Peridynamics model of viscoelasticity for shells and metasurfaces

Abstract

This paper develops a peridynamics shell viscoelasticity theory with a view to modeling creep deformation in shells and metasurfaces. The idea here is to use Simo’s assumption on the deformation field in the three-dimensional (3D) equation of motion and viscoelastic constitutive equations and integrate the thickness information out. A derivation is presented for the elastic constitutive equations for shell. 3D viscoelastic constitutive equation is dimensionally reduced to three constitutive equations for effective membrane stress resultant, effective stress couple resultant, and effective shear stress resultant. Three evolution equations for internal variables emerge in our shell formulation which are derived from the 3D evolution equation of internal variable. If the number of internal variables is p, the total number of degrees of freedom at a material point on the shell is 5 + 8p, viz., three displacement components, two incremental rotation components, six independent components of two 2 × 2 symmetric matrices for internal variables corresponding to effective membrane stress resultant and effective stress couple resultant, and two components for vector internal variable corresponding to effective shear stress resultant. A staggered solution strategy is adopted for the equations of motion and the evolution equations of the internal variables, and the update formulae for the effective membrane stress resultant, effective stress couple resultant, effective shear stress resultant, and internal variables are derived. Linearization of the shell governing equations is carried out, and the Newton-Raphson method is used at every time step for numerical implementation. Numerical simulations are performed on solid cylindrical shell and shell with hole subjected to various loading and boundary conditions and the results are validated with finite element method solutions obtained using ANSYS®. Creep deformation of metasurfaces is also furnished which attests to the efficacy of our proposal.