A beam theory for a new class of constitutive relation for elastic bodies

Abstract

A theory for two-dimensional (2D) beams is presented for the case of considering new constitutive equations, wherein the linearized (infinitesimal) strain tensor is assumed to be a nonlinear function of the Cauchy stress tensor. An incremental formulation is developed to solve numerically the resultant equations. Three constitutive equations for elastic bodies are considered: a model wherein we have strain-limiting behavior, a nonlinear model for rock, and a bimodular constitutive equation for rock (which can also be considered as a nonlinear model for rock). Two boundary value problems are studied: the deformation of a cantilever beam with a point load, and a three-point beam with a point load applied on the middle. The numerical results are compared with the predictions of the classical theory for beams for the case of linearized elastic solids. The results for the stresses can be notoriously different, when comparing the predictions of the nonlinear constitutive equations with the results obtained using the linearized theory of elasticity.

Approximate determination of shear stresses for a 2D beam made of a non-Green elastic solid

“…It is very important to indicate that the results obtained in this communication depend on the results obtained in Bustamante [19]. In that paper, an important assumption is that a plane surface remains plane when the beam deforms.…”

“…We can give some initial values to such reaction loads, and by demanding that the deflexion w(x) and its derivatives should satisfy the restrictions on the deflexion on such supports, we can iterate and find the actual values for such reaction loads. If one compares the magnitudes of the shear stresses from Figures 4-12 with the normal stresses due to bending, presented in 13,14,20,and 21 in Bustamante [19], which were obtained for the same types of beams (see Figure 2), it is possible to see that the shear stresses are about 100 times smaller than such normal stresses. That factor depends on the parameter ζ defined after equations (10), (19) and (20), which for the numerical examples examined here and in Bustamante [19], is of the order 0.05.…”

“…The bimodular model was proposed to use such experimental data within the context of constitutive equations of the form ε = f(T). For the fully 3D case, we need two Young's moduli and two Poisson's ratios (in tension and compression), plus an extra parameter that is used for the transition between those two regimes of behavior, which is called C. In equation ( 2) for the 1D counterpart of that 3D model (see ( 4)-( 8) in Bustamante and Ortiz [24] for the fully 3D model), we see only the two Young moduli plus the constant C. The main difference between the bimodular model ( 2) and the linearized model ( 3) is that for a 1D extension/compression of a bar the stiffness change with the sign of σ (see, for example, Figure 1(c) in Bustamante [19]), whereas for equation ( 3), the stiffness is constant in terms of the stress. It is necessary to indicate that nonlinear models for rock and similar materials have been proposed in the literature before (see, for example, the literature review in Bustamante and Rajagopal [15], Bustamante and Ortiz [24]).…”

“…In section 2, some basic equations are presented, in particular regarding the constitutive equations to be considered in this paper. In section 3, a summary of the main equations from Bustamante [19] is given. In section 4, a method to calculate the shear stresses is presented, which can be used when the strain is a function of the stresses.…”

“…That is not a problem for ε = f(T). In Exadaktylos et al [27], we see the application of such earlier models based on T = f(ε) for beams (see also the literature review in Bustamante [19]).…”