Isogeometric analysis of shear-deformable, in-plane functionally graded microshells by Mindlin’s strain gradient theory

This paper proposes a general strain-gradient and shear-deformable isogeometric microshell formulation based on the complete Mindlin’s form II strain gradient theory (SGT) and Reissner–Mindlin shell model for the static and dynamic analyses of in-plane functionally graded (IFG) microshell structures. The material properties are assumed to vary along in-plane directions and are effectively homogenized via the rule of mixture. Within the Galerkin weak form, tensor-based governing equations of motion expressed in natural curvilinear coordinates are first formulated and accordingly solved by a non-uniform rational basis spline-based isogeometric analysis (IGA) approach. As its general characteristics, the numerical formulation can not only accurately simulate the size-dependent behaviors of thin to moderately thick IFG microshells with arbitrary shapes and material gradation patterns but also effectively provide the predictions of different reduced SGT-based theories, i.e., the modified strain gradient, modified couple stress, and simplified strain gradient theories. These features are confirmed using selected examples related to static, free vibration, and transient dynamic problems. The presented formulation is expected to serve as a comprehensive and reliable instrument to assist the design of advanced thin-walled components and further interpret different aspects of their underlying theory.