Elastic Soft-Core Sandwich Plates: Critical Loads and Energy Errors in Commercial Codes Due to Choice of Objective Stress Rate

Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress rate the Jaumann rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the the Truesdell objective stress rate, which is work-conjugate to the GreenLagrangian finite strain tensor. The commercial codes should switch to this rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell rate and Green-Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell rate. Comparisons of the errors of various approaches are tabulated. 1 Motivation and Nature of Problem This paper is motivated by the design of large foam-core sandwich panels, intended for the cladding of a ribbed hull of light long ships with superior maneuverability and fuel-efficiency. In a laboratory test, one such panel failed at one third of the axial compressive load predicted by a standard commercial finite element program. Although the main cause of this gross underestimation of strength is probably the neglect of size effect due to cohesive delamination fracture, which was previously identified for cylindrical buckling of sandwich plates [7] and is the subject of a separate study, an important additional cause to be studied here appears to lie in two long-ignored flaws [2, 5] in the handling of finite strain by standard commercial codes [1, 15, 16, e.g.]: 1) One flaw of these code is the use of objective stress rates that are not work-conjugate to any finite strain tensor [2]. In the implicit updated Lagrangian analysis, it is the Jaumann rate Visiting Scholar, Northwestern University; Assistant Professor on leave from Czech Technical University in Prague. McCormick Institute Professor and W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, Illinois 60208; z-bazant@northwestern.edu (corresponding author) Graduate Research Assistant, Northwestern University. of Cauchy stress and, in the explicit analysis, the use of the Green-Naghdi rate. 2) Another flaw is that the bifurcation analysis uses the Jaumann rate of Kirchhoff stress. Although this rate is work-conjugate to the Hencky (or logarithmic) strain tensor, it cannot correctly capture the work of initial in-plane stresses in soft-in-shear highly orthotropic structures compressed in the strong direction. This work can be captured correctly only by the Truesdell objective stress rate [3, 4]. The former flaw can lead to major errors in volume changes of polymeric, ceramic and metallic foams, fiber-reinforced foams and other highly compressible porous materials such as loess, silt, tuff, snow, under-consolidated granular materials, light wood, honeycomb, osteoporotic bones and various biological tissues. But it is unimportant for elastic buckling of a sandwich, which is the only case to be studied here. The reason is that even though the foam core is highly compressible, its hydrostatic stress is negligible (except for inelastic buckling with delamination, or for indentation [6]). This study will focus on the latter flaw. Its seriousness has already been demonstrated for sandwich columns [3, 4] and for highly orthotropic columns [12, 13], but not for sandwich plates. The energetically correct form of the differential equations of equilibrium of sandwich plates will also be identified, and their solution will be compared to finite element analysis of two kinds–a two-dimensional analysis with sandwich-type elements having a linear strain profile across the core, and three-dimensional analysis in which the core thickness is subdivided into several elements. In contrast to finite elements for the entire cross section of sandwich column, this subdivision has already been shown to yield correct results regardless of the choice of objective stress rate; see [4] (and for elastomeric bearings see [28]). It will be verified whether this is also true for sandwich plates. A salient characteristic of sandwich plates is that the shear strain in a soft core is important for buckling. The shear buckling is a problem with a hundred-year controversial history. It requires using the stability criteria for a three-dimensional continuum, which were for half a century a subject of polemics. Although the polemics were resolved four decades ago, some authors still dispute various aspects. All the historical controversies can be traced to the arbitrariness in choosing the finite strain measure and to inattention to the work-conjugacy requirement, which means that the (doubly contracted) product of the incremental objective stress tensor with the incremental finite strain tensor must give a correct expression for the second-order work [5, ch.11]. How does the choice of strain measure affect the differential equations of equilibrium and the eigenvalues of compressed sandwich plates? And which choice is correct? These questions will be addressed first analytically, and then in the context of finite element analysis, with and without subdividing the core thickness into several layers of elements. The aim is to appraise the magnitude of errors and choose the best practical approach. 2 Review of Objective Stress Rates and Their Energy-Variational Basis A broad class of equally admissible finite strain measures which comprises virtually all of those ever used is represented by the Doyle-Ericksen tensors (m) = (U − I) /m, where m is a real parameter, I = unit tensor and U = right-stretch tensor [5]. The second-order approximation of these tensors is (m) ij = eij + 1 2 uk,iuk,j − αekiekj, eki = 12 (uk,i + ui,k) , α = 1− 1 2 m (1)