Learning nonlinear constitutive models in finite strain electromechanics with Gaussian process predictors

Abstract

This paper introduces a metamodelling technique that employs gradient-enhanced Gaussian process regression (GPR) to emulate diverse internal energy densities based on the deformation gradient tensor F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F}$$\\end{document} and electric displacement field D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{D}_0$$\\end{document}. The approach integrates principal invariants as inputs for the surrogate internal energy density, enforcing physical constraints like material frame indifference and symmetry. This technique enables accurate interpolation of energy and its derivatives, including the first Piola-Kirchhoff stress tensor and material electric field. The method ensures stress and electric field-free conditions at the origin, which is challenging with regression-based methods like neural networks. The paper highlights that using invariants of the dual potential of internal energy density, i.e., the free energy density dependent on the material electric field E0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{E}_0$$\\end{document}, is inappropriate. The saddle point nature of the latter contrasts with the convexity of the internal energy density, creating challenges for GPR or Gradient Enhanced GPR models using invariants of F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F}$$\\end{document} and E0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{E}_0$$\\end{document} (free energy-based GPR), compared to those involving F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{F}$$\\end{document} and D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{D}_0$$\\end{document} (internal energy-based GPR). Numerical examples within a 3D Finite Element framework assess surrogate model accuracy across challenging scenarios, comparing displacement and stress fields with ground-truth analytical models. Cases include extreme twisting and electrically induced wrinkles, demonstrating practical applicability and robustness of the proposed approach.