Natural States and Symmetry Properties of. Two-Dimensional Ciarlet-Mooney-Rivlin. Nonlinear Constitutive Models

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1 Natural States and Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Nonlinear Constitutive Models Alexei Cheviakov, Department of Mathematics and Statistics, Univ. Saskatchewan, Canada Jean-François Ganghoffer, LEMTA, Université de Lorraine, Nancy, France

2 Motivations Finding closed-form solutions in compressible hyperelasticity quite challenging: very few such solutions in the literature. Particular classes of exact solutions obtained for several models using Lie groups. Lie symmetries widely used in the analysis of contemporary elasticity models. In particular, classification of Lie point symmetries for 1D and D nonlocal elastodynamics in [Bluman, Anco, Cheviakov, Springer, 010; Bower. Applied Mechanics of Solids, 009]. Invariant solutions for radial motions of compressible hyperelastic spheres & cylinders [Capriz & Mariano, 007]. Similarity solutions for the motion of hyperelastic solids in [Cheviakov, GEM package, Computer Physics Communications, 007]. From a more formal viewpoint, Lie symmetries constitute a guide in the Lagrangian and Hamiltonian formalisms in continuum mechanics, especially for complex materials endowed with a microstructure [England & Spencer, 005]. For models admitting a variational formulation, one-to-one correspondence between variational Lie symmetries and local conservation laws through Noether's theorem. For non-variational models, this relation generally does not hold [Ericksen, 000].

3 Outline of the presentation Nonlinear dynamic equations for isotropic homogeneous hyperelastic materials considered in the Lagrangian formulation. Derive explicit criterion of existence of a natural state of a given constitutive law. Used to derive natural state conditions for some common constitutive relations. Equivalence transformations computed for D planar motions of Ciarlet-Mooney-Rivlin solids; yields reduction of the number of parameters in the constitutive law. Find special value of traveling wave speed for which nonlinear Ciarlet-Mooney-Rivlin equations admit an additional infinite set of point symmetries. Classification of point symmetries in dynamical case & for traveling wave coordinates. Perspectives.

4 Finite strain kinematics,t X x X F x Transformation gradient (tangent mapping) v d x dt velocity Use cartesian coordinates with flat space: Orientation preserving condition: Continuity equation: 0 /J F J : det 0 i ij ij i g F j : F j X ij

5 Boundary value problem in hyperelasticity Material (Lagrangian) format: x Div P R 0 tt X 0 R R X,t F P T.. P W F 0 F P F W T cons. linear momentum body forces per unit mass cons. angular momentum F hyperelastic model strain energy density per unit volume Physical (Eulerian) format: v div σ r e 0 t x e v x v X x X r r x,t σ σ,t (,t) d (,t) / dt T cons. linear momentum body forces per actual unit mass cons. angular momentum t σ. n σ P : J σ. F T traction vector, Cauchy stress first Piola-Kirchhoff stress T P. N nominal traction (force on reference surface element)

6 Boundary value problem in hyperelasticity For zero forcing, strong form obeys extremum principle: x F x V : W V dv R 1 K x: 0 t dv kinetic energy x H K V Hamiltonian L K - V Lagrangian L= l x, x, F dv Dx l, t t, Dxl, t, t in Lagrangian format. x x F x x F 0 potential energy t Euler-Lagrange equ.= momentum equ.

7 For incompressible materials: Hyperelastic constitutive models T T polar decomposition R..R :.,. 1,, 3 B B I : Tr, I :, I : Det /3 4/ F U V B F F V C:= F F U eigenvalues of U, V I : J I, I : J I, I : J reduced invariants to isolate volume changes W,, U I,I,I U I, I, I for isotropic materials 1 Ciarlet Mooney-Rivlin materials: W a I1 3 b I 3 ci3 d log I 3, a 0,b,c,d 0 General form of const. law for isotropic materials: S F. P I C C, I,I,I ,1, 0,1, 1 3 P F F X hydrostatic pressure T p 0 FW, p p(,t)

8 Constitutive relations and natural states Natural state F I σ 0 vanishing self-equilibrated stress (no external load) Neo-Hookean material fails to have natural states: W ai P F σ I Mooney-Rivlin materials (c=0=d) do not have natural states. Natural state requires external forces Stress tensors cannot be linear in F. 1 3 Theorem: a hyperelastic material with const. law given by W U I,I,I U U U has zero residual stress iff 0 when F I I1 I I3 Alternative: for W 1,, 3 tk 0 when 1 3 1: Biot stress vanishes k

9 Point symmetries and equivalence transformations k Consider PDE system R x,u, u,..., u 0, 1...N derivatives of order at most k n independent variables z z,...,z 1 n 1 m m dependent var. u(z) u (z),...,u (z) One-parameter Lie group of transformations of variables (z,u): one-to-one transformation acting in the m+n dimensional space (z,u) of the form: *i i i i z f z,u; z z,u O, i 1...n * u g z,u; z z,u O, 1...m z,u z,u i components define infinitesimal generator Y + i (z,u) transform -> transform of partial derivatives of u(z) given by prolongation formulas. One-parameter Lie group of point transformations = group of point symmetries of a PDE system iff its prolongation leaves invariant the solution manifold of the PDE in the space including u, z and all required partial derivatives of u(z). If u(z) is a solution, then u * (z * ) is also solution of the same system. Symmetry components found in algorithmic way from the determining equations: k k Y R x,u,..., u 0, 1...N on solutions x u

10 Point symmetries and equivalence transformations () Notion of equivalence transformations closely related to local symmetries. Equivalence transformations preserve differential structure of equations / modify constitutive and/or parameters of a DE model. Equivalence transformations used to reduce number of parameters of a given const. model. Interest: analyses involving classifications; construct exact solutions for new sets of constitutive functions/parameters, from known exact solutions for given constitutive functions/parameters. R x,u, u,..., u 0, k 1..N Consider family of PDE systems k with n independent variables z z,...,z n involving set of constitutive functions and/or parameters 1 m m dependent variables u(z) u z,...,u z K K,...,K 1 L One-parameter Lie group of equivalence transformations = Lie group of transformations i i z f z,u;, i 1...n u g z,u;, 1...m l l K G z,u;, l 1...L maps a given PDE syst. Into a new one with new constitutive functions.

11 Point symmetries and equivalence transformations (3) Equivalence transformations computed through sol. of determining equ. k Y k R x,u, u,... u 0, 1...N Prolongation of inifnitesimal generator for the one-parameter Lie group of point transformations z* f (z,u; ), u* g(z,u; ) global form x i Y z,u z,u i u local form i i i z * z x,u O, i 1..n i u * u x,u O, 1..m Need to specify variables on which the const. Functions / Parameters depend. Ex.1: if K 1 =a, K =b are constant parameters of Mooney-Rivlin model, Corresponding equivalence transformations will be of the form a G a,b;, b G a,b; 1 Ex.: for K 1 =Q(x), x independent var., equivalence transf. for Q(x) will be Qx G x,q(x); 1

12 Equivalence transformations for D Ciarlet-Mooney-Rivlin model 1 Strain energy density W a I1 3 b I 3 ci3 d log I 3, a 0,b,c,d 0 Theorem: D Ciarlet-Mooney-Rivlin model admits following equivalence transformations Includes scalings, translations and rotations of material coordinates (parameters 3, 4, 5, 7 ) Galilean transformations, transformations of eulerian coord. and time (parameters 1,, arbitrary functions f 1 (t), f (t)) Scaling of body density in ref. configuration ( ), transformations of parameters a,b,c,d of const. model (parameters, 3) 6

13 Equivalence transformations for D Ciarlet-Mooney-Rivlin model D motions: P P 1 P P 0 x,tt 1 0R 0, 0 x,tt 1 0R 0 X X X X 1 ij 1 W 0 0 X,X, P 0 X,X, i, j 1, Fij F11 F1 0 F F1 F Th.: Ciarlet-Mooney-Rivlin model in D depends only on 3 constitutive parameters D first Piola-Kirchhoff stress tensor takes the form d P 0 AF BJC C J A a b 0, B b c 0, d 0 F F F F F F1 F C F F1 cofactor matrix Previous Theorems can be used for direct analysis of D models.

14 Equivalence transformations for D Ciarlet-Mooney-Rivlin model Used to present equivalence and symmetry classification in a more compact form: Set of equivalence transformations of D Ciarlet-Mooney-Rivlin model given by 1,, 3, 4, 5, 6 arbitrary constants. f 1 (t),f (t) arbitrary functions. Action of equivalence transformations on essential parameters A, B, d is pure scalings.

15 Symmetry classification for time-dependent D Ciarlet-Mooney-Rivlin models Classify symmetries w.r. to constants A, B>0, d and types of density functions 1 0 X,X Cte Restrict to zero external body forces. Given modulo the previous equivalence transformations

16 Symmetry classification of D Ciarlet-Mooney-Rivlin model in traveling wave coordinates Assume that medium moves w.r. to observer at constant speed s>0 in direction X 1. i 1 i 1 x t,x,x w z,x, z X st, i 1, With this ansatz, one has Consider body density in ref. configuration of the form 0 0 X Assume no-forcing. Transformation of partial derivatives in PDE s: x X st,x w z,x s t z i 1 i 0

17 Full symmetry classification of PDE system in travelling wave coordinates modulo previous equivalence transformations

18 Full symmetry classification of PDE system in travelling wave coordinates modulo previous equivalence transformations General non-linear Ciarlet-Mooney-Rivlin model has special wave speed for which equations admit additional symmetries including infinite set of point symmetries, with generators Y,Y,Y 1 3 Special wave speed equal to the constant wave speed s* a b of linear neo-hookean version of dynamic equations x x x,tt A 1 X X x x x,tt A 1 X X A a b 0, B b c 0, d 0

19 Conclusion - Perspectives Existence of natural states for hyperelastic materials analyzed: question of high importance for consistency formulation of BVPs for numerical computations. Classical neo-hookean and Mooney-Rivlin models do not have a natural state / Hadamard materials may admit a natural state according to range of parameters. Point symmetries of the D Ciarlet-Mooney-Rivlin model in full dynamical setting classified with symbolic software. Such symmetries important to construct exact solutions and conservation laws. Computation of symmetry structure of elasticity models in non-planar D reductions (axial symmetry) & in 3D. Extension to anisotropic constitutive behavior: case of one and two families of fibers.

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