Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials

Abstract

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. AbstractThis paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials. At the microscale level, the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used. The geometry of the microstructure is random and characterized by a high number of random parameters. The method is based on a deterministic non-concurrent multiscale approach devoted to micro-macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale Cauchy Green strain states. To overcome the curse of dimensionality, due to the high number of involved random variables, the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion. Several strategies, based on novel algorithms dedicated to high stochastic dimension, are used and adapted for the class of multi-modal random variables which may characterize the potential. Numerical examples, at both small and large scales, allow analyzing the efficiency of the approach through comparisons with classical methods.

A computational stochastic multiscale methodology for MEMS structures involving adhesive contact

“…For the cases of non-negligible spatial correlation, the model can be extended, e.g. by coupling a Karhunen-Loève expansion with the PCE as developed in [31,32,36].…”

“…To reduce the computational cost, we consider here a stochastic model in the multiscale method. Although, such method was never developed for adhesive contact problems, we take advantage of previous works [31,32,33].…”

Analysis of the Nonlinear Stochastic Dynamics of an Elastic Bar With an Attached End Mass

“…The analysis of the stochastic dynamics of a highly nonlinear system, with three degrees of freedom, can be seen in [6]. The readers interested in structures built by heterogeneous hyperelastic materials is encouraged to consult [2]. To see the latest theoretical advances of stochastic modeling in structural dynamics, the reader is encouraged to consult the work of [18].…”

Multiscale modeling of failure in composites under model parameter uncertainty

“…The coordinates y are employed to parameterize the characteristic volume, . The response fields are written in terms of both macroscale and microscale coordinates as: ϕ(x, t) =φ(x, y(x), t) (7) in which ϕ denotes an arbitrary response field. The displacement field is decomposed through a two-scale asymptotic expansion with contributions from micro-and macroscopic scales as:…”

“…More recently, a Bayesian statistical approach was proposed to perform calibration of constituent material properties at the scale of the characteristic volume of a unidirectional ply based on ply-level experimental data [4]. The variability in the composite response can be linked to uncertainty at its microstructure using multiscale computational methods such as the -criterion [16], the nonconcurrent multiscale stochastic method [7], and homogenization-based methods such as stochastic finite elements [20], perturbation-based homogenization [30], and the multiscale spectral stochastic method [38], among others. Computational homogenization provides a rigorous basis for linking the fine and coarse scale response of composites and other heterogeneous materials, and has been successfully employed to compute effective elastic properties and failure in a deterministic setting (see e.g., [9,14,17,19,40]).…”