Fracture Mechanics Problems Research Articles

The effect of crack-tip constraint in some problems of fracture mechanics

The effect of in-plane and out-of-plane crack-tip constraint parameters, namely, the non-singular T-stress components and the parameter A, in some problems of elastic and elastic-plastic fracture mechanics is discussed. The three-dimensional crack tip stresses including the non-singular T-stress components are employed to describe the crack tip plastic zone, the failure assessment diagrams and the constraint-dependent fracture toughness for a solid with a finite notch/crack under mode I loading and different angles of surface crack growth under conditions of rolling-sliding. Theoretical and numerical aspects of the two-parameter J-A approach in elastic-plastic fracture mechanics are presented in connection with crack-tip constraint. It is shown that the parameter A can be introduced in a fracture criterion as a constraint parameter.

Formula omitted]-convergence for high order phase field fracture: Continuum and isogeometric formulations

We consider high order phase field functionals introduced in Borden et al. (2014) and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy. We take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in Sobolev spaces and in spaces of tensor product B-splines. In the latter convergence holds when the mesh size vanishes faster than the internal length of the phase-field model. On the theoretical level, this condition is natural since the size of the phase field layer, around the crack, itself scales like the internal length; on the numerical level, it should be satisfied by local h-refinement.Technically, convergence holds in the sense of Γ-convergence, with respect to the strong topology of L1, while the sharp crack energy is defined in GSBD2. The constraint on the phase field to take values in [0,1] is taken into account both in the Sobolev setting and in the iso-geometric setting; in the latter, it requires a special treatment since the projection operator on the space of tensor product B-splines is not Lagrangian (i.e., interpolatory).

A comparison between some fracture modelling approaches in 2D LEFM using finite elements

The finite element method has been widely used to solve different problems in the field of fracture mechanics. In the last two decades, new methods have been developed to improve the accuracy of the solution in 2D linear elastic fracture mechanics problems, such as the extended finite element method (XFEM) or the phantom node method (PNM). The goal of this work is to quantify the differences between some numerical approaches: standard finite element method (FEM), mechanical property degradation, interelemental crack method with multi-point constraints, XFEM and PNM. We explain the different techniques analysed together with their advantages and disadvantages. We compare these numerical techniques to model fracture using problems of reference with known solutions, evaluating their behaviour in terms of convergence with respect to the element size and accuracy of the stress intensity factor (SIF), stresses ahead the crack tip and crack propagation prediction. Some of the new techniques have shown a better accuracy in SIF calculation or stress fields ahead the crack tip and other lead to high errors in local results estimations. However, all methods reviewed here can predict crack propagation for the problems of reference of this work, showing good accuracy in crack orientation prediction.

Numerical determination of stress intensity factors: J-integral and modified virtual crack closure technique

Together with the development of numerical tools for stress-strain analysis, approaches to deal with fracture mechanics problems have been the object of continuous improvement. For linear-elastic fracture mechanics problems, the determination of Stress Intensity Factors (SIF) is available as a post-processing possibility in most commercial finite element software packages such as Ansys and Abaqus, allowing the users to perform straightforward assessments of cracked mechanical parts or structures. However, available techniques for SIF determination in post-processing of these commercial solutions are almost limited to the so-called displacement extrapolation and J-integral techniques. In the present work, the implementation of the J-integral technique is revised considering a new finite element software, labeled as Omicron. Furthermore, a more recent technique, the modified Virtual Crack Closure Technique (mVCCT) is also implemented and evaluated. For results assessment, finite element models were built in Omicron and SIFs were determined considering the J-integral and mVCCT approaches. From this study, it is concluded that the mVCCT implementation is simpler than the J-integral and it allows to determine accurately SIFs. Nevertheless, this technique is not straightly available in the current versions of the commercial packages for finite element modeling, which require separate post-processing for SIF determination.

The dynamical stress concentration near a spherical crack in a twice-truncated elastic cone

Engineering practice demands the solving of the inverse problems of fracture mechanics, with this aim, it is necessary to get an analytical solution of the corresponding direct problem. So, in this paper, research was made considering a twice-truncated elastic cone with a spherical crack inside in steady state torsional oscillation. As a result of the problem solving the eigen frequency values and dynamical stress intensity factor were investigated for different parameters of the cone and crack. The boundary resonance effect was found and analyzed; this gives the opportunity to use the derived results during nondestructive testing of the elastic body. The solution is constructed as the superposition of a continuous solution (the problem for an elastic cone without a crack under dynamical torsion), and a discontinuous one (the problem for a spherical crack). The former solution was derived with a great contribution from the G. Ya. Popov results [G. Ya. Popov, 1982]. The wave field of an elastic twice truncated cone was investigated. The eigen frequencies were calculated for different cone’s and crack’s geometrical parameters. The investigation demonstrated the fact of boundary resonance which is commonly used in the inverse problems of fracture mechanics.

Modeling of embedded and edge cracks in steel alloys by XFEM

This work aims at the modelling and simulating of the embedded and edge cracks in steel alloys by employing extended finite element method. Unlike FEM where the crack growth modelling is cumbersome due to the conformal meshing and mesh refinement, this technique allows modelling of all types of discontinuities independent of the mesh. Appropriate functions called enrichment functions are included to the standard FEM approximations so as to accommodate the influence of different discontinuities present in the domain. The level set method has been employed to keep track of discontinuities. The stress intensity factors have been obtained by employing the domain based interaction integral approach. Accurate and effective MATLAB codes have been developed on XFEM to model various types of cracks in engineering materials. Finally some numerical problems have been solved by the extended finite element method to reveal the capabilities of the proposed approach in modelling fracture mechanics problems.

The T-stress influence on the plastic zone size around the crack tip under the thermal load

A review of the fracture mechanics problems for estimating the safety of constructions with the crack-like defects is carried out. The need to develop a new methodology that combines the advantages of calculation and instrumental methods was revealed. The correlations between the most important fracture mechanics parameters and the characteristics of acoustic-emission signals proposed to use for estimating the parameters of stress-strain state. It is noted that considerable attention should be paid to the size and shape of the plastic zone in order to establish a theoretical relationship between the characteristics of acoustic signals and the fracture mechanics parameters. It is assumed that T-stresses are the cause of differences in the coefficients of the power dependence of the parameters of acoustic signals for the same values of the stress intensity factor. The task of finding the influence of T-stresses on the size of the plastic zone at the crack tip is set. A numerical modeling of thermal loading of a steel sheet with a central crack by the finite element method is carried out in order to determine the dependence of the crack tip plastic zone size on the stress intensity factor for several values of T-stress. Multiple calculations of the stress intensity factor and the T-stresses were performed for the several values of the cooling zone, using the parametric design language ANSYS APDL. By substituting the yield point of the material into the equations of linear elastic fracture mechanics, the specific volume and linear plastic zone size along the crack line were determined for the previously found values of the stress intensity factor and T-stress. A relationship between the fracture mechanics parameters and the plastic zone size at the crack tip was established using numerical methods. The assumptions that the T-stress influence to the plastic zone size are qualitatively confirmed. It is shown, that the negative values of T-stresses decrease the plastic zone size for a same value of the stress intensity factor.

Contact problems for cracks under impact loading

This paper concerns a fracture mechanics problem for elastic cracked materials under transient dynamic loading. The nonlinear contact problem for a linear crack under oblique Heaviside compression pulse is solved by the boundary integral equations method in the frequency domain, and the components of the solution are presented by the Fourier exponential series. The contact forces are calculated and the solution is analysed accounting for the friction. The dynamic stress intensity factors are computed at leading and trailing crack’s tips and compared with those obtained neglecting the crack closure and friction.

Transfer learning enhanced physics informed neural network for phase-field modeling of fracture

In this work, we present a new physics informed neural network (PINN) algorithm for solving brittle fracture problems. While most of the PINN algorithms available in the literature minimize the residual of the governing partial differential equation, the proposed approach takes a different path by minimizing the variational energy of the system. Additionally, we modify the neural network output such that the boundary conditions associated with the problem are exactly satisfied. Compared to the conventional residual based PINN, the proposed approach has two major advantages. First, the imposition of boundary conditions is relatively simpler and more robust. Second, the order of derivatives present in the functional form of the variational energy is of lower order than in the residual form used in conventional PINN and hence, training the network is faster. To compute the total variational energy of the system, an efficient scheme that takes as input a geometry described by spline based CAD model and employs Gauss quadrature rules for numerical integration, has been proposed. Moreover, we note that for obtaining the crack path, the proposed PINN has to be trained at each load/displacement step, which can potentially make the algorithm computationally inefficient. To address this issue, we propose to use the concept ‘transfer learning’ wherein, instead of re-training the complete network, we only re-train the network partially while keeping the weights and the biases corresponding to the other portions fixed. With this setup, the computational efficiency of the proposed approach is significantly enhanced. The proposed approach is used to solve six fracture mechanics problems. For all the examples, results obtained using the proposed approach match closely with the results available in the literature. For the first two examples, we compare the results obtained using the proposed approach with the conventional residual based neural network results. For both the problems, the proposed approach is found to yield better accuracy compared to conventional residual based PINN algorithms.

Analysis of Hertzian indentation fracture using a phase field approach

AbstractThe formation and further evolution of cracks caused by the compression of a stiff indenter onto the surface of an initially defect‐free brittle solid is a fascinating problem of fracture mechanics. Its prediction, however, is still a challenging task since crack nucleation is caused by a rather weak stress concentration in the contact near‐field. The present contribution focuses on phase field simulation of indentation fracture, including crack formation at some a priori unknown location outside of the contact region and the subsequent formation of a cone crack. While the phase field method, at first glance, appears to be a promising tool to simulate the current problem we elaborate critical issues and discuss essential modifications. Finally, the indentation fracture process is simulated showing the effect of varying indenter radii on crack initiation and the influence of Poisson's ratio on the angle of cone crack propagation in good agreement with experimental findings and other theoretical studies.

Estimation of the Strength of Plates with Cracks Based on the Maximum Stress Criterion in a Scale-Dependent Generalized Theory of Elasticity

The problem of the strength of a plate made of a brittle material with through mode I cracks is discussed. In contrast to the approach based on the singular solution of the classical theory of elasticity for a plane with a crack and on linear fracture mechanics, we propose to use nonsingular solutions obtained within the generalized elasticity theory and, as a result, to implement a method, conventional for the strength estimation of solids with stress concentration, based on the maximum stress criterion. The maximum stress is determined from a nonsingular solution of the generalized elasticity equations for a plane with a crack. The reported experimental results for plates with cracks under tension and bending confirm the solution obtained by the proposed method and allow it to be compared with a solution based on linear fracture mechanics. In fact, a new concept of fracture mechanics is put forward, which is free of singular solutions and allows the problems of fracture mechanics to be treated as problems of stress concentration. Comparison of the obtained analytical solutions with the experimental data has shown that the scale factor of generalized elasticity determines the critical state in fracture mechanics with no less accuracy than the critical stress intensity factor and therefore can be used as a fracture criterion. The resulting explicit nonsingular solutions allow the prediction of the stress concentration caused by a crack.

Minimization of the Stress State and Fracture Parameters of a Composite in Bending

A plane inverse fracture mechanics problem for a composite in bending is considered. Its binder is weakened by rectilinear cracks located arbitrarily near fibers. To determine the optimum tightness of fiber-binder connection that reduces stress concentrations at the interface of material joints and minimizes the fracture parameters (stress intensity factor) of the composite, a theoretical analysis is carried out on the basis of the principle of equal strength. A closed system of algebraic equations allowing one to obtain a solution to the optimum design problem for the composite in relation to the geometric and mechanical characteristics of the binder and fibers is constructed.

A New Approach to Non-Singular Plane Cracks Theory in Gradient Elasticity

A non-local solution is obtained here in the theory of cracks, which depends on the scale parameter in the non-local theory of elasticity. The gradient solution is constructed as a regular solution of the inhomogeneous Helmholtz equation, where the function on the right side of the Helmholtz equation is a singular classical solution. An assertion is proved that allows us to propose a new solution for displacements and stresses at the crack tip through the vector harmonic potential, which determines by the Papkovich–Neuber representation. One of the goals of this work is a definition of a new representation of the solution of the plane problem of the theory of elasticity through the complex-valued harmonic potentials included in the Papkovich–Neuber relations represented in a symmetric form, which is convenient for applications. It is shown here that this new representation of the solution for the mechanics of cracks can be written through one harmonic complex-valued potential. The explicit potential value is found by comparing the new solution with the classical representation of the singular solution at the crack tip constructed using the complex potentials of Kolosov–Muskhelishvili. A generalized solution of the singular problem of fracture mechanics is reduced to a non-singular stress concentration problem, which allows one to implement a new concept of non-singular fracture mechanics, where the scale parameter along with ultimate stresses determines the fracture criterion and is determined by experiments.

ОСОБЛИВОСТІ ВИКОРИСТАННЯ ІНТЕГРАЛІВ, НЕЗАЛЕЖНИХ ВІД КОНТУРУ ІНТЕГРУВАННЯ, В ЗАДАЧАХ МЕХАНІКИ РУЙНУВАННЯ

In recent decades, the Cherepanov-Rice J-integral has been considered to be the main energy criterion for fracture toughness in solving the problems of fracture mechanics. This parameter is successfully used in the consideration of elastic problems and within the framework of deformation theory of plasticity for stationary cracks. Unfortunately, the theoretical basis on which the J-integral is based does not allow it to be extended to solve problems requiring material inhomogeneity, the presence of temperature influences, an arbitrary load history, mass forces, crack development, and the like. The purpose of the paper is to review other integrals independent of the contour of integration, from the point of view of their theoretical justification, the possibility of experimentally determined and the ease of calculation. The article considers the J-integral, integrals of Wilson-Yu, Gurtins, Blackburn, Kishimoto, Ainsworth, Atluri.

2-D fracture mechanics problems by SGFEM

In this paper, an extensive investigation is done on Stable Generalized Finite Element Method (SGFEM) performance through the analysis of 2-D fracture mechanics problems. Condition number, Stress Intensity Factors (SIFs), global and local measures of the energy norm are used to study SGFEM conditioning and accuracy. Computational time is also briefly discussed. The method is compared with the standard Generalized/eXtended Finite Element Method (G/XFEM). Numerical experiments corroborate and complement the knowledge available in the literature so far, and demonstrate SGFEM accuracy in 2-D cracked problems. Modified Heaviside Functions, combined with other enrichment functions, are also studied in the simulations. A simple and yet generic implementation for SGFEM is described, under the Object Oriented strategy, in a open source software. The implementation can be used in 2-D and 3-D problems, and it allows to generalize the implementation of any type of enrichment function under the SGFEM approach.

A novel boundary element formulation for anisotropic fracture mechanics

A novel boundary element formulation for two-dimensional fracture mechanics is presented in this work. The formulation is based on the derivation of a supplementary boundary integral equation to be used in combination with the classic displacement boundary integral equation to solve anisotropic fracture mechanics problems via a single-region approach. The formulation is built starting from the observation that the displacement field for an anisotropic domain can be represented as the superposition of a vector field, whose components satisfy a suitably defined anisotropic Laplace equation, and the gradient of the Airy stress function. The supplementary boundary integral equation is then obtained using such representation into the integral expression of the aforementioned Laplace equation and employing the relationship between the stress function gradient and the boundary tractions. The supplementary equation neither requires the computation of hyper-singular integrals nor does it introduce additional variables for the problem, as it involves boundary displacements and tractions only. Numerical results are obtained for both uncracked and cracked bodies and show the accuracy and potential of the proposed approach.

On a Monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics

A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is also compared to two alternative algorithms for nonsmooth nonconvex optimization arising in optimal control and mathematical imaging.

Linear and Non-Linear Stress Analysis for the Prediction of Fracture Toughness for Brittle and Ductile Material using ASTM E399 and ASTM E1290 By ANSYS Program package

Prediction of fracture toughness value of most the metallic materials is a very important issue to be consider in designing of engineering structural components like pipes, flow tanks and pressure vessel where it can be used in the evaluation of critical crack length that is consider very important factor in non-destructive testing because during inspections of these components, the critical crack length is being compared with the minimum allowable flow size. Fracture toughness value can be predicted by using two different types of experimental tests such as ASTM-E399 (which is used to predict fracture toughness under plane strain condition for brittle materials which undergo to small plastic deformation) and ASTM-E1290 (which is used to predict fracture toughness for ductile materials which undergo to large plastic deformation). These experimental tests utilize specimens such as compact tension specimen CT and three-point bend specimen SENB which must be pre-cracked. However, manufacturing of pre-cracked in metallic structural components before testing is expensive and time-consuming process, thus in the current study, fracture toughness will be predicted by using economized method in both time and cost that is the finite elements method by using ANSYS PROGRAM Where, a three-dimensional model has been designed by using two different types of elements (plane-82) and (solid-95).Firstly, two-dimensional model mesh will be idealized by using element type (plane-82) due to ANSYSY PROGRAM can pick–up singularity for two dimensional model only, but in order to obtain more accurate results, a three dimensional model will be designed by utilizing Sweep Model with restriction with ANSYSY abilities in the treatment of fracture mechanics problems for three dimensional model analysis where the element length must be ranging from 1to4 in all directions. In elastic region, the fracture toughness is been predicted directly from ANSAYS PROGRAM using the specification in ASTM-E-399 for compact tension specimen for practical design problems. But in elastic-plastic region, the fracture toughness is been predicted by using the crack tip opening displacement model (CTOD-Model) using the specification in ASTM-E-1290 for three-point bend specimen for practical design problem. The critical value for crack tip opening displacement will be calculated after extracting load-displacement data from ANSAYS PROGRAM then using these data in (CTOD-Model) which is separated into two components, elastic and plastic, the elastic component has been estimated using Dug Dual-Model, while the plastic component will be estimated using Plastic Hinge-Model. However, the fracture toughness value that has been predicted by finite element method from non –linear elastic analyses and from non –linear elastic-plastic analyses gives excellent results which are very close to the experimental results with error ratio ranging between (10% to 14%).

Quasi-static analysis of mixed-mode crack propagation using the meshless local Petrov–Galerkin method

The efficiency of classical truly meshless local Petrov–Galerkin method with linear test function approximation in the crack growth problems of complex configurations is investigated in this article. Several unique and multiple (two) cracks are examined to show the accuracy of the proposed meshfree method by comparing the crack path with boundary element, finite element, element-free Galerkin and experimental results. Crack propagation under both mechanical and thermal loads and also mode I and mixed-mode conditions are evaluated. Effect of the functionality of material properties on crack paths and sensitivity of crack growth to boundary conditions are investigated. Classical maximum circumferential stress criterion is employed into the formulation of the MLPG method to predict crack growth direction. Despite the common numerical studies, a simple and straightforward automatic increment size determination method is introduced and implemented in the MLPG method. Besides, regular domain point distribution in the vicinity of the crack tip without the increased density of points is used. Various test problems show the computational accuracy and applicability of this method to linear elastic fracture mechanics problems.

Accurate modelling of instabilities caused by multi-site interface-crack onset and propagation in composites using the sequentially linear analysis and Abaqus

Even well-established non-linear FEM codes may present convergence issues when solving highly unstable problems, e.g. fracture mechanics problems, especially in the presence of a pronounced snap-back behaviour. In composites, at lamina level, the debonds occurring between the fibres and matrix, lead to multiple-crack configurations. Each individual debond growth is an unstable process. Then, when multiple debonds occur the problem becomes highly unstable. Moreover, experimental evidence shows that a sequence of multiple debonds originates a macro-crack that extends over the whole lamina thickness. In this article a new Python based numerical tool implementing a Sequentially Linear Analysis (SLA) procedure, which is able to call the FEM software Abaqus, is described and tested. The Linear Elastic Brittle Interface Model (LEBIM) is also included in Abaqus by means of a UMAT subroutine. Numerical results for a representative 100-fibre model show that the tool is able to adequately model simultaneous onset and propagation of multiple debonds resulting in a completely unstable process composed by a series of instability events. Eventually, these numerical results together with some previous experimental results for transverse failure loads for unidirectional carbon fibre-epoxy matrix plies allow us to estimate the tensile strength and critical fracture energy of the fibre-matrix interface by a simple inverse analysis.