Mixed-mode Problems Research Articles

An improved peridynamic approach for fatigue analysis of two dimensional functionally graded materials

This study investigates the fatigue behavior of functionally graded materials (FGMs) within an enhanced peridynamic (PD) framework that eliminates the need for surface and volume correction factors. FGMs offer mechanical advantages by varying material compositions and properties in one or more directions. The distribution of mechanical properties in FGMs is achieved by employing a power law. Comparative analysis is conducted between the PD predictions for reaction forces and outcomes obtained from the reference solutions for a quasi-static problem. Minimal divergence is observed between the PD and reference solutions, affirming the accuracy of the present PD formulation. Furthermore, the PD fatigue results for a Mode I Compact Tension (CT) specimen problem match the experimental findings. An experimental stop-hole problem is used to evaluate the accuracy of the PD fatigue code in mixed-mode circumstances. The load cycle determination is achieved via ABAQUS Computational Finite Element Method (CFEM) for the mixed mode problem. There is a good correlation between CFEM and the present PD outcomes. Furthermore, the effect of stop-hole existence on fatigue life in mixed mode is examined, revealing a significant retardation effect. Notably, this study adds a novel perspective by using the PD approach to investigate the fatigue analysis of FGMs and examine the impact of the stop hole on fatigue life- a component previously unexplored in the existing literature.

Fracture analysis of a curved crack in a piezoelectric plane under general thermal loading

The mixed-mode problem of a cracked piezoelectric plane subjected to a general in-plane thermal load is considered. The crack is modeled as distributed thermo-mechanical dislocations with unknown density which are determined by using the Fourier transform method. Furthermore, stress, electric displacement, and heat flux in the intact plane, under thermal load are obtained. These solutions are used to construct singular integral equations in the piezoelectric plane with an arc-shaped crack under thermo-mechanical loading. The integral equations are solved numerically to obtain dislocation density functions on a crack surface. These functions are used to determine field intensity factors for the crack. Finally, the effect of the crack length, the radius of the crack, and distance from the temperature disturbance is investigated under different loading conditions.

An examination of the size effect in quasi-brittle materials using a bond-based peridynamic model

This paper examines the size effect in quasi-brittle materials using a three-dimensional bond-based peridynamic model. This is the first time that the capability of a peridynamic model to capture the size effect in quasi-brittle materials has been examined. Correctly reproducing the size effect is an essential check on the validity of any computational model and it is demonstrated that a bond-based peridynamic model can accurately capture the failure stress of geometrically identical structures over a range of sizes. A systematic examination of geometrically similar notched and unnotched beams provides new insights into the predictive capabilities of the model, and mode I and mixed-mode problems are considered to examine the generality of the model. The model is validated using published experimental data, and the predictive accuracy is equivalent, if not superior, to well-established numerical methods whilst offering several benefits that justify further research and development. This study provides evidence that the length scale in a peridynamic model is a numerical parameter and not a material property.

The Effect of Inclusion on Crack Propagation Using Extended Finite Element Method

Numerical simulation is developed to investigate the effect of inclusion on crack propagation. In this study, the crack growth is modeled using extended finite element method (XFEM). Two-dimensional rectangular plate with single inclusion embedded off-centered is modeled. The specimen is subjected to uniaxial tension. The dimensions of the specimen are 40 mm x 80 mm and the radius of the inclusion is 10 mm. The specimen is pre-cracked with the length of an edge crack is 5 mm. The motion of the crack is modeled by XFEM based on traction-separation cohesive behavior for 2D mixed mode problem. In addition, enrichment procedure is used to implicitly determine predefined crack in XFEM framework. Two different inclusions, which are soft and hard inclusions, are considered on crack propagation scheme. The effects of soft and hard inclusions on crack propagation are studied and observed. The results showed that the trajectory of crack highly depends on inclusion inside the material. In the case of soft inclusion, propagation of the crack tended to approach the inclusion. Whereas in the case of hard inclusion, crack trajectory tended to move away from the inclusion. The mismatch of elastic modulus between inclusion and surrounded materials has significant effect on propagation of crack.

A note on mixed-mode fracture by the smoothing gradient damage model

ABSTRACT The objective of this technical note is to illustrate twofold. The first issue is devoted to the ability of the recently developed smoothing gradient-enhanced damage model (SGDM), which is associated with the modified von Mises equivalent strain in modelling mixed-mode fracture problems, in particular the common Nooru–Mohamed's test. We show that the developed approach works pretty well for this test, yielding appropriate structural damage response and crack paths for all three specimen sizes, i.e. and . The second issue is, in a similar manner, to discuss the less accuracy of another damage model, the localising gradient damage scheme recently studied by Shedbale et al. Int. J. Mech Sci 2021, 199:106410, in reproducing crack paths and structural responses for the same mixed-mode problem.

A new cohesive law for the simulation of crack propagation under cyclic loading. Application to steel- and concrete-FRP bonded interface

This paper presents a new cohesive law for modelling interfaces under mixed-mode cyclic loading. The formulation is based on the definition of a free energy function which governs the interface behaviour under monotonic loading, which is then extended to cyclic-driven decohesion through the introduction of a scalar damage variable, whose evolution in time is governed by a phenomenological rate equation. The cohesive model is formulated for a mixed-mode problem and then it is applied for the simulation of debonding phenomena occurring at the interface under a pure a shear stress state. Experimental results available in literature, related to single-lap shear tests, performed on both concrete and steel specimens reinforced by fibre reinforced composite (FRP), are used to validate the proposed model and to show its effectiveness to simulate very closely the observed experimental behaviour.

A self-contained element for modeling crack propagation in beams

The Cohesive Zone Model (CZM) has been used to model the propagation of cracks through beams and slabs, which is of practical application in a variety of fields. CZMs have been successfully incorporated into Finite Element (FE) models however, this involves significant pre-processing effort to develop a sufficiently fine mesh around cohesive elements, which also increases the required computational effort. To overcome this limitation, a self-contained’crack element,’ consisting of cohesive elements surrounded by pre-meshed bilinear bulk elements, was developed. A linear interpolation scheme to connect this crack element to adjoining beam elements, as well as a solution procedure using static condensation to solve crack propagation in beams was developed and implemented. Bilinear and non-linear CZMs were implemented to model the behavior of the cohesive elements. Under simple Mode I tension and flexure problems, the results were in-line with the expected behavior. It was found that the crack element was convergent with increasing mesh resolution, and needed to be only as long as its thickness, with the remainder of the beam being modeled by beam elements with just a handful of DOFs, reducing the size of the problem substantially. The element was also used to verify a theoretical model for the variation of modulus of rupture of a beam with its thickness, as well as a mixed-mode problem. In both those cases, the crack element showed good results.

Mixed mode analysis of multiple cracks in magneto-electro-elastic plane

In this paper, the mixed mode problem for a magneto-electro-elastic (MEE) material containing multiple cracks under in-plane magneto-electro-mechanical loads is analyzed. First, the solution to an generalized dislocation is obtained in the infinite MEE plane. The Fourier transform is employed to derive closed-form expressions for the stress, electric displacement and magnetic induction components. The solution to the problem is consequently reduced to derive singular integral equations for dislocation density functions on the surfaces of cracks. The equations are then solved numerically to determine field intensity factors for cracks. At the final step, the numerical results of calculated field intensity factors are presented for multiple curved cracks under certain ratios of the applied magnetical and electrical loads to mechanical loads. Also, the field intensity factors are calculated for various crack patterns with different geometries. Moreover, the analysis can be applied for any number of cracks with arbitrary arrangement in the MEE regions under in-plane condition.

Multiple mixed-mode cracks in a magneto-electro-elastic half-plane under in-plane loading

The mixed-mode problem of cracked magneto-electro-elastic (MEE) half-plane subjected to in-plane magneto-electro-mechanical loads is considered. The crack is modelled as distributed dislocations with unknown density and the closed form solution of the generalized dislocation is obtained by using the Fourier transform. Furthermore, stress, electric displacement and magnetic induction fields in the intact MEE half-plane, under magneto-electro-mechanical point forces are determined. These solutions are used to construct singular integral equations in the MEE half-plane with multiple cracks. The integral equations are solved numerically to obtain dislocation density functions on a crack surface. These functions are used to determine field intensity factors for cracks. The results show that the stress, electric displacement and magnetic induction intensity factors at the crack tips depend on crack geometry, patterns and the loading conditions. Moreover, the effect of the boundary of the MEE half-plane as well as interaction between cracks upon field intensity factors is investigated.

Comparison of some least-squares methods for the evaluation of notch stress intensities of sharp V-notched configurations

In the present work, three least-squares methods for the computation of the notch stress intensity factors (NSIFs) of sharp V-notched configurations are compared. The first method was proposed by Ju and Chung (2007) and the NSIFs are determined from the finite element (FE) stresses in a circular region around the notch tip. The second one was presented by Liu et al. (2008) and their method uses nodal FE stresses along a radial line from the notch tip. The third method was developed by Ayatollahi and Nejati (2011) and in their methodology the NSIFs are determined using the FE displacement field around the notch tip. The NSIFs of various mixed mode (I/II) problems are determined by applying all the three described methods. Finally, the results obtained using all the methods are compared with the results available in the literature.

A three-dimensional elastoplastic analysis of mixed-mode KI/KII around the crack front

Engineering problems that involve fatigue crack growth and fracture frequently can be studied by taking into account only mode-I features. However, many important problems that involve combined mode I and II loadings cannot be properly analyzed by a pure mode-I approach, which in particular may not be sufficient to estimate fracture toughness for practical purposes in such cases. Such mixed-mode problems involve crack orientation and/or load conditions that lead to combined local Stress Intensity Factors (SIFs) KI/KII around the crack front. Using multiaxial crack tip condition characterized by the crack inclination angle βin a mixed-mode KI/KII modified single edge tension SE(T) specimen, such mixed-mode effects on plastic zone shapes, volumes and plastic work UPL are taken into account to evaluate problems that involve fatigue and fracture.

Finite element analysis of the behaviour of a crack in the orthopedic cement

In this paper, the finite element method is used to analyse the crack behaviour in the orthopedic cement of the total hip replacement by computing the stress intensity factors (SIFs) arround the crack tip. In this work, three cases are studied: crack emanating from a cavity, interaction effect of the crack emanating from a cavity with another cavity and the interaction effect of two cracks emanatingfrom two cavities. The stress intensity factors under mixed mode problems at the crack tip are computed for three zones of prosthesis: proximal, median and distal. The obtained results show that the crack initiated from a micro-cavity in the distal zone of cement can be propagated at the same time by opening and shearing of its lips. It is contrary to that initiated in the proximal zone which cannot be propagated. The mechanical behaviour of cracks in the medial zone depends of the crack initiation position.

Asymptotics of eigenvalues of the nonlinear eigenvalue problem arising from the near mixed-mode crack-tip stress-strain field problems

In the present paper, approximate analytical and numerical solutions to nonlinear eigenvalue problems arising in nonlinear fracture mechanics in studying stress-strain fields near a crack tip under mixed-mode loading are presented. Asymptotic solutions are obtained by the perturbation method (the artificial small parameter method). The artificial small parameter is the difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear “undisturbed” problem. It is shown that the perturbation technique is an effective method of solving nonlinear eigenvalue problems in nonlinear fracture mechanics. A comparison of numerical and asymptotic results for different values of the mixity parameter and hardening exponent shows good agreement. Thus, the perturbation theory technique for studying nonlinear eigenvalue problems is offered and applied to eigenvalue problems arising in fracture mechanics analysis in the case of mixed-mode loading.

Асимптотика собственных значений нелинейной задачи на собственные значения, следующей из проблемы определения напряженно деформированного состояния у вершины трещины в условиях смешанного нагружения

Асимптотика собственных значений нелинейной задачи на собственные значения, следующей из проблемы определения напряженно деформированного состояния у вершины трещины в условиях смешанного нагружения

On the plastic zone correction of a Zener–Stroh crack interacting with a nearby inhomogeneity and an edge dislocation

The investigation on the interaction among a Zener–Stroh crack, a nearby inhomogeneity/inclusion and an extra edge dislocation has been carried out. A Zener–Stroh crack is formed by pileup of net dislocations. With the existence of a nearby inclusion, an extra dislocation near the crack may be either absorbed into the crack (so the crack grows bigger) or repelled away by the crack and the inclusion. The result depends on the force on the dislocation from the crack and inclusion. In our current study, the plastic zone correction at the crack tips is adopted, where the plastic zone size (PZS) is determined by our newly developed generalized Irwin approach since the stress fields around the crack tips are mixed modes. With the distributed dislocation method, the expressions for the stress intensity factor, PZS and crack tip opening displacement (CTOD) are obtained for different sets of geometric and material parameters. Numerical examples show that when the extra single edge dislocation is located near the sharp tip of the Zener–Stroh crack, the normalized PZS and normalized CTOD become very sensitive to the position of the edge dislocation. It is also observed that the mixed mode problem usually has larger PZS and CTOD comparing to the pure mode I loading problem.

Brittle failure in timber connections loaded parallel to the grain

An existing beam-on-elastic-foundation (BEF) model was used to determine the perpendicular-to-grain tensile stresses in timber members subjected to loading parallel to the grain by bolted connections. A set of relatively simple equations for the analysis of a Timoshenko beam of finite length on a Winkler foundation is given, and appropriate foundation stiffness values are discussed. While previous applications of the model have associated the foundation stiffness with the perpendicular-to-grain elastic strain in the timber, it is suggested that a fracture layer be introduced and the foundation stiffness be associated with the perpendicular-to-grain tensile strength and the mode I fracture energy of the wood. The latter estimation of the foundation stiffness, which leads to a so-called ‘quasi-non-linear fracture mechanics model’, has been applied with success to other problems where the BEF model has been used for the analysis of mode I fracture. An existing model for the analysis of the pure mode II fracture, which is also a quasi-non-linear fracture mechanics model based on similar assumptions as the proposed model for analysis of the mode I fracture, is reviewed. A simple way of handling mixed-mode fracture problems by means of the simple empirical interaction of the proposed pure mode I and pure mode II quasi-non-linear fracture mechanics models is discussed.

Brittle failure in timber connections loaded parallel to the grain

An existing beam-on-elastic-foundation (BEF) model was used to determine the perpendicular-to-grain tensile stresses in timber members subjected to loading parallel to the grain by bolted connections. A set of relatively simple equations for the analysis of a Timoshenko beam of finite length on a Winkler foundation is given, and appropriate foundation stiffness values are discussed. While previous applications of the model have associated the foundation stiffness with the perpendicular-to-grain elastic strain in the timber, it is suggested that a fracture layer be introduced and the foundation stiffness be associated with the perpendicular-to-grain tensile strength and the mode I fracture energy of the wood. The latter estimation of the foundation stiffness, which leads to a so-called ‘quasi-non-linear fracture mechanics model’, has been applied with success to other problems where the BEF model has been used for the analysis of mode I fracture. An existing model for the analysis of the pure mode II fracture, which is also a quasi-non-linear fracture mechanics model based on similar assumptions as the proposed model for analysis of the mode I fracture, is reviewed. A simple way of handling mixed-mode fracture problems by means of the simple empirical interaction of the proposed pure mode I and pure mode II quasi-non-linear fracture mechanics models is discussed.

An extended consecutive-interpolation quadrilateral element (XCQ4) applied to linear elastic fracture mechanics

A novel extended 4-node quadrilateral finite element (XCQ4) based on a consecutive-interpolation procedure (CIP) with continuous nodal stress for accurately modeling singular stress fields near crack tips of two-dimensional (2D) cracks in solids is presented. In contrast to the traditional approaches, the approximation functions constructed based on the CIP involve both nodal values and averaged nodal gradients as interpolation conditions. Our objective is to exhibit a pioneering extension of the recently developed CQ4 element enhanced by enrichment to precisely model 2D elastic crack problems, taking advantages of the strengths and making use all the desirable features of both techniques, the CIP and the local enriched partition of unity method. The stress intensity factors (SIFs) are estimated using the interaction integral. The accuracy and performance of the proposed XCQ4 and its numerical properties are illustrated by numerical examples, considering both single and mixed-mode problems with complicated configurations. Compared with reference solutions available in the literature and the conventional XQ4 results, it is found that the accuracy of the XCQ4 is high. Studies on the convergence rate of the SIFs in relative errors also reveal a better performance of the XCQ4 over the classical XQ4. The fracture parameters are found to be stable for different areas of integration paths around the crack tip. Further applications of the developed XCQ4 to other complex problems are potential.

Mapping Static and Dynamic Crack-Tip Deformations Using Reflection-Mode Digital Gradient Sensing: Applications to Mode-I and Mixed-Mode Fracture

The reflection-mode digital gradient sensing (r-DGS) method is extended for visualizing and quantifying crack-tip deformations in solids under quasi-static and dynamic loading conditions. The r-DGS technique employs digital image correlation principles to quantify two orthogonal surface slopes simultaneously in specularly reflective solids by measuring small deflections of light rays. Here, for the first time, r-DGS is implemented to study both mode-I and mixed-mode (I/II) problems and evaluate fracture parameters. Under dynamic loading conditions, r-DGS is employed in conjunction with high-speed digital photography to map surface slopes in edge cracked plates subjected to one-point impact. The measured surface slopes are used to successfully evaluate instantaneous stress intensity factor histories by combining measurements with the corresponding asymptotic crack-tip fields and performing over-deterministic least-squares analyses. Finite element computations are also used to complement experimental measurements as needed.

Finite strain fracture analysis using the extended finite element method with new set of enrichment functions

Summary Nonlinear fracture analysis of rubber-like materials is computationally challenging due to a number of complicated numerical problems. The aim of this paper is to study finite strain fracture problems based on appropriate enrichment functions within the extended finite element method. Two-dimensional static and quasi-static crack propagation problems are solved to demonstrate the efficiency of the proposed method. Complex mixed-mode problems under extreme large deformation regimes are solved to evaluate the performance of the proposed extended finite element analysis based on different tip enrichment functions. Finally, it is demonstrated that the logarithmic set of enrichment functions provides the most accurate and efficient solution for finite strain fracture analysis. Copyright © 2015 John Wiley & Sons, Ltd.