calculus of variations and applications to solid mechanics€¦ · calculus of variations and...

Calculus of Variations and applications toSolid Mechanics

Carlos Mora-Corral

January 25–29 2010Overview

C. Mora-Corral Calculus of Variations and Solid Mechanics

Outline of the course

1 Introduction to Solid Mechanics. The equations ofElasticity.

2 Hyperelasticity. Polyconvexity. Existence of minimizers.3 Constitutive equations. Isotropic materials.4 Quasiconvexity and lower semicontinuity.5 Phase transitions in crystal solids. The shape memory

effect.


IntroductionDescription of motion

The balance laws of continuum mechanicsNonlinear elasticity


Carlos Mora-Corral

January 25–29 2010Lecture 1




Outline

1 Introduction

2 Description of motion

3 The balance laws of continuum mechanics

Conservation of mass

ForcesBody forces

Surface forces

Conservation of linear momentum

Conservation of angular momentum

4 Nonlinear elasticity




Rubber, steel, wood, rock, metals. . . can all be modelled byelasticity theory, even though their chemical structures andmaximum reversible strains are very different.

Elasticity theory in the central model of Solid Mechanics. Inthis course we will deal with nonlinear theory, which is validfor any deformation, whether large or small. There is also thelinear theory, which consists of a linearization of the model ofthe nonlinear theory, and is valid only for small deformations.




Brief history1678 Hooke’s law1705 Jacob Bernouilli: elastica (elastic rod)1742 Daniel Bernouilli: elastica1744 L. Euler: elastica1821 Navier. Special case of linear elasticity via molecular model1822 Cauchy. Stress. Nonlinear and linear elasticity

For a long time the nonlinear theory was ignored.

1927 A. E. H. Love. Treatise on linear elasticity1950’s R. Rivlin. Exact solutions in incompressible nonlinear elas-

ticity (rubber)1960–80

Nonlinear theory clarified. J. L. Ericksen, C. Truesdell. . .

1980–today

Mathematical developments, applications to material sci-ence, biology.




Outline

1 Introduction




ForcesBody forces

Surface forces







For fluids, we commonly use the Eulerian or spatial descriptionof motion, fixing attention on a point ! in space, and studyinghow the velocity of the fluid v(!, t) varies with time t and spatialpoint !. Different particles of the fluid pass through ! at differenttimes. The Eulerian description is used mainly because in it thegoverning equations take a relatively simple form. However, thedescription is awkward when there are free boundaries, since apoint ! may be occupied by a fluid at some times, but not atothers.




For solids, it is more convenient to use the Lagrangian ormaterial description of motion, in which we fix attention on agiven particle (or material point) and study how it moves. In thisdescription, free boundaries are described automatically.

We label the material points by the positions x = (x1, x2, x3)they occupy in a reference configuration, in which the solidoccupies a region ! ! R3. We assume that ! is open andconnected with a sufficiently smooth boundary "!.




Let u(x, t) " R3 be the position occupied by the material pointx " ! at time t. Thus, u : !# (t1, t2)$ R3, for%& ' t1 < t2 ' &. In this first lecture, we suppose that u issufficiently smooth. Later, we will only assume that u isSobolev.

The deformation gradient is the differential of u with respect tox, denoted F = Du. In components,

Fi,! = ui,! ="ui

"x!, i,# = 1, 2, 3.

Notation: Greek indices for coordinates x!; Latin indices forcoordinates ui. Also, Einstein summation convention.




To avoid interpenetration of matter, we require that for eachtime t, u(·, t) is invertible on !. We also suppose that u(·, t) isorientation-preserving:

J := detF (x, t) > 0 for x " !, t " (t1, t2).




Conservation of massForcesConservation of linear momentumConservation of angular momentum

Outline

1 Introduction




ForcesBody forces

Surface forces








In what follows, E denotes an arbitrary open subset of ! withsufficiently smooth boundary "E. In a given motion u, thematerial points of E occupy the open region Et := u(E, t).






Denote by $(y, t) the density of the body at y " !t at time t.The mass inside the region P ! !t at time t is

m(P, t) =!

P$(y, t) dy.

The density of the body in the reference configuration at thepoint x is denoted $R(x). The mass of the material occupyingE ! ! is

m(E) =!

E$R(x) dx.





We assume the mass is conserved, i.e., for all E,

m(Et, t) = m(E).

Equivalently !

Et

$(y, t) dy =!

E$R(x) dx,

that is, !

E$(x, t) detDu(x, t) dx =

!

E$R(x) dx.

Hence !

E($J % $R) dx = 0 for all E,

i.e.,$J = $R.





ForcesBody Forces

Let b(y, t) " R3 be the force exerted per unit mass by theexternal world. Thus, if E ! ! then

!

Et

$b dy =!

E$Rb dx

is the total body force exerted on Et at time t.

Example: Gravity takes the form

b(y, t) = %ge3.





Surface forces

Cauchy’s hypothesis: There is a vector field s(y, t, n) " R3

(the Cauchy stress vector) with the following property:

Let S be a smooth oriented surface in !t with positive unitnormal n at y. Then s(y, t, n) is the force per unit area exertedacross S on the material on the negative side of S by thematerial on the positive side.

Thus, the resultant surface force on Et is given by!

"Et

s(y, t, n) da.

n = unit outward normal to "Et at y.da = surface area element = dH2(y).





The Piola–Kirchhoff stress vector sR(x, t,N) is parallel to theCauchy stress vector s, but measures the surface force per unitarea in the reference configuration, acting across the(deformed) surface having normal N in the referenceconfiguration.

What is the relationship between deformed and undeformedareas, i.e., between da and dA?





Given A " R3!3, the cofactor matrix A " R3!3 is

cof A =

"

######$

%%%%A22 A23

A32 A33

%%%% %%%%%A21 A23

A31 A33

%%%%

%%%%A21 A22

A31 A32

%%%%

%%%%%A12 A13

A32 A33

%%%%

%%%%A11 A13

A31 A33

%%%% %%%%%A11 A12

A31 A32

%%%%%%%%A12 A13

A22 A23

%%%% %%%%%A11 A13

A21 A23

%%%%

%%%%A11 A12

A21 A22

%%%%

&

''''''(.

If A is invertible, then

A"1 =(cof A)T

det A.

(Cramer’s rule).

Warning: this matrix is called adjoint by some authors, who callcofactor to its transpose.





Lemma (Piola’s identity)

Div(cof Du) = 0,

i.e., in components,

"

"x!(cof Du)i! = 0 (i = 1, 2, 3.

Proof.A computation using ui,!# = ui,#!.





Lemma

n da = JF"T N dA.

Proof.For any vector field %(x, t) we have

!

"Et

% · n da =!

Et

"%i

"yidy =

!

E%i,!(F"1)!iJ dx

=!

E%i,!(cof Du)i! dx =

!

E[%i(cof Du)i!],! dx

=!

"E% · JF"T N dA.





n da = JF"T N dA.

Thus

n =F"T N

|F"T N | , da = J |F"T N |dA = |(cof F )N |dA,

sR = J |F"T N |s.

The resultant surface force on Et, which is given by)"Et

s(y, t, n) da, can be expressed as!

"EsR(x, t,N) dA.






The velocity is defined as v(x, t) = u(x, t) = ""tu(x, t).

We suppose that for all E,

ddt

!

E$Rv dx =

!

"EsR(x, t,N) dA +

!

E$Rb dx

(Newton’s second law).

Theorem (A.L. Cauchy 1827)Conservation of linear momentum holds if and only if

(i) sR(x, t,N) = TR(x, t)N for a second order tensor TR(x, t).(ii) $Rv = Div TR + $Rb.

TR is called the Piola-Kirchhoff stress tensor.Equation (ii) is called Cauchy’s equation of motion.





NowsR dA = sda (definition of sR).

This implies

TRN dA = TRJ"1F T n da = sda,

i.e., s = Tn, where T := J"1TRF T is the Cauchy stress tensor.






ddt

!

Eu ) $Rv dx =

!

"Eu ) sR dA +

!

Eu ) $Rb dx (E ! !.

By Cauchy’s equation, this is equivalent to!

Eu )Div TR dx =

!

"Eu ) TRN dA.

In components,)E &ijk [uj(TR)k!,! % (ujTR k!),!] dx = 0,

iff)E &ijkuj,!(TR)k! dx = 0. As this is true for all E, this holds iff

&ijkuj,!(TR)k! = 0, iff

&ijk(TRF T )kj = 0, iff

TRF T is symmetric, iffT is symmetric.




Outline

1 Introduction




ForcesBody forces

Surface forces







Up to now, our discussion applies to all materials (fluids,solids. . . ) for which Cauchy’s stress hypothesis applies. Tospecify the material, and make the equations determinate, weneed a constitutive equation, expressing the Piola–Kirchhoff (orCauchy) stress tensor in terms of the motion.

The need of a constitutive equation can also be seen from thefollowing heuristic argument. Up to now, we have Cauchy’sequation of motion

$Rv = Div TR + $Rb

and the fact that T is symmetric. Cauchy’s equation has

9 unknowns: 3 for v and 6 for TR.3 equations.

Therefore, 6 additional equations are necessary to solve theequations.




Frame-indifference

Suppose initially that TR depends on x, y and the first partialderivatives of u with respect to x and t, i.e.,

TR = TR(x, y, F, v).

Two observes viewing a moving body will record differentmotions for the body, differing by the rigid motion representingthe movement of one observer relative to the other.

Let observer 1 have frame of reference with origin 0 andorthonormal basis {ei}. Let observer 2 have frame of referencewith origin c(t) and orthonormal basis {e#i (t)} with respect toobserver’s 1 frame. Both observers measure the same time.

We take {ei} and {e#i (t)} to have the same orientation. Soei = Q(t)e#i (t) for some Q(t) " SO(3).




Let the motion measured by the first observer be u(x, t).Let the motion measured by the second observer be u#(x, t).Then

u#(x, t) = Q(t) (u(x, t)% c(t)) .

Label quantities measured by the second observer with #. Thus

v#(x, t) =du#

dt= Q(t) (v(x, t)% c(t)) + Q(t) (u(x, t)% c(t)) ,

F #(x, t) = Dxu# = Q(t)F (x, t).

We now assume that the stress is frame-indifferent, i.e., thesecond observer measures the same stress vector sR as thefirst (but using the #-coordinates). Thus

s#R(x, t,N) = Q(t)sR(x, t,N), equivalently,T #R(x, t) = Q(t)TR(x, t).




HenceTR(x, y#, F #, v#) = Q(t)TR(x, y, F, v)

for all Q : (t1, t2)$ SO(3) and c : (t1, t2)$ R3. This isequivalent to saying that

TR = TR(x, F ) is independent of y, v

andTR(x, QF ) = QTR(x, F ) (Q " SO(3).




DefinitionA material is elastic if it has constitutive equation of the form

TR = TR(x, F ),

and hyperelastic if in addition

TR(x, F ) = DF W (x, F )

for some W : !# R3!3 $ R. In components,

(TR)i!(x, F ) ="W

"Fi!(x, F ) (x " !, F " R3!3, i,# " {1, 2, 3}.




From now on, we consider only hyperelastic materials that arealso homogeneous, i.e., W is independent of x, so that

TR(F ) = DW (F ).

These materials have the same material response at everypoint.

W : R3!3 $ R is called the stored-energy function.

From thermodynamics, we get the interpretation that!

!W (Du(x, t)) dx

is the elastic energy stored by the body at time t.




It is easy to see that

TR(QF ) = QTR(F ) iff W (QF ) = W (F )

(for all F " R3!3 and Q " SO(3)), and that, if this is the case,then

T is symmetric.

Hence, the conservation of angular momentum is automaticallysatisfied for hyperelastic materials.



Carlos Mora-Corral



What can we say about the solutions of the equation of motion?

The hyperbolic system

!Rv = Div DW + !Rb

was solved by T.J.R. Hughes, T. Kato & J.E. Marsden 1977under not totally realistic assumptions.

Simplification: leave out time. The elliptic system of equilibriumequations

!Div DW = !Rb

was solved by T. Valent 1978 when b is small, Dirichletboundary condition, using implicit function.

Both require too much regularity on the data. Mixed boundaryconditions are not allowed.

The orientation preserving is not covered properly:

detDu(x) > 0 "x # !.


Suppose that the body force is conservative:!R(x)b(x, y) = $yB(x, y) for some B : !% R3 & R3. Then, theequilibrium equations are the Euler–Lagrange equations of thefunctional !

![W (Du(x)) + B(x, u(x))] dx.

Instead of dealing with the equation, we focus the attention nowon finding minimizers of the functional.

This problem was solved by J.M. Ball 1977, and constituted thebeginning of the application of Calculus of Variations to SolidMechanics.


The direct method of the Calculus of Variations

PropositionLet X be a topological space. Let I : X & R ' {(} be

Coercive: for each M # R, the set {u # X : I[u] )M} issequentially compact.Sequentially lower semicontinuous: if lim

n!"un = u then

I[u] ) lim infn!"

I[un].

Then there exists a minimizer of I in X, i.e.,

*u0 # X such that I[u0] = infu#X

I[u].

Proof.Let un be a minimizing sequence. Then, for a subsequence,un & u0 # X, and I[u0] ) inf I.


Compactness and (semi-)continuity are antagonistic in atopological space, in the sense that

The stronger the topology is (i.e., the more open sets thereare), the more continuous functions there are, and thefewer compact sets there are.The weaker the topology is (i.e., the fewer open sets thereare), the fewer continuous functions there are, and themore compact sets there are.

Our original problem is to find minimizers on a set. This setdoes not have a priori a topology. The use of a topology is atrick that we use in order to find minimizers.

The idea is to construct a topology that is not too weak and nottoo strong so that there are enough compact sets and enough(semi-)continuous functions.


In Nonlinear elasticity, the central concept is the stress

TR = TR(x, Du(x)),

whereas in hyperelasticity, the central concept is the energy W ,from which the stress TR can be derived. The total energy of adeformation is !

!W (Du(x)) dx.

So the functional space where to look for minimizers shouldconsists of a space of functions with one derivative. C1 is not agood space because its topology is too strong (if we use thenorm topology) or too weak (if we use the pointwise topology).

If W (F ) + |F |p then the natural space is W 1,p(!, R3). In fact,the weak topology in W 1,p gives us a good compromisebetween compactness and (semi-)continuity.


Theorem (S. Banach 1932)

Let 1 < p <(. Let S be a bounded set in W 1,p. Then thereexists a sequence un # S and u #W 1,p such that

un " u in W 1,p.

This theorem suffices for compactness. What aboutsemicontinuity?


Proposition

Let 1 < p <(. Let W : R3$3 & R ' {(} be convex. Then

I[u] :=!

!W (Du(x)) dx

is sequentially weakly lower semicontinuous in W 1,p, i.e.,

un " u =, I[u] ) lim infn!"

I[un].

Proof.Dun " Du in Lp. By convexity,

W (Dun(x))!W (Du(x)) - DW (Du(x))(Dun(x)!Du(x)),

soI[un]! I[u] -

!

!DW (Du)(Dun !Du) dx& 0.


Proposition

There is no convex function W : R3$3+ & R satisfying

W (F )&( as det F & 0.

Proof.

"

##

#1

$

% =1 + #

2

"

#1

11

$

% +1! #

2

"

#!1

!11

$

%

Use convexity and send #& 0 to obtain a contradiction.

Convexity, in fact, would imply other physical inconsistencies.


Proposition (Y. Reshetnyak 1968)

Let un " u in W 1,p(!, R3).If p > 2 then cof Dun " cof Du in Lp/2.If p > 3 then det Dun " det Du in Lp/3.

Proof.If u # C2,

u1,1u

2,2 ! u1

,2u2,1 =

&u1u2

,2

',1!

&u1u2

,1

',2

.

Hence for all $ # C"c (!),!

!

&u1

,1u2,2 ! u1

,2u2,1

'$ dx =

!

!

&!u1u2

,2$,1 + u1u2,1$,2

'dx.

By density, this holds for all u #W 1,p.


The same proof gives:


Let un " u in W 1,p(!, R3).If p - 2 and cof Dun " v in Lp/2 then v = cof Du.If p - 3 then det Dun " w in Lp/3 then w = detDu.

There are counterexamples showing that these exponents areoptimal.

In Elasticity, asking p - 3 is too much. Apparently, for definingdet Du we would need u #W 1,p for p - 3. But there is a way toovercome this.


Proposition (J.M. Ball 1977)Let p - 2 and r > 1.

If u #W 1,p and cof Du # Lp! then det Du # L1.If un " u in W 1,p,

cof Dun " v in Lp!,

det Dun " w in Lr

then v = cof Du and w = detDu.

Proof.Using Laplace’s formula detF = F1i(cof F )1i

and Piola’s identity Div cof Du = 0we obtain det Du =

&u1(cof Du)1i

',i

for u # C2.Integrating by parts and using density of smooth functions,

!

!(detDu)$ dx = !

!

!u1(cof Du)1i$,i dx "$ # C"0 (!).


Theorem (J.M. Ball 1977)

Let p - 2, q - p%, r > 1, c1, c2 > 0. Let W : R3$3+ & R satisfy:

There exists a convex functionW : R3$3 % R3$3 % (0,()& [0,() such that

W (F ) = W (F, cof F,det F ) "F # R3$3+ .

W (F )&( as det F & 0.W (F ) - c1(|F |p + | cof F |q + (det F )r)! c2.

Let " be a subset of %! with positive area. Let $ : "& R3. Let

A :=(u #W 1,p(!, R3) : cof Du # Lq, detDu # Lr,

u = $ on ", det Du > 0).

Let I[u] :=!

!W (Du) dx, u # A.

Assume A .= !.Then there exists a minimizer of I in A.


We can add forces to the total energy:!

!f · u dx +

!

!!\"g · u da.


Analysis of strainMaterial symmetry

Incompressible elasticityConstitutive equations for rubber

Initial and boundary conditionsExamples of non-uniqueness of equilibrium solutions


Carlos Mora-Corral






Outline

1 Analysis of strain

2 Material symmetry

3 Incompressible elasticity

4 Constitutive equations for rubber

5 Initial and boundary conditions

6 Examples of non-uniqueness of equilibrium solutions





Preliminaries from linear algebraLocal state of strainInvariants

Outline


2 Material symmetry










Analysis of strainPreliminaries from linear algebra

R3!3 := {real 3!3 matrices}, R3!3+ := {A " R3!3 : detA > 0}

SO(3) := {R " R3!3 : RT R = 1,det R = 1} = {rotations}.

If a, b " R3, the tensor product a# b is the 3! 3 matrix withcomponents

(a# b)ij = aibj , i, j = 1, 2, 3.






Let A " R3!3 be symmetric, i.e., A = AT . Then:

1 All eigenvalues of A are real.2 The eigenvectors corresponding to distinct eigenvalues are

orthogonal.3 There is an orthonormal basis of R3 consisting of

eigenvectors of A.

Denote the orthonormal basis by {ei}3i=1. Thus

Aei = !iei, ei · ej = "ij , i, j = 1, 2, 3.

Then A = !iei # ei is the spectral decompostition of A.Equivalently,

A = QDQT ,

where D = diag(!1, !2, !3) and Q " SO(3).C. Mora-Corral Calculus of Variations and Solid Mechanics





Proposition (Square root)Let C be a positive definite symmetric 3! 3 matrix. Then thereexists a unique positive definite 3! 3 matrix U such thatC = U2.

We write U = C1/2.

Proposition (Polar decomposition)

Let F " R3!3+ . Then there exist positive definite symmetric

U, V " R3!3 and R " SO(3) such that F = RU = V R. Theserepresentations are unique.






When we apply this result to a deformation gradient F = Du,we call U, V the right and left stretch tensor.

C = U2 = F T F is the right Cauchy-Green strain tensor,

B = V 2 = FF T is the left Cauchy-Green strain tensor.

The eigenvalues v1, v2, v3 (> 0) of U (or V ) are called theprincipal stretches.






Local state of strain

Fix x, t. Then

u(x + z, t) = u(x, t) + F (x, t)z + o(|z|).

By polar decomposition, to first order in z the deformation u isgiven by a rotation followed by a stretching of amounts vi alongmutually orthogonal axes (or viceversa: first stretching and therotation).

Equivalently, since

F = RU = RQDQT = RDQT

where D = diag(v1, v2, v3) and R,Q, R " SO(3), then thedeformation is given by a rotation, followed by stretching alongthe coordinate axis, then another rotation.






Invariants

det(C $ !1) = $!3 + IC!2 $ IIC! + IIIC , and similarly for B.

But QCQT = diag(v21, v

22, v

23) for some Q " SO(3). So

det(C $ !1) = det Q(C $ !1)QT = det(QCQT $ !1)

= (v21 $ !)(v2

2 $ !)(v23 $ !)

= $!3 + (v21 + v2

2 + v23)!

2 $ (v21v

22 + v2

2v23 + v2

3v21)! + (v1v2v3)2.

Hence

IC = v21 + v2

2 + v23 = trC

IIC = v21v

22 + v2

2v23 + v2

3v21 =

12

!(trC)2 $ trC2

"

IIIC = (v1v2v3)2 = detC.






Note that

IC = IB = |F |2,IIC = IIB = | cof F |2,IIIC = IIIB = (detF )2.





Outline


2 Material symmetry









Material symmetry

Which initial linear deformations do not change W?

DefinitionThe symmetry set S of W is the set of H " R3!3

+ for which

W (F ) = W (FH) %F " R3!3+ .

S is a subgroup of R3!3+ .

Example: For a material with cubic symmetry,

S = P 24 = {rotations of a cube into itself}.

If S & SO(3) we say that W is isotropic, i.e.,

W (F ) = W (FR) %R " SO(3).C. Mora-Corral Calculus of Variations and Solid Mechanics




Examples of isotropic materials

Iron-Carbon alloy Galvanized steel Vulcanized rubber





Examples of non-isotropic materials

Wood Quartz





Theorem (R.S. Rivlin & J.L. Ericksen 1955)The following conditions are equivalent:

(i) W is isotropic.(ii) W (F ) = h(I, II, III) for some h : (0,')3 ( R.(iii) W (F ) = !(v1, v2, v3) for some symmetric ! : (0,')3 ( R.(iv) T (F ) = #01 + #1B + #2B2, where #0, #1, #2 are scalar

functions of I, II, III.





Outline


2 Material symmetry









Incompressible elasticity

For some materials (e.g., rubber), a large amount of energy isneeded to change the volume significantly. Such materials canbe modelled by incompressible elasticity, in which the motion isrequired to satisfy the constraint J = 1.





In incompressible elasticity, only the values of W (F ) fordet F = 1 are physically meaningful. In particular, the samematerial can be modelled using the stored-energy function

W (F ) := W (F )$ p(detF $ 1)

for any p " R. Now, when det F = 1,

(detF )"1DW (F )F T = (detF )"1#DW (F )F T $ p cof FF T

$

= T #(F )$ p1,

where T #(F ) := T #R(F )F T and T #R(F ) := DW (F ). So Wdetermines the Cauchy stress T only up to an arbitraryhydrostatic pressure p = p(x, t), i.e.,

T (F ) = $p1 + T #(F ).





Thus, in incompressible elasticity we have to solve the equationof motion

$Ru = Div (T #R(Du)$ p cof Du) + $RbdetDu = 1

%

for the unknowns u and p (Lagrange multiplier).





Modelled as incompressibleModelled as compressible

Outline


2 Material symmetry










Constitutive equations for rubberModelled as incompressible

Neo-Hookean material

W = #(I $ 3) = #(v21 + v2

2 + v23 $ 3),

# > 0.

Predicted by simplest statistical mechanical model oflong-chain molecules.






Mooney–Rivlin material

W = #(I $ 3) + %(II $ 3)

= #(v21 + v2

2 + v23 $ 3) + %

#(v2v3)2 + (v3v1)2 + (v1v2)2 $ 3

$

= #(v21 + v2

2 + v23 $ 3) + %

#v"21 + v"2

2 + v"23 $ 3

$,

#,% > 0.






Ogden’s models

W =N&

i=1

#i (vpi1 + vpi

2 + vpi3 $ 3)

+M&

i=1

%i ((v2v3)qi + (v3v1)qi + (v1v2)qi $ 3)

where #i, %i, pi, qi > 0 are constants.






Modelled as compressible

Add h(det F ) to above W , with h : (0,')( [0,') convex,h"1(0) = 1, and

limt$0

h(t) = limt%0

h(t) ='.





Initial conditionsBoundary conditions

Outline


2 Material symmetry










Initial and boundary conditionsInitial conditions

u(x, 0) = u0(x), x " ",

u(x, 0) = u1(x), x " ".






Boundary conditions

(a) Displacement

u(x, t) = u(x, t) x " &".

(b) Dead load traction

TR(x, t)N(x) = sr(x, t), x " &".

The dead loads maintain the same direction and magnituteper unit reference area irrespective of the deformations.

(c) Pressure

T (u(x, t), t)n(u(x, t)) = $p(t)n(u(x, t)), x " &".

(d) Mixed boundary conditions combining the above.






Boundary conditions

(a) Displacement

u(x, t) = u(x, t) x " &".

(b) Dead load traction

TR(x, t)N(x) = sr(x, t), x " &".

The dead loads maintain the same direction and magnituteper unit reference area irrespective of the deformations.

(c) Pressure

T (u(x, t), t)n(u(x, t)) = $p(t) n(u(x, t)), x " &".

(d) Mixed boundary conditions combining the above.





Outline


2 Material symmetry









Examples of non-uniqueness of equilibrium solutions



Carlos Mora-Corral



The two abstract ingredients for the direct method of Calculusof variations are:

Compactness.Lower semicontinuity.

Compactness is guaranteed by coercivity, whereas lowersemicontinuity is guaranteed by polyconvexity.

The argument for polyconvexity is:

A convex function of a w-continuous function is w-lowersemicontinuous.cof and det are w-continuous.

Are there more w-continuous functions?



Let W : R3!3 ! R be continuous. Let 1 " p " #. Suppose thatfor every sequence un ! u in W 1,p(!, R3) (weak" if p =#) wehave

W (Dun) ! W (Du) in D#(!).

Then

W (F ) = c0 + c1 · F + c2 · cof F + c3 detF $F % R3!3

for some c0, c3 % R, c1, c2 % R3!3.

This characterizes the w-continuous functions, but it does notcharacterize the w-lower semicontinuous functions.


Definition (C. B. Morrey 1952)

The function W % L$loc(R3!3) is called quasiconvex if

W (F ) "!

(0,1)3W (F + D"(x)) dx

for every F % R3!3 and every " %W 1,$0 ((0, 1)3, R3).

If D & R3 is any bounded domain, and we impose affineDirichlet boundary conditions

Fx, x % #D

then the affine map Fx is a minimizer of!

DW (Du) dx.

The following implications hold:

convex =' polyconvex =' quasiconvex =' rank-one convex


Theorem (C. B. Morrey 1952)

Let W : R3!3 ! R be continuous. Let

I[u] :=!

!W (Du(x)) dx, u %W 1,$(!, R3).

If I is w"slsc in W 1,$, i.e.,

I[u] " lim infn%$

I[un] whenever un"! u in W 1,$

then W is quasiconvex.


Theorem (C. B. Morrey 1952)

Let W : R3!3 ! R be continuous and quasiconvex. Define

I[u] :=!

!W (Du(x)) dx, u %W 1,$(!, R3).

Then I is w"slsc in W 1,$, i.e.,

I[u] " lim infn%$

I[un] whenever un"! u in W 1,$.


The above results also hold for W 1,p, under natural growthconditions on W .

To sum up,

W quasiconvex ('!

!W (Du) dx weakly lower semicont.

This seems to solve completely the direct method of Calculusof Variations, and in a certain sense it does.

There is, however, a big difficulty. There is no tractablecondition for quasiconvexity. Quasiconvexity is only marginallymore transparent that lower semicontinuity itself.

The chain of implications

polyconvex =' quasiconvex =' rank-one convex

is greatly useful.


Phase transformations in solidsInterfaces

Martensite to martensiteAustenite to martensite

MicrostructureShape-memory effect


Carlos Mora-Corral






Outline

1 Phase transformations in solids

2 Interfaces

3 Martensite to martensite

4 Austenite to martensite

5 Microstructure

6 Shape-memory effect





Phase transformations in solids

Example: cubic ! tetragonal transformation.

Model using elasticity with energy functional

I!(u) =!

!!(Du(x), ") dx,

where ! : R3!3 " (0,#)! R is smooth, bounded below andframe-indifferent:

!(F, ") = !(QF, ") $Q % SO(3),

and hence !(F, ") = !(U, "), where U = (F T F )1/2.

Take reference configuration to be undistorted austenite at" = "c. Assume cubic symmetry:

!(FR, ") = !(F, ") $R % P 24.C. Mora-Corral Calculus of Variations and Solid Mechanics




For " & "c, assume we have a transformation strain U = U(")symmetric positive definite.

The matrices {RURT : R % P 24} represent the N differentvariants of martensite.ExampleFor cubic ! tetragonal, we take

U(") = U1(") = diag(#2, #1, #1),

where #i = #i("), and there are just three variants:

U1 = diag(#2, #1, #1)U2 = diag(#1, #2, #1)U3 = diag(#1, #1, #2).





By adding a suitable function of " to !, we can suppose that

minF"R3!3

!(F, ") = 0.

So we assume that

K(") := {F % R3!3 : !(F, ") = 0}is given by

K(") =

"#######$

#######%

$(")SO(3) " > "c

SO(3) 'N&

i=1

SO(3)Ui("c) " = "c

N&

i=1

SO(3)Ui(") " < "c.

Experimentally $(") > 1: thermal expansion.C. Mora-Corral Calculus of Variations and Solid Mechanics