a b s t r a c tIn this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution.Crown
In this article, a robust algorithm for prediction of forming limit diagrams (FLD) has been presented. The presented model is based on the ''Marciniak and Kuczynski'' (M-K) theory. Solution to the system of equations has been obtained by applying the NewtonÕs method. Since the NewtonÕs method usually has nonconverging problem, a particular backtracking algorithm has been developed and applied. In this algorithm, a technique for step length selection in the frame of gradient descent method has been implemented. Also for the convergence criterion the so-called ''Armijo'' condition has been used. For verification of the results, BBC2000 yield function and Swift hardening law for AK steel metal have been used. To obtain the necking angle, the effect of groove orientation on the left-and right-hand sides of FLD has been considered. Finally, the predicted FLD has been compared with the published experimental results.
In this article, a constitutive model in the framework of continuum damage mechanics is proposed to simulate the elastic behavior of concrete in tension and compression states. We assume two parts for Gibbs potential energy function: elastic and damage parts. In order to obtain the elastic-damage constitutive relation with the internal variables, two damage thermodynamic release rates in tension and compression derived from the elastic part of Gibbs potential energy are introduced. Also, two anisotropic damage tensors (tension and compression) are defined which characterize the tensile and compressive behaviors of concrete. Furthermore, two different linear hardening rules for tension and compression states are adopted for characterizing the damage evolution. The spectral decomposition technique is used to resolve the stress tensor into tensile and compressive components. The accuracy and performance of the proposed model are validated by comparing the predicted results with different experimental data, such as monotonic uniaxial tension and compression tests, and monotonic biaxial compression test. As an application, an analytic closed-form solution for a concrete thick-walled cylinder is obtained. It is shown that two damages: tensile damage [Formula: see text] and shear damage [Formula: see text] propagate in the cylinder. These two damages introduce anisotropy in the elastic behavior of the concrete structure. The influence of these two damages is investigated on the stress field in the cylinder. It is found that effect of shear damage [Formula: see text] on radial and tangential stresses as well as the effect of tensile damage [Formula: see text] on radial stress are negligible, while the effect of tensile damage [Formula: see text] on the tangential stress in a concrete thick-walled cylinder is significant.
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