Nonlinear material modeling and design using statistical learning

A method for nonlinear material modeling and design using statistical learning is proposed to assist in the mechanical analysis of structural materials. Although the extension to other materials is straightforward, the scope of this paper is limited to materials with an underlined periodic microstructure. Conventional computational homogenization schemes are proven to underperform in analyzing the complex nonlinear behavior of such microstructures with finite deformations. Also, the higher computational cost of the existing homogenization schemes inspires the inception of a data-driven multiscale computational homogenization scheme. In this paper, a statistical nonlinear homogenization scheme is discussed to mitigate these issues using the Gaussian Process Regression (GPR) statistical learning technique. In the microscale, characteristic Representative Volume Elements (RVEs) are modeled, and the macroscale deformation is homogenized using periodic boundary conditions. Next, a data-driven model is trained for different strain states of an RVE using GPR. In the macroscale, the nonlinear response of the macroscopic structure is analyzed, for which the stresses and material response are predicted by the trained GPR model. This paper produces analytically tractable expressions for all the steps taken in relation to GPR learning, proofs of accuracy in energy, stress, and stress-tangent predictions