Mathematical Foundations of

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1 Mathematical Foundations of Elasticity Theory John Ball Oxford Centre for Nonlinear PDE J.M.Ball
2 Reading Ciarlet Antman Silhavy
3 Lecture note 2/1998 Variationalmodels for microstructure and phase transitions Stefan Müller Prentice Hall
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5 J.M. Ball, Some open problems in elasticity. In Geometry, Mechanics, and Dynamics, pages 359, Springer, New York, You can download this from my webpage
6 Elasticity Theory The central model of solid mechanics. Rubber, metals (and alloys), rock, wood, bone can all be modelled as elastic materials, even though their chemical compositions are very different. For example, metals and alloys are crystalline, with grains consisting of regular arrays of atoms. Polymers (such as rubber) consist of long chain molecules that are wriggling in thermal motion, often joined to each other by chemical bonds called crosslinks. Wood and bone have a cellular structure 6
7 A brief history 1678 Hooke's Law 1705 Jacob Bernoulli 1742 Daniel Bernoulli 1744 L. Euler elastica(elastic rod) 1821 Navier, special case of linear elasticity via molecular model (Dalton s atomic theory was 1807) 1822 Cauchy, stress, nonlinearand linear elasticity For a long time the nonlinear theory was ignored/forgotten A.E.H. Love, Treatise on linear elasticity 1950's R. Rivlin, Exact solutions in incompressiblenonlinear elasticity (rubber) Nonlinear theory clarified by J.L. Ericksen, C. Truesdell Mathematical developments, applications to materials, biology 7
8 Kinematics Label the material points of the body by the positions x Ω they occupy in the reference configuration. 8
9 Deformation gradient F =Dy(x,t), F iα = y i x α. 9
10 Invertibility To avoid interpenetration of matter, we require that for each t, y(,t) is invertible on Ω, with sufficiently smooth inverse x(, t). We also suppose that y(, t) is orientation preserving; hence J =detf(x,t)>0 for x Ω. (1) Bytheinversefunctiontheorem,ify(,t)isC 1, (1) implies that y(, t) is locally invertible. 10
11 11
12 Global inverse function theorem for C 1 deformations Let Ω R n be a bounded domain with Lipschitz boundary Ω (in particular Ω lies on one side of Ω locally). Let y C 1 ( Ω;R n ) with detdy(x)>0 for all x Ω and y Ω onetoone. Then y is invertible on Ω. Proof uses degree theory. cf Meisters and Olech, Duke Math. J. 30 (1963)
13 Notation M n n = {real n n matrices} M n n + = {F M n n :detf >0} SO(n) = {R M n n :R T R=1,detR=1} = {rotations}. If a R n, b R n, the tensor product a b is the matrix with the components (a b) ij =a i b j. [Thus (a b)c=(b c)a if c R n.]
14 Variationalformulation of nonlinear elastostatics We suppose for simplicity that the body is homogeneous, i.e. the material response is the sameateachpoint. Inthiscasethetotalelastic energy corresponding to the deformation y=y(x) is given by I(y)= W(Dy(x))dx, Ω wherew =W(F)isthestoredenergyfunction ofthematerial. WesupposethatW :M+ 3 3 R is C 1 and bounded below, so that without loss of generality W 0.
15 We will study the existence/nonexistence of minimizers of I subject to suitable boundary conditions. Issues. 1. What function space should we seek a minimizer in? This controls the allowable singularities in deformations and is part of the mathematical model. 2. What boundary conditions should be specified? 3. What properties should we assume about W?
16 The Sobolevspace W 1,p W 1,p ={y:ω R 3 : y 1,p < }, where y 1,p = { ( Ω [ y(x) p + Dy(x) p ]dx) 1/p if 1 p< esssup x Ω ( y(x) + Dy(x) ) if p= Dy is interpreted in the weak(or distributional) sense, so that Ω for all ϕ C 0 (Ω). y i x α ϕdx= Ω y i ϕ i x α dx
17 WeassumethatybelongstothelargestSobolev space W 1,1 =W 1,1 (Ω;R 3 ), so that in particu lar Dy(x) is well defined for a.e. x Ω, and I(y) [0, ].
18 y W 1, Lipschitz y W 1,1 (because every y W 1,1 is absolutely continuous on almost every line parallel to a given direction)
19 Cavitation y(x)= 1+ x x x y W 1,p for 1 p<3. Exercise: prove this.
20 Boundary conditions We suppose that Ω= Ω 1 Ω 2 N, where Ω 1, Ω 2 are disjoint relatively open subsets of Ω and N has twodimensional Hausdorff measure H 2 (N)=0 (i.e. N has zero area). We suppose that y satisfies mixed boundary conditions of the form y Ω1 =ȳ( ), where ȳ : Ω 1 R 3 is a given boundary displacement.
21 By formally computing d dτ I(y+τϕ) τ=0= d dτ W(Dy+τDϕ)dx t=0=0, Ω we obtain the weak form of the EulerLagrange equation for I, that is D FW(Dy) Dϕdx=0 ( ) Ω for all smooth ϕ with ϕ Ω1 =0. T R (F)=D F W(F) is the PiolaKirchhoff stress tensor. (A B=tr A T B)
22 If y, Ω 1 and Ω 2 are sufficiently regular then (*) is equivalent to the pointwise form of the equilibrium equations DivD F W(Dy)=0 in Ω, together with the natural boundary condition of zero applied traction D F W(Dy)N =0 on Ω 2, wheren =N(x)denotestheunitoutwardnormal to Ω at x.
23 Ω 1 Example: boundary conditions for buckling of a bar Ω 2 Ω2 Ω 1 1 ȳ(x)=x ȳ(x 1,x 2,x 3 )=(λx 1,x 2,x 3 ) y( Ω 2 ) traction free λ<1
24 Hence our variational problem becomes: Does there exist y minimizing in I(y)= Ω W(Dy)dx A={y W 1,1 :y invertible,y Ω1 =ȳ}? We will make the invertibility condition more precise later.
25 Properties of W To try to ensure that deformations are invertible, we suppose that (H1) W(F) as detf 0+. So as to also prevent orientation reversal we define W(F)= if detf 0. Then W :M 3 3 [0, ] is continuous.
26 Thus if I(y) < then detdy(x) > 0 a.e.. However this does not imply local invertibility (think of the map (r,θ) (r,2θ) in plane polar coordinates, which has constant positive Jacobian but is not invertible near the origin), nor is it clear that detdy(x) µ>0 a.e..
27 We suppose that W is frameindifferent, i.e. (H2) W(RF)=W(F) for all R SO(3),F M 3 3. In order to analyze (H2) we need some linear algebra.
28 Square root theorem Let C be a positive symmetric n n matrix. Then there is a unique positive definite symmetric n n matrix U such that (we write U =C 1/2 ). C=U 2 28
29 Formula for the square root Since C is symmetric it has a spectral decomposition C= n i=1 λ i ê i ê i. Since C>0, it follows that λ i >0. Then U = n i=1 λ 1/2 i ê i ê i satisfies U 2 =C. 29
30 Polar decomposition theorem Let F M+ n n. Then there exist positive definite symmetric U, V and R SO(n) such that F =RU =VR. These representations (right and left respectively) are unique. 30
31 Proof. Suppose F =RU. Then U 2 =F T F := C. Thus if the right representation exists U must be the square root of C. But if a R n is nonzero, Ca a = Fa 2 > 0, since F is nonsingular. Hence C > 0. So by the square root theorem, U =C 1/2 exists and is unique. Let R=FU 1. Then R T R=U 1 F T FU 1 =1 and detr=detf(detu) 1 =+1. The representation F =VR 1 is obtained similarly using B:=FF T, and it remains to prove R=R 1. ButthisfollowsfromF =R 1 ( R T 1 VR 1 ), and the uniqueness of the right representation. 31
32 Exercise: simple shear y(x)=(x 1 +γx 2,x 2,x 3 ). F = cosψ sinψ 0 sinψ cosψ cosψ sinψ 0 sinψ 1+sin2 ψ cosψ tanψ = γ 2. As γ 0+ the eigenvectors of U and V tend to 1 (e 2 1 +e 2 ), 1 (e 2 1 e 2 ),e 3. 32,
33 Hence (H2) implies that W(F)=W(RU)=W(U)= W(C). Conversely if W(F)= W(U) or W(F)= W(C) then (H2) holds. Thus frameindifference reduces the dependence ofw onthe9elementsoff tothe6elements of U or C.
34 Material Symmetry In addition, if the material has a nontrivial isotropy group S, W satisfies the material symmetry condition W(FQ)=W(F) for all Q S,F M ThecaseS=SO(3)correspondstoanisotropic material.
35 The strictly positive eigenvalues v 1,v 2,v 3 of U (or V) are called the principal stretches. Proposition 1 W is isotropic iff W(F)=Φ(v 1,v 2,v 3 ), where Φ is symmetric with respect to permutations of the v i. Proof. Suppose W is isotropic. Then F = RDQforR,Q SO(3)andD=diag(v 1,v 2,v 3 ). Hence W = W(D). But for any permutation P of 1,2,3 there exists Q such that Qdiag(v 1,v 2,v 3 ) Q T =diag(v P1,v P2,v P3 ). The converse holds since Q T F T FQ has the same eigenvalues (namely v 2 i ) as FT F.
36 (H1) and (H2) are not sufficient to prove the existence of energy minimizers. We also need growth and convexity conditions on W. The growth condition will say something about how fastw growforlargevaluesoff. Theconvexity condition corresponds to a statement of the type stress increases with strain. We return tothequestionofwhatthecorrectformofthis convexity condition is later; for a summary of older thinking on this question see Truesdell& Noll.
37 Why minimize energy? This is the deep problem of the approach to equilibrium, having its origins in the Second Law of Thermodynamics. We will see how rather generally the balance of energy plus a statement of the Second Law lead to the existence of a Lyapunov function for the governing equations.
38 Balance of Energy ( 1 ) d dt E 2 ρ R y t 2 +U dx = b y tdx E + t R y t ds + E E rdx q R NdS, (1) E for all E Ω, where ρ R =ρ R (x) is the density in the reference configuration, U is the internal energy density, b is the body force, t R is the PiolaKirchhoff stress vector, q R the reference heat flux vector and r the heat supply.
39 Second Law of Thermodynamics WeassumethisholdsintheformoftheClausius Duhem inequality d q R N r dt ηdx ds+ dx (2) E E θ Eθ for all E, where η is the entropy and θ the temperature.
40 The Ballistic Free Energy Suppose that the the mechanical boundary conditions are that y=y(x,t) satisfies y(,t) Ω1 = ȳ( ) and the condition that the applied traction on Ω 2 is zero, and that the thermal boundary condition is θ(,t) Ω3 =θ 0, q R N Ω\ Ω3 =0, where θ 0 >0 is a constant. Assume that the heat supply r is zero, and that the body force is given by b= grad y h(x,y),
41 Thusfrom(1),(2)withE=Ωandtheboundary conditions d dt Ω [ 1 Ω t R y t ds 2 ρ R y t 2 +U θ 0 η+h Ω Ω θ ( 1 θ 0 ) ] dx q R NdS = 0. So E = Ω[ 12 ρ R y t 2 +U θ 0 η+h ] dx is a Lyapunov function, and it is reasonable to suppose that typically (y t,y,θ) tends as t to a (local) minimizer of E.
42 For thermoelasticity, W(F, θ) can be identified with the Helmholtz free energy U(F,θ) θη(f, θ). Hence, if the dynamics and boundary conditions are such that as t we have y t 0 and θ θ 0, then this is close to saying that y tends to a local minimizer of I θ0 (y)= Ω [W(Dy,θ 0)+h(x,y)]dx. The calculation given follows work of Duhem, Ericksen and Coleman & Dill.
43 Of course a lot of work would be needed to justify this (we would need wellposedness of suitable dynamic equations plus information on asymptotic compactness of solutions and more; this is currently out of reach). Note that it is not the Helmoltz free energy that appears in the expression for E but U θ 0 η, where θ 0 is the boundary temperature. Forsomeremarksonthecasewhenθ 0 depends on x see J.M. Ball and G. Knowles, Lyapunov functions for thermoelasticity with spatially varying boundary temperatures. Arch. Rat. Mech. Anal., 92: , 1986.
44 Existence in one dimension To make the problem nontrivial we consider an inhomogeneous onedimensional elastic material with reference configuration Ω = (0, 1) and storedenergy function W(x, p), with corresponding total elastic energy 1 I(y)= [W(x,y x(x))+h(x,y(x))]dx, 0 where h is the potential energy of the body force.
45 We seek to minimize I in the set of admissible deformations A={y W 1,1 (0,1): y x (x)>0 a.e., y(0)=α,y(1)=β}, where α < β. (We could also consider mixed boundary conditions y(0) = α, y(1) free.) (Note the simple form taken by the invertibility condition.)
46 Hypotheses on W: We suppose for simplicity that W :[0,1] (0, ) [0, ) is continuous. (H1) W(x,p) as p 0+. As before we define W(x,p)= if p 0. (H2) is automatically satisfied in 1D. (H3) W(x,p) Ψ(p) for all p>0, x (0,1), where Ψ:(0, ) [0, ) is continuous with lim p Ψ(p) p =.
47 (H4) W(x,p) is convex in p, i.e. W(x,λp+(1 λ)q) λw(x,p)+(1 λ)w(x,q) for all p>0,q>0,λ (0,1),x (0,1). If W is C 1 in p then (H4) is equivalent to the stress W p (x,p) being nondecreasing in the strain p. (H5) h:[0,1] R [0, ) is continuous.
48 Theorem 1 Under the hypotheses (H1)(H5) there exists y that minimizes I in A. Proof. A is nonempty since z(x)=α+(β α)x belongs to A. Since W 0,h 0, 0 l=inf y A I(y)<. Let y (j) be a minimizing sequence, i.e. y (j) A, I(y (j) ) l as j.
49 We may assume that lim j 10 W(x,y (j) x )dx=l 1, lim j 10 h(x,y (j) )dx=l 2, where l=l 1 +l 2. Since Ω Ψ(y(j) x )dx M <, by the de la Vallée Poussin criterion(see e.g. Onedimensional variational problems, G. Buttazzo, M. Giaquinta, S. Hildebrandt, OUP, 1998 p 77) there exists a subsequence, which we continue to call y (j) x converging weakly in L 1 (0,1) to some z.
50 Lety (x)=α+ x 0 z(s)ds,sothaty x=z. Then y (j) (x)=α+ x 0 y (j) x (s)ds y (x) for all x [0,1]. In particular y (0)=α, y (1)=β. By Mazur s theorem, there exists a sequence z (k) = j=k λ (k) j y x (j) of finite convex combinations of the y x (j) converging strongly to z, and so without loss of generality almost everywhere.
51 By convexity 1 0 W(x,z(k) )dx 1 0 sup j k j=k 1 λ (k) j W(x,y (j) x )dx 0 W(x,y(j) x )dx. Letting k, by Fatou s lemma 1 0 W(x,y x)dx= 1 0 W(x,z)dx l 1. But this implies that y x(x) > 0 a.e. and so y A.
52 Also by Fatou s lemma 1 0 h(x,y )dx liminf j 1 0 h(x,y(j) )dx=l 2. Hence l I(y ) l 1 +l 2 =l. 1 2 So I(y )=l and y is a minimizer.
53 Discussion of (H3) We interpret the superlinear growth condition (H3) for a homogeneous material. y(x)=px 1/p 1 Total storedenergy = W(p) p. So lim p W(p) p = says that you can t get a finite line segment from an infinitesimal one with finite energy.
54 Simplified model of atmosphere y(1)=α x=1 Assume constant density ρ R and pressure p R in the reference configuration. Assume gas deforms adiabatically so that the pressure p and density ρ satisfy pρ γ =p R ρ γ R, where γ>1 is a constant. x=0 y(0)=0
55 The potential energy of the column is I(y) = 1 0 = ρ R g [ p R (γ 1)y x (x) γ 1+ρ Rgy(x) 1 0 where k= p R ρ R g(γ 1). [ k y x (x) γ 1+(1 x)y x(x) ] dx ] dx, We seek to minimize I in A={y W 1,1 (0,1):y x >0 a.e., y(0)=0,y(1)=α}.
56 Then the minimum ofi(y) on A is attained iff α α crit, where α crit = γ ( ) pr 1/γ. γ 1 ρ R g y α crit can be interpreted as the finite height of the atmosphere predicted by this simplified model (cf Sommerfeld). α α crit If α>α crit then minimizing sequences y (j) for I converge to the minimizer for α=α crit plus a vertical portion. 1 x
57 For details of the calculation see J.M. Ball, Loss of the constraint in convex variational problems, in Analyse Mathématiques et Applications, GauthierVillars, Paris, 1988, where a general framework is presented for minimizing general framework is presented for minimizing a convex functional subject to a convex constraint, and is applied to other problems such as ThomasFermi and coagulationfragmentation equations.
58 Discussion of (H4) For simplicity consider the case of a homogeneous material with storedenergy function W =W(p). The proof of existence of a minimizer used the direct method of the calculus of variations, the key point being that if W is convex then E(p)= 1 0 W(p)dx isweaklylowersemicontinuousinl 1 (0,1), i.e. p j pinl 1 (0,1)(that is 1 0 p (j) vdx 1 0 pvdx for all v L (0,1)) implies 1 W(p)dx liminf 0 j 1 0 W(p(j) )dx.
59 Proposition 2 (Tonelli) IfE isweaklylowersemicontinuousinl 1 (0,1) then W is convex. Proof. Define p (j) as shown. q p p (j) p (j) λp+(1 λ)q in L 1 (0,1) 10 W(p (j) )dx=λw(p)+(1 λ)w(q) W(λp+(1 λ)q) λ j 1 λ j 2 j 1 x
60 If W(x, ) is not convex then the minimum is in general not attained. For example consider the problem inf y(0)=0,y(1)= [ (yx 1) 2 (y x 2) 2 y x +(y 3 2 x)2 ] dx. 3/2 Then the infimum is zero but is not attained. slopes 1 and 2 1
61 More generally we have the following result. Theorem 2 If W :R [0, ] is continuous and I(y)= 1 0 [W(y x )+h(x,y)]dx attains a minimum on A={y W 1,1 :y(0)=0,y(1)=β} forallcontinuoush:[0,1] R [0, ) andall β then W is convex.
62 Let l=inf y A 10 W(y x )dx, m=inf y A 10 [W(y x )+ y βx 2 ]dx. Then l m. Let ε > 0 and pick z A with 10 W(z x )dx l+ε. Define z (j) A by z (j) (x)=β k j +j 1 z(j(x k j )) for k j x k+1,k=0,...,j 1. j
63 Then So z (j) (x) βx = β( k j x)+j 1 z(j(x k j )) I(z (j) ) = j+1 k j k j 1 j+1 k=0 j 1 j 1 1 k=0 l+2ε Cj 1 for k j x k+1. j W(z x (j(x k j ))dx+ 1 0 W(z x)dx+cj 2 for j sufficiently large. Hence m l and so l=m. 0 z(j) βx 2 dx
64 But by assumption there exists y with I(y )= 1 0 W(y x)dx+ 1 0 y βx 2 dx=l. Hence y (x)=βx and thus 10 W(y x )dx W(β) for all y A. Taking in particular y(x)= { px if 0 x λ qx+λ(p q) if λ x 1, with β=λp+(1 λ)q we obtain W(λp+(1 λ)q) λw(p)+(1 λ)w(q) as required.
65 There are two curious special cases when the minimum is nevertheless attained when W is not convex. Suppose that (H1), (H3) hold and that either (i) I(y)= 1 0 W(x,y x )dx, or (ii) I(y)= 1 0 [W(y x )+h(y)]dx. Then I attains a minimum on A.
66 (i)canbefoundinaubert&tahraoui(j.differential Eqns 1979) and uses the fact that inf y A 10 W(x,y x )dx=inf y A 10 W (x,y x )dx, where W is the lower convex envelope of W. W(x,p) W (x,p) p
67 (ii) I(y)= 1 0 [W(y x)+h(y)]dx Exercise. Hint: Write x=x(y) and use (i).
68 The EulerLagrange equation in 1D Let y minimize I in A and let ϕ C 0 (0,1). Formally calculating d I(y+τϕ) =0 dτ I(y+τϕ) τ=0=0 we obtain the weak form of the EulerLagrange equation 1 0 [W p(x,y x )ϕ x +h y (x,y)ϕ]dx=0.
69 However, there is a serious problem in making this calculation rigorous, since we need to pass to the limit τ 0+ in the integral [ ] 1 W(x,yx +τϕ x ) W(x,y x ) dx. 0 τ and the only obvious information we have is that 1 0 W(x,y x)dx<.
70 But W p can be much bigger than W. For example, when p is small and W(p) = 1 p then W p (p) = 1 p2 is much bigger than W(p). Or if p is large and W(p)=expp 2 then W p (p) =2pexpp 2 is much bigger than W(p). It turns out that this is a real problem and not just a technicality. Even for smooth elliptic integrands satisfying a superlinear growth condition there is no general theorem of the onedimensional calculus of variations saying that a minimizer satisfies the EulerLagrange equation.
71 Consider a general onedimensional integral of the calculus of variations 1 I(u)= f(x,u,u x)dx, 0 where f is smooth and elliptic (regular), i.e. f pp µ>0 for all x,u,p. Suppose also that lim p f(x,u,p) p = for all x,u.
72 Supposeu W 1,1 (0,1)satisfiestheweakform of the EulerLagrange equation 1 0 [f pϕ x +f u ϕ]dx=0 for all ϕ C 0 (0,1), so that in particular f p,f u L 1 loc (0,1). Then x f p = f u+const a where a (0,1), and hence (Exercise) u x is bounded on compact subsets of (0, 1). It follows that u is a smooth solution of the EulerLagrange equation d dx f p=f u on (0,1).
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77 For this problem one has that I(u +tϕ)= for t 0, if ϕ(0) 0. u u +tϕ Also if u (j) W 1, converges a.e. to u then I(u (j) ) (the repulsion property), explaining the numerical results.
78 Derivation of the EL equation for 1D elasticity Theorem 3 Let (H1) and (H5) hold and suppose further that W p exists and is continuous in (x,p) [0,1] (0, ),andthath y existsandiscontinuous in (x,y) [0,1] R. If y minimizes I in A then W p (,y x ( )) C 1 ([0,1]) and d dx W p(x,y x (x))=h y (x,y(x)) for all x [0,1].(EL)
79 Proof. Pick any representative of y x and let Ω j = {x [0,1] : 1 j y x(x) j}. Then Ω j is measurable, Ω j Ω j+1 and meas ( [0,1]\ j=1 Ω j) =0. Givenj, letz L (0,1)with Ω j zdx=0. For ε sufficiently small define y ε W 1,1 (0,1) by yx(x)=y ε x (x)+εχ j (x)z(x), y ε (0)=α, where χ j is the characteristic function of Ω j.
80 Then d dε I(yε ) ε=0 x ] = [W p (x,y x )z+h y χ jzds dx Ω[ 0 x ] = W p (x,y x ) h y(s,y(s))ds Ω j 0 =0. z(x) dx Hence x W p (x,y x ) h y(s,y(s))ds=c j in Ω j, 0 and clearly C j is independent of j.
81 Corollary 1 Let the hypotheses of Theorem 3 hold and assume further that lim max W p(x,p)= ( ). p 0+ x [0,1] Then there exists µ>0 such that y x (x) µ>0 a.e. x [0,1]. Proof. W p (x,y x (x)) C> by (EL).
82 Remark. (*) is satisfied if W(x,p) is convex in p for all x [0,1],0<p ε for some ε>0 and if Proof. (H1+) lim min W(x,p)=. p 0+ x [0,1] W p (x,p) W(x,ε) W(x,p). ε p
83 Corollary 2. If (*), (H3) and the hypotheses of Theorem 3 hold, and if W(x, ) is strictly convex then y has a representative in C 1 ([0,1]). Proof. W p (x,y x (x)) has a continuous representative. So there is a subset S [0,1] of full measure such thatw p (x,y x (x))is continuousons. We need to show that that if {x j },{ x j } S with x j x, x j x then the limits lim j y x (x j ), lim j y x ( x j ) exist and are finite and equal.
84 By (H3) and convexity, lim p min x [0,1] W p (x,p)=. Hencewemay assume that y x (x j ) z,y x ( x j ) z [µ, ) with z z. Hence W p (x,z)=w p (x, z), which by strict convexity implies z= z.
85 Existence of minimizersin 3D elastostatics
86 Minimize in I(y)= Ω W(Dy)dx A={y W 1,1 :detdy(x)>0 a.e., y Ω1 =ȳ}. (Notethatwehaveforthetimebeingreplaced the invertibility condition by the local conditiion detdy(x)>0 a.e., which is easier to handle.)
87 SofarwehaveassumedthatW :M [0, ) is continuous, and that (H1) W(F) as detf 0+, so that setting W(F) = if detf 0, we have that W : M 3 3 [0, ] is continuous, and that W is frameindifferent, i.e. (H2) W(RF)=W(F) for all R SO(3),F M 3 3. (In fact (H2) plays no direct role in the existence theory.)
88 Growth condition y=fx 1 F lim F W(F) F 3 = says that you can t get a finite line segment from an infinitesimal cube with finite energy.
89 We will use growth conditions a little weaker than this. Note that if W(F) C(1+ F 3+ε ) then any deformation with finite elastic energy Ω W(Dy(x))dx is in W 1,3+ε and so is continuous.
90 Convexity conditions ThekeydifficultyisthatW isneverconvex,so that we can t use the same method to prove existence of minimizers as in 1D. Reasons 1. Convexity of W is inconsistent with (H1) because M+ 3 3 is not convex.
91 A=diag(1,1,1) 1 W( 1 2 (A+B))= > 1 2 W(A)+1 2 W(B) 2 (A+B)=diag(0,0,1) detf <0 detf >0 B=diag( 1, 1,1)
92 2. IfW isconvex,thenanyequilibriumsolution (solution of the EL equations) is an absolute minimizer of the elastic energy Proof. I(z)= I(y)= Ω W(Dz)dx Ω W(Dy)dx. Ω [W(Dy)+DW(Dy) (Dz Dy)]dx=I(y). This contradicts common experience of nonunique equlibria, e.g. buckling.
93 Examples of nonunique equilibrium solutions Pure zero traction problem.
94 For an isotropic material could we assume instead that Φ(v 1,v 2,v 3 ) is convex in the principal stretches v 1,v 2,v 3? This is a consequence of the ColemanNoll inequality. While it is consistent with (H1) it is not in general satisfied for rubberlike materials, which are almost incompressible (v 1 v 2 v 3 =1), and so have nonconvex sublevel sets. v 2 v 1 v 2 =1 v 3 =1 nonconvex sublevel set Φ c v 1
95 Rankone matrices and the Hadamard jump condition N y piecewise affine Dy=A, x N >k Dy=B, x N <k x N =k Let C = A B. Then Cx = 0 if x N = 0. Thus C(z (z N)N) = 0 for all z, and so Cz=(CN N)z. Hence A B=a N Hadamard jump condition
96 More generally this holds for y piecewise C 1, with Dy jumping across a C 1 surface. N Dy + (x 0 )=A x 0 Dy (x 0 )=B A B=a N Exercise: prove this by blowing up around x using y ε (x)=εy( x x 0 ε ).
97 Rankone convexity W is rankone convex if the map t W(F +ta N) is convex for each F M 3 3 and a R 3,N R 3. (Same definition for M m n.) Equivalently, W(λF +(1 λ)g) λw(f)+(1 λ)w(g) if F,G M 3 3 with F G = a N and λ (0,1).
98 F Rankone cone Λ={a N :a,n R 3 } Rankone convexity is consistent with(h1) becausedet(f+ta N)islinearint,sothatM is rankone convex (i.e. if F,G M λf +(1 λ)g M ) with F G=a N then
99 A specific example of a rankone convex W is an elastic fluid for which W(F)=h(detF), with h convex and lim δ 0+ h(δ)=. Such a W is rankone convex because if F,G M+ 3 3 with F G=a N, and λ (0,1) then W(λF +(1 λ)g) = h(λdetf +(1 λ)detg) λh(detf)+(1 λ)h(detg) = λw(f)+(1 λ)w(g).
100 If W C 1 (M+ 3 3 ) then W is rankone convex ifft DW(F+ta N) a N isnondecreasing. The linear map y(x)=(f +ta N)x=Fx+ta(x N) represents a shear relative to Fx parallel to a planeπwithnormaln inthereferenceconfiguration, in the direction a. The corresponding stress vector across the plane Π is t R =DW(F +ta N)N, and so rankone convexity says that the component t R a in the direction of the shear is monotone in the magnitude of the shear.
101 If W C 2 (M+ 3 3 ) then W is rankone convex iff d 2 dt 2W(F +ta N) t=0 0, for all F M 3 3 +,a,n R3, or equivalently D 2 W(F)(a N,a N)= 2 W(F) F iα F jβ a i N α a j N β 0, (LegendreHadamard condition).
102 The strengthened version D 2 W(F)(a N,a N)= 2 W(F) F iα F jβ a i N α a j N β µ a 2 N 2, forallf,a,n andsomeconstantµ>0iscalled strong ellipticity. One consequence of strong ellipticity is that it implies the reality of wave speeds for the equations of elastodynamics linearized around a uniform state y=fx.
103 Equation of motion of pure elastodynamics ρ R ÿ=divdw(dy), where ρ R is the (constant) density in the reference configuration. Linearized around the uniform state y = Fx the equations become ρ R ü i = 2 W(F) F iα F jβ u j,αβ.
104 The plane wave is a solution if where u=af(x N ct) C(F)a=c 2 ρ R a, C ij = 2 W(F) F iα F jβ N α N β. Since C(F)>0 by strong ellipticity, it follows that c 2 >0 as claimed.
105 Quasiconvexity(C.B. Morrey, 1952) Let W :M m n [0, ] be Borel measurable. W is said to be quasiconvex at F M m n if the inequality W(F +Dϕ(x))dx W(F)dx Ω Ω holds for any ϕ W 1, 0 (Ω;R m ), and is quasiconvexifitisquasiconvexateveryf M m n. Here Ω R n is any bounded open set whose boundary Ω has zero ndimensional Lebesgue measure.
106 Remark W 1, 0 (Ω;R m ) is defined as the closure of C0 (Ω;Rm ) in the weak* topology of W 1, (Ω;R m ) (and not in the norm topology  why?). That is ϕ W 1, 0 (Ω;R m ) if there exists a sequence ϕ (j) C0 (Ω;Rm ) such that (j) (j) ϕ ϕ, Dϕ Dϕ in L. Sometimes the definition is given with C 0 replacing W 1, 0. This is the same if W is finite and continuous, but it is not clear (to me) if itisthesameifw iscontinuousandtakesthe value +.
107 Setting m=n=3 we see that W is quasiconvex if for any F M 3 3 the pure displacement problem to minimize I(y)= W(Dy(x))dx Ω subject to the linear boundary condition y(x)=fx, x Ω, has y(x)=fx as a minimizer.
108 Proposition 3 Quasiconvexity is independent of Ω. Proof. Suppose the definition holds for Ω, and let Ω 1 be another bounded open subset of R n suchthat Ω 1 hasndimensionalmeasurezero. BytheVitalicoveringtheoremwecanwriteΩ as a disjoint union Ω= i=1 (a i +ε i Ω 1 ) N, where N is of zero measure.
109 Ω Ω 1 Let ϕ W 1, 0 (Ω 1 ;R n ) and define ϕ(x)= { εi ϕ( x a i ε i ) for x a i +ε i Ω 1 0 otherwise.
110 Then ϕ W 1, 0 (Ω;R n ) and W(F +Dϕ(x))dx Ω = a i +ε i Ω 1 W(F +Dϕ i=1 = = i=1 ( measω measω 1 ( x ai ε i ) )dx ε n i Ω 1 W(F +Dϕ(x))dx ) Ω 1 W(F +Dϕ(x))dx (measω)w(f).
111 Another form of the definition that is equivalent for finite continuous W is that Q W(Dy)dx (measq)w(f) foranyy W 1, suchthatdyistherestriction to a cube Q (e.g. Q=(0,1) n ) of a Qperiodic map on R n with Q Dydx=F. One can even replace periodicity with almost periodicity(see J.M. Ball, J.C. Currie, and P.J. Olver. Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Functional Anal., 41: , 1981).
112 Theorem 4 If W is continuous and quasiconvex then W is rankone convex. Remark This is not true in general if W is not contin uous. As an example, define for given nonzero a,n W(0)=W(a N)=0,W(F)= otherwise. Then W is clearly not rankone convex, but it isquasiconvexbecausegivenf 0,a N there is no ϕ W 1, 0 with F +Dϕ(x) {0,a N}
113 Proof We prove that W(F) λw(f (1 λ)a N)+(1 λ)w(f+λa N) for any F M m n,a R m,n R m,λ (0,1). Without loss of generality we suppose that N =e 1. We follow an argument of Morrey. Let D=( (1 λ),λ) ( ρ,ρ) n 1 and let D ± j be the pyramid that is the convex hull of the origin and the face of D with normal ±e j.
114 1 λ x j Dϕ=λa e 1 (1 λ)a e 1 D 1  ρ ρ 1 λ(1 λ)a e j ρ 1 λ(1 λ)a e j D 1 + ρ D j + D j λ x 1 Let ϕ W 1, 0 (D;R m ) be affine in each D ± j with ϕ(0)=λ(1 λ)a. The values of Dϕ are shown.
115 By quasiconvexity (2ρ) n 1 W(F) (2ρ)n 1 λ W(F (1 λ)a e 1 ) n + (2ρ)n 1 (1 λ) W(F +λa e 1 ) n n (2ρ) n 1 + [W(F +ρ 1 λ(1 λ)a e j ) 2n j=2 +W(F ρ 1 λ(1 λ)a e j )] SupposeW(F)<. Thendividingby(2ρ) n 1, letting ρ 0+ and using the continuity of W, we obtain W(F) λw(f (1 λ)a e 1 )+(1 λ)w(f +λa e 1 ) as required.
116 Now suppose that W(F (1 λ)a e 1 ) and W(F+λa e 1 )arefinite. Theng(τ)=W(F+ τa e 1 ) lies below the chordjoining the points ( (1 λ),g( (1 λ))),(λ,g(λ)) whenever g(τ)<, and since g is continuous it follows that g(0)=w(f)<. g (1 λ) λ
117 Corollary 3 If m = 1 or n = 1 then a continuous W : M m n [0, ] is quasiconvex iff it is convex. Proof. If m = 1 or n = 1 then rankone convexity is the same as convexity. If W isconvex (forany dimensions) then W is quasiconvex by Jensen s inequality: 1 measω W ( Ω 1 measω W(F +Dϕ)dx Ω (F +Dϕ)dx ) =W(F).
118 Theorem 5 Let W :M m n [0, ] be Borel measurable, and Ω R n a bounded open set. A necessary condition for I(y)= W(Dy)dx Ω to be sequentially weak* lower semicontinuous on W 1, (Ω;R m ) is that W is quasiconvex. Proof. Let F M m n,ϕ W 1, 0 (Q;R m ), where Q=(0,1) n.
119 Given k write Ω as the disjoint union Ω= (a (k) j +ε (k) j=1 j Q) N k, wherea (k) j R m, ε (k) j <1/k,measN k =0,and define y (k) (x)= Fx+ε (k) j ϕ F x x a(k) j ε (k) j for x a (k) j +ε (k) j Q otherwise.
120 Then y (k) Fx in W 1, as k and so (measω)w(f) liminf = j=1 = = j=1 a (k) j +ε (k) j Q ε (k)n j ( measω measq Q ) k I(y(k) ) W(F +Dϕ W(F +Dϕ)dx Q W(F +Dϕ)dx x a (k) j ε (k) j )dx
121 Theorem 6 (Morrey, AcerbiFusco, Marcellini) Let Ω R n be bounded and open. Let W : M m n [0, ) be quasiconvex and let 1 p. If p< assume that 0 W(F) c(1+ F p ) for all F M m n. Then I(y)= Ω W(Dy)dx is sequentially weakly lower semicontinuous(weak* if p= ) on W 1,p. Proof omitted. Unfortunately the growth conditionconflictswith (H1),so wecan tusethis to prove existence in 3D elasticity.
122 Theorem 7 (Ball & Murat) Let W :M m n [0, ] be Borel measurable, and let Ω R n be bounded open and have boundary of zero ndimensional measure. If I(y)= [W(Dy)+h(x,y)]dx Ω attains an absolute minimum on A={y :y Fx W 1,1 0 (Ω;Rm )} for all F and all smooth nonnegative h, then W is quasiconvex. Proof (Exercise: use the same method as Theorem 2 and Vitali.)
123 Theorem 8 (van Hove) Let W(F)=c ijkl F ij F kl be quadratic. Then W is rankone convex W is quasiconvex. Proof. Let W be rankone convex. Since for any ϕ W 1, 0 Ω [W(F +Dϕ) W(F)]dx= Ω c ijklϕ i,j ϕ k,l dx we just need to show that the RHS is 0. Extend ϕ by zero to the whole of R n and take Fourier transforms.
124 By the Plancherel formula c ijklϕ i,j ϕ k,l dx = Ω R nc ijklϕ i,j ϕ k,l dx = 4π 2 R nre[c ijklˆϕ i ξ j ˆϕ k ξ l ]dξ 0 as required.
125 Null Lagrangians When does equality hold in the quasiconvexity condition? That is, for what L is Ω L(F +Dϕ(x))dx= Ω L(F)dx for all ϕ W 1, 0 (Ω;R m )? We call such L quasiaffine.
126 Theorem 9 (Landers, Morrey, Reshetnyak...) If L : M m n R is continuous then the following are equivalent: (i) L is quasiaffine. (ii) L is a (smooth) null Lagrangian, i.e. the EulerLagrange equations DivD F L(Du) = 0 hold for all smooth u. (iii) L(F)=constant + d(m,n) k=1 c k J k (F), where J(F) = (J 1 (F),...,J d(m,n) (F)) consists of all the minors of F. (e.g. m = n = 3: L(F)=const.+C F +D coff +edetf). (iv) u L(Du) is sequentially weakly continuous from W 1,p L 1 for sufficiently large p (p>min(m,n) will do).
127 Ideas of proofs. (i) (iii) use L rankone affine (Theorem 4). (iii) (iv) Take e.g. J(Du) = u 1,1 u 2,2 u 1,2 u 2,1 if u is smooth. So if ϕ C 0 (Ω) Ω J(Du) ϕdx= = (u 1 u 2,2 ),1 (u 1 u 2,1 ),2 Ω [u 1u 2,1 ϕ,2 u 1 u 2,2 ϕ,1 )dx. True for u W 1,2 by approximation.
128 If u (j) u in W 1,p,p>2, then Ω J(Du(j) )ϕdx = Ω [u(j) 1 u(j) Ω J(Du)ϕdx 2,1 ϕ,2 u (j) 1 u(j) 2,2 ϕ,1]dx since u (j) 1 u 1 in L p, u (j) 2,1 u 2,1 in L p, and since J(Du (j) ) is bounded in L p/2 it follows that J(Du (j) ) J(Du) in L p/2. For the higherorder Jacobians we use induction based on the identity (u 1,...,u m ) m (x 1,...,x m ) = s=1 ( 1) s+1 x s ( u 1 (u 2,...,u m ) (x 1,...,ˆx s,...,x m ) )
129 For example, suppose m=n=3, p 2, u (j) u in W 1,p,cofDu (j) bounded in L p, detdu (j) χ in L 1. Then χ=detdu. (iv) (i) by Theorem 5. (i) (ii) d dt Ω L(F +Dϕ+tDψ)dx =0 t=0 Ω for all ϕ,ψ C 0 DL(F +Dϕ) Dψdx=0. }{{} Du
130 Polyconvexity Definition W is polyconvex if there exists a convex function g:r d(m,n) (, ] such that W(F)=g(J(F)) for all F M m n. e.g. W(F)=g(F,detF) if m=n=2, W(F)=g(F,cofF,detF) if m=n=3, with g convex.
131 Theorem 10 Let W : M m n [0, ] be Borel measurable and polyconvex, with g lower semicontinuous. Then W is quasiconvex. Proof. Writing 1 fdx= fdx, Ω measω Ω Ω W(F +Dϕ(x))dx = Jensen g Ω ( g(j(f +Dϕ(x)))dx Ω = g(j(f)) = W(F). ) J(F +Dϕ)dx
132 Remark If m 3,n 3 then there are quadratic rankone convex W that are not polyconvex. Such W cannot be written in the form W(F)=Q(F)+ N αl J (l) 2 (F), l=1 whereq 0isquadraticand thej (l) 2 are2 2 minors (Terpstra, D. Serre).
133 Examples and counterexamples We have shown that W convex W polyconvex W quasiconvex W rankone convex. The reverse implications are all false if m > 1,n > 1, except that it is not known whether W rankone convex W quasiconvex when n m=2. W polyconvex W convex since any minor is polyconvex.
134 Example (Dacorogna & Marcellini) m=n=2 W γ (F)= F 4 2γ F 2 detf, γ R, where F 2 =tr(f T F). It is not known for what γ the function W γ is quasiconvex.
135 W γ is convex γ polyconvex γ 1 quasiconvex γ 1+ε for some (unknown) ε>0 rankone convex γ Numerically (DacorognaHaeberly) 1+ε= In particular W quasiconvex W polyconvex (see also later).
136 Theorem 11 (Sverak 1992) If n 2,m 3 then W rankone convex W quasiconvex. Sketch of proof. Itisenoughtoconsiderthecasen=2,m=3. Consider the periodic function u:r 2 R 3 u(x)= 1 2π sin2πx 1 sin2πx 2 sin2π(x 1 +x 2 ).
137 Then Du(x) = L:= cos2πx cos2πx 2 cos2π(x 1 +x 2 ) cos2π(x 1 +x 2 ) r 0 0 s t t :r,s,t R a.e.
138 L is a 3dimensional subspace of M 3 2 and the only rankone lines in L are in the r,s,t directions (i.e. parallel to 0 0, 0 1, or 0 0 ). 1 1 Hence g(f)= rst is rankone affine on L.
139 However (0,1) 2g(Du)dx= 1 4 <0=g violating quasiconvexity. For F M 3 2 define ( (0,1) 2Dudx ), f ε,k (F)=g(PF)+ε( F 2 + F 4 )+k F PF 2, where P :M 3 2 L is orthogonal projection. Can check that for each ε > 0 there exists k(ε)>0 such that f ε =f ε,k(ε) isrankoneconvex, and we still get a contradiction.
140 So is there a tractable characterization of quasiconvexity? This is the main roadblock of the subject. Theorem 12 (Kristensen 1999) For m 3,n 2 there is no local condition equivalent to quasiconvexity (for example, no condition involving W and any number of its derivatives at an arbitrary matrix F). Idea of proof. Sverak s W is locally quasiconvex, i.e. it coincides with a quasiconvex function in a neighbourhood of any F.
141 This might lead one to think that no characterization of quasiconvexity is possible. On the other hand Kristensen also proved Theorem 13 (Kristensen) For m 2,n 2 polyconvexity is not a local condition. For example, one might contemplate a characterization of the type W quasiconvex W is the supremum of a family of special quasiconvex functions(including null Lagrangians).
142 Existence based on polyconvexity Wewillshowthatitispossibletoprovetheexistence of minimizers for mixed boundary value problems if we assume W is polyconvex and problems if we assume W is polyconvex and satisfies (H1) and appropriate growth conditions. Furthermore the hypotheses are satisfied by various commonly used models of natural rubber and other materials.
143 Theorem 14 Suppose that W satisfies (H1) and the hypotheses (H3) W(F) c 0 ( F 2 + coff 3/2 ) c 1 for all F M 3 3, where c 0 >0, (H4)W ispolyconvex,i.e. W(F)=g(F,cofF,detF) for all F M 3 3 for some continuous convex g. Assume that there exists some y in A={y W 1,1 (Ω;R 3 ):y Ω1 =ȳ} with I(y)<, where H 2 ( Ω 1 )>0 and ȳ: Ω 1 R 3 ismeasurable. Thenthereexists a global minimizer y of I in A.
144 Theorem 14 is a refinement (weakening the growth conditions) of J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal., 63: , 1977 (see also J.M. Ball. Constitutive inequalities and existence theorems in nonlinear elastostatics. In R.J. Knops, editor, Nonlinear Analysis and Mechanics, HeriotWatt Symposium, Vol. 1. Pitman, 1977.) due to S. Müller, T. Qi, and B.S. Yan. On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré, Analyse Nonlinéaire, 11:217243, 1994.
145 Proof of Theorem 14 Togiveareasonablysimpleproofwewillcombine (H1), (H3), (H4) into the single hypothesis W(F)=g(F,cofF,detF) forsomecontinuousconvexfunctiong:m M 3 3 R R + with g(f,h,δ) < iff δ>0 and g(f,h,δ) c 0 ( F p + H p )+h(δ), for all F M 3 3, where p 2, 1 p + 1 p = 1, c 0 > 0 and h : R [0, ] is continuous with h(δ)< iff δ>0 and lim δ h(δ) δ =.
146 Letl=inf y A I(y)andlety (j) beaminimizing sequence for I in A, so that lim j I(y(j) )=l. Then since by assumption l < we may as sume that l+1 I(y (j) ) ( c0 [ Dy (j) p + cofdy (j) p ] for all j. Ω +h(detdy (j) ) ) dx
147 Lemma 1 There exists a constant d>0 such that ( Ω z p dx d Ω Dz p dx+ for all z W 1,p (Ω;R 3 ). Ω 1 zda p ) Proof. Suppose not. Then there exists z (j) such that 1= for all j. ( Ω z(j) p dx j Ω Dz(j) p dx+ Ω 1 z (j) da p )
148 Thus z (j) is bounded in W 1,p and so there is a subsequence z (j k) z in W 1,p. Since Ω is Lipschitz we have by the embedding and trace theorems that z (j k) z strongly in L p, z (j k) z in L 1 ( Ω). In particular Ω z p dx=1.
149 But since p is convex we have that Ω Dz p dx liminf k Ω Dz(j k) p dx=0. Hence Dz=0 in Ω, and since Ω is connected it follows that z = constant a.e. in Ω. But also we have that Ω 1 zda=0, and since H 2 ( Ω 1 )>0 it follows that z=0, contradicting Ω z p dx=1.
150 By Lemma 1 the minimizing sequence y (j) is bounded in W 1,p and so we may assume that y (j) y in W 1,p for some y. But also we have that cofdy (j) is bounded in L p and that Ω h(detdy(j) )dx is bounded. So we may assume that cofdy (j) H in L p and that detdy (j) δ in L 1. Bytheresultsontheweakcontinuityofminors we deduce that H=cofDy and δ=detdy.
151 Let u (j) =(Dy (j),cofdy (j),detdy (j) ), u=(dy,cofdy,detdy )). Then u (j) u in L 1 (Ω;R 19 ). Butg isconvex,andsousingmazur stheorem as in the proof of Theorem 1, I(y )= g(u)dx liminf Ω j Ω g(u(j) )dx = lim j I(y (j) )=l. But y (j) Ω1 =ȳ y Ω1 in L 1 ( Ω 1 ;R 3 ) and so y A and y is a minimizer.
152 Incompressible elasticity Rubber is almost incompressible. Thus very large forces, and a lot of energy, are required to change its volume significantly. Such materials are well modelled by the constrained theory of incompressible elasticity, in which the deformation gradient is required to satisfy the pointwise constraint detf =1.
153 Existence of minimizersin incompressible elasticity Theorem 15 Let U = {F M 3 3 : detf = 1}. Suppose W : U [0, ) is continuous and such that (H3) W(F) c 0 ( F 2 + coff 3 2) c 1 for all F U, (H4) W ispolyconvex,i.e. W(F)=g(F,cofF) for all F U for some continuous convex g. Assume that there exists some y in A={y W 1,1 (Ω;R 3 ):detdy(x)=1 a.e.,y Ω1 =ȳ} with I(y)<, where H 2 ( Ω 1 )>0 and ȳ: Ω 1 R 3 ismeasurable. Thenthereexists a global minimizer y of I in A.
154 Proof. For simplicity suppose that W(F)=g(F,cofF) for some continuous convex g:m 3 3 M 3 3 R, where g(f,h) c 0 ( F p + H q ) c 1 forallf,h,where p 2,q p and c 0 >0. Lettingy (j) beaminimizingsequencetheonly newpointistoshowthattheconstraintissatisfied. But since detdy (j) =1 we have by the weak continuity properties that for a subsequence detdy (j) detdy in L, so that the weak limit satisfies detdy=1.
155 Models of natural rubber 1. Modelled as an incompressible isotropic material. Constraint is detf =v 1 v 2 v 3 =1. NeoHookean material Φ=α(v 2 1 +v2 2 +v2 3 3), whereα>0isaconstant. Thiscanbederived from a simple statistical mechanics model of longchain molecules.
156 MooneyRivlin material. Φ=α(v 2 1 +v2 2 +v2 3 3) +β((v 2 v 3 ) 2 +(v 3 v 1 ) 2 +(v 1 v 2 ) 2 3), where α > 0,β > 0 are constants. Gives a better fit to biaxial experiments of Rivlin & Saunders. Ogden materials. Φ= N i=1 α i (v p i 1 +vp i 2 +vp i 3 3) + M i=1 where α i,β i,p i,q i are constants. β i ((v 2 v 3 ) q i+(v 3 v 1 ) q i+(v 1 v 2 ) q i 3),
157 e.g. for a certain vulcanised rubber a good fit is given by N =2,M =1,p 1 =5.0, p 2 =1.3,q 1 =2,α 1 = ,α 2 =4.8, β 1 =0.05kg/cm 2. The high power 5 allows a better modelling of the tautening of rubber as the longchain molecules are highly stretched and the crosslinks tend to prevent further stretchand the crosslinks tend to prevent further stretching.
158 2. Modelled as compressible isotropic material Add h(v 1 v 2 v 3 ) to above Φ, where h is convex, h(δ) as δ 0+, and h has a steep minimum near δ=1.
159 Convexity properties of isotropic functions Let R n + ={x=(x 1,...,x n ) R n :x i 0}. Theorem 16 (Thompson & Freede) Let n 1 and for F M n n let W(F)=Φ(v 1,...,v n ), where Φ is a symmetric realvalued function of the singular values v i of F. Then W is convex on M n n iff Φ is convex on R n + and nondecreasing in each v i.
160 Proof. Necessity. If W is convex then clearly Φ is convex. Also for fixed nonnegative v 1,...,v k 1,v k+1,...,v n g(v)=w(diag(v 1,...,v k 1, v,v k+1,...,v n )) is convex and even in v. But any convex and even function of v is nondecreasingfor v>0. The sufficiency uses von Neumann s inequality  for the details see Ball (1977).
161 Lemma 2 (von Neumann) Let A,B M n n have singular values v 1 (A)... v n (A), v 1 (B)... v n (B) respectively. Then vi n max tr(qarb)= (A)v i (B). Q,R O(n) i=1
162 ApplyingTheorem16weseethatifp 1then Φ p (F)=v p 1 +vp 2 +vp 3 is a convex function of F. Since the singular values of coff are v 2 v 3,v 3 v 1,v 1 v 2 it also follows that if q 1 Ψ q (F)=(v 2 v 3 ) q +(v 3 v 1 ) q +(v 1 v 2 ) q is a convex function of coff.
163 Hence the incompressible Ogden material Φ= N i=1 α i (v p i 1 +vp i 2 +vp i 3 3) + M i=1 β i ((v 2 v 3 ) q i+(v 3 v 1 ) q i+(v 1 v 2 ) q i 3), is polyconvex if the α i 0,β i 0, p 1... p N 1,q 1... q M 1. And in the compressible case if we add a convex function h=h(detf) of detf then under the same conditions the storedenergy function is polyconvex.
164 It remains to check the growth condition of Theorems 14, 15, namely W(F) c 0 ( F 2 + coff 3/2 ) c 1 for all F M 3 3, where c 0 >0. This holds for the Ogden materials provided p 1 2,q and α 1>0,β 1 >0. This includes the case of the MooneyRivlin material, but not the neohookean material. In the incompressible case, Theorem 15 covers the case of the storedenergy function if p 3. Φ=α(v p 1 +vp 2 +vp 3 3)
165 For the neohookean material (incompressible orwithh(detf)added)itisnotknownifthere exists an energy minimizer in A, but it seems unlikely because of the phenomenon of cavitation.
166 In particular the storedenergy function W(F)=α( F 2 3)+h(detF) is not W 1,2 quasiconvex (same definition but with the test functions ϕ W 1,2 0 instead of W 1, 0 ). To see this consider the radial deformation y: B(0,1) R 3 given by where R= x. y(x)= r(r) R x,
167 Since y i = r(r) R x i it follows that y i,α = r(r) R δ iα+ ( r r R R ) xi x α R, that is ( ) Dy(x)= r R 1+ r r x R R x R. In particular Dy(x) 2 =r 2 +2 ( r(r) R ) 2.
168 Set F =λ1 where λ>0. Then W(λ1)=3α(λ 2 1)+h(λ 3 ). On the other hand, if we choose r 3 (R)=R 3 +λ 3 1, then y(x)=λx for x =1 and ( detdy(x)=r r 2=1. R)
169 Then B(0,1) [α( Dy(x) 2 3)+h(detDy(x))]dx =4π 1 0 R2 α ( R 3 +λ 3 1 R 3 which is of order λ 2 for large λ. ) 4 3 ( R 3 +λ 3 ) h(1) dr, R 3 Hence B(0,1) W(Dy(x))dx< for large λ. B(0,1) W(λ1)dx
170 SinceW 1,2 quasiconvexityisnecessaryforweak lower semicontinuity of I(y) in W 1,2 this suggests that the minimum is not attained. (For a further argument suggesting this see J.M. Ball, Progress and Puzzles in Nonlinear Elasticity, Proceedings of course on Poly, Quasiand RankOne Convexity in Applied Mechanics, CISM, Udine, 2010.
171 However there is an existence theory that covers the neohookean and other cases of polyconvex energies with slow growth, due to M. Giaquinta, G. Modica and J. Souček, Arch. Rational Mech. Anal. 106 (1989), no. 2, using Cartesian currents. In the previous examusing Cartesian currents. In the previous examplethiswouldgiveastheminimizery(x)=λx, i.e. the function space setting does not allow cavitation. A simpler proof of this result is in S. Müller, Weak continuity of determinants and nonlinear elasticity, C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), no. 9,
172 The EulerLagrange equations Suppose that W C 1 (M+ 3 3 ). Can we show that the minimizer y in Theorem 14 satisfies the weak form of the EulerLagrange equa tions? As we have seen the standard form of these are formally obtained by computing d dτ I(y+τϕ) τ=0= d dτ W(Dy+τDϕ)dx t=0=0, Ω for smooth ϕ with ϕ Ω1 =0.
173 This leads to the weak form D FW(Dy) Dϕdx=0 Ω for all smooth ϕ with ϕ Ω1 =0. As we have seen, the problem in deriving this weakformisthat DW(Dy) canbebiggerthan W(Dy), and that we do not know if detdy(x) µ>0 a.e. It is an open problem to give hypotheses under which this or the above weak form can be proved.
174 However, it turns out to be possible to derive other weak forms of the EulerLagrange equations by using variations involving compositions of maps. We consider the following conditions that may be satisfied by W: (C1) D F W(F)F T K(W(F)+1) for all F M 3 3 +, where K>0 is a constant, and (C2) F T D F W(F) K(W(F)+1) for all F M 3 3 +, where K>0 is a constant.
175 As usual, denotes the Euclidean norm on M 3 3,forwhichtheinequalities F G F G and FG F G hold. But of course the conditions are independent of the norm used up to a possible change in the constant K. Proposition 4 Let W satisfy (C2). Then W satisfies (C1).
176 Proof SinceW isframeindifferentthematrixd F W(F)F T is symmetric (this is equivalent to the symmetry of the Cauchy stress tensor T = (detf) 1 T R (F)F T ). To prove this, note that d =D dt W(exp(Kt)F) t=0=d F W(F) (KF)=0 for all skew K. Hence D F W(F)F T 2 = [D F W(F)F T ] [F(D F W(F)) T ] = [F T D F W(F)] [F T D F W(F)] T F T D F W(F) 2, from which the result follows.
177 Example Let W(F)=(F T F) detf. Then W is frameindifferent and satisfies (C1) but not (C2). Exercise: check this. As before let A={y W 1,1 (Ω;R 3 ):y Ω1 =ȳ}.
178 We say that y is a W 1,p local minimizer of I(y)= W(Dy)dx Ω in A if I(y)< and I(z) I(y) for all z A with z y 1,p sufficiently small.
179 Theorem 17 For some 1 p < let y A W 1,p (Ω;R 3 ) be a W 1,p local minimizer of I in A. (i) Let W satisfy (C1). Then Ω [D FW(Dy)Dy T ] Dϕ(y)dx=0 for all ϕ C 1 (R 3 ;R 3 ) such that ϕ and Dϕ are uniformly bounded and satisfy ϕ(y) Ω1 =0 in the sense of trace. (ii) Let W satisfy (C2). Then Ω [W(Dy)1 DyT D F W(Dy)] Dϕdx=0 for all ϕ C 1 0 (Ω;R3 ).
180 We use the following simple lemma. Lemma 3 (a) If W satisfies (C1) then there exists γ>0 such that if C M+ 3 3 and C 1 <γ then D F W(CF)F T 3K(W(F)+1) for all F M (b) If W satisfies (C2) then there exists γ>0 such that if C M+ 3 3 and C 1 <γ then F T D F W(FC) 3K(W(F)+1) for all F M
181 Proof of Lemma 3 We prove (a); the proof of (b) is similar. We first show that there exists γ >0 such that if C 1 <γ then W(CF)+1 3 (W(F)+1) for all F M For t [0,1] let C(t)=tC+(1 t)1. Choose γ (0, 1 6K )sufficientlysmallsothat C 1 <γ implies that C(t) 1 2 for all t [0,1]. This is possible since 1 = 3<2.
182 For C 1 <γ we have that W(CF) W(F) 1 d = 0 dt W(C(t)F)dt 1 = D FW(C(t)F) [(C 1)F]dt 0 1 = D F W(C(t)F)(C(t)F) T ((C 1)C(t) 1 )dt 0 K 1 2Kγ 0 [W(C(t)F)+1] C 1 C(t) 1 dt 1 0 (W(C(t)F)+1)dt.
183 Let θ(f)=sup C 1 <γ W(CF). Then W(CF) W(F) θ(f) W(F) 2Kγ(θ(F)+1). Hence (θ(f)+1)(1 2Kγ) W(F)+1, from which W(CF) (W(F)+1) follows.
184 Finally, if C 1 < γ we have from (C1) and the above that D F W(CF)F T = D F W(CF)(CF) T C T as required. K(W(CF)+1) C T 3K(W(F)+1),
185 Proof of Theorem 17 Given ϕ as in the theorem, define for τ sufficiently small Then y τ (x):=y(x)+τϕ(y(x)). Dy τ (x)=(1+τdϕ(y(x)))dy(x) a.e. x Ω. and so y τ A. Also detdy τ (x) > 0 for a.e. x Ω and lim τ 0 y τ y W 1,p=0.
186 Hence I(y τ ) I(y) for τ sufficiently small. But 1 τ (I(y τ) I(y)) 1 d ds W((1+sτDϕ(y(x)))Dy(x))dsdx = 1 τ Ω 0 1 = DW((1+sτDϕ(y(x)))Dy(x)) Ω 0 [Dϕ(y(x))Dy(x)] ds dx.
187 Since by Lemma 3 the integrand is bounded by the integrable function 3K(W(Dy(x))+1) sup z R 3 Dϕ(z), we may pass to the limit τ 0 using domi nated convergence to obtain Ω [D FW(Dy)Dy T ] Dϕ(y)dx=0, as required.
188 (ii) This follows in a similar way to (i) from Lemma 3(b). We just sketch the idea. Let ϕ C 1 0 (Ω;R3 ). Forsufficientlysmall τ >0the mapping θ τ defined by θ τ (z):=z+τϕ(z) belongstoc 1 (Ω;R 3 ), satisfies detdθ τ (z)>0, and coincides with the identity on Ω. By the global inverse function theorem θ τ is a diffeomorphism of Ω to itself.
189 Thus the inner variation y τ (x):=y(z τ ), x=z τ +τϕ(z τ ) defines a mapping y τ A, and Dy τ (x)=dy(z τ )[1+τDϕ(z τ )] 1 a.e. x Ω. Since y W 1,p it follows easily that y τ y W 1,p 0 as τ 0.
190 Changing variables we obtain I(y τ )= from which Ω W(Dy(z)[1+τDϕ(z)] 1 ) det(1+τdϕ(z))dz, Ω [W(Dy)1 DyT D F W(Dy)] Dϕdx=0 follows from (C2) and Lemma 3 using dominated convergence.
191 Interpretations of the weak forms TointerpretTheorem17(i), wemakethefollowing Invertibility Hypothesis. y is a homeomorphism of Ω onto Ω :=y(ω), Ω is a bounded domain, and the change of variables formula Ω f(y(x))detdy(x)dx= Ω f(z)dz holds whenever f :R 3 R is measurable, provided that one of the integrals exists.
192 Theorem 18 Assume that the hypotheses of Theorem 17 and the Invertibility Hypothesis hold. Then Ω σ(z) Dϕ(z)dz=0 for all ϕ C 1 (R 3 ;R 3 ) such that ϕ y( Ω1 ) =0, wherethecauchy stress tensor σ is definedby σ(z):=t(y 1 (z)), z Ω andt(x)=(detdy(x)) 1 D F W(Dy(x))Dy(x) T. Proof. Since by assumption y( Ω) is bounded, we can assume that ϕ and Dϕ are uniformly bounded. The result then follows straighforwardly.
193 Thus Theorem 17 (i) asserts that y satisfies the spatial (Eulerian) form of the equilibrium equations. Theorem 17(ii), on the other hand, involves the socalled energymomentum tensor E(F)=W(F)1 F T D F W(F), F and is a multidimensional version of the Du Bois Reymond or Erdmann equation of the onedimensional calculus of variations, and is the weak form of the equation DivE(Dy)=0.
194 The hypotheses (C1) and (C2) imply that W has polynomial growth. Proposition 4 Suppose W satisfies (C1) or (C2). Then for some s>0 W(F) M( F s + F 1 s ) for all F M
195 Proof Let V M 3 3 be symmetric. For t 0 d dt W(etV ) = (D F W(e tv )e tv ) V From this it follows that = (e tv D F W(e tv )) V K(W(e tv )+1) V. W(e V )+1 (W(1)+1)e K V. Now set V = lnu, where U = U T > 0, and denote by v i the eigenvalues of U.
196 Since lnu =( it follows that 3 i=1 (lnv i ) 2 ) 1/2 3 i=1 lnv i, e K lnu (v K 1 +v K ( C( 1 )(vk 2 +v K 2 )(vk 3 +v K 3 ) 3 3 K K 3 i=1 3 i=1 v K i + v 3K i + i=1 3 i=1 v K i ) 3 v 3K i ) C 1 [ U 3K + U 1 3K ], where C>0,C 1 >0 are constants.
197 We thus obtain W(U) M( U 3K + U 1 3K ), where M = C 1 (W(1)+1). The result now follows from the polar decomposition F = RU of an arbitrary F M+ 3 3, where R SO(3), U =U T >0.
198 IfW =Φ(v 1,v 2,v 3 )isisotropicthenboth(c1) and(c2) are equivalent(exercise) to the condition that (v 1 Φ,1,v 2 Φ, 2,v 3 Φ,3 ) K(Φ(v 1,v 2,v 3 )+1) for all v i > 0 and some K > 0, where Φ,i = Φ/ v i. Now for p 0,q 0 3 i=1 v i (v p v 1 +vp 2 +vp 3 ) i =p(vp 1 +vp 2 +vp 3 ), 3 i=1 v i ((v 2 v 3 ) q +(v 3 v 1 ) q +(v 1 v 2 ) q ) v i =2q((v 2 v 3 ) q +(v 3 v 1 ) q +(v 1 v 2 ) q )
199 And 3 i=1 v i h(v 1 v 2 v 3 ) v i =3v 1v 2 v 3 h (v 1 v 2 v 3 ). Hence both (C1) and (C2) hold for compressible Ogden materials if p i 0,q i 0,α i 0,β i 0, and h 0, for all δ>0. δh (δ) K 1 (h(δ)+1)
200 Exercise. Work out a corresponding treatment of weak forms of the EulerLagrange equation in the incompressible case.
201 Existence of minimizerswith body and surface forces Mixed displacementtraction problems. Suppose that Ω= Ω 1 Ω 2, where Ω 1 Ω 2 =. Considerthe deadload boundary conditions: y Ω1 = ȳ t R Ω2 = t R, where t R (x)=d F W(Dy(x))N(x) is the Piola Kirchhoff stress vector, N(x) is the unit outward normal to Ω and t R L 1 ( Ω 2 ;R 3 ).
202 Suppose the body force b is conservative, so that b(y)= grad y Ψ(y), where Ψ = Ψ(y) is a realvalued potential. The most important example is gravity, for which b = ρ e, where e = (0,0,1), where which b = ρ R e 3, where e 3 = (0,0,1), where thedensityinthereferenceconfigurationρ R > 0 is constant. In this case we can take Ψ(y)= gy 3.
203 Consider the functional I(y)= Ω [W(Dy)+Ψ(y)]dx R yda. Ω 2 t Then formally a local minimizer y satisfies [D FW(Dy) Dϕ b(y) ϕ]dx R ϕda=0 Ω Ω2 t for all smooth ϕ with ϕ Ω1 =0, and thus DivD F W(Dy)+b = 0 in Ω, t R Ω2 = t R.
204 Theorem 19 Suppose that W satisfies (H1) and (H3) W(F) c 0 ( F 2 + coff 3/2 ) c 1 for all F M 3 3, where c 0 >0, (H4)W ispolyconvex,i.e. W(F)=g(F,cofF,detF) for all F M 3 3 for some continuous convex g. Assume further that Ψ is continuous and such that Ψ(y) d 0 y s d 1 for constants d 0 > 0,d 1 > 0,1 s < 2, and that t R L 2 ( Ω 2 ;R 3 ).
205 Assume that there exists some y in A={y W 1,1 (Ω;R 3 ):y Ω1 =ȳ} with I(y)<, where H 2 ( Ω 1 )>0 and ȳ: Ω 1 R 3 ismeasurable. Thenthereexists a global minimizer y of I in A. Sketch of proof. We need to get a bound on a minimizing sequence y (j). By the trace theorem there is a constant c 2 >0 such that Ω [ Dy 2 + y 2 ]dx d 0 y 2 da Ω 2 for all y W 1,2.
206 Using Lemma 1 and the boundary condition on Ω 1, we thus have for any y A, I(y) c 0 2 Ω Dy 2 dx+c 0 +m Ω y 2 dx d 0 Ω cofdy 3 2dx Ω y s dx 1 [ε 1 t 2 2 R +ε y ]da+const. 2 Ω2 Thus choosing a small ε I(y) a 0 Ω [ Dy 2 + y 2 + cofdy 3 2]dx a 1 for all y A and constants a 0 > 0,a 1, giving the necessary bound on y (j).
207 Pure traction problems. Suppose Ω 1 = and that b=b 0 is constant. Then choosing ϕ= const. we find that a necessary condition for a local minimum is that Ω b 0dx+ Ω 2 t R da=0, sayingthatthetotalappliedforceonthebody is zero. If this condition holds then I is invariant to the addition of constants, and it is convenient to remove this indeterminacy by minimizing I subject to the constraint Ω ydx=0.
208 Wethengettheexistenceofaminimizerunder the same hypotheses as Theorem 19, but using the Poincaré inequality Ω y 2 dx C ( Ω Dy 2 dx+ ( Ω ydx ) 2 ). It is also possible to treat mixed displacement pressure boundary conditions (see Ball 1977), which are conservative.
209 Invertibility Recall the Global Inverse Function Theorem, that if Ω R n is a bounded domain with Lipschitz boundary Ω and if y C 1 ( Ω;R n ) with detdy(x)>0 for all x Ω and y Ω onetoone,theny isinvertibleon Ω. Can we prove a similar theorem for mappings in a Sobolev space?
210 Before discussing this question let us note an amusing example showing that failure of y to be C 1 at just two points of the boundary can invalidate the theorem. y Rubber sheet stuck to rigid wires Inner wire twisted through π about vertical axis Yellow region double covered
211 A. Weinstein, A global invertibility theorem for manifolds with boundary, Proc. Royal Soc. Edinburgh, 99 (1985) shows that a local homeomorphism from a compact, connected manifold with boundary to a pact, connected manifold with boundary to a simply connected manifold without boundary is invertible if it is onetoone on each component of the boundary.
212 Results for y W 1,p,p>n, (so that y is continuous). J.M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Royal Soc. Edinburgh 90a(1981)
213 Theorem 20 Let Ω R n be a bounded domain with Lipschitz boundary. Let ȳ : Ω R n be continuous in Ω and onetoone in Ω. Let p > n and let y W 1,p (Ω;R n ) satisfy y Ω = ȳ Ω, detdy(x)>0 a.e. in Ω. Then (i) y( Ω)=ȳ( Ω), (ii) y maps measurable sets in Ω to measurable sets in ȳ( Ω), and the change of variables formula A f(y(x))detdy(x)dx= ȳ(a) f(v)dv
214 holds for any measurable A Ω and any measurable f :R n R, provided one of the integrals exist, (iii) y is onetoone a.e., i.e. the set S={v ȳ( Ω):y 1 (v) contains more than one element} has measure zero, (iv) if v ȳ(ω) then y 1 (v) is a continuum contained in Ω, while if v ȳ(ω) then each connectedcomponentofy 1 (v)intersects Ω.
215 y y 1 (v) y 1 ( v) v v Note that ȳ(ω) is open by invariance of domain. Proof of theorem uses degree theory and change of variables formula of Marcus and Mizel. Examples with complicated inverse images y 1 (v) can be constructed.
216 Theorem 21 Let the hypotheses of Theorem 20 hold, let ȳ(ω) satisfy the cone condition, and suppose that for some q>n Ω (Dy(x)) 1 q detdy(x)dx<. Then y is a homeomorphism of Ω onto ȳ(ω), and the inverse function x(y) belongs to W 1,q (ȳ(ω);r n ). The matrix of weak derivatives x( ) is given by Dx(v)=Dy(x(v)) 1 a.e. in ȳ(ω). If further ȳ(ω) is Lipschitz then y is a homeomorphism of Ω onto ȳ( Ω).
217 Note that formally we have Ω (Dy(x)) 1 q detdy(x)dx = ȳ(ω) Dx(v) q dv. Idea of proof. Get the inverse as the limit of a sequence of mollified mappings. Suppose x( ) is the inverse of y. Let ρ ε be a mollifier, i.e. ρ ε 0, suppρ ε B(0,ε), R n ρ ε (v)dv=1,anddefine x ε (v)= ȳ(ω) ρ ε(v u)x(u)du.
218 Changing variables we have x ε (v)= Ω ρ ε(v y(z))zdetdy(z)dz. In this way the mollified inverse is expressible directly in terms of y, and one can show that for any smooth domain D ȳ(ω) we have Dx ε(v) q dv M <, D where M is independent of sufficiently small ε. Then we can extract a weakly convergent subsequenceinw 1,p (D;R n )foreveryd,giving a candidate inverse.
219 For the pure displacement boundaryvalue problem with boundary condition y Ω =ȳ Ω for which the existence of minimizers was proved intheorem14,wegetthatanyminimizerisa homeomorphism provided we strengthen (H3) by assuming that W(F) c 0 ( F p + coff q +(detf) s ) c 1, where p>3,q>3,s> 2q q 3, and that ȳ satisfies the hypotheses of Theorem 21.
220 With this assumption we have that for any y A with I(y)<, Ω Dy(x) 1 σ detdy(x)dx = Ω cofdy(x) σ (detdy(x)) 1 σ dx c ( cofdy q +(detdy) (1 σ) ( q σ ) )dx Ω <, where σ= q(1+s) q+s >3, since (1 σ) ( q σ ) = s.
221 An interesting approach to the problem of invertibility(i.e. noninterpenetration of matter) in mixed boundaryvalue problems is given in P.G. Ciarlet and J. Nečas, Unilateral problems in nonlinear threedimensional elasticity, Arch. Rational Mech. Anal., 87:(1985) They proposed minimizing I(y)= W(Dy)dx Ω subject to the boundary condition y Ω1 = ȳ and the global constraint detdy(x)dx volume(y(ω)), Ω
222 If y W 1,p (Ω;R 3 ) with p>3 then a result of Marcus & Mizel says that Ω detdy(x)dx= y(ω) cardy 1 (v)dv. sothat theconstraintimpliesthat y isonetoone almost everywhere. They showed that IF the minimizer y is sufficiently smooth then this constraint corresponds to smooth selfcontact.
223 They then proved the existence of minimizers satisfying the constraint for mixed boundary conditions under the growth condition W(F) c 0 ( F p + coff q +(detf) s ) c 1, with p > 3,q p p 1,s > 0. (The point is to show that the constraint is weakly closed.)
224 Results in the space A + p,q(ω)={y:ω R n ;Dy L p (Ω;M n n ), cofdy L q (Ω;M n n ),detdy(x)>0 a.e. in Ω}, following V. Šverák, Regularity properties of deformations with finite energy, Arch. Rat. Mech. Anal. 100(1988) For the results we are interested in Šverák assumes p>n 1,q p p 1, but Qi, Müller, Yan show his results go through with p>n 1,q n n 1, which we assume. Notice that then,since (detf)1=f(coff) T,wehave detf coff n 1, n so that detdy L 1 (Ω).
225 Infact,ifDy L n 1,cofDy L n n 1 thendetdy belongs to the Hardy space H 1 (Ω) (Iwaniec & Onninen 2002). If y A + p,q(ω) with p>n 1,q n 1 n then it is possible to define for every a Ω a setvalued image F(a,y), and thus the image F(A) of a subset A of Ω by F(A)= a A F(a,y).
226 Furthermore Theorem 22 Assume p n. (i)yhasarepresentativeỹwhichiscontinuous outside a singular set S of Hausdorff dimension n p. (ii) H n 1 (F(a))=0 for all a Ω. (iii) For each measurable A Ω, F(A) is measurable and L n (F(A)) detdy(x)dx. A In particular L n (F(S))=0.
227 We suppose that Ω is C and for simplicity thatȳisadiffeomorphismofsomeopenneighbourhood Ω 0 of Ω onto ȳ(ω 0 ). Now suppose that y A + p,q(ω) with y Ω =ȳ Ω. Given v ȳ( Ω), let G(v)={x Ω:v F(x)}. Thus G(v) consists of all inverse images of v.
228 Theorem 23 (i)foreachv ȳ( Ω)thesetG(v)isanonempty continuum in Ω. (ii) For each measurable A ȳ( Ω) the set G(A)= v A G(v) is measurable and L n (A)= detdy(x)dx. dx. G(A) (iii) Let T ={v ȳ( Ω);diamG(v)>0}. Then H n 1 (T)=0. Thus we can define the inverse function x(v) for all v T, and Šverák proves that x( ) W 1,1 (ȳ(ω)).
229 Regularity of minimizers Open Problem: Decide whether or not the global minimizer y in Theorem 14 is smooth. Here smooth means C in Ω, and C up to Here smooth means C in Ω, and C up to the boundary (except in the neighbourhood of points x 0 Ω 1 Ω 2 where singularities can be expected).
230 Clearly additional hypotheses on W are needed for this to be true. One might assume, for example, that W :M R is C, and that W is strictly polyconvex (i.e. that g is strictly convex). Also for regularity up to the boundary we would need to assume both smoothness of the boundary (except perhaps at Ω 1 Ω 2 ) and that ȳ is smooth. The precise nature of these extra hypotheses is to be determined.
231 The regularity is unsolved even in the simplest special cases. In fact the only situation in which smoothness of y seems to have been proved is for the pure displacement problem with small boundary displacements from a stressfree state. For this case Zhang (1991), stressfree state. For this case Zhang (1991), following work of Sivaloganathan, gave hypotheses under which the smooth solution to the equilibrium equations delivered by the implicit function theorem was in fact the unique global minimizer y of I given by Theorem 14.
232 An even more ambitious target would be to somehow classify possible singularities in minimizers of I for generic storedenergy functions W. If at the same time one could associate with each such singularity a condition on W that prevented it, one would also, by imposing all such conditions simultaneously, possess a set of hypotheses implying regularity. Itispossibletogoalittlewayinthisdirection.
233 Jumps in the deformation gradient y piecewise affine N Dy=A, x N >k Dy=B, x N <k x N =k A B=a N. When can such a y with A B be a weak solution of the equilibrium equations?
234 Theorem 24 Suppose that W :M+ 3 3 R has a local minimizer F 0. Then every piecewise affine map y as above is C 1 (i.e. A = B) iff W is strictly rankone convex, i.e. the map t W(F +ta N) is strictly convex for each F M 3 3 and a R 3,N R 3. J.M. Ball, Strict convexity, strong ellipticity, and regularity in the calculus of variations. Proc. Camb. Phil. Soc., 87(1980)
235 Sufficiency. Suppose y is a weak solution. Then (DW(A) DW(B))N =0. Let θ(t)=w(b+ta N). Then θ (1)>θ (0) and so (DW(A) DW(B)) a N =a (DW(A) DW(B))N >0, a contradiction. The necessity use a characterization of strictly convexc 1 functionsdefinedonanopenconvex subset U of R n.
236 Theorem 25 A function ϕ C 1 (U) is strictly convex iff (i) there exists some z U with ϕ(w) ϕ(z)+ ϕ(z) (w z) for w z sufficiently small, and (ii) ϕ is onetoone. The sufficiency follows by applying this to ϕ(a)=w(b+a N). the assumption implying that ϕ is onetoone.
237 Proof of Theorem 25 for the case when U = R n and ϕ(z) as z. z Let z R n. Let w R n minimize h(v)=ϕ(v) ϕ(z) v. By the growth condition w exists and so h(w)= ϕ(w) ϕ(z)=0. Hence w=z, the minimum is unique, and so ϕ(v)>ϕ(z)+ ϕ(z) (v z) for all v z, giving the strict convexity.
238 Discontinuities in y An example is cavitation, which we know is prevented(together with other discontinuities) e.g. by W(F) c 0 F n c 1.
239 Counterexamples to regularity
240
241 Quasiconvexity and partial regularity
242
243 In view of the above and other counterexamples for elliptic systems, if minimizers are smooth itmustbeforspecialreasonsapplyingtoelasticity. Plausible such reasons are: (a) the integrand depends only on Dy (and perhaps x), and not y perhaps x), and not y (b) the dimensions m=n=3 are low, (c) the frameindifference of W. (d) invertibility of y.
244 2D incompressible example. Minimize I(y)= B Dy 2 dx, where B =B(0,1) is the unit disc in R 2, and y:b R 2, in the set of admissible mappings A={y W 1,2 (B;R 2 ):detdy=1 a.e.,y B =ȳ}, whereinpolarcoordinatesȳ:(r,θ) ( 1 2 r,2θ). Then there exists a global minimizer y of I in A. (Note that A is nonempty since ȳ A.) But since by degree theory there are no C 1 maps y satisfying the boundary condition, it is immediate that y is not C 1.
245 Solid phase transformations Displacive phase transformations are characterizedbyachangeofshapeinthecrystallattice at a critical temperature. e.g. cubic tetragonal
246 Energy minimization problem for single crystal
247 Frameindifference requires ψ(rf,θ)=ψ(f,θ) for all R SO(3). If the material has cubic symmetry then also ψ(fq,θ)=ψ(f,θ) for all Q P 24, where P 24 is the group of rotations of a cube.
248 Energywell structure K(θ)={F M Assume :F minimizes ψ(,θ)} austenite martensite Assuming the austenite has cubic symmetry, andgiventhetransformationstrainu 1 say,the N variantsu i arethedistinctmatricesqu 1 Q T, where Q P 24.
249 Cubic to tetragonal (e.g. Ni 65 Al 35 ) N =3 U 1 =diag(η 2,η 1,η 1 ) U 2 =diag(η 1,η 2,η 1 ) U 3 =diag(η 1,η 1,η 2 )
250 Exchange of stability
251 Atomistically sharp interfaces for cubic to tetragonal transformation in NiMn Baele, van Tenderloo, Amelinckx
252 Macrotwins in Ni 65 Al 35 involving two tetragonal variants (Boullay/Schryvers) 252
253 Martensitic microstructures in CuAlNi (Chu/James) 253
254 There are no rankone connections between matrices A,B in the same energy well. In general there is no rankone connection between A SO(3) and B SO(3)U i.
255 Theorem 26 Theorem 26 Let D = U 2 V 2 have eigenvalues λ 1 λ 2 λ 3. Then SO(3)U and SO(3)V are rankone connected iff λ 2 = 0. There are exactly two solutions up to rotation provided λ 1 < λ 2 = 0<λ 3, and the corresponding N s are orthogonal iff tru 2 =trv 2, i.e. λ 1 = λ 3.
256
257 Gradient Young measures y (j) :Ω R m 257
258 The Young measure encodes the information on weak limits of all continuous functions of Dy (j k). Thus f(dy (j k) ) ν x,f. In particular Dy (j k) νx =Dy(x).
259 Simple laminate A B=a N Dy (j) λa+(1 λ)b Young measure ν x =λδ A +(1 λ)δ B
260 260
261 (Classical) austenitemartensite interface in CuAlNi (CH Chu and R.D. James)
262 Gives formulae of the crystallographic theory of martensite (Wechsler, Lieberman, Read) 24 habit planes for cubictotetragonal
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