Path-independent Integrals Research Articles

Meshless variational method applied to fracture mechanics with functionally graded materials

With a variation in crack length, the stress intensity factors are obtained by a path independent contour integral in this work. The Chebyshev polynomials are employed using the finite block method for Functionally Graded Materials (FGMs) with two-dimensional fracture problems. By the mapping technique, a block of quadratic type is transformed from Cartesian coordinate to a normalised coordinate with 8 nodes. The new governing equations in terms of displacements are deduced in the mapping domain. All coefficients of Chebyshev polynomials are determined by considering governing equations, boundary conditions and connecting conditions for two blocks. The accuracy and convergence of the FBM with Chebyshev polynomials is demonstrated with four examples and comparison has been implemented with analytical solutions and different numerical approaches.

Energy release rate for cracks in hydrogels undergoing finite deformations

The rupture of hydrogels and swellable elastomers involves large deformations, and there exists a large literature devoted to their experimental characterization including methods for measuring and enhancing fracture toughness. Analytical investigations of the fracture of hydrogels have recognized the importance of large deformations and the contributions of liquid flow, but they have largely been restricted to plane-strain formulations that ignore through-thickness effects. In this paper, boundary–initial value problems for cracked specimens are solved in plane-strain and three dimensions for both permeable and impermeable boundary conditions and various rates of loading. The transient stress, strain and chemical potential fields near the crack tip/front are found to be notably different than the asymptotic solutions of linear poroelasticity and large deformation plane-strain formulations. The energy release rate is computed using a poroelastic path-independent integral ( J ∗ ), and generally that also is a function of the liquid flow. It is shown for moderately thin three-dimensional specimens that liquid flow is largely in the out-of-plane direction under permeable boundaries and largely in-plane for impermeable boundaries; thus, liquid flow makes larger contributions to the energy release in the latter. In agreement with experiments, the energy release rate tends to be larger at higher loading rates due to the contributions of liquid flow. Finally, criteria for crack growth based on the critical stretch ahead of the crack are adopted to predict the critical energy release rate as a function of solid volume fraction, and the possibility of a non-monotone dependence of energy release on solid volume fraction is uncovered. The methods presented in this paper can be utilized to analyze a wide variety of problems in the rupture of hydrogels including applications to soft tissues and fibrous gels.

Analysis of closed branched and intersecting cracks by the boundary element method

The boundary element method (BEM) and a computer code for the analysis of plates with closed branched and intersecting cracks are developed. The BEM enables simple and accurate modelling of cracked plates by using boundary elements. Contact tractions between crack surfaces are computed using an iterative procedure. Stress intensity factors (SIFs) are determined using the path-independent integral. Three numerical examples are studied: a star-shaped crack in a square plate, multiple interacting cracks in an infinite plate and randomly distributed and intersecting cracks in a square plate. The examples demonstrate the simplicity of numerical modelling, the accuracy of the method and the possible applications. The influences of load directions, distances between cracks and the contact of the crack surfaces on SIF are investigated. For the plate with randomly distributed cracks, the effective elastic properties are additionally computed by considering or neglecting contact of crack surfaces. The results show that the importance of the contact procedure depends on how the cracked material is loaded.

Singular enrichment for multi-material corners with application to assessing the risk of fracture in semiconductor devices

A computational method for multi-material corners that enables accurate estimations of generalized stress intensity factors with relatively coarse and uniform discretizations is described in this paper. Specifically, explicitly modeled geometries of material junctions, crack tips and debonded interfaces are isoparametrically and hierarchically enriched to construct approximations with the known local behavior. The analysis procedure is termed enriched isogeometric analysis since the enrichments are constructed directly (explicitly) on the parametric spline geometry of the vertex or crack face unlike in procedures where the influence of the enriching entity is implicitly inferred on the underlying mesh. The developed method allows direct extraction of generalized stress intensity factors without needing a posteriori evaluation of path independent integrals for decisions on crack propagation. The numerical implementation is first validated through analysis of a bi-material corner and growth of an inclined crack in a homogeneous solid. Next, a detailed analysis of a representative back end of line structure of a semiconductor chip with multiple singular corners is carried out to identify critical corners susceptible to crack initiation. The developed procedure demonstrates rapid convergence to the solution stress intensity factors with relatively fewer degrees of freedom, even with uniformly refined coarse discretizations.

On the interaction of a solute atmosphere with a circular void in two-dimensional elasticity: A statistical mechanics approach

Solid solutions are of critical importance to many technological applications. While continuum frameworks have been developed and invoked in the past for studying behavior of solid solutions, recent studies point to the importance of image stresses in finite systems, and in particular, suggest that the homogenized self-stress field of the solutes, which enter the chemical potential in the prevalent continuum models, should be excluded in a correct treatment. The present work is concerned with the study of a solute atmosphere in the vicinity of a void in two-dimensional elasticity wherein the solute atmosphere is treated as a distribution of solute rods extended indefinitely normal to the plane of analysis. A statistical mechanics framework is adopted based on the Gibbs canonical ensemble, and the well-established path-independent integrals of solid mechanics are used for examining configurational forces on the void induced by the presence of a solute atmosphere. Numerical Monte Carlo simulations fully accounting for the interaction energy between the solute rods are used for calculation of the ensemble averages of the energy release rates associated with void translation and expansion, and semi-analytical solutions are further developed in the limit of dilute solute concentrations and large temperatures. Effect of interaction between the stress fields of a solute atmosphere and an external loading on the configurational forces is also investigated through both numerical and analytical calculations. It is shown that the interaction between the image stress field of a given concentration of solute rods and the stress field of an equi-biaxial loading results in an added contribution to the energy release rate of void expansion which does not fade away even when the solid dimensions grow indefinitely. Numerical simulations are further used to examine the effect of loading-induced solute redistribution on the energetic driving force for void expansion.

Asymptotic path-independent integrals for the evaluation of crack-tip parameters in a neo-Hookean material

In this paper, we develop new asymptotic path-independent integrals for the evaluation of the crack tip parameters in a 2D neo-Hookean material. The new integrals are of both J-integral and interaction energy integral type and rely on the separation of the asymptotic boundary value problem into independent problems for each of the deformed coordinates. Both the plane stress and plane strain cases are considered. The integrals developed are used to compute the amplitude parameters of the asymptotic crack tip fields, which allows for direct extraction of these parameters from numerical results. A long strip with an edge crack under mixed loading modes is considered for both homogeneous and biomaterial cases. It is found that the asymptotic J-integrals produce good results for the first-order parameters while the interactions integrals produce good results for both the first and second-order parameters.

Experimental determination of generalized stress intensity factors from full-field measurements

Unidirectional tensile tests of PMMA square hole specimens result in mixed mode crack initiation at the square hole corners, highlighted by inclined cracks with respect to the V-notch bisector. Generalized stress intensity factors (GSIF) of the opening and shear modes at a V-notch are derived experimentally in a direct manner from digital image correlation (DIC) displacement and strain fields by means of a path independent integral. With increasing square hole side, the opening mode GSIF increases whereas the shear mode one decreases so that the mode mixity globally decreases, together with the crack deflection with respect to the V-notch bisector. GSIFs predictions obtained by means of finite element calculations are in good agreement with those determined experimentally.

Estimation of stress intensity factors from crack flank displacements

In the present numerical investigation, the efficacy of a recently proposed displacement based method for estimation of mixed mode (I/II) stress intensity factors of 2D configurations are demonstrated using additional benchmark examples. The proposed displacement based technique is simple, efficient and robust and can be used for the fatigue crack growth simulations. This technique uses the crack opening and crack sliding displacements obtained from the finite element analysis to determine the mode I, mode II and mixed mode (I/II) stress intensity factors (SIFs). This technique can provide the sign of the mode II stress intensity factors which enables it to be used for the fatigue crack growth simulations involving numerous computational steps. Quarter point elements (QPEs) are employed around the crack tip. Convergence of the present method is studied and the performance is assessed by comparing the extracted SIFs using other methods such as displacement extrapolation, J-integral and interaction integral techniques. The present study indicates that very accurate values of SIFs can be computed using the proposed technique and the accuracy of the SIF solutions are of the order of the path independent integrals

Computation of generalized stress intensity factors of 3D singularities

The 2D stress intensity factor at the tip of a crack plays an essential role in the models of brittle fracture to predict the growth of a pre-existing crack. The coupled criterion, based on a twofold condition in energy and stress, extends the results to the initiation of a new crack at any 2D singular points. The generalized stress intensity factor, sometimes baptized notch stress intensity factor, is the relevant parameter for failure predictions. This criterion has been recently extended to the 3D case. In order to be able to implement it, it is necessary to have a means of calculating the 3D stress intensity factors. To this aim, a method based on a path independent integral is developed. It is an extension of the 2D approach relying on the so-called dual displacement field, also called complementary field. An algorithm is proposed to calculate the 3D singular field as well as its dual counterpart and the post-processing procedure to implement the path independent integral in a FE code is described. Examples of 3D corner singularities are chosen to check the independence of the integral with respect to the chosen paths. The ability of the method is tested through an analysis of the way the generalized stress intensity factors vary along the edge of a slanted V-notch in a 3-point bending specimen as a function of the orientation of the wedge. Finally, the method is applied to a bi-material corner and compared to experimental results from the literature.

Modeling of bonded elastic structures by a variational method: Theoretical analysis and numerical simulation

The paper deals with an equilibrium problem of two bodies adhesively bonded to each other along the part of interface between them. There is a crack on the rest part of the interface. The bonding between the bodies is described by “spring type” condition modeling a soft and thin material layer. We also impose non-penetration conditions and Tresca’s friction conditions on the interface including both the adhesive layer and the crack. The non-penetration condition excludes mutual penetration of bodies. A formula for the derivative of the energy functional with respect to the crack length is obtained. It is shown that the derivative can be represented as a path-independent integral (J-integral). Moreover, a non-overlapping domain decomposition method for the bonded structure is proposed and its convergence is studied theoretically and numerically. The numerical study shows the efficiency of the proposed method and the importance of the non-penetration condition.

Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application

• Noether symmetries are discussed by standard techniques not for the Lagrangian density of an elastic body. • The deep-level symmetries of an elastic field from arbitrary functions included in general solution. • Conserved quantities with arbitrary integer order spatial derivative or integral. • An invariant finite physical parameter can be obtained for arbitrary integer order singularity field. The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.

3D Path independent integral for thermoelastic and magnetostriction problem

In this work, a universal path independent integral Ju that represents the energy release rate or flux during crack extension in a homogeneous and isotropic material, has been derived for straight crack subjected to multiple loads. This version of integral includes the effect of elastic strain, Eigen strain, thermal strain, body forces and magnetic strain. Particularly focus is made on thermo-mechanical and magnetic strains. The 3D J-integral expression represents the energy release rate in the combined thermoelastic field with magnetostriction characteristics. This integral is applicable to the real field problems concerning medium in an electric field where the multiphysics of heat generation and magnetic field are involved. 3D finite element analysis of edge cracked magnetostrictive specimen has been carried out to demonstrate the efficacy of the integral expressions.

New insight on physical meaning of fracture criteria for growing cracks

Because plastic unloading is inevitable during crack growth in elastic plastic materials, the J integral is not applicable anymore, and several path-independent integrals are thus proposed by different authors to overcome this limitation. In this paper, we start from the crack tip energy flux. We first derive two important formulas and then find that these path-independent integrals are in fact identical to each other and consistent with the crack tip energy flux. That is, the crack tip energy flux is more essential for growing cracks in elastic plastic materials. In addition, a force such as crack tip parameter, which is conjugated to crack tip velocity, is proposed by us on the perspective that crack tip energy flux can be considered as the power dissipated by crack tip due to crack growth. This parameter is proved to be the absolute value of the projection of the crack tip configurational force vector into the direction of the nominal crack growth direction, and thus, this parameter is the thermodynamic crack driving force. Moreover, this parameter can represent crack tip stress and displacement fields of growing cracks, and thus, a fracture criterion based on this parameter is proposed. The critical value of this parameter corresponds to the rate of energy dissipated in the fracture process zone per unit crack extension.

Transient Dynamic Stress Intensity Factor of FGM Plates Using the State Space and MLPG Methods

In this paper, a new effective method named state-space is extended and utilized to analyze FGM dynamic fracture problems, discretized by the meshless local Petrov–Galerkin method. Both the moving least square and the direct method have been applied to estimate the shape function and to impose the essential boundary conditions. The visibility criterion is used to simulate the displacement and stress field around the crack tip. Normalized dynamic stress intensity factors are calculated using the path-independent integral, J*, which is formulated for the non-homogeneous material. Two methods commonly used in solving such discretized problems, namely, the central difference method and the Newmark method, are compared with the proposed method. Both homogeneous and non-homogeneous (FGM) center-cracked plates under uniform tensile impact load are used to examine the accuracy and the computational time of the new method. The results of the implementation of these methods are compared with existent solutions. The results show that the state-space method saves substantial computational time for a given accuracy. The center-cracked plate made of FGM with two different material gradations (along and normal to the crack length) and four different FGM zones under the effect of uniform tensile impact load are considered, and the dynamic stress intensity factors are studied under the influence of various non-homogeneity ratios.

Path-independent integrals in electrochemomechanical systems with flexoelectricity

Recently, a larger flexoelectric-like response in a bending semiconductor was found by Narvaez et al. This fact indicates that the electrochemical processes have a great influence on flexoelectricity. Thus, study on the coupling between the electrochemical process and the flexoelectric effect is crucial. In this paper, the classical Noether theorem is applied to the dissipative electrochemomechanical processes with consideration of the flexoelectric effect and the strain gradient. The necessary and sufficient conditions for obtaining the symmetry transformation are derived. Several conservation integrals, including J- and L-integrals, are obtained. Then, the relation of the J- and L-integrals with the energy release rate is verified. We also show that J-integral here can reduce to the result in previous work. Finally, a Mode III crack problem taking the flexoelectric effect into account is discussed to show the path-independence of the J-integral.

A new analytical generalization of the J and G-theta integrals for planar cracks in a three-dimensional medium

This paper focuses on the theoretical and numerical development of the J-integral concept for three-dimensional problems. A new integral parameter, developed for the real three-dimensional case, computes the combined energy release rate for an arbitrary crack front. A path-independent integral is verified in both the static and crack propagation cases. Generalization of the J3D-integral toward its Gθ3D implementation form is performed. The efficiency of this proposed integral is then compared with a number of J integrals available in the literature. Lastly, the paper offers a finite element implementation along with its numerical validation.

Application of material forces and path independent integrals to the prediction of crack initiation and crack paths

Next to the shape of a growing crack, it is of importance to predict the initiation of new cracks, especially in structures with stress concentrations due to e.g. holes or notches. In the past, different criteria for crack initiation have been discussed. Furthermore, cracks growing in the vicinity of such stress concentrations in general are deflected and therefore complex crack paths arise. The prediction of crack initiation and the shape of crack paths becomes even more complicated if a material exhibits anisotropic constitutive properties or fracture toughnesses.In this paper, a unified approach of material forces for the prediction of the initiation and growth of a crack is presented. The material or configurational forces as crack driving quantity are strongly related to path-independent integrals, e.g. the Jk-integral. A novel methodology is presented for accurately calculating material forces at cracks employing local numerical data. Material tractions at a round U-notch are investigated and related to crack initiation.

A new three-dimensional J-integral formulation for arbitrary load history and finite deformation

In this paper, a new formulation of three-dimensional J-integral for the evaluation of elastic-plastic fracture problem is presented. It is known that the J-integral represents the energy release rate per unit crack extension. The J-integral is a path-independent integral and can be computed on arbitrary integral path or domain. This property requires the assumption of proportional loading when an elastic-plastic material is considered. Because of this assumption, J-integral loses path-independent property under a non-proportional loading condition. We present a new formulation of three-dimensional J-integral representing the energy dissipation inside a small but finite domain in the vicinity of crack front. The dissipated energy includes the energy released by crack extension and the deformation energy that dissipates in the process zone. This formulation is the extension of the three-dimensional J-integral using equivalent domain integral method and derived without any assumptions on the deformation history. Therefore, it is possible to evaluate the J-integral for problems subject to any load histories. Finally, the problems of hyperelastic and large deformation cyclic elastic-plastic analysis using finite element method are presented. They show that the proposed method can be applied to non-proportional loading problem.

Path independent integrals in equilibrium electro-chemo-elasticity

By using the Noether's first theorem, this paper constructed two types of path-independent integrals in equilibrium electro-chemo-elasticity and proved their uniqueness. These path-independent integrals are the electro-chemo-elastic extensions of the classical J- and l-integrals in elasticity. Similar to their elastic counterparts, the electro-chemo-elastic J- and l-integrals represent energy release when a crack or a cavity undergoes a translation and rotation, respectively. Also shown in this paper is that the M-integral in elasticity cannot be extended to electro-chemo-elasticity. Results of this study established a theoretical foundation for energy conservation laws in equilibrium electro-chemo-elasticity. Such conservation laws are useful in modeling various phenomena in electro-chemo-elastic systems. In addition, the path-independent integrals obtained here provide a theoretical tool for understanding and a practical tool for numerical evaluation of singular fields.

Conservation laws and path-independent integrals in mechanical-diffusion-electrochemical reaction coupling system

In this study, the conservation laws οf dissipative mechanical-diffusion-electrochemical reaction system are systematically obtained based on Noether's theorem. According to linear, irreversible thermodynamics, dissipative phenomena can be described by an irreversible force and an irreversible flow. Additionally, the Lagrange function, L and the generalized Hamilton least-action principle are proposed to be used to obtain the conservation integrals. A group of these integrals, including the J-, M-, and L-integrals, can be then obtained using the classical Noether approach for dissipative processes. The relation between the J-integral and the energy release rate is illustrated. The path-independence of the J-integral is then proven. The J-integral, derived based on Noether's theorem, is a line integral, contrary to the propositions of existing published works that describe it both as a line and an area integral. Herein, we prove that the outcomes are identical, and identify the physical meaning of the area integral, a concept that was not explained previously. To show that the J-integral can dominate the distribution of the corresponding field quantities, an example of a partial, stress-diffusion coupling process is disscussed.