Quasi-static anti-plane shear crack propagation in nonlinear strain-limiting elastic solids using phase-field approach

Yoon, Heesung; Lee, Sanghyun; Mallikarjunaiah, S. M.

doi:10.1007/s10704-020-00501-y

Cited by 12 publications

(11 citation statements)

References 63 publications

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“…As β increases, the SED reduces for the same r , indicating that the strain-limiting model under variation of β takes effect to keep the SED bounded without possibly becoming infinitely large. This result suggests that one can utilize the SED combined with the crack-surface energy for formulating BVPs to study the evolution of crack under mechanical loading [49, 50] or thermal loading [51].…”

Section: Computational Resultsmentioning

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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Section: Computational Resultsmentioning

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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“…These implicit constitutive relations have been explored in many studies to describe the state of stress-strain in the neighborhood of cracktip, [24][25][26][27][28][29][30][31][32][33] for the behavior of thermoelastic bodies, 34,35 viscoelastic materials, [36][37][38][39][40][41][42] and quasi-static crack evolution. 43,44 Utilizing the implicit constitutive relation in Rajagopal 45,46 and Rajagopal and Saccomandi 47 and the density-dependent moduli derived with a simplified parameter of the model, we address the preferential or anisotropic stiffness of a material with in the sense of tension/compaction that is, change in void: dilation and compaction. Another goal is to weigh up the applicability of the density-dependent material moduli model for the quasi-static or dynamic propagation of a network of cracks under mechanical/thermal loading.…”

Section: Introductionmentioning

confidence: 99%

“…Instead of an explicit porosity, that is, one of the state variables based on the mixture theory 3,48,49 that can also be saturated with fluid inside, a pure solid matrix with implicit porosity is considered through the change of volumetric strain. We employ a computational model that is physically and mathematically consistent with the framework developed in Yoon et al 43 and Lee et al 44 a universal approach based on the Newton's method and standard finite element method (FEM). This approach yields a numerically stable and efficient algorithm against the serious nonlinear problem of our interest.…”

Section: Introductionmentioning

confidence: 99%

Finite element solution of crack-tip fields for an elastic porous solid with density-dependent material moduli and preferential stiffness

Yoon,

Mallikarjunaiah,

Bhatta

2024

Advances in Mechanical Engineering

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In this paper, the finite element solutions of crack-tip fields for an elastic porous solid with density-dependent material moduli are presented. Unlike the classical linearized case in which material parameters are globally constant under a small strain regime, the stiffness of the model presented in this paper can depend upon the density with a modeling parameter. The proposed constitutive relationship appears linear in the Cauchy stress and linearized strain independently. From a subclass of the implicit constitutive relation, the governing equation is bestowed via the balance of linear momentum, resulting in a quasi-linear partial differential equation (PDE) system. Using the classical damped Newton’s method, the sequence of linear problems is then obtained, and the linear PDEs are discretized through a bilinear continuous Galerkin-type finite element method. We perform a series of numerical simulations for material bodies with a single edge-crack subject to a variety of loading types (i.e. the pure mode-I, II, and mixed-mode). Numerical solutions demonstrate that the modeling parameter in our proposed model can control preferential mechanical stiffness with its sign and magnitude along with the change of volumetric strain. This study can provide a mathematical and computational foundation to further model the quasi-static and dynamic evolution of cracks, utilizing the density-dependent moduli model and its modeling framework.

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“…In particular, the results in (24-26; 30; 34; 35) predict crack-tip stress singularity similar to the classical LEFM model for static crack problems. This implies that the crack-tip is a singular energy sink hence one can utilize a local fracture criterion to study the evolution of cracks within the nonlinear strain-limiting relation (35)(36)(37). Thus, the objective of this study is to offer a reasonable rationale to use the nonlinear strain-limiting models and describe a reliable thermoelastic response particularly around the crack-tip.…”

Section: Introductionmentioning

confidence: 99%

“…To that end, we take as a starting point constructing the boundary value problem within nonlinear stress-strain relationship, of which one special case reduces to the classical linearized thermoelasticity constitutive model. For future work, this study can be a stepping stone to develop physically meaningful models for the crack evolution under thermo-mechanical loading such as by using the phase-field regularization (36)(37)(38).…”

Section: Introductionmentioning

confidence: 99%

A finite element model for a coupled thermo-mechanical system: nonlinear strain-limiting thermoelastic body

Yoon,

Vasudeva,

Mallikarjunaiah

2021

Preprint

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We investigate a specific finite element model to study the thermoelastic behavior of an elastic body within the context of nonlinear strain-limiting constitutive relation. As a special subclass of implicit relations, the thermoelastic response of our interest is such that stresses can be arbitrarily large, but strains remain small, especially in the neighborhood of crack-tips. Thus, the proposed model can be inherently consistent with the assumption of the small strain theory. In the present communication, we consider a two-dimensional coupled system -linear and quasilinear partial differential equations for temperature and displacements, respectively. Two distinct temperature distributions of the Dirichlet type are considered for boundary condition, and a standard finite element method of continuous Galerkin is employed to obtain the numerical solutions for the field variables. For a domain with an edge-crack, we find that the near-tip strain growth of our model is much slower than the growth of stress, which is the salient feature compared to the inconsistent results of the classical linearized description of the elastic body. Current study can provide a theoretical and computational framework to develop physically meaningful models and examine other coupled multiphysics such as an evolution of complex network of cracks induced by thermal shocks.

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Quasi-static anti-plane shear crack propagation in nonlinear strain-limiting elastic solids using phase-field approach

Cited by 12 publications

References 63 publications

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

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

Finite element solution of crack-tip fields for an elastic porous solid with density-dependent material moduli and preferential stiffness

A finite element model for a coupled thermo-mechanical system: nonlinear strain-limiting thermoelastic body

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