existence and time-discretization for the finite-strain souza–auricchio constitutive model for...

Continuum Mech. Thermodyn. (2012) 24:63–77DOI 10.1007/s00161-011-0221-x

ORIGINAL ARTICLE

Sergio Frigeri · Ulisse Stefanelli

Existence and time-discretization for the finite-strainSouza–Auricchio constitutive model for shape-memoryalloys

Received: 14 June 2011 / Accepted: 17 November 2011 / Published online: 29 November 2011© Springer-Verlag 2011

Abstract We prove the global existence of solutions for a shape-memory alloys constitutive model at finitestrains. The model has been presented in Evangelista et al. (Int J Numer Methods Eng 81(6):761–785, 2010) andcorresponds to a suitable finite-strain version of the celebrated Souza–Auricchio model for SMAs (Auricchioand Petrini in Int J Numer Methods Eng 55:1255–1284, 2002; Souza et al. in J Mech A Solids 17:789–806,1998). We reformulate the model in purely variational fashion under the form of a rate-independent process.Existence of suitably weak (energetic) solutions to the model is obtained by passing to the limit within aconstructive time-discretization procedure.

Keywords Shape-memory alloys · Finite-strain · Phase transformations in solids · Rate-independence ·Existence · Time-discretization

Mathematics Subject Classification (1991) 74C15 · 74N30

1 Introduction

Shape-memory alloys (SMAs) show abrupt diffusionless stress- and temperature-driven martensitic phase-changes. At the macroscopic level, the result is an amazing thermomechanical behavior. At certain (suitablyhigh) temperature regimes, SMAs are superelastic, namely, they completely recover comparably large defor-mations during mechanical loading-unloading cycles. At lower temperatures, severely deformed specimensregain their original shape after a thermal treatment. This is the so-called shape-memory effect [26].

The specific thermomechanical behavior of SMAs is not present (at least to a comparable extent) in mate-rials traditionally used in engineering and is at the basis of a variety of innovative applications to aeronautical,structural, earthquake, and biomedical technologies [19,20], just to mention a few hot topics. This fact moti-vated an intense research toward a comprehensive and effective description of the complex thermomechanicalbehavior of SMAs. The relevant engineering literature on SMAs modeling is vast and a whole menagerie of

Communicated by Andreas Öchsner.

S. FrigeriDipartimento di Matematica ’F. Casorati’, Università di Pavia, v. Ferrata 1, 27100 Pavia, ItalyE-mail: [email protected]: http://www.imati.cnr.it/sergio

U. StefanelliIMATI - CNR, v. Ferrata 1, 27100 Pavia, Italy

U. Stefanelli (B)WIAS, Mohrenstr. 39, Berlin 10115, GermanyE-mail: [email protected]: http://www.imati.cnr.it/ulisse

64 S. Frigeri, U. Stefanelli

models has been advanced having ambitions for different ranges of applicability (from lab single-crystal exper-iments to commercially exploitable tools) and different abilities to fit particular experiments and to explainmicrostructures, stress/strain relations, or hysteresis. In the macroscopic-phenomenological setting, the readershall be minimally referred to [12,25,28,31,33,35,51–53,57] and to the survey [54].

We are here concerned with a phenomenological, internal-variable-type model for polycrystalline shape-memory bodies originally advanced in the small-strain regime by Souza et al. [55] and combined with finiteelements by Auricchio and Petrini [6–8], hence referred to as Souza–Auricchio in the following. This modeldescribes very efficiently both the superelastic and the shape-memory behavior. Moreover, it shows a remark-able robustness with respect to parameters and discretizations despite its simplicity: the constitutive behaviorof the material is determined by eight easily fitted material parameters [5,11] (note that linearized thermo-plas-ticity with linear hardening already requires five material parameters). The Souza–Auricchio model admitsa sound mathematical treatment. It has been analyzed from the viewpoint of existence and approximation ofsolutions of the isothermal three-dimensional quasi-static evolution problem in [5] and convergence rates forspace-time discretizations are discussed in [46,47]. Results in the non-isothermal case have also been obtained[22,29,30,48,45]. Moreover, extensions of the original model to residual plasticity [9,10,21], more realis-tic non-symmetric behaviors and transformation-dependent material parameters [11], and the ferromagneticshape-memory effect [3,4,14,15,56] are also available.

This note is focused on the finite-strain version of the Souza–Auricchio model advanced by Evangelistaet al. [23,24] and considered also in Arghavani et al. [1]. Note that shape-memory devices are generallyengineered in order to exploit martensitic-transformation-induced inelastic strains up to 8%. This sets clearlybeyond the possible reach of linearized elasticity theory calling for a finite-strain analysis instead. In [24] theauthors propose a finite-strain choice for the free energy of the material, a specific flow rule for the internalvariable, and formally verify the recovery of the small-strain Souza–Auricchio model upon strain lineariza-tion. Moreover, some comments on algorithmical aspects of a possible fully implicit time-discretization of theconstitutive relation and the integration of the latter in some finite elements test is reported.

A first issue of this paper is that of reformulating the finite-strain model of [24] in a purely variational fash-ion. This complements the analysis of [24] where material relations (in particular, the flow rule) are presentedvia the classical complementary formalism. The variational reformulation of the model bears a clear math-ematical advantage as it provides the possibility of recasting the constitutive relation in terms of symmetricright Cauchy-Green tensors only.

Secondly, this novel variational reformulation of the model allows us to settle the material constitutiveproblem within the now quite developed theory of rate-independent processes. This in turn provides the pos-sibility of proving the convergence of time discretizations of the material constitutive relation. In particular,the material constitutive relation is proved to admit a suitably defined energetic solution [49]. Note that noconvergence nor existence proof is reported in the former [24].

We directly develop our analysis in the non-isothermal regime by, however, letting the temperature of thebody being given a priori. This assumption seems justified in case of a shape-memory body which is thin in atleast one direction and for slowly varying loads. In this case, one could assume that the heat which is mechan-ically produced within the sample gets immediately transferred to the surrounding heat bath. This perspectiveis followed in [22,48,45] in the small-strain regime. On the contrary, a fully thermomechanically coupledproblem for the small-strain model is solved in one spatial dimension in [29,30]. Note that the smoothness onthe given temperature here assumed is weaker than the corresponding one from [48,45] exactly in the samespirit as in [22]. It would indeed be interesting to consider the fully thermomechanically coupled problem, i.e.,coupling the constitutive relation with three-dimensional quasi-static mechanics and temperature evolution.On the other hand, this seems presently out of reach, even in the comparably simpler small-strain case [29,30].

Before moving on, let us briefly review the literature on phenomenological SMA constitutive modeling atfinite strains. A first reference in this direction is Auricchio and Taylor [13], where a pseudo-elastic consti-tutive model is presented. The finite-strain model from [13] reduces to the so-called RL model by Ranieckiand Lexcellent [52] upon strain linearization. Other extensions of the RL model to finite strains are to befound in Ziolkowski [58], where thermo-mechanical coupling effects are also considered, and Müller andBruhns [50], where an additive decomposition if the deformation gradient is performed. Finite-strain modelsbased on multiplicative decomposition instead are in Helm and Haupt [28] and Reese and Christ [18,53].Eventually, the small-strain model by Panico and Brinson [32] has been extended to the finite-deformationsrealm in Arghavani et al. [2].

To our knowledge, the mathematical treatment here presented is unprecedented with respect to the above-reviewed literature as none of the mentioned finite-strain constitutive SMA models has already obtained

Existence and time-discretization for the finite-strain 65

consideration from the viewpoint of existence of solutions nor convergence proofs of time discretizations.On the other hand, we shall mention the mathematical results in Mainick and Mielke [40] and Mielke et al.[41–44] where existence of solutions for quasistatic incremental problems and energetic evolutions in finite-strain elastoplasticity (including higher order gradient terms though) are discussed. Note that here we limitourselves in considering the finite-dimensional situation of the constitutive relation. In particular, we shalltackle the full PDE quasistatic evolution problem elsewhere.

The paper is organized as follows. In Sect. 2 we recall the basics of the model from [24] and commenton dissipativity whereas Sect. 3 is devoted to the variational reformulation of the model. Then, Sect. 4 bringsto the convergence proof of time-incremental approximations and, in particular, to the existence of energeticsolutions.

2 The model

The aim of this section is that of recalling the model from [24]. We limit ourselves to mention the essentialmodeling features and notation. The reader is instead referred to [24] for additional details and comments aswell as some numerical evidence of the performance of the model.

2.1 Tensors

We shall use bold capital letters for 2-tensors in Rd×d (d = 2, 3) and double capitals (e.g., T) for 4-tensors. Let

Rd×d be the space of 2-tensors endowed with the natural scalar product A:B .= Ai j Bi j (summation conven-

tion) and the corresponding norm |A|2 .= A:A for all A, B ∈ Rd×d . The symbol R

d×dsym denotes the subspace

of symmetric tensors. The space Rd×dsym is orthogonally decomposed as R

d×dsym = R

d×ddev ⊕ R 1, where R 1 is the

subspace spanned by the identity 2-tensor 1 and Rd×ddev is the subspace of symmetric and deviatoric tensors. In

particular, for all A ∈ Rd×dsym , we let tr A

.= A:1 and dev A.= A − (tr A)1/d so that A = dev A + (tr A)1/d .

On the other hand, the identity 4-tensor I is given by Ii j�k.= δi�δ jk and we have I:A = A:I = A. We shall

use the classical notation

GL+(d).= {A ∈ R

d×d | det A > 0},SL(d)

.= {A ∈ Rd×d | det A = 1}, SO(d)

.= {A ∈ SL(d) | A� A = AA� = 12}where A� is the transpose of the tensor A ∈ R

d×d .The 2-tensors T:B and B:T are classically defined by (B:T)ik .= B jl Tj�ik and (T:B)ik .= Tik�j B�j , respec-

tively. Given the smooth function B �→ A(B) we shall denote by ∂B A the 4-tensor defined by (∂B A)ik j�.=

∂B j� Aik or, equivalently,

∂B A:C .= d

dαA(B + αC)

∣∣∣α=0

∀C ∈ Rd×d .

In particular, we have that

(∂B A:C)ik = ∂B j� AikC j�, (C:∂B A)ik = C j� ∂Bik A j�

for all C ∈ Rd×d . Note that the associative property (C:∂B A):D = C:(∂B A:D) holds for every C, D ∈ R

d×d ,and we can write C:∂B A:D without ambiguity. Note also that ∂A A = I.

The 4-tensor T1:T2 is defined by (T1:T2)i jk�.= (T1)i jmn(T2)mnk�. In particular, if A is a function of B

and C is a function of D (all 2-tensors), we have

(∂B A:∂DC)ik j� = ∂Bmn Aik∂D j�Cmn .

We have that A:(T1:T2) = (A:T1):T2 and (T1:T2):B = T1:(T2:B) for every A, B ∈ Rd×d so that brackets

can be omitted. Finally, the product derivative formulas

∂A(BC):D = (∂A B:D)C + B(∂AC:D) (2.1)

B:∂A(C D) = B D�:∂AC + C� B:∂A D (2.2)

will turn out to be useful in the following.


2.2 Deformation gradient

Let ϕ : � → Rd denote the deformation of the SMA sample from the reference configuration � ⊂ R

d . Weshall classically decompose the deformation gradient Dϕ as [34]

Dϕ = F = Fel Ftr

where Fel stands for the elastic deformation gradient whereas Ftr denotes the inelastic (or transformation)part and is constrained to belong to SL(d) in order to ensure the inelastic deformation to be volume preserving.

In order to formulate the model, we will rely on the right Cauchy-Green deformation tensor

C.= F� F

and the corresponding elastic and inelastic analogs

Cel.= F�

el Fel, C tr.= F�

tr Ftr.

In particular, one shall note that C tr ∈ SLsym(d).= SL(d) ∩ R

d×dsym .

2.3 Free energy

The free energy of the body is assumed to be decomposed as

ψ.= ψel(Cel)+ ψtr(C tr, θ). (2.3)

Here, θ denotes the absolute temperature of the sample, ψel is the elastic part of the free energy whereas ψtrstands for the inelastic part.

As for the elastic part of the energy, we shall simply assume it to be smooth far from the set wheredet F = det Fel ≤ 0, frame indifferent, and isotropic. In particular, ψel is derived from an elastic potentialW ∈ C1(GL+(d)) via ψel(Cel)

.= W (Fel) with W : Rd×d → [0,∞] such that

W (RFel) = W (Fel) ∀R ∈ SO(d), Fel ∈ Rd×d , (2.4)

W (Fel R) = W (Fel) ∀R ∈ SO(d), Fel ∈ Rd×d , (2.5)

|∂Fel W (Fel)F�el | ≤ c0(W (Fel)+ 1). (2.6)

The above assumptions are nothing but frame-indifference (2.4), isotropy (2.5), and the controllability of theKirchhoff tensor in terms of the energy (2.6). In particular, they are completely compatible with the standardpolyconvexity frame. Note that the functional W can be additionally asked to fulfill

W (Fel) → ∞ for det Fel → 0+.

The specific form for the inelastic energy ψtr is chosen as

ψtr(C tr, θ).= b(θ − θM )

+|Etr| + h

2|Etr|2 + IεL (Etr) (2.7)

where Etr is the inelastic Green-St.Venant tensor

Etr.= 1

2(C tr − 12),

b > 0, θM > 0 is the critical martensite-austenite transition temperature at zero stress, h > 0 is a kinematichardening modulus, and IεL is the indicator function of the set {Etr ∈ R

d×d | |Etr| ≤ εL} with εL > 0representing a measure of the maximal strain obtainable via martensitic variants reorientation. In particular,we have that

IεL (Etr).=

{

0 if |Etr| ≤ εL∞ else.


We shall stick to the specific form (2.7) for the inelastic energy for the sake of reference with the originalmodel in [24]. Let us, however, mention that the mathematical considerations developed hereafter apply toa much broader class of energies, possibly including kinematic hardening finite-strain plasticity. Althoughclose from the mathematical viewpoint, the present SMA model and kinematic hardening plasticity differunder two respects. A first difference is the occurrence of the constraint |Etr| ≤ εL on the inelastic responseof the material. In particular, all energetic solutions fulfill this constraint pointwise everywhere in time as aconsequence of the finiteness of the energy. The second difference relies on the linear growth in C tr of thethermo-mechanical coupling term b(θ − θM )

+|Etr| in the inelastic energy, see (2.7).Throughout this work we shall assume the absolute temperature t �→ θ(t) to be given. This in particular

motivates the absence of the classical purely caloric term cV θ(1 − ln θ) in the expression of the free energy(2.3) (cV being the specific heat). Note that, even by including such term into the free energy, the model isnot directly suited for describing fully coupled thermomechanical evolutions as the choice (2.7) (which ismotivated in the isothermal frame) would prescribe the non-monotone temperature-entropy relation

s.= −∂θψ = cV ln θ − bH(θ − θM )|Etr|

where H stands for the classical Heaviside function. Further details on this issue are to be found in [29,30].

2.4 Clausius-Duhem inequality

Let us complement the analysis of [24] and provide some detail in the direction of the Clausius-Duheminequality for the body. In particular, this reduces to check that, at least for smooth evolutions,

− ψ − sθ + S:12

C − q · ∇θθ

≥ 0 (2.8)

where s is the entropy, S is the second Piola-Kirchhoff stress tensor, and q is the heat flux. With the aid of(2.3), we manipulate the term −ψ + S:C/2 in the above left-hand side as follows

−ψ + S:12

C = − d

dt

(

ψel(F−�tr C F−1

tr )+ ψtr(C tr, θ)) + S:1

2C

= − ∂θψθ − ∂Celψel:(− F−�

tr F�tr F−�

tr C F−1tr + F−�

tr C F−1tr − F−�

tr C F−1tr Ftr F−1

tr

)

− ∂C trψtr:C tr + S:1

2C

= − ∂θψθ +(

S − 2F−1tr ∂Celψel F

−�tr

)

:12

C

+ ∂CelψelCel:F−�tr F

�tr + Cel∂Celψel:Ftr F−1

tr − ∂C trψtr:C tr

= − ∂θψθ +(

S − 2F−1tr ∂Celψel F

−�tr

)

:12

C

+(

2F−1tr Cel∂Celψel F

−�tr − 2∂C trψtr

)

:12

C tr (2.9)

where we used the coaxiality of Cel and ∂Celψel (which in turn follows from isotropy (2.5)), the identity C tr =F

�tr Ftr + F�

tr Ftr, and the short-hand notation ∂Celψel = ∂Celψel(F−�tr C F−1

tr ) and ∂C trψtr = ∂C trψtr(C tr, θ)(here and in the following). We shall systematically make use of the symbol ∂ for the subdifferential of aconvex function [16]. In particular, given any φ : R

d×d → (−∞,∞] convex, we denote its subdifferential∂φ as

B ∈ ∂φ(A) ⇐⇒ φ(A) < ∞ and B:(T − A) ≤ φ(T )− φ(A) ∀T ∈ Rd×d .

Note incidentally that, as ψtr is non-smooth, the symbol ∂C trψtr represents a selection in the convex set∂C trψtr(C tr, θ).


2.5 Constitutive relations and flow rule

We let the second Piola-Kirchhoff stress tensor S be given by

S.= 2F−1

tr ∂Celψel F−�tr

and define

T.= F−1

tr M F−�tr − A (2.10)

where

M.= 2Cel∂Celψel, A

.= 2∂C trψtr (2.11)

are the so-called Mandel stress and back stress, respectively [53]. Owing to the computation in (2.9), the tensorT is the thermodynamic force associated with the internal variable C tr for we have that

2F−1tr Cel∂Celψel F

−�tr − 2∂C trψtr = F−1

tr M F−�tr − A = T .

Let us stress that both the Mandel stress M and the back stress A are symmetric (M by the coaxiality of Cel

and ∂Celψel and A by the current choice of ψtr) so that T = F−1tr M F−�

tr − A is symmetric as well.The dissipative evolution of the material is prescribed via the associative flow rule

C tr = ζ ∂T f (Ftr, T ). (2.12)

for some yield function f : Rd×d × R

d×dsym → R, such that for all Ftr ∈ R

d×d the function f (Ftr, ·) is convexand positively 1-homogeneous ( f (Ftr, λT ) = λ f (Ftr, T ) for all λ ≥ 0), along with the classical Kuhn-Tuckercomplementarity conditions

ζ ≥ 0, f ≤ 0, ζ f = 0. (2.13)

Following [23,24], we choose the yield function f as

f (Ftr, T ) = |dev (FtrT F�tr )| − r (2.14)

where r > 0 plays the role of a given yield stress.Along with this choice, the flow rule (2.12)–(2.13) reads

C tr ∈

⎧

⎪⎪⎨

⎪⎪⎩

ζ F�tr

dev (FtrT F�tr )

|dev (FtrT F�tr )|

Ftr if dev (FtrT F�tr ) �= 0,

ζ F�tr {B ∈ R

d×ddev ∩ R

d×dsym : |B| ≤ 1}Ftr if dev (FtrT F�

tr ) = 0.

(2.15)

Note in particular that C tr ∈ Rd×dsym . Hence, all evolutions t �→ C tr(t) fulfilling the latter flow rule and starting

from a symmetric initial datum stay in Rd×dsym for all times.

2.6 Dissipativity

The above choices for constitutive relations and flow rule entail the validity of the Clausius-Duhem inequality(2.8), at least for smooth evolutions. Indeed, we may equivalently rewrite (2.12)–(2.13) or (2.15) as

T ∈ ∂C trR(Ftr, C tr) (2.16)

where R(Ftr, ·) .= f ∗(Ftr, ·) is the classical Legendre conjugate of f (Ftr, ·). Namely, by the 1-homogeneityof f , the function R(Ftr, ·) is the support function of the convex set {T ∈ R

d×dsym | f (Ftr, T ) ≤ 0}. In particular,

R(Ftr, C tr).= sup{T :C tr | f (Ftr, T ) ≤ 0}.


Namely, R(Ftr, 0) = 0 and we have that, for all (Ftr, C tr) with R(Ftr, C tr) < ∞,

∂C trR(Ftr, C tr)

.={

T ∈ Rd×dsym | T :(B − C tr) ≤ R(Ftr, B)− R(Ftr, C tr) ∀B ∈ R

d×d}

.

Since the right-hand side of (2.9) equals sθ + T :C tr/2 by definition of the entropy s.= −∂θψ and we have

T :C tr(2.16)≥ R(Ftr, C tr)− R(Ftr, 0) = R(Ftr, C tr) ≥ 0,

by choosing the classical Fourier flux q.= −κ∇θ, κ > 0 we have that

− ψ − sθ + S:1

2C − q · ∇θ

θ≥ 1

2R(Ftr, C tr)+ κ∇θ · ∇θ

θ≥ 0.

In particular, the Clausius-Duhem inequality holds true.

3 Variational reformulation

As already remarked, one of the main issues of this paper is that of reformulating the constitutive problem ina purely variational fashion. This yields a clear advantage with respect to the original formulation of [23,24].In particular, this variational setting allows us to prove the convergence of the time-discretization scheme.Moreover, we shall be using the variables θ,C (given) and C tr (unknown) only and the model turns out to benot directly depending on Ftr. This possibility is indeed the effect of the overall isotropy and frame-indiffer-ence assumptions and, although not new in the setting of finite-strain plasticity, it was not yet pointed out inconnection with shape-memory alloys.

The first step in the direction of reformulating variationally the model consists in avoiding the use of thetensor Cel in the definition of the free energy ψ (2.3). By recalling that

Ftr = RC1/2tr = C1/2

tr R for some R, R ∈ SO(d) (3.1)

we have that

ψel(Cel) = ψel(F−�tr C F−1

tr ) = ψel(R−�C−1/2tr CC−1/2

tr R−1) = ψel(C−1/2tr CC−1/2

tr ). (3.2)

The last equality above is a consequence of (2.4)–(2.5). Indeed, let for brevity S = C−1/2tr CC−1/2

tr = O� DOwith O ∈ SO(d) and D diagonal and positive definite. Hence

ψel(R−�SR−1) = ψel(R−� O� DO R−1) = W (R−� O� D1/2 O R−1)(2.4)+(2.5)= W (O� D1/2 O) = ψel(S)

whence (3.2) follows. Let us now define

ψ(C,C tr, θ).= ψel(C

−1/2tr CC−1/2

tr )+ ψtr(C tr, θ).

We compute R(Ftr, C tr) from the specific form of f (Ftr, T ) in (2.14) as

R(Ftr, C tr) = sup{

T :C tr∣∣ T ∈ R

d×dsym , |dev (FtrT F�

tr )| ≤ r}

= sup{

T :C tr∣∣ T ∈ R

d×dsym , |dev (C1/2

tr T C1/2tr )| ≤ r

}

= sup{

B:C−1/2tr C trC

−1/2tr

∣∣ B ∈ R

d×dsym , |dev B| ≤ r

}

={

r |dev (C−1/2tr C trC

−1/2tr )| if tr (C−1/2

tr C trC−1/2tr ) = 0

∞ else.

={

r |C−1/2tr C trC

−1/2tr | if tr (C−1/2


∞ else.


Hence, one defines the dissipation R : Rd×dsym ×R

d×dsym → [0,∞] as (the meaning of the factor 1/2 will become

clear in the proof of Theorem 3.1 below)

R(C tr, C tr).=

{ r

2|C−1/2

tr C trC−1/2tr | if tr (C−1/2


∞ else.

The core of this section resides in the following result where we prove that one can rewrite the constitutivematerial relation from (2.10)–(2.13) for the internal variable C tr in terms of R and ψ only.

Theorem 3.1 (Constitutive relation) The constitutive material relation from (2.10)–(2.13) can be equiva-lently restated as

∂C trR(C tr, C tr)+ ∂C trψ(C,C tr, θ) � 0. (3.3)

Proof Indeed, relations (2.12) and (2.13) are equivalent to (2.16) and hence to

2∂C trR(C tr, C tr) � T .

The statement of the theorem will follow by proving directly that

− 1

2T ∈ ∂C trψ(C,C tr, θ) = ∂C trψel(C

−1/2tr CC−1/2

tr )+ ∂C trψtr(C tr, θ). (3.4)

Let us work out the two terms in the above right-hand side separately. We shall preliminarily observe that,for all B ∈ R

d×d , we have the following

∂Ftr F−Ttr :B = −F−�

tr B� F−�tr , ∂Ftr F−1

tr :B = −F−1tr B F−1

tr , (3.5)

(∂C tr F�tr :B)� = ∂C tr Ftr:B, (∂C tr Ftr:B)� = ∂C tr F�

tr :B, (3.6)

∂Ftr(F−�tr C F−1

tr ):B = −F−�tr B� F−�

tr C F−1tr − F−�

tr C F−1tr B F−1

tr

= −F−�tr B�Cel − Cel B F−1

tr = ∂F�tr(F−�

tr C F−1tr ):B�, (3.7)

B = ∂C tr C tr:B = ∂C tr(F�tr Ftr):B = (∂C tr F�

tr :B)Ftr + F�tr (∂C tr Ftr:B). (3.8)

In particular, relation (3.8) entails that, for all B ∈ Rd×d ,

F−�tr B F−1

tr = F−�tr (∂C tr F�

tr : B)+ (∂C tr Ftr : B)F−1tr . (3.9)

Hence, given B ∈ Rd×d one has that

∂C trψel(C−1/2tr CC−1/2

tr ):B = ∂Celψel:∂Ftr(F−�tr C F−1

tr ):∂C tr Ftr:B= −∂Celψel:F−�

tr (∂C tr F�tr :B)F−�

tr C F−1tr − ∂Celψel:F−�

tr C F−1tr (∂C tr Ftr:B)F−1

tr

= −∂CelψelCel:F−�tr (∂C tr F�

tr :B)− Cel∂Celψel:(∂C tr Ftr:B)F−1tr

(3.9)= −Cel∂Celψel:[F−Ttr (∂C tr F�

tr :B)+ (∂C tr Ftr:B)F−1tr ]

= −Cel∂Celψel:F−Ttr B F−1

tr = −F−1tr Cel∂Celψel F

−�tr :B

= −1

2F−1

tr M F−�tr :B. (3.10)

In the second identity above we have used (3.7) by replacing B by ∂C tr Ftr:B.Namely, we have finally checked that

∂C trψ = ∂C trψel(C−1/2tr CC−1/2

tr )+ ∂C trψtr(C tr, θ) = −1

2(M − A) ≡ −1

2T

and (3.4) holds. ��


3.1 Dissipation distance

Let us close this Section by observing that the dissipation functional R defines a left-invariant Finsler metric.Hence, one can coordinate to R a global metric by letting D : R

d×d × Rd×d → [0,∞] be defined by

D(C tr,0,C tr,1).= inf

⎧

⎨

⎩

1∫

0

R(C tr, C tr)

∣∣∣ C tr ∈ C1([0, 1]; R

d×d), C tr(0) = C tr,0, C tr(1) = C tr,1

⎫

⎬

⎭.

Note in particular that

D(C tr,0,C tr,1) = D(1,C tr,1C−1tr,0) ∀C tr,0, C tr,1 ∈ R

d×d ,

D(1,C tr,1) < ∞ �⇒ C tr,1 ∈ SL(d).

The interested reader is referred to [17,36,37] for additional details.

4 Existence of energetic solutions

We shall now turn our attention to the existence of suitably weak solutions to the constitutive material relation(3.3). To this aim, assume to be given t �→ θ(t) > 0 and t �→ C(t) ∈ R

d×dsym and set for the sake of notational

simplicity

e(t,C tr).= ψ(C(t),C tr, θ(t))

throughout. In order to make precise the solution notion we shall be dealing with we need to introduce thestable states S(t) at time t ∈ [0, T ] as

S(t) .= {C tr ∈ SLsym(d) | e(t,C tr) < ∞ and e(t,C tr) ≤ e(t, B)+ D(C tr, B) ∀B ∈ SLsym(d)}. (4.1)

An energetic solution [38,49] corresponds to a trajectory C tr : [0, T ] → Rd×d such that C tr(0) = C tr,0, where

C tr,0 is some given initial datum, and

C tr(t) ∈ S(t) ∀t ∈ [0, T ], (4.2)

e(t,C tr(t))+ Diss [0,t](C tr) = e(0,C tr(0))+t∫

0

∂t e(s,C tr(s))ds ∀t ∈ [0, T ] (4.3)

where the total dissipation Diss [0,t](C tr) in the time interval [0, t] is given by

Diss [0,t](C tr).= sup

{N

∑

i=1

D(C tr(ti−1),C tr(t

i ))

∣∣∣ 0 = t0 < t1 < · · · < t N = t

}

the supremum being taken among all partitions of the interval [0, t]. Note that Diss [0,T ](C tr) < ∞ impliesthat C tr(t) ∈ SL(d) for all times. Moreover, from the finiteness of the energy for all times from (4.2), wededuce that all energetic solutions t �→ C tr(t) are such that

|Etr(t)| = 1

2|C tr(t)−1| ≤ εL ∀t ∈ [0, T ].

In particular, the maximal-inelastic-strain constraint is fulfilled at all times. Let us now formulate the assump-tions on C and θ as

C ∈ W 1,1(0, T ; Rd×dsym ), det C(t) ≥ α0 > 0 ∀t ∈ [0, T ], (4.4)

θ ∈ W 1,1(0, T ). (4.5)

Theorem 4.1 (Existence of energetic solutions) Let (4.4)–(4.5) hold and C tr,0 ∈ S(0). Then, there exist anenergetic solution of (3.3).

The proof of this theorem is developed in the remainder of this section and follows by adapting the by-nowclassical argument of [49]. In particular, we shall proceed by time-discretization. An interesting by-productof this procedure relies in the possibility of approximating the limit problem constructively. Note that somenumerical computation for this model has been provided in [24]. Here, we complement the discussion from[24] by rigorously proving convergence of the numerical scheme.


4.1 Incremental minimization

Assume to be given a partition of [0, T ] which we identify with the corresponding vector τ = (τ 1, . . . , τ Nτ )of strictly positive time-steps. Note that we indicate with superscripts the elements of a generic vector. Inparticular, τ j represents the j th component of the vector τ (and not the j th power of the scalar τ ).

We let t0τ

.= 0 and define recursively

t iτ

.= t i−1τ + τ i , I i

τ.= [t i−1

τ , t iτ ) for i = 1, . . . , Nτ .

Let the symbol τ.= maxi=1,...,Nτ τ

i denote the maximum time-step (fineness of the partition). We will makeuse of the notation (B0

τ , . . . , BNττ ) for elements in (Rd×d)Nτ +1 and let Bτ : [0, T ] → R

d×d be the correspond-ing piecewise constant interpolant on the intervals I i

τ , namely Bτ (t).= Bi−1

τ for all t ∈ I iτ , i = 1, . . . , Nτ

and Bτ (T ).= BNτ

τ .As we are given (C, θ) : [0, T ] → R

d×dsym × (0,∞), we define (C i

τ , θiτ )

.= (C(t iτ ), θ(t

iτ )). Then, we look

for a vector (C0tr,τ , . . . ,C Nτ

tr,τ ) ∈ (SLsym(d))Nτ +1 such that

C0tr,τ = C tr,0, (4.6)

C itr,τ ∈ Arg min

{

e(tiτ ,C tr)+ D(C i−1

tr,τ ,C tr)

∣∣∣ C tr ∈ SLsym(d)

}

for i = 1, . . . , N τ . (4.7)

Hence, the solution C tr,τ is obtained by sequentially minimizing on SLsym(d)

C tr �→ ψel(C−1/2tr C i

τ C−1/2tr )+ ψtr(C tr, θ

iτ )+ D(C i−1

tr,τ ,C tr).

Such incremental solution clearly exists as the above functional turns out to be coercive and lower semicontin-uous and SLsym(d) is closed. In particular, Ei

tr,τ.= (C i

tr,τ−1)/2 is such that |Eitr,τ | ≤ εL for all i = 1, . . . , Nτ .

4.2 Discrete stability and estimates

Let us show that C itr,τ ∈ S(t i

τ ) for i = 1, . . . , Nτ . By using the minimality (4.7) and the triangle inequalityfor D, we have that, for all B ∈ SLsym(d),

e(t iτ ,C i

tr,τ )+ D(C i−1tr,τ ,C i

tr,τ ) ≤ e(t iτ , B)+ D(C i−1

tr,τ , B)

≤ e(t iτ , B)+ D(C i−1

tr,τ ,C itr,τ )+ D(C i

tr,τ , B)

whence stability follows. Furthermore, again by minimality (4.7) the following inequality holds

e(t iτ ,C i

tr,τ )+ D(C i−1tr,τ ,C i

tr,τ ) ≤ e(t i−1τ ,C i−1

tr,τ )+t iτ∫

t i−1τ

∂se(s,C i−1tr,τ )ds.

By summing up we get the discrete upper energy estimate for the piecewise constant C tr,τ

e(tmτ ,C tr,τ (t

mτ ))+ Diss[t i

τ ,tmτ ](C tr,τ )

≤ e(t iτ ,C tr,τ (t

iτ ))+

tmτ∫

t iτ

∂se(s,C tr,τ (s))ds for i ≤ m. (4.8)

Our next aim is that of controlling the power of external actions. In particular, for all given C tr ∈ K.= {C tr ∈

SLsym(d) | |C tr − 12| ≤ 2εL} (recall (2.7)), we have

∂t e(t,C tr) = ∂Cψel(C−1/2tr C(t)C−1/2

tr ):C(t)+ ∂tψtr(C tr, θ(t)). (4.9)


Due to assumption (4.5) and to the explicit form (2.7) of the inelastic energy ψtr(C tr, θ), we have that t �→ψtr(C tr, θ(t)) ∈ W 1,1(0, T ) and

∂tψtr(C tr, θ(t)) = bH(θ(t)− θM )|Etr|θ (t) for a.e. t ∈ (0, T ) (4.10)

where H is the Heaviside function. As for the first term in the right-hand side of (4.9) we compute by (2.2)

∂Cψel(C−1/2tr C(t)C−1/2

tr ) = ∂Aψel(A)|A=C−1/2tr CC−1/2

tr:∂C(C

−1/2tr C(t)C−1/2

tr )

= C−1/2tr ∂Aψel(A)|A=C−1/2

tr C(t)C−1/2tr

C−1/2tr . (4.11)

Let us observe that assumption (4.4) entails in particular that C ∈ L∞(0, T ; Rd×dsym ). Moreover, we have that

C tr ∈ K ⇒ |C tr| + |C−1tr | ≤ cK

where cK > 0 depends on data only. From the smoothness of the square root [27, p. 23] an analogous boundholds true for |C1/2

tr |+ |C−1/2tr |. In particular, given the above bounds, one can check that |Fel|+ |F−1

el | is alsoa priori bounded in terms of data and |C(t)|. Now, we remark that, for all B ∈ R

d×d , one has

∂Fel W (Fel)F�el :B = Fel∂Celψel(Cel)F�

el :(B + B�).Hence, we can control the first term in the right-hand side of (4.9) as follows (recall (3.1), which impliesC−1/2

tr CC−1/2tr = R�Cel R)

∣∣∂Cψel(C

−1/2tr C(t)C−1/2

tr ):C(t)∣∣= ∣

∣Fel∂Aψel(A)|A=C−1/2tr C(t)C−1/2

trF�

el :F−�el C−1/2

tr C(t)C−1/2tr F−1

el

∣∣

= |Fel R�∂Celψel(Cel)RF�el : F−�

el C−1/2tr C(t)C−1/2

tr F−1el |

= |Fel∂Celψel(Cel)F�el : F−�

el RC−1/2tr C(t)C−1/2

tr RT F−1el |

= 1

2|∂Fel W (Fel)F�

el : F−Tel RC−1/2

tr C(t)C−1/2tr RT F−1

el |

≤ 1

2c0(W (Fel)+ 1)|F−�

el RC−1/2tr C(t)C−1/2

tr RT F−1el |

≤ c(ψel(C−1/2tr C(t)C−1/2

tr )+ 1)|C(t)|.where c depends on data only. Hence, we can conclude that

|e(t,C tr)| ≤ γ (t)(1 + e(t,C tr)) (4.12)

where t �→ γ (t).= ce(|C(t)| + |θ (t)|) ∈ L1(0, T ) and ce > 0 depends on data only. Moreover, note that for

almost all times t ∈ [0, T ] the power C tr �→ ∂t e(t,C tr) turns out to be continuous on K as a consequence ofthe smoothness of ψel and the continuity of the inverse on SL(d) and of the square root in R

d×dsym .

Now, from the minimality (4.7) we deduce that the piecewise constant C tr,τ satisfies C tr,τ (t) ∈ K forevery t ∈ [0, T ]. Moreover, owing to the bound (4.12), the discrete upper energy estimate (4.8) entails viaGronwall (see [38]) that

supτ

(

supt∈[0,T ]

e(t,C tr,τ (t))+ Diss[0,T ](C tr,τ )

)

< ∞ (4.13)

Let us now take a sequence of partitions τ k such that τ k → 0 as k → ∞. Let Cktr

.= C tr,τ k be the correspond-ing discrete solutions. From the bound (4.13) and Helly’s selection principle (see, e.g., [39]) we can deducethe existence of a (not relabeled) subsequence Ck

tr and of a function C tr ∈ BVD([0, T ]; SLsym(d)).= {t ∈

[0, T ] �→ C tr(t) ∈ SLsym(d) | Diss[0,T ](C tr) < ∞} such that

Cktr(t) → C tr(t) as k → ∞ ∀t ∈ [0, T ].

In particular, as |Cktr(t)−1| ≤ 2εL for all times t ∈ [0, T ], the latter convergence entails that |C tr(t)−1| ≤ 2εL

for all t ∈ [0, T ] as well. Hence, the limit function C tr fulfills the constraint |Etr| = |C tr−1|/2 ≤ εL for alltimes. We now aim at showing that C tr is an energetic solution of (3.3), namely that it satisfies the stabilitycondition (4.2) and the energy balance (4.3).


4.3 Stability of the limit function

We first show that the set S[0,T ].= {(t,C tr) ∈ [0, T ] × SLsym(d) : C tr ∈ S(t)} is closed in [0, T ] × SLsym(d).

Let us take (tn,C(n)tr ) ∈ S[0,T ] such that tn → t∗ in [0, T ] and C(n)

tr → C∗tr in SLsym(d). Since C(n)

tr ∈ S(tn),we have e(tn,C(n)

tr ) < ∞ and

e(tn,C(n)tr ) ≤ e(tn, C tr)+ D(C(n)

tr , C tr) ∀C tr ∈ SLsym(d).

By passing to the limit for n → ∞ and using the continuity of ψ (restricted to Rd×dsym × K ×R), we immediately

get that (t∗,C∗tr) ∈ S[0,T ]. Let us now fix t ∈ [0, T ] and define st

k.= max{t i

τ k| t i

τ k≤ t}. We have that st

k → t

and Cktr(s

tk) = Ck

tr(t) → C tr(t) as k → ∞. Since Cktr(s

tk) ∈ S(st

k), by the closedness of S[0,T ] we have thatC tr(t) ∈ S(t).

4.4 Upper energy estimate

Along with the above notation, the discrete upper energy estimate (4.8) can be rewritten as

e(stk,Ck

tr(stk))+ Diss[0,st

k ](Cktr) ≤ e(0,C tr,0)+

stk∫

0

∂se(s,Cktr(s))ds.

By observing that Diss[0,stk ](C

ktr) = Diss[0,t](Ck

tr), passing to the limit as k → ∞, and exploiting the bound(4.12), relation (4.13), the lower semicontinuity of the functional Diss[0,t], and the Lebesgue’s DominatedConvergence Theorem, we deduce the upper energy estimate

e(t,C tr(t))+ Diss[0,t](C tr) ≤ e(0,C tr,0)+t∫

0

∂se(s,C tr(s))ds ∀t ∈ [0, T ].

4.5 Lower energy estimate

Let us take a sequence of partitions τm of the interval [0, t], with τm → 0 as m → ∞. Since C tr(t) ∈ S(t),for every t ∈ [0, T ], we have (setting for brevity Nm

.= Nτm and t jm

.= t jτm )

e(t jm,C tr(t

jm))+ D(C tr(t

j−1m ),C tr(t

jm))

= e(t j−1m ,C tr(t

jm))+

t jm∫

t j−1m

∂se(s,C tr(tj

m))ds + D(C tr(tj−1

m ),C tr(tj

m))

≥ e(t j−1m ,C tr(t

j−1m ))+

t jm∫

t j−1m

∂se(s,C tr(tj

m))ds.

Summing up for j = 1, . . . , Nm we deduce that

e(t,C tr(t))− e(0,C tr,0)+ Diss[0,t](C tr)

≥Nm∑

j=1

[

e(t jm,C tr(t

jm))− e(t j−1

m ,C tr(tj−1

m ))]

+Nm∑

j=1

D(C tr(tj−1

m ),C tr(tj

m))

≥t∫

0

∂se(s,C(m)tr (s))ds, (4.14)


where C(m)tr is defined as C

(m)tr (t) = C tr(t

jm) for every t ∈ (t j−1

m , t jm]. By recalling (4.9)–(4.10), the integral on

the right-hand side is given by the sum of two contributions, namely

t∫

0

∂se(s,C(m)tr (s))ds =

t∫

0

bH(θ(s)− θM )|E(m)tr (s)|θ (s)ds

+t∫

0

∂Cψel((C(m)tr (s))−1/2C(C

(m)tr (s))−1/2)

∣∣∣C=C(s)

: C(s)ds

where E(m)tr = (C

(m)tr − 12)/2. As C tr ∈ BVD([0, T ]; SLsym(d)), we have that C tr is continuous on [0, T ]

with the exception of at most a countable set of points [39, Theorem 3.3]. Hence, we have C(m)tr (t) → C tr(t)

for a.e. t ∈ (0, T ). We can then apply Lebesgue’s Dominated Convergence Theorem to conclude that

t∫

0

bH(θ(s)− θM )|E(m)tr (s)|θ (s)ds →t∫

0

bH(θ(s)− θM )|Etr(s)|θ (s)ds

Next, by recalling the continuity of the power term and assumption (4.4), we may use again Lebesgue’s theoremand show that

t∫

0

∂Cψel((C(m)tr (s))−1/2C(s)(C

(m)tr (s))−1/2):C(s)ds

→t∫

0

∂Cψel((C tr(s))−1/2C(s)(C tr(s))

−1/2):C(s)ds.

Hence, the integral on the right-hand side of (4.14) converges to

t∫

0

bH(θ(s)− θM )|Etr(s)|θ (s)ds +t∫

0

∂Cψel((C tr(s))−1/2C(s)(C tr(s))

−1/2):C(s)ds

and, by taking the limit m → ∞, we get the desired lower energy estimate

e(t,C tr(t))+ Diss[0,t](C tr) ≥ e(0,C tr,0)+t∫

0

∂se(s,C tr(s))ds ∀t ∈ [0, T ].

In particular, we have proved that C tr fulfills both the stability (4.2) and the energy equality (4.3). Namely, C tris an energetic solution.

Acknowledgments The Authors are indebted to Alexander Mielke for some interesting discussion on this model. U.S. acknowl-edges the partial support of FP7-IDEAS-ERC-StG Grant #200497 (BioSMA), the CNR-AVCR joint project SmartMath, and theAlexander von Humboldt Foundation.

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existence and time-discretization for the finite-strain souza–auricchio constitutive model for...

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