Framework Of Gradient Elasticity Research Articles

A phase field model for fracture based on the strain gradient elasticity theory with hybrid formulation

In this paper, a novel phase field (PF) model for fracture is developed in the framework of strain gradient elasticity. The strain energy decomposition methods initially proposed for linear elastic fracture problems are extended to the gradient elasticity situation. The PF model is numerically implemented through Abaqus subroutine UEL (user defined element) with nine-node quatrilateral C1-continous element. The finite element implementation is validated by studying the stress field near the static crack tip. When the length-scale parameter of the gradient elasticity is zero, the stress field is consistent with the analytical solution predicted by linear elastic fracture mechanics (LEFM). For non-zero length-scale parameters, the Cauchy stress at the crack tip is found to be non-singular and the numerical results exhibit less mesh sensitivity in the crack tip region compared with the linear elastic case. After that, four different strain energy decomposition methods (Method I-IV) are compared and discussed by studying Mode I fracture behavior under tensile load. It is found that Method I can properly characterize the toughening effect of gradient elasticity. Based on Method I, the influences of the fracture length-scale parameter lc and the strain gradient length-scale parameter l on the characteristics of the smeared crack model and the load–displacement curves are systematically discussed. The specimen size effect of Mode I crack propagation is also investigated. It is found that the length-scale parameter l has a larger influence on the load–displacement curve as the specimen size decreases. Finally, crack propagation behavior in pure shear test, three point bending test and tensile test of a notched plate with hole is investigated. It is demonstrated that the proposed PF model can well predict the curvilinear crack propagation path and properly characterize the effect of gradient elasticity. Thus, the PF model developed in this paper can be applied to study complex fracture behaviors in the framework of gradient elasticity, where the effect of internal structure on the mechanical responses become significant.

Non-local criteria for the borehole problem: Gradient Elasticity versus Finite Fracture Mechanics

Two nonlocal approaches are applied to the borehole geometry, herein simply modelled as a circular hole in an infinite elastic medium, subjected to remote biaxial loading and/or internal pressure. The former approach lies within the framework of Gradient Elasticity (GE). Its characteristic is nonlocal in the elastic material behaviour and local in the failure criterion, hence simply related to the stress concentration factor. The latter approach is the Finite Fracture Mechanics (FFM), a well-consolidated model within the framework of brittle fracture. Its characteristic is local in the elastic material behaviour and non-local in the fracture criterion, since crack onset occurs when two (stress and energy) conditions in front of the stress concentration point are simultaneously met. Although the two approaches have a completely different origin, they present some similarities, both involving a characteristic length. Notably, they lead to almost identical critical load predictions as far as the two internal lengths are properly related. A comparison with experimental data available in the literature is also provided.

Comparison between two nonlocal criteria: A case study on pressurized holes

Two nonlocal approaches are applied to the borehole geometry, i.e. a circular hole in an infinite elastic medium subjected to internal pressure. The former approach lays in the framework of Gradient Elasticity (GE), which results nonlocal in the strict sense, being based on a nonlocal constitutive relationship. Changing the stress field as the geometry (i.e., the radius of the hole) varies, the related stress concentration factor can be thought as the critical failure parameter. The latter approach is the Finite Fracture Mechanics (FFM), well-consolidated in the framework of brittle fracture. Whereas the model belongs to classical linear elasticity, it reveals nonlocal in a loose sense: the failure condition is no more punctual, but achieved when two average requirements on the stress and the energy ahead of the notch tip are simultaneously fulfilled. Τhe two approaches, although different, present some similarities, both involving a characteristic length. It will be shown that the GE and FFM predictions are in excellent agreement when the two lengths are properly defined.

Comparison between the Mori-Tanaka and generalized self-consistent methods in the framework of anti-plane strain inclusion problem in strain gradient elasticity

In the present paper, two different averaging schemes are compared in the framework of the anti-plane inclusion problem of the strain gradient elasticity. Effective longitudinal shear modulus of fiber-reinforced composites was evaluated based on the Mori-Tanaka method (MTM) within the direct approach and based on the generalized self-consistent method (GSCM) within an energy approach. It is known that, in gradient theories, the non-uniform fields are realized inside the inhomogeneity even in the presence of a uniform strain or stress field prescribed at the infinity. Therefore, in MTM, an averaged concentration tensor is utilized. GSCM is used together with Eshelby's formula, which evaluates the total strain energy of the composite representative fragment by a particular surface integral. Thus, in this energy-based approach, one does not need to use averaged field quantities. The two mentioned approaches were compared and validated by finite element modeling in the framework of the simplified strain gradient elasticity theory. It was shown that both methods capture the inclusion size effects, however the quantitative predictions of these methods are significantly different for high volume fraction of small inclusions. It was shown that MTM underestimates the effective modulus compared to numerical simulations, while GSCM provides a sufficiently good solution. Thus, it seems that energy-based averaging schemes are preferable in the framework of gradient elasticity. Nevertheless, the MTM can be useful for the effective properties approximate evaluation of composites with small volume fraction of inclusions.

A new multi‐scale dispersive gradient elasticity model with micro‐inertia: Formulation and ‐finite element implementation

SummaryMotivated by nano‐scale experimental evidence on the dispersion characteristics of materials with a lattice structure, a new multi‐scale gradient elasticity model is developed. In the framework of gradient elasticity, the simultaneous presence of acceleration and strain gradients has been denoted as dynamic consistency. This model represents an extension of an earlier dynamically consistent model with an additional micro‐inertia contribution to improve the dispersion behaviour. The model can therefore be seen as an enhanced dynamic extension of the Aifantis' 1992 strain‐gradient theory for statics obtained by including two acceleration gradients in addition to the strain gradient. Compared with the previous dynamically consistent model, the additional micro‐inertia term is found to improve the prediction of wave dispersion significantly and, more importantly, requires no extra computational cost. The fourth‐order equations are rewritten in two sets of symmetric second‐order equations so that ‐continuity is sufficient in the finite element implementation. Two sets of unknowns are identified as the microstructural and macrostructural displacements, thus highlighting the multi‐scale nature of the present formulation. The associated energy functionals and variationally consistent boundary conditions are presented, after which the finite element equations are derived. Considerable improvements over previous gradient models are observed as confirmed by two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

Cracks in Strain Gradient Elasticity-distributed Dislocation Technique

The mode III fracture analysis of graded cracked plane in the framework of classical and strain gradient elasticity is presented in this work. Solutions to the problem of screw dislocation in plane are available for classical and strain gradient elasticity theories. Different approaches for the formulation of the strain gradient theory, especially considering the boundary conditions, result in singular and nonsingular stress fields at the crack tip. One of the applications of the dislocation is the analysis of cracked medium via the Distributed Dislocation Technique (DDT). The DDT has been applied extensively in the framework of the classical elasticity. In this article, this technique is generalized for the nonsingular strain gradient elasticity formulation available in the literature. For a system of interacting cracks in classical elasticity, DDT results in a system of Cauchy singular integral equations. In the framework of the gradient elasticity, due to the regularization of the classical singularity, a system of nonsingular integral equations is obtained. Plane with one crack is studied and the singular stress distribution in the classical elasticity is compared with the nonsingular stress components in gradient elasticity theories.

A nonlocal formulation based on a novel averaging scheme applicable to nanostructured materials

A novel averaging process for modeling the mechanical behavior of heterogeneous fine grained or nanostructured materials is presented. The representative volume element (RVE) is considered as an assembly of heterogeneous system of grains which are assumed to be small. Heterogeneity may have different origin; they can be isotropic with different mechanical properties, they can be a collection of the same crystal structure with different orientation or they can be a collection of grains with radically different material properties and different crystal structures. The averaging process is explained for an RVE which consists of two isotropic materials with different material properties, for the sake of simplicity. The resulting constitutive equation is of nonlocal character, and it is shown that the equation can be reduced to a gradient type. Opening of Mode III crack in small scale yielding condition within the framework of gradient elasticity is summarized. Further, single-grit scratching experiments on a range of metals and ceramics are considered. Finally, it is indicated that the results of the analysis of a groove created in a scratch test can be used to determine the material parameters in the constitutive equations of gradient dependent elasticity, and the advantages of gradient dependent theories in understanding material removal mechanisms is outlined.