In this article, the authors presented a unified solution for the dynamic analysis of laminated composite annular, circular, and sector plate with general boundary conditions. The first-order shear deformation theory is employed to formulate the theoretical model. Regardless of the shapes of the plates and the types of boundary conditions, each displacement and rotation component of the elements is expanded as an improved Fourier series expansion which is composed of a double Fourier cosine series and several auxiliary functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries and to accelerate the convergence of series representations. Since the displacement fields are constructed adequately smooth throughout the entire solution domain, an exact solution is obtained based on the Rayleigh–Ritz procedure by the energy functions of the plates. The accuracy, reliability, and versatility of the current solution is fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions. Some new results of free vibration analysis for composite laminated annular sector plate, circular sector plate, annular plate, and circular plate are presented, which may be served as benchmark solution for future computational methods. The effects of the sector angles, layer numbers, and boundary spring stiffness on vibration characteristics of the plates are reported. In addition, the force vibration analysis of the plates is also studied. The influence of the boundary spring stiffness, layer number, orthotropic stiffness ration, and fiber orientation angle on dynamic characteristics of the plates is investigated.
We used a physically motivated internal state variable plasticity/damage model containing a mathematical length scale to idealize the material response in finite element simulations of a large-scale boundary value problem. The problem consists of a moving striker colliding against a stationary hazmat tank car. The motivations are (1) to reproduce with high fidelity finite deformation and temperature histories, damage, and high rate phenomena that may arise during the impact accident and (2) to address the material postbifurcation regime pathological mesh size issues. We introduce the mathematical length scale in the model by adopting a nonlocal evolution equation for the damage, as suggested by Pijaudier-Cabot and Bazant in the context of concrete. We implement this evolution equation into existing finite element subroutines of the plasticity/failure model. The results of the simulations, carried out with the aid of Abaqus/Explicit finite element code, show that the material model, accounting for temperature histories and nonlocal damage effects, satisfactorily predicts the damage progression during the tank car impact accident and significantly reduces the pathological mesh size effects.
The Bammann, Chiesa, and Johnson (BCJ) material model predicts unlimited localization of strain and damage, resulting in a zero dissipation energy at failure. This difficulty resolves when the BCJ model is modified to incorporate a nonlocal evolution equation for the damage, as proposed by Pijaudier-Cabot and Bazant (1987, "Nonlocal Damage Theory," ASCE J. Eng. Mech., 113, pp. 1512-1533. In this work, we theoretically assess the ability of such a modified BCJ model to prevent unlimited localization of strain and damage. To that end, we investigate two localization problems in nonlocal BCJ metals: appearance of a spatial discontinuity of the velocity gradient in any finite, inhomogeneous body, and localization of the dissipation energy into finite bands. We show that in spite of the softening arising from the damage, no spatial discontinuity occurs in the velocity gradient. Also, we find that the dissipation energy is continuously distributed in nonlocal BCJ metals and therefore cannot localize into zones of vanishing volume. As a result, the appearance of any vanishing width adiabatic shear band is impossible in a nonlocal BCJ metal. Finally, we study the finite element (FE) solution of shear banding in a rectangular plate, deformed in plane strain tension and containing an imperfection, thereby illustrating the effects of imperfections and finite size on the localization of strain and damage.
It has been long time established that application of damage delocalization method to softening constitutive models yields numerical results that are independent of the size of the finite element. However, the prediction of real-world large and small scale problems using the delocalization method remains in its infancy. One of the drawbacks encountered is that the predicted load versus displacement curve suddenly drops, as a result of excessive smoothing of the damage. The present paper studies this unwanted effect for a delocalized plasticity/damage model for metallic materials. We use some theoretical arguments to explain the failure of the delocalized model considered, following which a simple remedy is proposed to deal with it. Future works involve the numerical implementation of the new version of the delocalized model in order to assess its ability to reproduce real-world problems. c ⃝The addition of characteristic length scales to constitutive models involving softening through damage delocalization method is very well known to remove the pathological mesh size effects in the finite element (FE) solution of problems involving these constitutive models. 1-4 Another closely related technique which consists of incorporating gradient terms in the evolution equation of the parameter(s) governing softening yields the same conclusions, although its numerical implementation into FE codes is not an easy task compared to that of the delocalization technique. A complete review of this technique and its associated numerical implementation can be found in Ref. 5. Despite these successes, nonlocal or gradient models have not yet reached a situation where they are applicable to small or large scale structure problems. For example, Enakoutsa et al. 1 have demonstrated that the use of nonlocal Gurson model 6 does eliminate spurious mesh size effects in FE simulations of ductile rupture of typical pre-cracked Ta specimens, but fails to reproduce the experimental load versus displacement curve, i.e., the predicted load-displacement curve remains quasi-stationary for some time and decreases abruptly. According to these authors, this undesirable feature is due to excessive smoothing of the damage distribution in the ligament ahead of the crack tip of the specimen. They provided a theoretical explanation of this phenomenon based on such as crude assumptions as unboundness of the body considered and homogeneity of the mechanical fields. Namely, they showed that the nonlocal evolution equation for the damage is qualitatively similar to some diffusion equations which result in an excessive smoothing of the damage. Following this theoretical analysis, they proposed a simple remedy to deal with the execesa) Corresponding author. sive smoothing of the damage. It consists of adopting the nonlocal concept for the logarithm of the damage instead of the damage itself; this has the avantage to eliminate the analogy between the nonlocal evolution equation and a diffusion equation. Good agreements between experimental and numerical results ...
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