4.1 Polyconvexity of the Stored-energy Function

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4.1.1 Physical requirements

Basic physical considerations imply that extreme strains must lead to infinite stress (Antman

1983). The word “extreme” applies equally well to large compressions as well as to large

expansions. If the material is extremely compressed such that it is on the verge of being

annihilated, J →0, large stresses should result. Large stresses should equally well be

required to expand a material beyond bounds (J →∞). The notion of “extreme strains”

The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt

2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8

178 HYPERELASTIC MATERIALS

F

_L/L

Loading

Unloading

Figure 4.1 Force-stretch diagram for a hyperelastic material in a uniaxial test

can be further concretized by looking at the invariants of C in terms of the principal values

_1, _2 and _3 (cf Equations (1.121)–(1.123)):

I1 = _1 + _2 + _3 (4.3)

I2 = _1_2 + _1_3 + _2_3 (4.4)

I3 = _1_2_3. (4.5)

Recall that the eigenvalues of C are the squares of the stretch in the principal direction.

Indeed, one finds, using Equation (1.132) and defining the norm for vectors and tensors of

rank two by _N_ =

√

N · N and _A_ =

√

A : A respectively,

_i = Ni · FT · F · Ni = _F · Ni_2. (4.6)

Consequently, _i ≥ 0. Furthermore, the eigenvalues are the solution of the characteristic

equation

_3 − I1_2 + I2_ − I3 = 0. (4.7)

If at least one _i →0, then I3 → 0 must apply in order to satisfy the above equation. The

other way around, if _1_2_3 → 0, then at least one _i → 0. Accordingly, a small value

of I3 is equivalent to small extreme strains. If at least one _i →∞, then I1→∞ since

_i ≥ 0. The inverse is also true: if I1→∞, at least one _i →∞. Accordingly, a large

value of I1 is equivalent to large extreme strains. If I1→∞, then I1 + I2 + I3→∞since

I1, I2 ≥ 0, and I1 + I2 + I3 cannot be large unless at least one _i →∞, which implies

that I1→∞. Consequently,

I1→∞ ⇔ I1 + I2 + I3→∞. (4.8)

Summarizing,

“small” extreme strains ⇔ I3 → 0 (4.9)

“large” extreme strains ⇔ I1 + I2 + I3→∞. (4.10)

HYPERELASTIC MATERIALS 179

In treatises on stored-energy functions, the invariants of C are frequently written as a

function of F. One has

_F_ =

√

F : F = _I1 (4.11)

_CofF_ = _tr[(CofF)T · (CofF)]

= J_tr(F

−1 · F

−T )

= J_tr(C

−1)

= J

_ 1

_1

+ 1

_2

+ 1

_3

= J

_I2

I3

= _I2 (4.12)

det F = J = _I3. (4.13)

Recall that

F

−1 = (CofF)T

detF

(4.14)

which was used in the derivation of Equation (4.12). Equations (4.9) and (4.10) can now

be replaced by

“small” extreme strains ⇔ det F → 0 (4.15)

“large” extreme strains ⇔ _F_ + _CofF_ + detF →+∞.

(4.16)

“Large” stresses basically mean

____

∂_

∂C

____→+∞.

(4.17)

If _(C,X) is continuous on a closed interval [a, b] and differentiable within the open

interval (a, b), then the mean value theorem states that

sup

C∈(a,b)

____

∂_

∂C

____

≥

__(b) − _(a)_

_b − a_ . (4.18)

This means that _ →+∞is sufficient for _∂_

∂C

_ → +∞. Summarizing, the requirements

for _ are

_(C,X)→+∞ if det F → 0+ (4.19)

_(C,X)→+∞ if (_F_ + _CofF_ + det F)→+∞.

(4.20)

180 HYPERELASTIC MATERIALS

Figure 4.2 Gurtin’s experiment

For simplicity, the temperature dependence is dropped from _(C, θ,X). Recall that the

deformation gradient F belongs to the set of 3 × 3 matrices with a positive determinant,

that is,

F ∈ M3

+. (4.21)

Equation (4.20) is sometimes replaced by the coerciveness inequality, which reads

_(C,X) ≥ α __F_p + _CofF_q + (detF)r _ + β,

α, p, q, r > 0, F ∈ M3

+,X ∈ V0. (4.22)

This condition plays a major role in proving the existence of a solution. The present

section essentially follows (Ciarlet 1993) and is based on the research by John Ball (see,

for instance, (Ball 1977)). Here, only the main results will be quoted. For proofs and further

reading, the reader is referred to (Ciarlet 1993).

The existence of a solution immediately calls into mind the uniqueness problem. Contrary

to linear problems, nonlinear problems can have infinitely many solutions. Merely

one example is given here: a beam under torsion fixed at its ends and with stress-free sides

(Figure 4.2, Gurtin’s experiment). There are infinitely many solutions to this problem,

each differing by a torsion angle of a multiple of 2π from the others. Consequently, the

solution is physically not unique and accordingly a numerical uniqueness is not desirable

either.

4.1.2 Convexity

To proceed, some basic mathematical concepts of convexity have to be explained. Indeed,

convexity plays a major role in the derivation of stored-energy functions satisfying Equations

(4.19) and (4.22).

Definition 4.1.1 A subset of a vector space is convex if, for any two elements a and b

belonging to the subset, the closed interval [a, b] also belongs to the subset.

HYPERELASTIC MATERIALS 181

The interval [a, b] consists of all points a + λ(b − a), where λ ∈ [0, 1]. For example,

consider the vector space over R of all 3 × 3 matrices M3. The matrices with positive

determinant (M3

+) form a nonconvex subset. Indeed, A = Diag(−3, 2,−1) ∈ M3

+ and B =

Diag(2,−3,−1) ∈ M3

+ but Diag(−1,−1,−2) = A + B _∈ M3

+. Here, Diag(−3, 2,−1) is

a diagonal 3 × 3 matrix with elements −3,2 and −1.

Definition 4.1.2 The closed convex hull co U of a subset U of a vector space V is the

smallest closed convex subset of V that contains U.

One can prove (Ciarlet 1993),

co M3

+ = M3 (4.23)

co{(F, CofF, detF) ∈ M3

+ ×M3

+ × R+} = M3 ×M3 × (0,∞). (4.24)

Note that CofF ∈ M3

+ since

det(CofF) = (det F)2 (4.25)

because of Equation (4.14) and the properties of determinants (Gradshteyn and Ryzhik

1980).

Definition 4.1.3 A function f : U ⊂ V → R defined on a convex subset U of a vector space

V is convex on U if

∀a, b ∈ U, λ ∈ [0, 1] : f [λa + (1 − λ)b] ≤ λf (a) + (1 − λ)f (b). (4.26)

The following theorem, formulated here for the special case of an inner product space, can

be used to prove the convexity of a function:

Theorem 4.1.4 Let f : U → R be a function defined and twice differentiable over a convex

subset U of an inner product vector space. The function f is convex on U if and only if

f

__

(a) · (b − a, b − a) ≥ 0, ∀a, b ∈ U. (4.27)

(Ciarlet 1993).

The second derivative is a bilinear mapping of its arguments, that is, it has two arguments

and is linear in each of them. In our applications, the bilinear mapping reduces to a classical

inner product. As an example, consider

f : A ∈ M3 → _A_2. (4.28)

Since _A_2 = A : A = AijAij (in rectangular coordinates), one finds

∂AijAij

∂Akl

= Akl + Akl = 2Akl (4.29)

and for the second derivative

2

∂Akl

∂Amn

= 2δkmδln. (4.30)

182 HYPERELASTIC MATERIALS

Accordingly, ∀B,

f

__

(A)klmnBklBmn = 2BklBkl = 2_B_2. (4.31)

Since _B_ ≥ 0, f is convex.

On the other hand,

f : A ∈ M3 → _CofA_2 (4.32)

and

g : A ∈ M3 →detA = I3A (4.33)

are not convex. Indeed,

A := Diag(3, 1, 1) ∈ M3 (4.34)

B := Diag(1, 3, 1) ∈ M3 (4.35)

f (A) = 19 = f (B) (4.36)

g(A) = 3 = g(B) (4.37)

C = λA + (1 − λ)B (4.38)

f (C) = 19 + 16λ − 32λ3 + 16λ4 (4.39)

g(C) = 2 + 4λ − 4λ2. (4.40)

Accordingly,

λf (A) + (1 − λ)f (B) = 19 (4.41)

λg(A) + (1 − λ)g(B) = 3 (4.42)

but

f (C)|

λ=0.01 > 19 (4.43)

g(C)|

λ=0.01 > 3. (4.44)

This concludes the proof.

An important example of a convex function is

f : (x,A) ∈ R

+ ×M3 →

_A_2

x2/3 . (4.45)

The proof given here goes back to (Hartmann and Neff 2003). First consider

g : (x, y) ∈ R

+ × R →f (x) · g(y). (4.46)

R

+ × R is a convex domain. According to Theorem 4.1.4, g is convex if and only if

_x

y

_T        f

__

(x)g(y) f

_

(x)g

_

(y)

f

_

(x)g

_

(y) f (x)g

__

(y)

_x

y

_ ≥ 0, ∀x, y. (4.47)

HYPERELASTIC MATERIALS 183

This implies that the 2 × 2 matrix in Equation (4.47) must be positive semidefinite. A matrix

is positive semidefinite if and only if all eigenvalues are not negative. The eigenvalues of

a 2 × 2 symmetric matrix

            a11 a12

a12 a22

 (4.48)

are

λ1,2 = 12

           

(a11 + a22) ± _(a11 + a22)2 − 4(a11a22 − a2

12)

 (4.49)

which are positive if and only if

_ a11a22 − a2

12

≥0 and

a11 ≥ 0 (or a22 ≥ 0).

(4.50)

Accordingly, we require

_ f

__

(x)g(y) ≥ 0

f

__

(x)g(y)f (x)g

__

(x) ≥ [f

_

(x)g

_

(x)]2.

(4.51)

Let

_ f (x) := x

−α, α≥ 0

g(y) := yp

(4.52)

then Equations (4.51) are equivalent to

α + 1

α

≥ p

p − 1

⇒ α ≤ p − 1. (4.53)

For instance, for p = 2 and α = 2/3,

g : (x, y) ∈ R

+ × R → y2

x2/3 (4.54)

is a convex function. Substituting _A_ for y, one obtains, using the Cauchy–Schwarz

condition,

f [λx1 + (1 − λ)x2, λA1 + (1 − λ)A2] =

_λA1 + (1 − λ)A2_2

[λx1 + (1 − λ)x2]3/2

≤ [λ_A1_ + (1 − λ)_A2_]2

[λx1 + (1 − λ)x2]3/2

≤ λ

_A1_2

x

3/2

1

+ (1 − λ)

_A2_2

x

3/2

2

. (4.55)

The last step is a consequence of the convexity of g. This concludes the proof that the

function f in Equation (4.45) is convex. In a similar way, one can prove that

f : (x,A) ∈ R

+ ×M3 →

_A_3

x2 (4.56)

is convex by choosing p = 3 and α = 2.

The convexity of a function can be extended to nonconvex subsets:

184 HYPERELASTIC MATERIALS

Definition 4.1.5 A function f

∗ : U →R is convex if there exists a convex function

f : co U → R such that f

∗

(a) = f (a) ∀a ∈ U.

Accordingly,

f

∗ : F ∈ M3

+ → _F_2 = I1 (4.57)

is a convex function since f defined by Equation (4.28) is convex in M3 = co M3

+.

A convex function f : R

+ →R can be extended to a convex function

f : R → R ∪ {+∞} by defining

f (x) = f (x), x ∈ R

+

= +∞, x∈ R\R

+

. (4.58)

Convexity is a very nice property and it would be advantageous if we could take simple

functions such as in Equation (4.57) to be stored-energy functions. Unfortunately, this is

not possible (for a proof, see (Ciarlet 1993)):

Theorem 4.1.6 Let X ∈ V0 and _ : F ∈ M3

+ →_(X,C) ∈ R be convex. Then:

1. Equation (4.19) is not satisfied.

2. The eigenvalues σi of the resulting Cauchy stress satisfy σ1 + σ2 ≥ 0, σ1 + σ3 ≥ 0,

σ2 + σ3 ≥ 0 at any X ∈ V0.

Accordingly, for a convex function, there is no constraint to prevent the annihilation of

material, and some stress states, such as uniform hydrostatic pressure, cannot be simulated.

Therefore, convex functions are unsuitable as stored-energy functions.

4.1.3 Polyconvexity

To solve this problem, John Ball (Ball 1977) had the idea of relaxing the convexity requirement

to polyconvexity, which is defined as follows:

Definition 4.1.7 A stored-energy function ˆ_ : V0 ×M3

+ → R is polyconvex, if for each X ∈

V0 there exists a convex function

_ : V0 ×M3 ×M3 × (0,+∞)→ R (4.59)

such that

ˆ _(X, F) = _(X, F, CofF, detF) ∀F ∈ M3

+. (4.60)

Using this definition, both _CofF_2 and detF are polyconvex (the latter because

f : x ∈ R

+ →x is convex). On the basis of Equation (4.45), one also finds that

f : F ∈ M3

+ →

_F_2

(det F)2/3 (4.61)

HYPERELASTIC MATERIALS 185

is polyconvex. Notice that the expression _F_2/(det F)2/3 is the first invariant of C satisfying

C = C

(detF)2/3 (4.62)

and

I 3 := I3C

= detC = 1. (4.63)

Accordingly, C contains the isochoric motion of C. In hyperelastic applications and von

Mises plasticity, the total motion is frequently split into an isochoric part and a volumetric

part. Equation (4.61) is now equivalent to

f : F ∈ M3

+ → I1C (4.64)

which is a polyconvex function.

Since f (I ) = 3, the function

g : F ∈ M3

+ →I 1 − 3 (4.65)

is a convex residual stress-free stored-energy potential. Furthermore, one can prove that

I 1 − 3 ≥ 0. Indeed (Schr¨oder and Neff 2001)

3I2 − I 2

1

= (_1_2 + _1_3 + _2_3) − (_1 + _2 + _3)2 (4.66)

= _1_2 + _1_3 + _2_3 − _21

− _22

− _23

(4.67)

= −12

_(_1 − _2)2 + (_2 − _3)2 + (_3 − _1)2 ≤ 0. (4.68)

Accordingly,

I 2

1

≥ 3I2. (4.69)

Notice that this only applies if the eigenvalues are real, which is guaranteed since C is

symmetric. Equation (4.69) can also be obtained by requiring the solution of the characteristic

equation to be real (cf the explicit solution of a cubic equation in (Abramowitz and

Stegun 1972)).

Equation (4.69) also applies to the inverse of C unless detC = 0:

I 2

1C

−1 ≥ 3I2C

−1 . (4.70)

Recall that the eigenvalues of the inverse of a matrix are the inverse of the eigenvalues.

Accordingly,

I1C

−1 = I2

I3

(4.71)

I2C

−1 = I1

I3

(4.72)

186 HYPERELASTIC MATERIALS

and

I3C

−1 = 1

I3

. (4.73)

Consequently, Equation (4.70) is equivalent to

I 2

2

≥ 3I1I3. (4.74)

Equations (4.66) and (4.74), together with Equations (4.11) to (4.13), yield (recall that all

eigenvalues and invariants are strictly positive unless material annihilation is accepted)

I 4

1

≥ 27I1I3 (4.75)

⇓

_F_4 ≥ 3

√

3_F_(det F) (4.76)

⇓

I 1 =

_F_2

(det F)2/3

≥ 3 (4.77)

which completes the proof. Accordingly, the function g = I 1 − 3 is a positive, polyconvex,

residual stress-free stored-energy function.

Now, the following theorem will be used:

Theorem 4.1.8 If

f : x ∈ V0 → f (x) ∈ R

+ (4.78)

is convex and

g : y ∈ R

+ →g(y) (4.79)

is monotonic increasing and convex, then

g ◦ f : x ∈ V0 → (g ◦ f ) (x) (4.80)

is convex.

Proof. f is convex means

f [λa + (1 − λ)b] ≤ λf (a) + (1 − λ)f (b). (4.81)

Hence, since g is monotonic increasing,

g{f [λa + (1 − λ)b]} ≤ g[λf (a) + (1 − λ)f (b)] (4.82)

≤ λg[f (a)] + (1 − λ)g[f (b)] (4.83)

due to the convexity of g.

HYPERELASTIC MATERIALS 187

Let us apply this theorem to f = I 1 − 3. Choosing g(y) = yi, i ≥ 1 one finds that

h : F ∈ M3

+ →(I 1 − 3)i, i≥ 1 (4.84)

is polyconvex. This function is frequently used in stored-energy functions for rubber materials.

The second invariant of C satisfies

I 2 =

_CofF_2

(det F)4/3 . (4.85)

This function is not polyconvex. However,

f : F ∈ M3

+ →

_CofF_3

(det F)2

= I

3/2

2 (4.86)

is polyconvex because of Equation (4.56). Equations (4.66) and (4.74) reveal that

I 4

2

≥ 27I2I 2

3 (4.87)

⇓

I

3/2

2

I3

≥ 3

√

3. (4.88)

Using the same reasoning as for I 1, one finds that

h : F ∈ M3

+ → _I

3/2

2

− 3

√

3_i

, i≥ 1 (4.89)

is polyconvex. Furthermore h(I ) = 0, since I2I = 3, and consequently the initial configuration

is stress-free. Terms of the kind in Equation (4.89) are only recently being used in

stored-energy functions (see (Hartmann and Neff 2003) and (D¨uster et al. 2003)).

Notice that the basic norm properties in conjunction with Theorem 4.1.8 can be used to

prove that f : A ∈ M3 → _A_2 in Equation (4.28) is convex. Indeed, the norm properties

guarantee that

_λA1 + (1 − λ)A2_ ≤ λ_A1_ + (1 − λ)_A2_. (4.90)

Consequently, _A_ is convex and also positive. Application of Theorem 4.1.8 with

g(y) = yi , i ≥ 1 shows that

f : A ∈ M3 → _A_i, i≥ 1 (4.91)

is convex.

To prove that I 2 is not polyconvex, the following definitions are introduced:

Definition 4.1.9 A twice differentiable function _(A),A ∈ M3 leads to an elliptic system

if and only if

∀A ∈ M3, ∀ξ , η ∈ R3 : _

__

(A) · (ξ ⊗ η, ξ ⊗ η) ≥ 0. (4.92)

188 HYPERELASTIC MATERIALS

Definition 4.1.10 A function _(A),A ∈ M3 is rank-one convex if

f : t ∈ R → _(A + t (ξ ⊗ η)) (4.93)

is convex ∀A ∈ M3, ∀ξ , η ∈ R3.

One can prove (Dacorogna 1989),

Theorem 4.1.11 1. For sufficiently smooth functions _, one has

_ leads to an elliptic system

_

_ is rank-one convex

2. _ is polyconvex ⇒ _ is rank-one convex

Notice that for ellipticity, the direction the second derivative is projected on is a rank-one

matrix (ξ ⊗ η), whereas for convexity, this direction can have an arbitrary rank (a general

matrix A, Theorem 4.1.4). Accordingly, convexity implies rank-one convexity. Recall that

the rank of a matrix is the dimension of its image. Since

(ξ ⊗ η) · ζ = ξ (η · ξ ) (4.94)

the rank of ξ ⊗ η is one. The concept of rank-one convexity is somewhat simpler than

convexity. Therefore, invoking Theorem 4.1.11(2), it is mainly used to prove that a function

is not polyconvex. Let us apply this to prove that I 2 in Equation (4.85) is not polyconvex.

Rank-one convexity implies that

f : t ∈ R →

_Cof[A + t (ξ ⊗ η)]_2

det[A + t (ξ ⊗ η)]4/3 (4.95)

is convex. The expression det[A + t (ξ ⊗ η)] is linear in t. This can be seen by applying a

coordinate rotation (which leaves the determinant unchanged since it is an invariant of its

argument) such that ξ coincides with a basis vector. Then the term t (ξ ⊗ η) leads to a linear

change of just one row in A. Since the cofactors of a matrix are the minor determinants,

the numerator is a sum of squares of linear relations. This leads to a quadratic relation with

positive coefficients for the quadratic term and the constant term:

f (t) = λ21

t2 + λ2t + λ23

(λ4t + λ5)4/3 . (4.96)

The second derivative of a function g = aαbβ has the form

g

__ = aα−2bβ−2[α(α − 1)(a

_

)2b2 + αaa

__

b2 + β(β − 1)(b

_

)2a2 + βbb

__

a2 + 2αaa

_

βbb

_].

(4.97)

Taking

α = γ/2 (4.98)

a = λ21

t2 + λ2t + λ23

(4.99)

b = λ4t + λ5 (4.100)

HYPERELASTIC MATERIALS 189

the term in the square brackets yields

λ24

λ41

t4[(β + γ )2 − (β + γ )]. (4.101)

This function is only convex for (β + γ )2 − (β + γ ) ≥ 0, that is, β + γ ≤ 0 or β + γ ≥ 1.

Since for Equation (4.96), γ = 2 and β = −4/3, f in Equation (4.95) is not rank-one

convex and consequently not polyconvex (Theorem 4.1.11).

Finally, note that all convex functions are polyconvex, but not vice versa.

4.1.4 Suitable stored-energy functions

The polyconvexity concept plays an important role in the existence theorems. Indeed, John

Ball proved that a solution exists if

1. the stored-energy function is polyconvex

2. Equation (4.19) applies:

lim

detF→0+

_(C,X) = +∞ (4.102)

3. and the coerciveness inequality is satisfied, Equation (4.22).

For details, the reader is referred to (Ciarlet 1993). These are sufficient but not necessary

conditions.

An important class of materials satisfying these conditions is evoked by the following

theorem:

Theorem 4.1.12 Let _ be a stored-energy function of the form

F ∈ M3

+ → _(F) =

M

_

i=1

aiI1(Cγi /2)

+

N

_

j=1

bj I2(C

δj /2

)

+  __I3C_ (4.103)

where ai > 0, γi ≥ 1, bj > 0, δj ≥ 1 and  : (0,+∞)→R is a convex function, then _

is polyconvex and satisfies

_(F) ≥ α __F_p + _CofF_q_ +  __I3C_ (4.104)

∀F ∈ M3

+ with α > 0, p = maxi(γi ), q = maxj (δj ).

If, in addition, limδ→0+ (δ) = +∞ (Equation (4.19)), the material is called an Ogden

material. An Ogden material satisfies the conditions in the existence theorem by Ball.

Accordingly, a solution exists. Notice that both ai > 0 and bj > 0 apply. Hence, both I1C

and I2C must be present, together with I3C because of Equation (4.19). In Equation (4.103),

I1(Cγi /2) and I2(C

δj /2

)

are defined by

I1(Cγi /2) :=_

j

_

γi/2

j (4.105)

I2(C

δj /2

)

:=_

k,l

k_=l

(_k_l)δj /2 . (4.106)

190 HYPERELASTIC MATERIALS

Notice that Theorem 4.1.12 only gives information on how to construct polyconvex functions

that have the desired property. This does not mean that any function not in the form of

Equation (4.103) is inappropriate. Yet, in most cases, no existence results will be available.

As an example of a well-known stored-energy function that is not polyconvex, consider

the St.Venant–Kirchhoff potential (Equation (1.440)):

_ = 12

λ(trE)2 + μtr(E2). (4.107)

For the proof, the reader is referred to (Ciarlet 1993). Using Equations (1.444) and (1.445)

together with

2I1E = −3 + I1C (4.108)

4I2E = 3 − 2I1C + I2C (4.109)

8I3E = −1 + I1C − I2C + I3C (4.110)

it is clear that _ does not depend on I3C and Equation (4.19) cannot be satisfied. Furthermore,

the stress obtained by differentiating Equation (4.107)

S = λ(trE)G_ + 2μE (4.111)

can be inverted to yield

E = 1

E

[−ν(trS)G + (1 + ν)S] (4.112)

which implies uniqueness. As explained previously, uniqueness is not desirable for large

strains. Although the use of Equation (4.107) will yield better results than the use of

infinitesimal strains, it should not be used for large strains. Its field of operation is often

called large deformation–small strains, which emphasizes the good performance for large

rotations (shell applications).

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