32 = 52 + n2 is used for simplicity. Equations (7.2.20), (7.2.21) imply p(x, Y. z, t) = $06 + me + 172) + X(C(f€ + 9n) + 0.5 + 0277) + +Y(C 00 when detF —* 0, and for incompressible ones the integrand takes infinite value when the incompressibility con- straint is violated. Thus the limited partial regularity presented in this work (viz., r, z E CI(D) a.e.) gives a nontrivial example of regularity problem solved for the integrand with physically relevant behavior. New approaches are needed for further advances in regularity analysis for variational problems in nonlinear elasticity. 2. Polyconvexity employed in this work implies quasiconvexity, which is essentially 92 equivalent to weak lower semicontinuity. Many important for applications integrands do not enjoy this property. Hence the problem of quasiconvexification (QC) arises. At the present no effective systematic QC technique is known, but for specific integrands some approaches have been deveIOped (see, for example, [20]). The integrand of the reduced functional obtained in axisymmetric settings in this work depends on the rectangular matrix Du, u = (5,17, 2), and under suitable assumptions may be amenable to the method developed for such functions in [49]. In [49] the problem of constructing the semiconvex envelope 1 for so called invariant integrands2 depending on m x n, m > n, matrices is reduced to the same problem for an associated function defined on n x n matrices. If the strain energy density depends only on the first invariant II, it is easy to verify that the integrand for the reduced axisymmetric variational problem is invariant, so that the method in [49] applies with appropriate modifications. The case of nonconvex anisotropic problem is more challenging. To show this, consider the following simple example of the stored energy function modeling so called fiber reinforced material (see [28] for more detail) W = Wiso(Ila 12) + Waniso(14)i (8H02) where Wm represents the stored energy due to deformation of the isotropic incom- pressible matrix material, and Wm,” accounts for the effect of reinforcing. A partic- 1this is a unifying term for convex, polyconvex, quasiconvex, and rank one convex envelopes 2A function f : me" —’ R is said to be invariant if f(QAR) = f(A) for each A 6 MM", Q e 50(m), R e SO(n). 93 ularly simple form for the reinforcement term is f(14) = 7(14 - 1)2/2, (803) where 'y > 0 is a positive constant depending on the fiber material, 14 = T C a is the pseudo-invariant of C, and a unit vector a represents the preferred direction of the fibers. This form has been used by a number of authors to analyze different aspects of the theory of transversely isotropic materials (see, for example, [46] and references therein). Even if neo-Hookean response is chosen for a matrix material, Wm = u(Il — 3)/2, (8.0.4) constructing the QC envelope for the overall strain energy density (8.0.2) is a chal- lenging task. The method developed in [20] does not apply since the reinforcement term cannot be represented as a function of the singular values (principal stretches) alone. Moreover, justification of the dimensionality reduction for isotropic integrands described in Section 2.2 needs to be modified for this anisotrOpic case. Yet another problem is to choose an appropriate functional space to accommodate different gowth exponents for partial derivatives with respect to R and Z (the pseudo-invariant 1.; contains partial derivatives with respect to Z of power four that may be different from the growth exponents of the isotropic part). A new relaxation theorem that does not make use of the bound (8.0.1) is also needed for this and other physically relevant situations. 3. Dimensionality reduction of three-dimensional problems due to an assumption that the problems under consideration have certain types of symmetry is crucial in 94 this work. It is desirable to justify this reduction using a more elegant and systematic approach than the straightforward and tedious computation in Section 2.2 or ad hoc considerations in Chapters 5 and 7. The derivation of lower dimensional theories for domains which are thin in one or more directions has a long history (detailed account on this issue and further references can be found in [3]). Recently results rigorously justifying some classical theories for rods, plates, and membranes (see, for example, [38]) were obtained using F-convergence technique. Another approach, based on the principle of virtual power as a starting point, was developed by Antman [3], and its application for derivation of the governing equation for TIE motions in a rod- like bodies may be mathematically tractable. 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