Calculus of Variations and applications to Solid Mechanics · Introduction Description of motion The balance laws of continuum mechanics Nonlinear elasticity Calculus of Variations

Calculus of Variations and applications toSolid Mechanics

Carlos Mora-Corral

April 4–8 2011Overview

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. Relaxation.

Regularity. The scalar case.5 Phase transitions in crystal solids. The shape memory

effect.


IntroductionDescription of motion

The balance laws of continuum mechanicsNonlinear elasticity


Carlos Mora-Corral

April 4–8 2011Lecture 1. Introduction to Solid Mechanics. The equations of Elasticity




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




Outline

1 Introduction




ForcesBody forces

Surface forces







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 R. Hooke’s law.1705 Jacob Bernouilli: elastica (elastic rod).1742 Daniel Bernouilli: elastica.1744 L. Euler: elastica.1821 C.-L. Navier. Special case of linear elasticity via molecular

model.1822 A.-L. Cauchy. Stress. Nonlinear and linear elasticity.

For a long time the nonlinear theory was ignored.

1927 A. E. H. Love. Treatise on linear elasticity.1950’s R. Rivlin. Exact solutions in incompressible nonlinear

elasticity (rubber).1960–80 Nonlinear theory clarified. J. L. Ericksen, C. Trues-

dell. . .1980–today Mathematical developments, applications to material

science, biology.




Solid Mechanics is a meeting place for physics, materialsscience, PDEs, analysis, algebra, geometry, computation. . .




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$R b 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

detA.

(Cramer’s rule).

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





Lemma (Piola’s identity)

Div(cofDu) = 0,

i.e., in components,

"

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

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





Change of variables fomula for surface integrals

Lemma

n da = JF"TN dA.

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

!

"Et

% · n da =

!

Et

"%i

"yidy =

!

E%i,!(F

"1)!iJ dx

=

!

E%i,!(cofDu)i! dx =

!

E[%i(cofDu)i!],! dx

=

!

"E% · JF"TN dA.





n da = JF"TN dA.

Thus

n =F"TN

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

sR = J |F"TN |s.

The resultant surface force on Et can be expressed as!

"EsR(x, t,N) dA =

!

"Et

s(y, t, n) da.






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

We suppose that for all E,

d

dt

!

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 = s da (definition of sR).

This implies

TRN dA = TRJ"1F Tn da = s da,

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






d

dt

!

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(TRFT )kj = 0, iff

TRFT is symmetric, iff

T 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 argument. Up to now, we have Cauchy’s equation ofmotion

$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) = at+d and orthonormal basis {e#i } with respectto observer’s 1 frame. Both observers measure the same time.

We take {ei} and {e#i } to have the same orientation. Soei = Q(t)e#i for some Q " 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 (u(x, t)% c(t)) .

Label quantities measured by the second observer with #. Thus

v#(x, t) =du#

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

F #(x, t) = Dxu# = QF (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) = QsR(x, t,N), equivalently,T #R(x, t) = QTR(x, t).




Hence

TR(x, y#, F #, v#) = Q TR(x, y, F, v), i.e.,

TR(x,Q(y % ct% d), QF,Q(v % c)) = Q TR(x, y, F, v),

for all Q " SO(3) and a, d " R3. This is equivalent to saying that

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

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




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

TR = TR(x, F ),

and hyperelastic if in addition

TR(x, F ) = DFW (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

April 4–8 2011Lecture 2. Hyperelasticity. Polyconvexity. Existence of minimizers


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

The hyperbolic system

!Rv = DivDW + !Rb

was shown to have existence by T.J.R. Hughes, T. Kato &J.E. Marsden 1977 under not totally realistic assumptions.

Simplification: leave out time. The elliptic system of equilibriumequations

!DivDW = !Rb

was shown to have existence by T. Valent 1978, when b is small,Dirichlet boundary 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, but thefewer compact sets there are.The weaker the topology is (i.e., the fewer open sets thereare), the fewer continuous functions there are, but 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 atool 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 shouldconsist 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 provides 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 detF & 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 cofDun " cofDu in Lp/2.If p > 3 then detDun " detDu in Lp/3.

Proof.If u # C2,

u1,1u2,2 ! u1,2u

2,1 =

&u1u2,2

',1!&u1u2,1

',2.

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

!

!

&u1,1u

2,2 ! u1,2u

2,1

'$ 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 cofDun " v in Lp/2 then v = cofDu.If p - 3 then detDun " w in Lp/3 then w = detDu.

Counterexamples show that these exponents are optimal.

In Elasticity, asking p - 3 is too restrictive. Apparently, in orderto define detDu we would need u # W 1,p for p - 3. But there isa way to overcome this.


Proposition (J.M. Ball 1977)Let p - 2.

If u # W 1,p and cofDu # Lp! then detDu # L1.If un " u in W 1,p,

cofDun " v in Lp! ,

detDun " w in L1

then v = cofDu and w = detDu.

Proof.Using Laplace’s formula detF = F1i(cof F )1iand Piola’s identity Div cofDu = 0we obtain detDu =

&u1(cofDu)1i

',i

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

!(detDu)$ dx = !

!

!u1(cofDu)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, detF ) "F # R3$3+ .

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

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

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

u = $ on ", detDu > 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

April 4–8 2011Lecture 3. Constitutive equations. Isotropic materials





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 : RTR = 1, detR = 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 =!3

i=1 !iei # ei is the spectral decomposition 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 TF 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 thenrotation).

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, v22, v

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

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

= (v21 $ !)(v22 $ !)(v23 $ !)

= $!3 + (v21 + v22 + v23)!2 $ (v21v

22 + v22v

23 + v23v

21)!+ (v1v2v3)

2.

Hence

IC = v21 + v22 + v23 = trC

IIC = v21v22 + v22v

23 + v23v

21 =

1

2

"(trC)2 $ trC2

#

IIIC = (v1v2v3)2 = detC.






In terms of F , we have

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(thus large forces) is needed to change the volume significantly.Such materials can be modelled by incompressible elasticity, inwhich the motion is required to satisfy the constraint J = 1.





In incompressible elasticity, only the values of W (F ) fordetF = 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 detF = 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 cofDu) + $Rb

detDu = 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 + v22 + v23 $ 3),

# > 0.

Predicted by simplest statistical mechanical model oflong-chain molecules.






Mooney–Rivlin material

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

= #(v21 + v22 + v23 $ 3) + %$(v2v3)

2 + (v3v1)2 + (v1v2)

2 $ 3%

= #(v21 + v22 + v23 $ 3) + %$v"21 + v"2

2 + v"23 $ 3

%,

#,% > 0.






Ogden’s models

W =N'

i=1

#i (vpi1 + vpi2 + vpi3 $ 3)

+M'

i=1

%i ((v2v3)qi + (v3v1)

qi + (v1v2)qi $ 3)

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

Example: For a certain vulcanized rubber a good fit is given by

N = 2 , M = 1 , p1 = 5.0 , p2 = 1.3 , q1 = 2 ,

#1 = 0.0024 , #2 = 4.8 , %1 = 0.05 .






Modelled as compressible

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

limt$0

h(t) = limt%&

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.





Outline


2 Material symmetry









Examples of non-uniqueness of equilibrium solutions


Lower semicontinuityRelaxationRegularity

The scalar case


Carlos Mora-Corral

April 4–8 2011Lecture 4. Quasiconvexity and lower semicontinuity. Relaxation.

Regularity. The scalar case.



The scalar case

Outline

1 Lower semicontinuity

2 Relaxation

3 Regularity

4 The scalar case



The scalar case

Lower semicontinuity

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?C. Mora-Corral Calculus of Variations and Solid Mechanics


The scalar case


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.



The scalar case

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 convexC. Mora-Corral Calculus of Variations and Solid Mechanics


The scalar case

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.



The scalar case

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 scalar case

The above results also hold for W 1,p, under p-growth conditionson W .

To sum up,

W quasiconvex ('!

!W (Du) dx s.w.l.s.c.

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.C. Mora-Corral Calculus of Variations and Solid Mechanics


The scalar case

Convexity and rank-one convexity are local properties. For C2

functions, they can be characterized with a pointwise inequality.

Polyconvexity and quasiconvexity are not local properties(J. Kristensen 1999, 2000).



The scalar case

Outline


2 Relaxation

3 Regularity

4 The scalar case



The scalar case

Relaxation

We saw that if W has p-growth conditions, then

W is quasiconvex ('!

!W (Du) dx is s.w.l.c. in W 1,p.

Under the same assumptions, the following relaxation resultholds (B. Dacorogna 1982)

inf

!

!W (Du) dx = min

!

!W qc(Du) dx,

where W qc is the quasiconvexification of W , i.e., the largestquasiconvex function less that W :

W qc(F ) := sup{V (F ) : V quasiconvex, V " W}.



The scalar case

Outline


2 Relaxation

3 Regularity

4 The scalar case



The scalar case

Regularity

When are minimizers smooth? This is mainly an open problem.

Examples show that minimizers of"!W (Du) dx with

W % C$(Rn!n,R) may have singularities.

This happens even in the 1D case u : R ! R for convex W .The reason is that the Euler–Lagrange equation (which iselliptic if W is convex) may not be satisfied for minimizers.

Under similar assumptions on W (smooth, strict quasiconvexity,p-growth), the following partial regularity result holds (L.C. Evans1986):

There exists a closed set E of measure zero such thatthe minimizer is C$(! \ E).



The scalar case

Outline


2 Relaxation

3 Regularity

4 The scalar case



The scalar case

The scalar case

Most of the theory of nonlinear elasticity, done for functionsu : R3 ! R3, admits straightforward generalizations to functionsu : Rn ! Rn.

In Calculus of Variations, the expression the scalar case refersto functions u : Rn ! R or u : R ! Rn.

In this case, all concepts of convexity are equivalent:

convex (' polyconvex (' quasiconvex (' rank-one convex.



The scalar case

In particular, the following results hold:

Existence:

W convex ('!

!W (Du) dx is s.w.l.s.c.

Relaxation:

inf

!

!W (Du) dx = min

!

!W c(Du) dx,

where W c is the convexification of W , i.e., the largest convexfunction less that W .


Phase transformations in solidsInterfaces

Martensite to martensiteAustenite to martensite

MicrostructureShape-memory effect


Carlos Mora-Corral

April 4–8 2011Lecture 5. Phase transitions in crystal solids. The shape memory effect





Outline

1 Phase transformations in solids

2 Interfaces

3 Martensite to martensite

4 Austenite to martensite

5 Microstructure

6 Shape-memory effect





Outline


2 Interfaces



5 Microstructure






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 TF )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




Outline


2 Interfaces



5 Microstructure






InterfacesLemma (Hadamard’s jump condition)

Let N % R3 \ {0}, k % R, A,B % R3!3. If u : R3 ! R3 iscontinuous with Du = A in {x ·N > k} and Du = B in{x ·N < k} then there exists a % R3 such that A(B = a)N .

When is Hadamard’s jump condition satisfied for A,B % K(")?

First suppose that A,B % SO(3)U with U = UT > 0. ThenA = R1U , B = R2U and R1U (R2U = a)N . Hence

RT2 R1 = 1+RT

2 a) U#1N.

So RT2 R1 has two linear independent axes of rotation. Hence

RT2 R1 = 1 and A = B. Therefore, no rank-one connection

(interface) between an energy well and itself.C. Mora-Corral Calculus of Variations and Solid Mechanics




Next suppose we have two distinct energy wells: SO(3)U andSO(3)V with U = UT > 0, V = V T > 0. When are theyrank-one connected?

Same as asking when SO(3) and SO(3)F are rank-oneconnected, F := V U#1.

Theorem (J.M. Ball & R.D. James 1987)

Let F % R3!3+ \ SO(3). Then, the wells SO(3) and SO(3)F are

rank-one connected iff the middle eigenvalue of F TF is 1.





Outline


2 Interfaces



5 Microstructure






Martensite to martensite

Twins are rank-one connections (interfaces) between twodifferent variants of martensite, i.e., between SO(3)U andSO(3)V , where V = QUQT for some Q % P 24.

By the theorem above, these wells are rank-one connected iffthe middle eigenvalue of (V U#1)TV U#1 is 1. Butdet(V U#1)TV U#1 = 1, and so wells are rank-one connected iff

det((V U#1)TV U#1 ( 1) = 0, iffdet(V 2 ( U2) = 0, iffdet(V ( U) = 0.

In this case, given any matrix in SO(3)U , there are exactly tworank-one connections (twins) to SO(3)V .





Example (Twins for cubic ! tetragonal)Take U1 = diag(#2, #1, #1) and U2 = diag(#1, #2, #1). Thendet(U1 ( U2) = 0, so wells are rank-one connected. The twinsare U1 and RU2 with

R =

'

(cos " ( sin " 0sin " cos " 00 0 1

)

* , cos " =2#1#2#21 + #22

, sin " = ±#22 ( #21#21 + #22

.





Outline


2 Interfaces



5 Microstructure






Austenite to martensite

Austenite transforms to martensite via a more complicatedinterface.

The minimizing sequence u(j) satisfies

I!c(u(j)) =

!

boundarylayer

%(Du(j), "c) dx (!j$%

0.





Lemma

The minimizing sequence u(j) in the picture satisfies

u(j)&&

+['A+ (1( ')B]x, x ·m * 0

x, x ·m & 0in W 1,%

loc (R3,R3).

Therefore, to find all austenite-martensite transitions of thistype we need to solve

'A+ (1( ')B = 1+ b)m

for A,B,', b,m.





Outline


2 Interfaces



5 Microstructure






Microstructure

In the austenite!martensite transition we have constructed aminimizing sequence and calculated its w&-W 1,% limit:

u(j)&& u.

An explicit calculation shows that u is not a minimizer. Hence Iis not w&-lower semicontinuous. Hence W is not quasiconvex.

As W is not quasiconvex, the existence of minimizers is notguaranteed. In fact, in most cases, minimizers do not exist.This leads to the formation of microstructure. An explicitexample of non-attainment of the minimum is to put affineboundary conditions Fx, with F a convex combination of tworank-one connected matrices in two different wells.





Since minimizers do not exist, the right object to study in orderto understand the “minimum” are the minimizing sequences.

(The mathematical theory of Young measures can also be used tounderstand the “minimum”).

Minimizing sequences predict infinitely fine microstructure. Inpractise, the microstructure is extremely fine but not infinitelyfine. The microstructure can be as fine as 6 atoms wide.





1 mm

Martensite!martensite twinning in Ni-Al.(Courtesy of C. Chu & R. D. James)





Twins in a Ni-Al alloy (Transmission electron microscopy)Boullay, Schryvers & Ball, Acta Materiala 2003





Austenite!martensite interface in Cu82.5Al14.0Ni3.5(Courtesy of R.J. James, U. Minnesota)





Outline


2 Interfaces



5 Microstructure






Shape-memory effect

The wire is initially martensite. As you bend it, themicrostructure changes to accommodate the bending, withoutdamage to the material. When heating above "c the minimumenergy is given by

Du(x) % $(")SO(3) $x % !,

which implies (J. Liouville 1850)

Du(x) = $(")R $x % !

for some constant R % SO(3).





Conclusion

The shape-memory effect is an exceptional phenomenom.Currently, about 30 such metal alloys are known, of which about15 have industrial applications: Cu-Al-Ni, Al-Ni-Ti, Mn-Cu. . .

Mathematically, this exceptionality is given by the followingnecessary conditions on W for the shape-memory effect to takeplace:

1 W is not quasiconvex.2 W has one well at high temperatures, and several wells at

low temperatures.3 The martensitic wells are rank-one connected (middle

eigenvalue 1 condition).





Experimentally, eigenvalues in [0.99, 1.01] are OK, but notoutside this range. It is the possibility of the fine tuning ofpercentages in alloys that allows (some of) them to reach theeigenvalue 1 condition: Cu69.0Al27.5Ni3.5, Ni30.5Ti49.5Cu20.0,Cu68Zn15Al17. . .


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