Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materialsThis work was supported by the German Science Foundation (Deutsche Forschungsgemeinschaft, DFG) under Schu 1978/4-1 and Schu 1978/4-2.

# Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials††thanks: This work was supported by the German Science Foundation (Deutsche Forschungsgemeinschaft, DFG) under Schu 1978/4-1 and Schu 1978/4-2.

A. Wöstehoff Helmut Schmidt University, Department of Mechanical Engineering, Holstenhofweg 85, 22043 Hamburg, Germany (arne.woestehoff@hsu-hh.de).    T. Schuster Saarland University, Department of Mathematics, Campus, 66123 Saarbrücken, Germany (thomas.schuster@num.uni-sb.de), corresponding author.
###### Abstract

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behaviour of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. Thus results on unique solvability and continuous dependence from the initial values are of large interest in materials research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied, for a certain class of hyperelastic materials where the first Piola-Kirchhoff tensor is written as a conic combination of finitely many given tensors.

Key words. Cauchy’s equation of motion, hyperelastic materials, uniqueness and stability, Cordes condition, Gronwall’s lemma

AMS subject classifications. 35A01, 35A02, 35L20, 35L70, 74B20

## 1 Introduction

Cauchy’s equation of motion follows from conservation of mass and momentum and reads as

 ρ(x)¨u(t,x)=divP(t,x)+ρ(x)f(t,x),t≥0,x∈Ω \hb@xt@.01(1.1)

where is a bounded domain, denotes the mass density, is the vector of particle displacement, is the first Piola-Kirchhoff stress tensor and an external body force. It describes the discplacement that a particle in position at time perceives under stress and external force . If we specifically investigate the behavior of elastic materials we furthermore need a constitutive law which states a connection between the stress tensor and the position as well as the deformation gradient , i.e.

 P(t,x)=^P(x,Ju(t,x)). \hb@xt@.01(1.2)

Actually the stress-strain law (LABEL:eq-CL) characterizes elastic materials. A special class of such materials are hyperelastic materials, where the constitutive function can be expressed as a derivative of a stored energy function ,

 ^P(x,Y)=∇YC(x,Y),Y∈R3×3,detY>0. \hb@xt@.01(1.3)

Here, the derivative is to be understood componentwise. The class of hyperelastic materials comprehends isotropic materials, Mooney-Rivlin materials, neo-Hookean materials and even elastic fluids. Combining (LABEL:eq-Cauchy), (LABEL:eq-CL) and (LABEL:eq-hyper) yields the equation of motion for hyperelastic materials

 ρ(x)¨u(t,x)−div∇YC(x,Ju(t,x))=ρ(x)f(t,x). \hb@xt@.01(1.4)

For detailled derivations of Cauchy’s equation of motion and introductions to elastic and hyperelastic materials we refer to the standard textbooks [2, 7, 14] to name only a few.

Since equation (LABEL:eq-motion-hyper) models the behavior of hyperelastic materials, this equation has many applications ranging from engineering to biomedical research. E.g. composite materials like carbon-fibre reinforced epoxy are of growing interest in aircraft construction or wind power stations and thus has a deep, economic impact. Dveloping autonomous structural helath monitoring (SHM) systems for such materials is a current and vivid research field to which not only engineers but also mathematicians and computer scientists contribute. Understanding the behavior of composites and developing numerical solvers for the inverse problems which arise in SHM demand for a deep analysis of (LABEL:eq-motion-hyper) equipped with appropriate initial- and boundary values, see also . Existence- and uniqueness results for special cases, especially for the linearized Cauchy equation, can be found in standard references on systems of hyperbolic equations such as [8, 10, 16, 17]. In  the authors deal with existence and uniqueness of a global solution in nonlinear elasticity and they further prove continuous dependence of the solution from initial values. The existence of weak solutions of the linearized version of (LABEL:eq-motion-hyper) can also be proven by means of evolution equations, see . Of course this list is by far not complete. We prove a novel existence- and uniqueness result where it is important to know how the arising constants of the stability estimates depend on the underlying differential operator. This result, which is the main result of the entire article and stated in Theorem LABEL:MainTheorem, relies on a specific class of constitutive functions which are assumed to be conic combinations of fintely many, given tensors, i.e. we suppose that

 ^P(x,Y)=∂YC(x,Y)=N∑K=1αKdiv∇YCK(x,Y),

where and , are given. This setting is inspired by the article of Kaltenbacher and Lorenzi . There the authors also assume such a conic combination but their results do hold for scalar displacements, contant mass density and homogeneous engery functions only, whereas our results are valid for systems of equations in any dimension and spatially variable functions and . This is why we do not only consider the three-dimesnional case, even if that case might be the most prominent case in view of applications, but formulate our setting for arbitrary domains with sufficiently smooth boundary and displacements in . Equipped with appropriate initial- and homogeneous Dirichlet boundary values this gives the system

 ρ(x)¨u(t,x)−N∑K=1αKdiv∇YCK(x,Ju(t,x))=ρ(x)f(t,x) \hb@xt@.01(1.5)

for and along with the boundary conditions

 u(t,x)=0,t∈[0,T],x∈∂Ω \hb@xt@.01(1.6)

and given initial values

 u(0,⋅)=u0∈H2(Ω,Rn),˙u(0,⋅)=u1∈H1(Ω,Rn). \hb@xt@.01(1.7)

We will prove existence, uniqueness and continuous dependence from the given initial-boundary values, if has a -boundary and the solution as well as the given functions satisfy boundedness estimates for derivatives up to the order and , respectively. The assertions are stated in Theorem LABEL:MainTheorem. The crucial difficulty of the proof is to show that the constants involved to the stability estimates are uniformly bounded with respect to the coefficients .
The proof is performed in several steps. First we need a generalization of the Cordes condition. To this end we extend a result stated in  (Section 3). The next three main steps of the proof are derivations of upper bounds of the solutions and their derivatives corresponding two different sets of initial values , coefficients , and forces , which are outlined in Sections 4.1, 4.2 and 4.3. The concluding step of this extensive proof is described in Section 4.4.

## 2 Preliminaries and main result

Throughout the entire article, denotes a bounded, open and convex domain with -boundary and is a fixed time interval with . Furthermore we suppose that is a function satisfying estimates

 ρmin≤infx∈Ωρ(x)≤supx∈Ωρ(x)≤ρmax

for constants . The divergence of a function is the mapping defined by

 divf(t,x):=(d∑j=1∂∂xjfi(t,x))i=1,…,n

and the Jacobian of a function is

 Ju(t,x):=(∂∂xjui(t,x))i=1,…,n,j=1,…,d.

The derivative with respect to time is always denoted by a dot like , .

By we denote the Sobolev space endowed with the norm

 ∥⋅∥W2,2γ0(Ω,Rn):=(n∑k=1∥uk∥W2,2γ0(Ω))1/2:=(n∑k=1∫Ωd∑ℓ=1d∑j=1(∂ijuk(x))2dx)1/2

turning into a Banach space whose norm is equivalent to the -norm. Especially, there is a constant , such that for all .

Before stating and proving the main result it is necessary to confine the nonlinearity of the PDE-system (LABEL:PDE). To this end we require for every the existence of constants , , and , satisfying

 κK∥Y∥2F≤CK(x,Y)≤μK∥Y∥2F \hb@xt@.01(2.1)

and

 κK∥H∥2F≤⟨⟨H|∇Y∇YCK(x,Y)H⟩⟩≤μK∥H∥2F \hb@xt@.01(2.2)

for all and almost all . Here, denotes the inner product of -matrices and the trace of ; as is known this inner product induces the Frobenius norm . Furthermore we assume for any the existence of constants , such that the functions and their derivatives are bounded as

 ∥∂Ypq∂Yij∂YkℓCK∥L∞(Ω×Rn×d) ≤μK \hb@xt@.01(2.3) ∥∂Yab∂Ypq∂Yij∂YkℓCK∥L∞(Ω×Rn×d) ≤μK \hb@xt@.01(2.4) ∥∂ℓ∂YkℓCK∥L∞(Ω×Rn×d) ≤μK \hb@xt@.01(2.5) ∥∂Yij∂ℓ∂YkℓCK∥L∞(Ω×Rn×d) ≤μK \hb@xt@.01(2.6) ∥∂ℓ∂Yij∂YkℓCK∥L∞(Ω×Rn×d) ≤μK \hb@xt@.01(2.7) ∥∂Ypq∂ℓ∂Yij∂YkℓCK∥L∞(Ω×Rn×d) ≤μK \hb@xt@.01(2.8)

for any and . Additionally, let be three times continuously differentiable for almost all , and let

 ∂Yij∂ℓ∂YkℓC(x,Y)=∂ℓ∂Yij∂YkℓC(x,Y) \hb@xt@.01(2.9)

for any and . E.g. (LABEL:Bound35)–(LABEL:Premise311) are fulfilled if . Finally we assume that the body force appearing on the right-hand side of (LABEL:PDE) is to be an element of , which is a the set of all satisfying and which is equipped with the norm

 ∥f∥W1,1((0,T),L2(Ω,Rn)) :=∥f∥L1((0,T),L2(Ω,Rn))+∥˙f∥L1((0,T),L2(Ω,Rn))
###### Theorem 2.1

Let , be two solutions of problem (LABEL:PDE),(LABEL:BoundaryValues), (LABEL:InitialValues) corresponding to the parameters and initial values , , respectively. Furthermore assume that

 ∥∂ℓ∂ju∥L∞((0,T),L2(Ω,Rn))≤M0∥∂ℓ∂j~u∥L∞((0,T),L2(Ω,Rn))≤M0∥∂ℓ˙u∥L∞((0,T)×Ω)≤M1∥∂ℓ˙u∥L∞((0,T)×Ω)≤M1 \hb@xt@.01(2.10) and ∥∂ℓ∂j˙uk∥L∞((0,T)×Ω)≤M2∥∂ℓ∂j˙~uk∥L∞((0,T)×Ω)≤M2∥∂ℓ∂juk∥L∞((0,T)×Ω)≤M3∥∂ℓ∂j~uk∥L∞((0,T)×Ω)≤M3 \hb@xt@.01(2.11)

hold for any and all . If, in addition, the dimensions and satisfy

 nd−2nd−1μ<κ

where and , and if there are constants and , so that and , then there exist constants , , and such that the stability estimate

 ≤[∥(˙u−˙~u)(t,⋅)∥2L2(Ω,Rn)+κ(α)∥(Ju−J~u)(t,⋅)∥2L2(Ω,Rn×d)+ ≤+∥(¨u−¨~u)(t,⋅)∥2L2(Ω,Rn)+κ(α)∥(J˙u−J˙~u)(t,⋅)∥2L2(Ω,Rn×d)+ ≤+∥(u−~u)(t,⋅)∥2H2(Ω,Rn)]1/2 ≤¯¯¯¯C0[μ(α)∥u0−~u0∥2H2(Ω,Rn)+∥u1−~u1∥H1(Ω,Rn)]1/2+ ≤+¯¯¯¯C1∥f−~f∥W1,1((0,T),L2(Ω,Rn))+¯¯¯¯C2∥α−~α∥∞ \hb@xt@.01(2.13)

is valid for all . Thereby the constants , , and only depend on , , , , ,

 ¯¯¯¯C(α) :=N∑K=1αKμK(N∑K=1αKκK)−1, \hb@xt@.01(2.14) and \hb@xt@.01(2.15) ^C(α) :=^K1−√1−εN∑K=1αKμK(N∑K=1αKκK)−2,

where is a constant whose existence is ensured by inequality (LABEL:Dimensions). Moreover, the constants , , and are uniformly bounded if with bounded.

The principal techniques to prove this theorem take advantage of a lemma by Gronwall on the first hand and use a generalization of a known result by Maugeri, Palagachev and Softova in , the so-called Cordes condition, on the other hand. To this end we conclude this section by stating Gronwall’s lemma as we need it in our proof. The generalization of the Cordes condition is subject of section 3.

###### Lemma 2.2 (Gronwall)

Let and be nonnegative functions. If satisfies

 ψ(τ)≤a+∫τ0b(t)ψ(t)dt+∫τ0k(t)ψ(t)pdt

for all with constants and , then

 ψ(τ)≤exp(∫τ0b(t)dt)[a1−p+(1−p)∫τ0k(t)exp((p−1)∫t0b(σ)dσ)dt]1/(1−p)

for all .

A proof of this version can be found for example in .

## 3 The Cordes condition

In this section we prove the mentioned generalization of a result accomplished in . More on the Cordes condition can be found in the original articles [3, 4].

###### Theorem 3.1

Let and for and . Additionally, let there be with in such a way that

for all and allmost all as well as

 n∑k=1d∑ℓ=1n∑i=1d∑j=1a2kℓij(x)(n∑k=1d∑ℓ=1akℓkℓ(x))−2≤1nd−1+ε \hb@xt@.01(3.1)

for allmost all . Then the Dirichlet problem

 n∑k=1d∑ℓ=1n∑i=1d∑j=1akℓij(x)ek∂ℓjui(x)=f(x),u∈H2(Ω,Rn)∩H10(Ω,Rn)

with denoting the th standard basis vector in admits a unique solution for every . Moreover, this solution fulfills

 ∥u∥H2(Ω,Rn)≤C(α)∥f∥L2(Ω,Rn) \hb@xt@.01(3.2)

with

 C(α):=^Kesssupx∈Ωα(x)1−√1−ε

and

 α(x):=n∑k=1d∑ℓ=1akℓkℓ(x)(n∑k=1d∑ℓ=1n∑i=1d∑j=1a2kℓij(x))−1.

Proof. To prove Theorem LABEL:theorem:Cordes we follow the lines of the according proof in . Let be the differential operator

 Lu:=n∑k=1d∑ℓ=1n∑i=1d∑j=1akℓij(x)ek∂ℓjui(x).

Due to the premises is strictly positive, since putting with being the Kronecker symbol reveals . Thus, is equivalent to . The idea is to analyze the operator defined by , where denotes the unique solution of the Poisson problem

 ΔU=αf+Δw−αLw∈L2(Ω,Rn),U∈W2,2γ0(Ω,Rn). \hb@xt@.01(3.3)

Existence and uniqueness of a solution of (LABEL:eq-poisson) can be seen by applying standard results as shown e. g. in  or  to the th component

 Δ(Uk)=αfk+Δ(wk)−αd∑ℓ=1n∑i=1d∑j=1akℓij∂ℓjwi∈L2(Ω)%.

We focus now at the properties of and want to show that this mapping is a contraction. For this purpose, we draw on the famous Miranda-Talenti estimate

 \hb@xt@.01(3.4)

A proof of (LABEL:eq-miranda) can also be found in . Let . Then using (LABEL:eq-miranda) and the Cauchy-Schwarz inequality yields

 ≤∥Tw1−Tw2∥2W2,2γ0(Ω,Rn)=n∑k=1∫Ωd∑ℓ=1d∑j=1[∂ℓj(U1,k(x)−U2,k(x))]2dx ≤∫Ωn∑k=1{[Δ(U1(x)−U2(x))]k}2dx=∥Δ(w1−w2)−αL(w1−w2)∥2L2(Ω,Rn) ≤∫Ω[n∑k=1d∑ℓ=1n∑i=1d∑j=1(δℓjδki−α(x)akℓij(x))2][d∑ℓ=1n∑i=1d∑j=1(∂ℓj(w1,i(x)−w2,i(x)))2]dx.

The expression of the first factor of the integrand can be estimated as

 ≤n∑k=1d∑ℓ=1n∑i=1d∑j=1(δℓjδki−α(x)akℓij(x))2 =nd−2α(x)n∑k=1d∑ℓ=1akℓkℓ(x)+α2(x)n∑k=1d∑ℓ=1n∑i=1d∑j=1a2kℓij(x) ≤nd−(nd−1+ε)=1−ε,

where we made use of the Cordes condition (LABEL:Cordes). We summarize that

 ∥Tw1−Tw2∥2W2,2γ0(Ω,Rn) ≤∫Ω(1−ε)d∑ℓ=1n∑i=1d∑j=1(∂ℓj(w1,i(x)−w2,i(x)))2dx =(1−ε)∥w1−w2∥2W2,2γ0(Ω,Rn)

what proves that in fact is a contraction in , since . Due to the Banach fixed-point theorem, has a unique fixed-point, i. e. there exists a unique satisfying . The definition of implies , which is equivalent to .

It remains to varify (LABEL:EstimateDueToCordes). We have already shown that

 ∥U1−U2∥2W2,2γ0(Ω,Rn)≤∥Δ(U1−U2)∥2L2(Ω,Rn)

Setting , with the unique fixed-point of , and yielding we infer

 ∥w∥W2,2γ0(Ω,Rn) ≤∥Δw∥L2(Ω,Rn)≤∥αf∥L2(Ω,Rn)+∥Δw−αLw∥L2(Ω,Rn) ≤esssupx∈Ωα(x)∥f∥L2(Ω,Rn)+√1−ε∥u∥W2,2γ0(Ω,Rn),

where we again used (LABEL:eq-miranda). The assertion finally follows from the equivalence of the norms and .

## 4 Proof of Theorem LABEL:MainTheorem

Before we start with the proof of Theorem LABEL:MainTheorem we note that we may replace in estimate (LABEL:MainEstimate) by the equivalent, weighted norm and get

 ≤[∥(˙u−˙~u)(t,⋅)∥2L2ρ(Ω,Rn)+κ(α)∥(Ju−J~u)(t,⋅)∥2L2(Ω,Rn×d)+ ≤+∥(¨u−¨~u)(t,⋅)∥2L2ρ(Ω,Rn)+κ(α)∥(J˙u−J˙~u)(t,⋅)∥2L2(Ω,Rn×d)+ ≤+∥(u−~u)(t,⋅)∥2H2(Ω,Rn)]1/2 ≤¯¯¯¯C0[μ(α)∥u0−~u0∥2H2(Ω,Rn)+∥u1−~u1∥H1(Ω,Rn)]1/2+ ≤+¯¯¯¯C1∥f−~f∥W1,1((0,T),L2(Ω,Rn))+¯¯¯¯C2∥α−~α∥∞.

The proof is subdivided in four parts:

1. We deduce an upper bound for the norm of , which depends on , , and (Section 4.1).

2. We show an upper bound for the time-derivative which additionally depends on (Section 4.2).

3. We prove an upper bound for the -norm of depending on , , and and other norms of derivatives of (Section 4.3).

4. We summarize the results so far and finish the proof (Section 4.4).

### 4.1 An upper bound for u−~u

To derive our aim to bound the norm of we at first prove some intermediate results. Thereby the key role will play Gronwall’s lemma LABEL:Gronwall.

###### Lemma 4.1

We have

 ≤∥˙u(τ,⋅)∥2L2ρ(Ω,Rn)+2N∑K=1αKκK∥Ju(τ,⋅)∥2L2(Ω,Rn×d)

Proof. Multiplying equation (LABEL:PDE) by and integrating over gives

 =∂t∥˙u(t,⋅)∥2L2ρ(Ω,Rn)−2N∑K=1αK⟨˙u(t,⋅)∣∣div∇YCK(⋅,Ju(t,⋅))⟩L2(Ω,Rn) =2⟨ρ(⋅)˙u(t,⋅)|f(t,⋅)⟩L2(Ω,Rn). \hb@xt@.01(4.1)

Using the divergence theorem and the chain rule yields

 =⟨˙u(t,⋅)∣∣div∇YCK(⋅,Ju(t,⋅))⟩L2(Ω,Rn) =n∑k=1∫Ω˙uk(t,x)div[eTk∇YCK(x,Ju(t,x))]dx =∂t[−∫ΩCK(x,Ju(t,x))dx].

If we use this reformulation in (LABEL:eq-help1), we see that

 =∂t{∥˙u(t,⋅)∥2L2ρ(Ω,Rn)+2N∑K=1αK∫ΩCK(x,Ju(t,x))dx}