Data-Driven Finite Elasticity

Abstract

We extend to finite elasticity the Data-Driven formulation of geometrically linear elasticity presented in Conti et al. (Arch Ration Mech Anal 229:79–123, 2018). The main focus of this paper concerns the formulation of a suitable framework in which the Data-Driven problem of finite elasticity is well-posed in the sense of existence of solutions. We confine attention to deformation gradients $F \in L^p(\Omega ;{\mathbb {R}}^{n\times n})$ and first Piola-Kirchhoff stresses $P \in L^q(\Omega ;{\mathbb {R}}^{n\times n})$, with $(p,q)\in (1,\infty )$ and $1/p+1/q=1$. We assume that the material behavior is described by means of a material data set containing all the states (F, P) that can be attained by the material, and develop germane notions of coercivity and closedness of the material data set. Within this framework, we put forth conditions ensuring the existence of solutions. We exhibit specific examples of two- and three-dimensional material data sets that fit the present setting and are compatible with material frame indifference.

Appendix A. Traces of Sobolev Spaces

We use standard properties of Sobolev spaces and their traces, see for example [5, 17, 20]. For the convenience of the reader we recall here the basic definitions and the facts used in the preceding analyses.

Definition A.1

Let $\Omega \subseteq {\mathbb {R}}^n$ be a bounded Lipschitz set, $p\in (1,\infty )$. For $f\in L^p(\partial \Omega ;{\mathbb {R}}^n)$ we define

$$\begin{aligned} {[}f]_{1-1/p,p}^p:= \int _{\partial \Omega \times \partial \Omega } \frac{|f(x)-f(y)|^p}{|x-y|^{n+1}} \mathrm{d}{\mathcal {H}}^{n-1}(x) \mathrm{d}{\mathcal {H}}^{n-1}(y) , \end{aligned}$$

(A.1)

and set

$$\begin{aligned} W^{1-1/p,p}(\partial \Omega ;{\mathbb {R}}^n):=\{f\in L^p(\partial \Omega ;{\mathbb {R}}^n): [f]_{1-1/p,p}^p<\infty \}, \end{aligned}$$

(A.2)

equipped with the norm $\Vert f\Vert _{L^p(\partial \Omega )}+[f]_{1-1/p,p}$. Furthermore, the dual space is denoted by

$$\begin{aligned} W^{-1+1/p,q}(\partial \Omega ;{\mathbb {R}}^n) = (W^{1-1/p,p}(\partial \Omega ;{\mathbb {R}}^n))^*, \end{aligned}$$

(A.3)

where $q>1$ is defined by the condition $1/p+1/q=1$.

It is readily checked that $W^{1-1/p,p}(\partial \Omega ;{\mathbb {R}}^n)$ is a reflexive Banach space [5, Sect. 6.8].

Lemma A.2

Let $\Omega \subseteq {\mathbb {R}}^n$ be a bounded Lipschitz set, $p\in (1,\infty )$. Than

• (i) There is a linear continuous operator $B:W^{1,p}(\Omega ;{\mathbb {R}}^n)\rightarrow W^{1-1/p,p}(\partial \Omega ;{\mathbb {R}}^n)$ such that $B\varphi =\varphi |_{\partial \Omega }$ for any $\varphi \in C^1({\bar{\Omega }};{\mathbb {R}}^n)$.

• (ii) There is a linear continuous operator $\mathop {Ext}: W^{1-1/p,p}(\partial \Omega ;{\mathbb {R}}^n)\rightarrow W^{1,p}(\Omega ;{\mathbb {R}}^n)$ such that $B\mathop {Ext}u=u$ for any $u\in W^{1-1/p,p}(\partial \Omega ;{\mathbb {R}}^n)$.

Proof

See [5, Ths. 6.8.13 and 6.9.2] $\quad \square$

Definition A.3

Let $\Omega \subseteq {\mathbb {R}}^n$ be a bounded Lipschitz set, $q\in (1,\infty )$. We define

$$\begin{aligned} E^q(\Omega ):=\{ v\in L^q(\Omega ;{\mathbb {R}}^{n\times n}): \mathrm {div\,}v\in L^q(\Omega ;{\mathbb {R}}^n)\} , \end{aligned}$$

(A.4)

and endow it with the norm $\Vert v\Vert _{E^q}:=\Vert v\Vert _{L^q}+\Vert \mathrm {div\,}v\Vert _{L^q}$.

It is easily verified that $E^q$ is a reflexive Banach space, and that $C^\infty ({\bar{\Omega }})\cap E^q(\Omega )$ is dense in v, see, for example, [17, Th. 1.1] (the different exponent makes no difference in the proof).

Lemma A.4

Let $\Omega \subseteq {\mathbb {R}}^n$ be a bounded Lipschitz set, $q\in (1,\infty )$. There is a linear continuous operator $B_\nu :E^q(\Omega )\rightarrow W^{-1/q,q}(\partial \Omega ;{\mathbb {R}}^n)$ such that $B_\nu \varphi =\varphi |_{\partial \Omega }\cdot \nu$, where $\nu$ is the outer normal to $\partial \Omega$, for any $\varphi \in C^1({\bar{\Omega }};{\mathbb {R}}^n)\cap E^q(\Omega )$. Furthermore,

$$\begin{aligned} \langle B_\nu v, Bu\rangle = \int _\Omega v\cdot Du \,\mathrm{d}x + \int _\Omega u \cdot \mathrm {div\,}v \,\mathrm{d}x , \end{aligned}$$

(A.5)

for all $v\in E^q(\Omega )$, $u\in W^{1,p}(\Omega ;{\mathbb {R}}^n)$, where $\langle \cdot ,\cdot \rangle$ denotes the duality pairing between $W^{1/q,p}(\partial \Omega ;{\mathbb {R}}^n)$ and $W^{-1/q,q}(\partial \Omega ;{\mathbb {R}}^n)$.

Proof

The proof follows from the same argument used in [17, Th. 1.2 and Rem. 1.3]

$\square$