In the field of computational mechanics, there has been a very challenging problem, which is the characterization of heterogeneous media with randomly distributed material properties. In reality, no material is homogeneous and deterministic in nature and it has been well-known that randomness in microstructures and properties of materials could significantly influence scatter of structural response at larger scales. Therefore, stochastic characterization of heterogeneous materials has increasingly received attention in various engineering and science fields. In order to deal with this challenging problem, two major challenges need to be addressed: 1) developing an efficient modeling technique to discretize the material uncertainty in the stochastic domain and 2) developing a robust and general inverse identification computational framework that can estimate parameters related to material uncertainties.
In this dissertation, two major challenges have been addressed by proposing a robust inverse analysis framework that can estimate parameters of material constitutive models based on a set of limited global boundary measurements and combining the framework with a general stochastic finite element analysis tool. Finally a new stochastic inverse analysis framework has been proposed, which has a novel capability of modeling effects of spatial variability of both linear and nonlinear material properties on macroscopic material and structural response. By inversely identifying statistical parameters (e.g. spatial mean, spatial variance, spatial correlation length, and random variables) related to spatial randomness of material properties, it allows for generating statistically equivalent realizations of random distributions of linear and nonlinear material properties and their applications to the development of probabilistic structural models.
First, a robust inverse identification framework, called the Self-Optimizing Inverse Method (Self-OPTIM), has been developed. Unlike other signal matching approaches used in model updating, in the course of two parallel finite element simulations, Self-OPTIM automatically minimizes an implicit objective function defined as a function of internal full-field stresses and strains. The performance of Self-OPTIM has been proven by both numerical and experimental verifications. Second, in the stochastic finite element method (SFEM), the spatially varying random material properties, such as linear elastic modulus, Poisson’s ratio and initial yield strength, are discretized by the Karhunen-Loève expansion. This computational tool allows for Monte Carlo simulations considering material uncertainties and their propagations through forward simulations. By combining SFEM with Self-OPTIM, a novel algorithm named as stochastic Self-OPTIM for stochastic material characterization of heterogeneous media with randomly distributed material properties is invented. Its performance has been verified by applying to reconstruction problems and stochastic inverse identifications of statistical parameters of random fields related to material properties. To demonstrate promising applications of the stochastic Self-OPTIM, a probabilistic low-cycle fatigue life prediction model has been developed with considerations of spatial variability of Young’s modulus, Poisson ratio and the initial yield strength. For this purpose, a probabilistic surrogate model has also been developed by using the polynomial chaos expansion.