Equivalent Domain Integral Method Research Articles

Artificial neural networks-based J-integral prediction for cracked bodies under elasto-plastic deformation state –cyclic loading

The study builds upon a prior modeling framework designed for monotonic loading to compute the cyclic J-integral encompassing a crack tip experiencing elastic–plastic deformation due to cyclic loads. The proposed modeling approach extends the authors' earlier time-efficient method for computing the J-integral by combining artificial neural networks (ANNs) with finite element (FE) analyses. The ANN-FE integration allows efficient and accurate prediction of elastic–plastic stresses, strains and displacement fields for a crack body under cyclic loadings from the linear elastic FE solution. In the context of cyclic loading, stress, strain, and displacement fields near the crack tip of stainless steel (SS304) are determined by FE analyses under both elastic and elasto-plastic states for various crack sizes and R-ratios. ANNs models are developed to establish the nonlinear relationship between these two states. Consequently, the proposed approach enables the prediction of the elasto-plastic cyclic J-integral through the equivalent domain integral method, based on the elastic FE solution bypassing the need of a complicated elasto-plastic FE solution under cyclic loadings. The results show that this approach can effectively predict elasto-plastic cyclic J-integral while avoiding the complex computation of elastic–plastic deformation fields around the crack tip during cyclic loadings.

Three-dimensional J-integral evaluation for thermomechanically loaded cracks and temperature-dependent incremental plasticity

In this paper, the J-integral is derived for temperature-dependent elastic–plastic materials described by incremental plasticity. It is implemented using the equivalent domain integral method for assessment of three-dimensional cracks based on results of finite-element calculations. The J-integral considers contributions from inhomogeneous temperature fields and temperature-dependent elastic and plastic material properties as well as from gradients in the plastic strains and the hardening variables. Different energy densities are considered, the Helmholtz free energy and the stress-working density, providing a physical meaning of the J-integral as a fracture criteria for crack growth. Results obtained for a plate with two different crack configurations each loaded by a cool-down thermal shock show domain-independence of the incremental J-integral for different energy densities even for high temperature gradients and significant temperature-dependence of the yield stress and the hardening exponent in the presence of large scale yielding. Hence, the derived J-integral is an appropriate parameter for the assessment of cracks in thermomechanically loaded components.

The energy release rate of crack growth in an irradiation-induced thermo-diffusion-mechanical (I-TDM) coupling system

The material cracking behavior in the reactor is generated under the irradiation effect accompanied by thermal expansion, fission product diffusion, and mechanical load. In this study, the energy release rate for crack growth under irradiation has been deduced synthetically according to the thermodynamically consistent method and numerically implemented by the finite element method (FEM). Variation in the total energy was obtained based on the principle of minimum potential energy in which the dissipative behavior can be characterized by fission energy, irreversible heat flow, and diffusion of fission products. Through calculating the variation in the total energy with respect to crack length, the energy release rate for crack propagation was analytically represented. Additionally, the total energy release rate for deflective cracks was also derived to predict the crack kinking. Furthermore, the numerical implementation of the presented model was performed by FEM and the equivalent domain integral method. Effects of irradiation on the physical fields and the energy release rate near the crack tip were investigated and analyzed in such a complex I-TDM coupling system. This study can be developed to investigate fracture problems, assess structural integrity, and evaluate material strength of irradiated materials.

Artificial neural networks-based [formula omitted]-integral prediction for cracked bodies under elasto-plastic deformation state –monotonic loading

Artificial neural networks (ANNs) integrated with a finite element (FE)-based equivalent domain integral method are developed to compute J-integral at the vicinity of crack tips through a time-efficient approach. Robust ANN models are trained to establish nonlinear relationships between FE predicted elastic and elasto-plastic stress, strain, and displacement fields of stainless steel (SS304). Subsequently, elastic–plastic J-integral can be determined by using only elastic FE analysis solution rather than computationally expensive elasto-plastic FE analysis solution. The results show that well-trained ANN models can efficiently and accurately determine J-integral around the crack tips on the basis of numerical elastic FE analysis solution.

Fracture analysis employing equivalent domain integral method and nodal integration techniques based on reproducing kernel particle method

Fracture analysis employing equivalent domain integral method and nodal integration techniques based on reproducing kernel particle method

Theory and calculation of the mixed-mode fracture for coupled chemo-mechanical fracture mechanics

In this paper, a J-integral formula and numerical calculation method are presented, which confirm the path-independence of a J-integral, considering the mixed-mode fracture in a chemo-mechanical coupled medium. The Jk-integral denotes a two-dimensional vector consisting of the components J1 and J2. Using the Equivalent Domain Integral (EDI) method, the chemo-mechanical coupled J2-integral, with an equivalent domain form that includes both a surface integral and a line integral, is deduced using a poroelastical model. Based on a semi-analytical method, an auxiliary J¯2-integral is derived to develop a finite element calculation program. By choosing two characteristic lengths, near the crack tip, the mixed-mode fracture parameters, KI, KII and Ts (T stress) at the crack tip, can be calculated. The mixed-mode fracture problems, which are related to center-cracked and edge-cracked plates, are numerically studied for certain chemo-mechanical coupled boundary conditions. The accuracy of this method is verified with numerical examples, which shows the path-independence of the J-integral for chemo-mechanical coupling. The stress fields of two different cracked hydrogel plates are analyzed with different boundary conditions under the chemo-mechanical coupled mixed-mode fracture. This work provides new insights into the mixed-mode fracture mechanics of a chemo-mechanical coupled medium.

Investigation of the [formula omitted]-integral and SIFs by block-based Galerkin free element method

In this paper, a block-based Galerkin free element method is presented for the calculation of J-integral and mix-model stress intensity factors. Based on the free element collocation method, the Galerkin weak-form is constructed to achieve more accurate results. By absorbing the advantages of SFEM and FBM, the sub-domain mapping technique is used to simplify the geometry and to avoid the appearance of the deformity element. For fractural parameters, the Equivalent Domain Integral (EDI) method is used to predict the J-integral, and the interaction integral method is used to obtain the stress intensity factors. Some numerical examples are given to demonstrate the accuracy, convergence and stability of the proposed method for the computation of fractural parameters.

On the Computational Derivation of Bond-Based Peridynamic Stress Tensor

The concept of ‘contact stress’, as introduced by Cauchy, is a special case of a nonlocal stress tensor. In this work, the nonlocal stress tensor is derived through implementation of the bond-based formulation of peridynamics that uses an idealised model of interaction between points as bonds. The method is sufficiently general and can be implemented to study stress states in problems containing stress concentration, singularity, or discontinuities. Two case studies are presented, to study stress concentration around a circular hole in a square plate and conventionally singular stress fields in the vicinity of a sharp crack tip. The peridynamic stress tensor is compared with finite element approximations and available analytical solutions. It is shown that peridynamics is capable of capturing both shear and direct stresses and the results obtained correlate well with those obtained using analytical solutions and finite element approximations. A built-in MATLAB code is developed and used to construct a 2D peridynamic grid and subsequently approximate the solution of the peridynamic equation of motion. The stress tensor is then obtained using the tensorial product of bond force projections for bonds that geometrically pass through the point. To evaluate the accuracy of the predicted stresses near a crack tip, the J-integral value is computed using both a direct contour approximation and the equivalent domain integral method. In the formulation of the contour approximation, bond forces are used directly while the proposed peridynamic stress tensor is used for the domain method. The J-integral values computed are compared with those obtained by the commercial finite element package Abaqus 2018. The comparison provides an indication on the accurate prediction of the state of stress near the crack tip.

The concept and numerical evaluation of M-integral based on domain integral method in cracked viscoelastic materials

A new form of M-integral associated with time dependence parameters, is presented herein for viscoelastic materials. Based on the equivalent domain integral method, this time-dependent M-integral is numerically implemented as an effective and accepted fracture mechanical parameter for damage induced by crack growth in viscoelastic materials. Based on the linear viscoelastic model defined through Prony series, the conservation of the time-dependent M-integral for viscoelasticity is verified by applying user defined Python scripts. The results show that the newly proposed time-dependent M-integral can be successfully calculated numerically. Furthermore, numerical examples are given to demonstrate the validity and relevance of the time-dependent M-integral in viscoelastic material. In particular, the variations of the time-dependent M-integral for different loading rates are considered, which show that the crack growth behavior over a period of time in viscoelastic material can be evaluated based on the value of the time-dependent M-integral. In addition, the time-dependent M-integral is calculated to assess the damage degree induced by two collinear cracks.

Comprehensive Comparisons of Two-and Three-Dimensional Numerical Estimation of Stress Intensity Factors and Crack Propagation in Linear Elastic Analysis

This paper presents a comprehensive comparison of two finite element software in two and three dimensions. The first is a two-dimensional crack growth simulation program developed as source code by using Visual Fortran Language whereas the second is ANSYS mechanical program (3D structural analysis for fracture mechanic analysis). The procedure consists of computing stress intensity factors (SIFs), the crack growth path and the stresses and strain distributions via an incremental analysis of the crack extension, considering two and three-dimensional analysis. Comparisons between these two software were performed for different case studies for stress intensity factors as well as crack growth trajectory. This source code includes the mesh generator based on the advancing front method as well as all the pre and post process for the crack growth simulation under linear elastic fracture mechanics theory with a user-friendly interface. The maximum circumferential stress criterion was used for prediction of the crack growth in isotropic materials under mixed-mode loading. Furthermore, the equivalent domain integral method has been used for calculating the stress intensity factors values during crack growth. Verification of the predicted crack path and stress intensity factors values were validated with relevant experimental data and numerical results obtained by other researchers with a good agreement.

Fracture Analysis of Cast Iron Materials with Cracks Based on Elastoplastic Extended Finite Element Method

Based on the ABAQUS platform for finite element analysis, the extended finite element method (XFEM) considering elastoplastic constitutive relationship is developed, by which the displacement discontinuity across the crack surface and the strong nonlinearity near the crack tip can be described more accurately. The strip specimens with unilateral cracks and central cracks under uniaxial tension are simulated using the XFEM and the FEM, respectively. The J-integral across the crack on each increment is calculated using the equivalent domain integral method (EDIM), the interaction integral method and the FEM, respectively. The effects of mesh size and mesh shape near the crack tip, element type and different calculation methods on the accuracy of J-integral are analyzed. The simulation results show that the XFEM is more accurate than the FEM with the same element size and type. The fracture tests of cast iron specimens with unilateral cracks under uniaxial tension are performed, and the J-integral criterion is valid to predict the fracture initiation in numerical simulation. The critical value of J-integral is calculated using the EDIM of the XFEM. The comparisons demonstrate that the simulated elastoplastic load–displacement curves with the XFEM are in good agreement with the experimental results.

Numerical assessment of cyclic J-integral △J for predicting fatigue crack growth rate

The cyclic J-integral is a significant mechanical parameter to describe the elastic-plastic fatigue crack growth rate. A equivalent domain integral method based on finite element method was proposed for evaluating the cyclic J-integral. This numerical method was verified by comparing the results of cyclic J-integral with J-integral evaluated by using the built-in functionality of ABAQUS under monotonic tensile load. Base on this numerical method, the uniqueness of extreme value of cyclic J-integral was studied through a kinked crack and a specific code was also developed to simulate the crack propagation in 304 stainless steel (304SS) compact tension (CT) specimen under mode I loading. The expression of crack growth rate was established based on the cyclic J-integral obtained by the specific code and this expression is found to be in good agreement with the experimental results. The research contents in this paper may contribute to the damage tolerance assessment of components.

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.

Mesh-independent equivalent domain integral method for J-integral evaluation

The equivalent domain integral method is a reliable tool for J-integral computation in two- and three-dimensional elastic and elastic-plastic fracture mechanics problems. A variant of this method that is independent of finite element mesh is presented. Finite element solution of a boundary value problem is performed on a mesh composed of arbitrary elements. Nodal results are approximated by the moving least squares method that does not require knowledge of mesh topology. Domain integrals are evaluated on a background mesh of hexahedral elements. The mesh has the polar structure with the refinement towards the crack front. Elements of the background mesh are generated in the coordinate system associated with the crack front and then transformed to the global system. Domain integration for each background element is performed once during computations. Evaluation of J-integral for multiple domains is achieved by multiplication of an element domain integral with multiple domain weight functions. Performance of the proposed algorithm is demonstrated by the examples of three-dimensional cracks using meshes of both hexahedral and tetrahedral elements.

A coupled FEM–Bernstein approach for computing the J k integrals

The stress intensity factors of mixed mode I–II crack loading are often evaluated numerically via FEM implementation combined with the equivalent domain integral method to derive the J k integrals. Mesh refinement or singular elements are classic techniques for reproducing the sharp stress gradient due to material fracture. In this context, high-order Bernstein shape functions are proposed for a model that couples a radial spectral expansion for the stress field around crack tip with a finite element mesh in the outer regions. Lagrange multipliers are used to enforce the coupling in displacements along the FEM–Bernstein approximations interface, in which non-matching nodal patterns are allowed.

Thermal fracture analysis of orthotropic functionally graded materials using an equivalent domain integral approach

A new computational method based on the equivalent domain integral (EDI) is developed for mode I fracture analysis of orthotropic functionally graded materials (FGMs) subjected to thermal stresses. By using the constitutive relations of plane orthotropic thermoelasticity, generalized definition of the J-integral is converted to an equivalent domain integral to calculate the thermal stress intensity factor. In the formulation of the EDI approach, all the required thermomechanical properties are assumed to have continuous spatial variations through the functionally graded medium. Developed methodology is integrated into a fracture mechanics research finite element code FRAC2D using graded finite elements that possess cubic interpolation. Steady-state and transient temperature distribution profiles in orthotropic FGMs are computed using the finite elements based heat transfer analysis software HEAT2D. EDI method is validated and domain independence is demonstrated by comparing the numerical results obtained using EDI to those calculated by an enriched finite element method and to those available in the literature. Single and periodic edge crack problems in orthotropic FGMs are examined in order to study the influences of principal thermal expansion coefficient and thermal conductivity components, relative crack length and crack periodicity on the thermal stress intensity factors. Numerical results show that among the three principal thermal expansion coefficient components, the in-plane component perpendicular to the crack axis has the most significant influence on the mode I stress intensity factor. Gradation profile of the thermal expansion coefficient parallel to the crack axis is shown to have no effect on the outcome of the fracture analysis.

An Equivalent Domain Integral Method for Fracture Analysis of Functionally Graded Materials Under Thermal Stresses

This paper presents the formulation and finite element implementation of the equivalent domain integral (EDI) for fracture analysis of functionally graded materials (FGMs) under thermal stresses. By carrying out the neccesary modifications resulting from material nonhomogeneity and thermal strains, the generalized J-integral is converted to an equivalent domain integral around the crack tip for both plane stress and plane strain problems of thermoelasticity. The developed procedure is integrated in a fracture analysis code FRAC2D using graded and cubic finite elements in order to calculate the stress intensity factor under mode I steady-state and transient thermal loading conditions. Temperature distribution profiles in FGMs are calculated using the finite elements based heat transfer analysis code HEAT2D. Comparisons of the computed thermal stress intensity factors to the results available in the literature and to those calculated by an enriched finite element method show that developed EDI approach produces highly accurate results and possesses the required domain independence. Detailed parametric analyses are performed in order to examine the influences of material property variation profiles and geometrical parameters on the mode I stress intensity factors. It is shown that variation profiles of the thermomechanical parameters such as Poisson's ratio, thermal expansion coefficient and thermal conductivity significantly influence both the amplitude of the stress intensity factors and the transient crack closure behavior.

Error estimation for the finite element evaluation of [formula omitted] and [formula omitted] in mixed-mode linear elastic fracture mechanics

A discretization error estimator for the finite element evaluation of the strain energy release rates (SERRs) G I , G II is presented for mixed-mode bidimensional problems of the linear elastic fracture mechanics (LEFM). The estimator is related to one of the most efficient energetic methods: the equivalent domain integral method (EDI). A continuum approach of the shape design sensitivity analysis (SDSA) is applied to the fracture mechanics problem in combination with the field decomposition technique to obtain separate estimates of the discretization error for each mode. The error estimator enables an a posteriori improvement of G I , G II for a given finite element mesh. The improvement is achieved by adding the estimated errors to the previously calculated values of G I , G II by means of the discrete analytical stiffness derivative method (DASD). This is verified through numerical examples based on the Westergaard's problem and a finite domain problem.

A Three Dimensional Numerical Investigation of the T* integral along a Curved Crack Front

The T ∗ e integral was calculated numerically along an extending, tunneling crack front in an 8 mm thick, aluminum three-point bend (3PB) specimen, us- ing a numerical model driven by experimentally obtained surface displacements. The model provided input to a contour integration for the T ∗ e integral, via the Equivalent Domain Integral (EDI) method with incremental plastic- ity. Validity of the analysis was ensured by the agreement of the T ∗ e integral obtained on the surface (plane stress) and the plane stress values from previous studies. T ∗ e was observed to decrease from the outer surface of the speci- men to the more constrained mid-plane. This difference became more pronounced as the crack grew. keyword: T ∗ e integral, 3-D fracture, CTOA, stable

Dynamic J Integral and Dynamic Stress Intensity Factor Distributions along Naturally and Dynamically Propagating Three-Dimensional Fracture Fronts.

To the best of the authors' knowledge, any fracture parameter such as the dynamic J integral for a naturally and dynamically propagating crack front has not been evaluated. In our previous experimental study, high-speed photographs of dynamically propagating crack fronts in DCB specimens were firstly recorded. In this paper, first, to overcome difficulties in three-dimensional dynamic fracture simulation, a three-dimensional moving finite element method together with an automatic element control method is developed using a mapping technique. Next, to make it possible to evaluate the dynamic J integral along the dynamically propagating curved crack front, an equivalent domain integral method of the dynamic J integral is developed. Furthermore, to accurately evaluate dynamic stress intensity factors along the curved crack front, the component separation method of the dynamic J integral is also developed. Based on these simulation technologies, the generation-phase simulations are carried out, using the experimentally recorded histories of three-dimensional dynamic fracture events in 20mm thick DCB specimens. The distributions of the dynamic J integral and stress intensity factor along the actual dynamic fracture fronts are firstly elucidated. Based on these results, pertinent mechanism of three-dimensional fracture is also discussed.