𝒞¹ continuous discretization of gradient elasticity

Abstract

For the modeling of size effects, gradient continua can be applied. In this contribution, strain gradients are used in a nonlinear hyperelastic material model. The resulting partial differential equations are solved numerically in terms of a Galerkin scheme requiring C 1 continuity of the shape functions. The performance of three C 1 continuous finite elements and the C 1 Natural Element method is compared. motivationThe modeling of materials with size effects is an issue of high interest. Size effects can be observed in various types of materials such as concrete, metals and composite materials, an overview on the topic is found in [1]. Within the classical Boltzmann theory these effects can not be captured. To overcome this problem, extended models like gradient elasticity can be used. This theory which was established by Mindlin [4] and Toupin [7] incorporates the strain gradient into the definition of the free energy. The introduction of strain gradients and an additional material parameter, associated with the material microstructural length, allows for the modeling of size effects which was proven by Aifantis [1]. The crux in handling gradient elasticity is that a C 1 continuous approximation of the deformation is required. This increases the complexity of the numerical methods compared to the usual C 0 continuous interpolation. Several alternatives like implicit methods and mixed element formulations have been developed to avoid the requirement of C 1 continuity, e.g. [3,5]. However since these alternatives come along with different numerical problems, it is worth studying C 1 methods and evaluating their numerical behavior. This was e.g. done by Askes et al. [2] who used the Element Free Galerkin method. Zervos et al.[8] applied two C 1 finite elements, the Bell triangle and the Bogner-Fox-Schmitt element to linear gradient elasticity and compared the results to implicit methods. In our contribution, we compare the performance of the Argyris, the Clough-Tocher (HCT) and the isoparametric BognerFox-Schmidt (BFS) element with the C 1 Natural Element Method (C 1 NEM) [6] in nonlinear gradient elasticity. basic equationsIn gradient elasticity the free energy density depends not only on the deformation gradient but additionally on its gradient. By introducing the Piola stresses P := ∂W/∂F and the double stress Q := ∂W/∂G balance of momentum then results asFirst and second order boundary conditions are prescribed as t P = t P , ϕ = ϕ and t Q = t Q , ∇ N X ϕ =φ N using the abbreviations numerical methodsThe Argyris element with 21 degrees of freedom and the HCT macro element with 12 resulting degrees of freedom are chosen to represent the class of subparametric finite elements. The degrees of freedom of these elements are the normal derivatives at the center of the edges as well as the displacements and the derivatives up to the second order at the corner nodes. Furthermore, the isoparametric quadrilateral BFS element is selected whose degrees of freedom are the displacement, the first derivatives and the mix...