On the local identifiability of constituent stress–strain laws for hyperelastic composite materials

Abstract

For natural composite materials such as biological tissues, mechanically characterizing the individual constituents and elucidating their roles in supporting structural integrity has remained experimentally challenging since the constituents can often not be isolated without impairment and require non-standard testing devices. Adopting an inverse viewpoint, we examine, in this article, macroscopic samples whose constituent architecture is accessible and investigate whether it is possible to conclude on the stress–strain behaviour of the individual constituents based on experimental measurements from standard material tests. Focussing on isotropic hyperelastic composites, a direct discretization of the constituents’ strain energy densities in terms of global shape functions is explored. In order to assess the local deterministic identifiability of material parameters near an initial estimate, we adopt a sensitivity-based criterion and determine feasible combinations of candidate experiments without recourse to experimental measurements. Both the local identifiability and attainable reidentification accuracy are investigated in detail for a composite truss and a composite sheet whose force–strain responses in uniaxial or biaxial tension tests, respectively, are mainly determined by the constituents’ volume fractions.

Appendix: Evaluating the sensitivities of predicted force resultants

In this section, we address the semi-analytical evaluation of the sensitivities $\partial F_i(\varepsilon ; {{\mathbf {w}}})/\partial {{\mathbf {w}}}$ for the ith force–strain curve $F_i(\varepsilon ; {{\mathbf {w}}})$ and a particular imposed strain $\varepsilon \in {{\mathcal {E}}}_i$ [5, 19]. Restricting the attention to a displacement-based finite element formulation, we first recall that the Galerkin approximation of the principle of virtual work leads to the following system of nonlinear equations

$$\begin{aligned} {{\mathbf {f}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}}) - {{\mathbf {f}}}^{(ext)}({{\mathbf {d}}}) - \begin{pmatrix} {\mathbf {0}} \\ {\hat{{{\mathbf {a}}}}} \end{pmatrix} = {\mathbf {0}}, \end{aligned}$$

(60)

where ${{\mathbf {f}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}})$ denotes the vector of discrete internal forces, ${{\mathbf {f}}}^{(ext)}({{\mathbf {d}}})$ is the vector of discrete external forces and ${{\mathbf {d}}}$ summarizes all displacement degrees of freedom. In order to account for Dirichlet boundary conditions, ${{\mathbf {d}}}$ is commonly decomposed, after potential reordering, into degrees of freedom ${\hat{{{\mathbf {d}}}}}$ that are prescribed as ${\hat{{{\mathbf {d}}}}}_{\varepsilon }$ on the Dirichlet boundary,

$$\begin{aligned} {\hat{{{\mathbf {d}}}}} = {\hat{{{\mathbf {d}}}}}_{\varepsilon }, \end{aligned}$$

(61)

and into remaining unknown degrees of freedom ${\tilde{{{\mathbf {d}}}}}$,

$$\begin{aligned} {{\mathbf {d}}}= \begin{pmatrix} {\tilde{{{\mathbf {d}}}}} \\ {\hat{{{\mathbf {d}}}}} \end{pmatrix}. \end{aligned}$$

(62)

Here, the subscript $\varepsilon$ indicates that the prescribed values of ${\hat{{{\mathbf {d}}}}}$ are set with reference to the externally imposed strain $\varepsilon$. The order of displacement degrees of freedom in Eq. (62) is also reflected in the ordering of the entries in Eq. (60); for subsequent reference, we specifically record the block-partitioned representation of the discrete internal force vector

$$\begin{aligned} {{\mathbf {f}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}}) = \begin{pmatrix} {{\mathbf {f}}}_{{\tilde{{{\mathbf {d}}}}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}}) \\ {{\mathbf {f}}}_{{\hat{{{\mathbf {d}}}}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}}) \end{pmatrix}. \end{aligned}$$

(63)

The Dirichlet boundary conditions in Eq. (61) are enforced by the discrete reaction forces ${\hat{{{\mathbf {a}}}}}$ in Eq. (60). For given constitutive parameters ${{\mathbf {w}}}$ and prescribed Dirichlet displacements ${\hat{{{\mathbf {d}}}}}_{\varepsilon }$, Eqs. (60) and (61) can be solved for the unknown displacement degrees of freedom ${\tilde{{{\mathbf {d}}}}}$ and the discrete reaction forces ${\hat{{{\mathbf {a}}}}}$ that establish equilibrium. The latter are particularly relevant as they determine the predictions of cumulative forces $F_i(\varepsilon ; {{\mathbf {w}}})$ on the Dirichlet boundary. By consequence, the sensitivities $\partial F_i(\varepsilon ; {{\mathbf {w}}})/\partial {{\mathbf {w}}}$ may be computed in terms of the sensitivities of the discrete reaction forces $\partial {\hat{{{\mathbf {a}}}}}({{\mathbf {w}}})/\partial {{\mathbf {w}}}$.

If we consider $\varepsilon$ and, hence, ${\hat{{{\mathbf {d}}}}}_{\varepsilon }$ as given, then the equilibrium solution ${{\mathbf {d}}}= {{\mathbf {d}}}^{\star }({{\mathbf {w}}})$ is smoothly parameterized by the constitutive parameters ${{\mathbf {w}}}$. Computing the gradient of Eq. (60) at ${{\mathbf {d}}}= {{\mathbf {d}}}^{\star }({{\mathbf {w}}})$ with respect to ${{\mathbf {w}}}$ and taking into account $\partial {\hat{{{\mathbf {d}}}}}^\star ({{\mathbf {w}}})/\partial {{\mathbf {w}}}= {\mathbf {0}}$ by Eq. (61) yields the following two-step scheme for $\partial {\hat{{{\mathbf {a}}}}} ({{\mathbf {w}}})/\partial {{\mathbf {w}}}$,

$$\begin{aligned} \left. {{\mathbf {k}}}_{{\tilde{{{\mathbf {d}}}}}{\tilde{{{\mathbf {d}}}}}}({{\mathbf {d}}}; {{\mathbf {w}}})\right| _{{{\mathbf {d}}}^{\star }({{\mathbf {w}}})} \frac{\partial {\tilde{{{\mathbf {d}}}}}^{\star }({{\mathbf {w}}})}{\partial {{\mathbf {w}}}} = - \left. \frac{\partial {{\mathbf {f}}}_{{\tilde{{{\mathbf {d}}}}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}})}{\partial {{\mathbf {w}}}}\right| _{{{\mathbf {d}}}^{\star }({{\mathbf {w}}})} \end{aligned}$$

(64)

and

$$\begin{aligned} \frac{\partial {\hat{{{\mathbf {a}}}}}({{\mathbf {w}}})}{\partial {{\mathbf {w}}}} = \left. \frac{\partial {{\mathbf {f}}}_{{\hat{{{\mathbf {d}}}}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}})}{\partial {{\mathbf {w}}}}\right| _{{{\mathbf {d}}}^{\star }({{\mathbf {w}}})} + \left. {{\mathbf {k}}}_{{\hat{{{\mathbf {d}}}}}{\tilde{{{\mathbf {d}}}}}}({{\mathbf {d}}})\right| _{{{\mathbf {d}}}^{\star }({{\mathbf {w}}})} \frac{\partial {\tilde{{{\mathbf {d}}}}}^{\star }({{\mathbf {w}}})}{\partial {{\mathbf {w}}}}, \end{aligned}$$

(65)

where ${{\mathbf {k}}}({{\mathbf {d}}}; {{\mathbf {w}}})$ represents the tangent stiffness matrix

$$\begin{aligned} \begin{aligned} {{\mathbf {k}}}({{\mathbf {d}}}; {{\mathbf {w}}})&= \begin{pmatrix} {{\mathbf {k}}}_{{\tilde{{{\mathbf {d}}}}}{\tilde{{{\mathbf {d}}}}}}({{\mathbf {d}}}; {{\mathbf {w}}}) &{}{{\mathbf {k}}}_{{\tilde{{{\mathbf {d}}}}}{\hat{{{\mathbf {d}}}}}}({{\mathbf {d}}}; {{\mathbf {w}}}) \\ {{\mathbf {k}}}_{{\hat{{{\mathbf {d}}}}}{\tilde{{{\mathbf {d}}}}}}({{\mathbf {d}}}; {{\mathbf {w}}}) &{}{{\mathbf {k}}}_{{\hat{{{\mathbf {d}}}}}{\hat{{{\mathbf {d}}}}}}({{\mathbf {d}}}; {{\mathbf {w}}}) \end{pmatrix} \\&= \frac{\partial }{\partial {{\mathbf {d}}}} \left( {{\mathbf {f}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}}) - {{\mathbf {f}}}^{(ext)}({{\mathbf {d}}})\right) . \end{aligned} \end{aligned}$$

(66)

Equations (64) and (65) were previously developed by Schmidt et al. [19] and show that, at an equilibrium solution ${{\mathbf {d}}}^\star ({{\mathbf {w}}})$, the evaluation of the sensitivities $\partial {\hat{{{\mathbf {a}}}}} ({{\mathbf {w}}})/\partial {{\mathbf {w}}}$ requires the solution of n additional linear systems. Furthermore, the sensitivities of the discrete internal force vector $\partial {{\mathbf {f}}}^{(int)}({{\mathbf {d}}}; {{\mathbf {w}}})/\partial {{\mathbf {w}}}$ appear here; we compute these analytically.