Most commercial finite-element programs use the Jaumann (or co-rotational) rate of Cauchy stress in their incremental (Riks) updated Lagrangian loading procedure. This rate was shown long ago not to be work-conjugate with the Hencky (logarithmic) finite strain tensor used in these programs, nor with any other finite strain tensor. The lack of work-conjugacy has been either overlooked or believed to cause only negligible errors. Presented are examples of indentation of a naval-type sandwich plate with a polymeric foam core, in which the error can reach 28.8 per cent in the load and 15.3 per cent in the work of load (relative to uncorrected results). Generally, similar errors must be expected for all highly compressible materials, such as metallic and ceramic foams, honeycomb, loess, silt, organic soils, pumice, tuff, osteoporotic bone, light wood, carton and various biological tissues. It is shown that a previously derived equation relating the tangential moduli tensors associated with the Jaumann rates of Cauchy and Kirchhoff stresses can be used in the user's material subroutine of a black-box commercial program to cancel the error due to the lack of work-conjugacy and make the program perform exactly as if the Jaumann rate of Kirchhoff stress, which is work-conjugate, were used.
The finite-volume direct averaging micromechanics (FVDAM) theory for periodic heterogeneous materials is extended by incorporating parametric mapping into the theory’s analytical framework. The parametric mapping enables modeling of heterogeneous microstructures using quadrilateral subvolume discretization, in contrast with the standard version based on rectangular subdomains. Thus arbitrarily shaped inclusions or porosities can be efficiently rendered without the artificially induced stress concentrations at fiber/matrix interfaces caused by staircase approximations of curved boundaries. Relatively coarse unit cell discretizations yield effective moduli with comparable accuracy of the finite-element method. The local stress fields require greater, but not exceedingly fine, unit cell refinement to generate results comparable with exact elasticity solutions. The FVDAM theory’s parametric formulation produces a paradigm shift in the continuing evolution of this approach, enabling high-resolution simulation of local fields with much greater efficiency and confidence than the standard theory.
Abstract:The typical cause of flexural failure of prestressed beams is compression crushing of concrete, which is a progressive softening damage. Therefore, according to the amply validated theory of deterministic (or energetic) size effect in quasi-brittle materials, a size effect must be expected. A commercial finite-element code, A TEN A, with embedded constitutive equations for softening damage and a localization limiter in the form of the crack band model, is calibrated by the existing data on the load-deflection curves and failure modes of prestressed beams of one size. Then this code is applied to beams scaled up and down by factors of 4 and I /2. It is found that the size effect indeed takes place. Within the size range of beam depths of approximately 152-I ,220 mm, the size effect represents a nominal strength reduction of about 30% to 35%. In the interest of design economy and efficiency, a size effect correction factor could be introduced easily into the current code design equation. However, this is not really necessary for safety since the safety margin required by the code is exceeded for the normal practical size range if the hidden safety margins are taken into account. The mildness of the size effect in the normal size range is explained by the fact that the compression softening zone occupies a large portion of the beam and that, at peak load, the normal stress profiles across the softening zone exhibit only a minor stress reduction below the strength limit. Fitting the type 2 size effect Jaw to the data can provide a simple extrapolation to much deeper beams, for which a stronger size effect is expected. But the extrapolation has some degree of uncertainty because of higher scatter of the test data used for calibrations.
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.
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