A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies

It is well known that the linearized theory of elasticity admits the logically inconsistent solution of singular strains when applied to certain naive models of fracture while the theory is a first-order approximation to finite elasticity in the asymptotic limit of infinitesimal displacement gradient. Meanwhile, the strain-limiting models, a special subclass of nonlinear implicit constitutive relations, predict uniformly bounded strain in the whole material body including at the strain-concentrator such as a crack tip or reentrant corner. Such a nonlinear approximation cannot be possible within the standard linearization procedure of either Cauchy or Green elasticity. In this work, we examine a finite-element discretization for several boundary value problems to study the state of stress–strain in the solid body of which response is described by a nonlinear strain-limiting theory of elasticity. The problems of notches, oriented cracks, and an interface crack in anti-plane shear are analyzed. The numerical results indicate that the linearized strain remains below a value that can be fixed a priori, therefore, ensuring the validity of the nonlinear model. In addition, we find high stress values in the neighborhood of the crack tip in every example, thereby suggesting that the crack tip acts as a singular energy sink for a stationary crack. We also calculate the stress intensity factor (SIF) in this study. The computed value of SIF in the nonlinear strain-limiting model is corresponding to that of the classical linear model, and thereby providing a tenet for a possible local criterion for fracture. The framework of strain-limiting theories, within which the linearized strain bears a nonlinear relationship with the stress, can provide a rational basis for developing physically meaningful models to study a crack evolution in elastic solids.

Thermal Post-Buckling Strength Prediction and Improvement of SMA Bonded Carbon Nanotube-Reinforced Shallow Shell Panel: A Nonlinear FE Micromechanical Approach

This article reported first-time the post-buckling temperature load parameter values of nanotube-reinforced polymeric composite panel and their improvement by introducing the functional material (shape memory alloy, SMA) fibre. The temperature load values of nanotube composite and SMA activation are modelled using the single-layer type higher-order kinematic model in association with isoparametric finite element technique. To ensure the effective properties of SMA bonded nanotube composite under the elevated temperature, a hybrid micromechanical material modelling approach is adopted (Mori-Tanaka scheme and rule of mixture). The present structural geometry distortion under elevated temperature is modelled through the nonlinear strain kinematics (Green-Lagrange), whereas the strain reversal achieved with the help of marching technique (inclusion of material nonlinearity). Owing to the importance of geometrical distortion of the polymeric structure, the current model includes all of the nonlinear strain terms to accomplish the exact deformation. Further, to compute the post-buckling responses, the governing nonlinear eigenvalue equations are derived by Hamilton's principle. The numerical solution accuracy is verified with adequate confirmation of model consistency. The material model applicability for different structural configurations including important individual/combined parameter tested through a series of examples. Moreover, the final understanding relevant to the post-buckling characteristics of the polymeric structure and SMA influences is highlighted in details considering the prestrain, recovery stress and their volume fractions.