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

Abstract

Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NAS-TRAN, use as the objective stress rate the Jaumann rate of Cauchy (or true) stress, which has two flaws; It does not conserve energy since it 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 eqidvalent 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 Truesdell objective stress rate, which is work-conjugate to the Green-Lagrangian finite strain tensor. The commercial codes should switch to this rate. The classical differential equations for buckling of elastic softcore 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 inplane 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.

How to Realize Volume Conservation During Finite Plastic Deformation

“…Some of the objective stress rates, including the Jaumann rate, are not work-conjugate to a strain tensor (Bažant 1971, Ji et al 2013, causing energy conservation problems. Due to this problem, an improper choice of the objective stress rates may lead to large errors when the material is highly compressible or highly anisotropic (Bažant et al, 2012;Ji et al, 2013;Vorel et al, 2013;Bažant and Vorel, 2014;Vo rel and Bažant, 2014). It is recommended by Bažant that the Truesdell objective stress rate should be used instead of the commonly used Jaumann rate or the Green-Naghdi rate (Bažant and Vorel 2014,).…”

Modeling damage growth using the crack band model; effect of different strain measures

“…(11) is small for most engineering problems.If the continuum strain is finite, the practical application of the constitutive model in commercial FEM software is not so straight forward, and largely depends on the choice of stress and strain measures chosen by the particular software. Some of these issues have been addressed in recent publications [25,26,27]. While, the crack strain can extend to large values when small elements are used.…”

Energy-Conservation Error Due to Use of Green–Naghdi Objective Stress Rate in Commercial Finite-Element Codes and Its Compensation

“…Another problem with energy conservation was shown to be caused by the use of tangential moduli not associated by work, particularly by combining the Jaumann rate of Kirchhoff stress with a constant shear modulus in small-strain shear buckling [6][7][8][9]. Such combination causes serious energy errors for materials with very strong orthotropy; for example for fiber-reinforced polymers, homogenized foam-composite sandwich plates, wood and some biologic tissues, or for constitutive models of isotropic materials in which damage is represented by smearing of parallel cracks.…”

“…Recently, though, it was discovered that significant energy errors arise for rigid foams (polymeric, metallic, and ceramic), honeycomb, certain soils (loess, silt, underconsolidated and organic soils), some rocks (pumice, tuff), light wood, carton osteoporotic bone, and various biologic tissues. These errors afflict the most popular commercial software-ABAQUS (version 6.8), ANSYS (version 12.0), LS-DYNA (included in ANSYS V12.0), and probably others (ATENA, version 4 [3] is one exception, and the open source code ooFEM version 2.1 [4] is another) [5,6].…”