Mixed Tetrahedral Elements for the Analysis of Structures with Material and Geometric Nonlinearities

Abstract

A new mixed tetrahedral element, particularly suited for the analysis of structures exhibiting nonlinear material and geometric behavior, is here presented. Its derivation is based on a Hu-Washizu type formulation, including also rotation and skewsymmetric stress fields as independent variables, instrumental to equip the element with nodal rotational degrees of freedom. A Gauss-point-discontinuous interpolation is selected for the total strain field, in order to account for its possibly highly nonlinear spatial distribution due to inelastic strains. Accordingly, the resulting tetrahedron can properly describe inelastic effects occurring over a space scale smaller than the element size. An original and efficient iterative procedure is proposed to perform the element state determination. Mixed variational formulation and finite element approximationLet B ⊂ R 3 be the domain occupied by a three-dimensional continuum body whose kinematics is described by displacement u, infinitesimal strain ε and rotation ω. Denote by σ [resp. τ ] the symmetric [resp. skew-symmetric] part of the stress tensor field, and suppose the material constituting the body to be endowed with a convex (free or incremental) energy density ψ. As usual, the boundary of the domain is partitioned in the two disjoint sets Γ D and Γ N , where Dirichlet and Neumann conditions are imposed, respectively. Without loss of generality, homogeneous Dirichlet condition on Γ D are assumed. Once introduced the following functional spaces:the equilibrium problem is stated adopting the Hu-Washizu functional F : V × W × E × S × T → R such that (e.g., see [1]):where µ is a typical stiffness modulus, γ is an arbitrary positive non-dimensional parameter, D and L are the differential operators for deriving, respectively, strain and rotation from displacement field, and F ext is the load potential contribution. To proceed with finite element formulation on a given tetrahedral mesh B = e B e , interpolation spaces consistent with those ones listed in (1) have to be introduced. Switching to vector notation and denoting by χ e the characteristic function associated to finite element B e , interpolation spaces for displacements and stresses are selected to be:where a ∈ R n a collects the nodal DOFs, including three Allman's rotations per node, and β e ∈ R n β [resp. α e ∈ R n α ] are the interpolation parameters of symmetric [resp. skew-symmetric] stress at element level. In particular, the displacement shape functions N reported in [2] and the element stress shape functions P e and Q e proposed in [3] are adopted, whereas linear Lagrangian interpolation, given by N ω , is assumed for the rotational field. Distinctive feature of the present formulation is a Gauss-point-discontinuous strain interpolation. Denoting by e e g ∈ R 6 the strain at Gauss point g of the element e, say x e g , the following choice is made (e.g., see [4]):where δ x − x e g is a Dirac delta measure centered at x e g . The substitution of (3) and (4) into the functional (2) yields its finite element a...

An Overview of Mixed Finite Elements for the Analysis of Inelastic Bidimensional Structures

“…That difficulty is even more pronounced if a strain interpolation space of large dimension is involved, as desirable for a more accurate description of the strain field at element level. A possible remedy is offered by the iterative procedure, here referred to as nodal-force-based algorithm, presented in [67] and inspired by the techniques discussed in [97,93,63,84,66]. Relying on a piecewise constant strain interpolation, that procedure allows for independent solution of as many material state update problems as the number of element subdomains, thus reducing the computational cost which stems from coupled nonlinear constitutive equations, and mitigating convergence difficulties of Newton's method.…”