Bayesian identification of Mean-Field Homogenization model parameters and uncertain matrix behavior in non-aligned short fiber composites

Abstract

We present a stochastic approach combining Bayesian Inference (BI) with homogenization theories in order to identify, on the one hand, the parameters inherent to the model assumptions and, on the other hand, the composite material constituents behaviors, including their variability. In particular, we characterize the model parameters of a Mean-Field Homogenization (MFH) model and the elastic matrix behavior, including the inherent dispersion in its Young's modulus, of non-aligned Short Fibers Reinforced Polymer (SFRP) composites. The inference is achieved by considering as observations experimental tests conducted at the SFRP composite coupons level. The inferred model and material law parameters can in turn be used in Mean-Field Homogenization (MFH)-based multi-scale simulations and can predict the confidence range of the composite material responses.

Bayesian inference of non-linear multiscale model parameters accelerated by a Deep Neural Network

“…More details on the construction of π P (p(θ, φ)) from a can be found in [8,16]. Since a constant fiber aspect ratio Ar is assumed, the value of v ω (k) , in Eq.…”

“…In the considered short-fiber reinforced material, the fiber aspect ratio is not a uniform value. It has however been shown in [16] that the distribution of aspect ratio along the injection direction (θ = π 2 , φ = 0) is representative of the in-plane directions, while the content of fibers along out-of-plane direction is neglectable. Therefore, the approximation in Eq.…”

“…For each simulation, the strain ε M increases from zero to a given value and the material parameters are taken randomly from the given ranges listed in Table 1. The bounds of E 0 and A r are chosen with respect to the identification process conducted [16] for the same material system, but limited to the linear range, in which the inferred values are within these ranges. The bounds of the other parameters correspond to extreme values by lack of prior knowledge.…”

“…Compared to the hundreds of thousands of simulations required for a MCMC random walk, the total computation time is reduced drastically. been estimated from the mass fraction measured using the pyrolysis technique, see details in [16]. The second-order orientation tensors a thickness and the volume fraction v thickness I when considering the full thickness are reported in Table 2.…”

“…The inference of multi-scale model parameters is more difficult because of its inherent computational cost making the evaluation of the likelihood time consuming when considering a random sampling process. Although in [16] the authors have inferred the two-step MFH parameters of SFRP, this was limited to the linear range because of the computational cost bottleneck: when considering a two-step homogenization as a non-linear multiscale model, one simple analysis needs around one minute of computation but hundreds of thousands of simulations are required for the sampling process. In order to make BI computationally affordable in the non-linear range, in this work, the two-step MFH model is substituted by a surrogate model during the random sampling process.…”