Non-parametric material state field extraction from full field measurements

Abstract

Data-driven computations propose a completely new paradigm to the computational mechanics research community and to experimentalists. Classically, admissible material states can only be obtained experimentally for homogeneous stress/strain configurations or using a parametric optimization of material laws based on heterogeneous tests. Data-driven algorithms aim at circumventing these limitations. However, data-driven algorithms require a large database of admissible material states, otherwise extrapolation is required and some limitations of the classical constitutive equation based approach remain. In this paper, an inverse data-driven approach based on full field measurements is presented. The main idea is to extract, with no assumption on the constitutive equations, rich (i.e. heterogeneous and multiaxial) material state fields from displacement fields and external load measurements. The capability of the proposed method to extract databases of admissible material states and to evaluate stress fields without parametric constitutive equations is illustrated through three examples dedicated to non-linear elasticity, plasticity and dynamics.

Appendix: Self-balanced stress field computation

The elaboration of the basis is performed using a singular value decomposition of the elementary internal force vector. Using the notations introduced in the paper, any elementary stress field having one component c activated in one single element k gives rise to finite element internal forces

$$\begin{aligned} \mathbf {f}_j^{k,c}= \sum _e w_e {\mathbf {B}_{ej}}^T\cdot \varvec{\sigma }_k^c \quad \forall j,k,c . \end{aligned}$$

(23)

A singular value decomposition of this set of internal force vector is carried out to obtain the combination of elementary stress $\varvec{\sigma }^s$ giving rise to vanishing internal force vector. If $N^n$ denotes the number of nodes in the finite element mesh, the internal force vector has $2N^n$ components in 2D. The number of independent elementary stress being $3N^e$ ($N^e$ being the number of quadrature points), the number of self-balanced stress fields is $3N^e-2N^n$ in 2D. $\mathbf {L}_e^{s}$ is simply computed as the singular vectors of the matrix gathering all $\mathbf {f}_j^{k,c}$, corresponding the $3N^e-2N^n$ smallest singular values.