Numerical identification of constitutive parameters in reduced-order bi-dimensional models for pantographic structures: application to out-of-plane buckling

Abstract

Mechanical properties are investigated for a class of microstructured materials with promising applications. Specifically, we consider a composite material with orthogonal, mutually interconnected fibers building a pantographic substructure. In order to predict the behavior of such a system in three-dimensional continuum, a reduced-order model is introduced by means of a bi-dimensional elastic surface accurately describing large deformations. The properties of this reduced-order model are characterized by an elastic energy density that involves second space derivatives of the displacement for capturing the resistance of twisted and bent fibers in plane as well as out of plane. For determining the coefficients in the elastic energy of the reduced-order model, we utilize a numerical inverse analysis and make use of ad hoc computational experiments performed by a direct numerical simulation on the microscale with detailed modeling of the pantographic substructure. This reduced-order model represents a homogenized material on macro-scale with its substructure on microscale. The homogenized model is capable of describing materials response at a significantly less computational cost than the direct numerical simulations.

A Appendix: Sensitivity of identified parameters

The identification process made possible to find a set of stiffnesses characterizing the mechanical behavior of the pantographic structure. The subsequent step of our investigation consists in estimating the sensitivity of the objective function upon changes in the constitutive parameters.

Fig. 20

In-plane bias extension test, strain energy varying $K_{e}$, $K_{g}$ and $K_{s}$

To this aim, further simulations were carried out using the macroscopic model. The values of identified stiffnesses were changed individually, and the results were compared with those obtained using such unchanged stiffnesses.

The in-plane bias extension test and the three stiffnesses $K_e$, $K_{g}$ and $K_{s}$ that characterize its mechanical behavior were initially considered. As mentioned before, in each simulation mechanical parameters were increased or decreased individually by $\pm \,10\%$ and $\pm \,20\%$. The outcome of such analysis is that there is one mechanical parameter which characterizes most the response under extension, that is, the stiffness $K_{s}$ related to the shear strain. Indeed, it is possible to notice that a change in $K_{s}$ produces the greatest differences on every contribution to the objective function in Eq. (18) (see Figs. 20b, 22b, 24b). In Figs. 21b, 23b and 25b, the relative differences of contributions to the objective function with respect to the unchanged identified stiffnesses are plotted.

In Fig. 22a–c, the angle $\theta$ is plotted varying the parameters $K_e$, $K_{s}$ and $K_{g}$, respectively, in a neighborhood of the identified stiffnesses. In Fig. 23a–c, the relative difference for the angle $\theta$ is plotted varying, respectively, $K_e$, $K_{s}$ and $K_{g}$. From this last figure, it can be observed that the relative difference of $\theta$ is comparable for similar relative changes in the stiffnesses $K_e$ and $K_{s}$, while relative differences are much less when $K_{g}$ changes.

It is remarkable that the angle $\phi$ seems to depend on all the three stiffnesses in a similar way, as they produce comparable effects in each parametric study (see Figs. 24, 25).

Sensitivity analysis varying the constitutive stiffnesses has been carried out for the out-of-plane bias shear test too. As for the previous case, we want to evaluate the changes in the quantities involved in the objective function when $K_{n}$ and $K_{t}$ are varied of a certain amount. Also in this case, we consider relative changes of stiffnesses of $\pm \,10\%$ and $\pm \,20\%$. In Fig. 26, we observe that the magnitude of strain energy depends similarly both on $K_{n}$ and on $K_{t}$. Furthermore, the sensitivity of the model is relevant for these two stiffnesses as shown by plots in Fig. 27. Indeed, relative changes of the stiffnesses, say of $X\%$, produce relative differences in the related quantities appearing in the objective function of about $X/2\%$.

Fig. 21

In-plane bias extension test, relative changes in strain energy varying $K_{e}$, $K_{g}$ and $K_{s}$

Fig. 22

In-plane bias extension test, values of the angle $\theta$ varying $K_{e}$, $K_{g}$ and $K_{s}$

Fig. 23

In-plane bias extension test, relative changes in the angle $\theta$ varying $K_{e}$, $K_{g}$ and $K_{s}$

Fig. 24

In-plane bias extension test, values of the angle $\psi$ varying $K_{e}$, $K_{g}$ and $K_{s}$

Fig. 25

In-plane bias extension test, relative changes in the angle $\phi$ varying $K_{e}$, $K_{g}$ and $K_{s}$

Fig. 26

Out-of-plane bias test, strain energy varying $K_{n}$ and $K_{t}$

Fig. 27

Out-of-plane bias test, relative changes in strain energy varying $K_{n}$ and $K_{t}$