Pure bending of an elastic prismatic beam made of a material with density-dependent material parameters

Průša, Vít; Rajagopal, K. R.; Wineman, Alan S.

doi:10.1177/10812865221081519

Cited by 12 publications

(1 citation statement)

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“…The current work can be extended to study 3D quasi-static or dynamic crack evolution under thermal or mechanical loading using a diffuse-interface (famously known as the phase field) type crack regularization [49]. A 3D investigation to resolve singularities in porous elastic solid, of which material moduli depend upon the density, is also an interesting future study [64][65][66][67].…”

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confidence: 99%

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

Gou,

Mallikarjunaiah

2023

Mathematics and Mechanics of Solids

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We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

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confidence: 99%