Investigation of fracture characterizations of functionally graded materials

In this paper, the mode I crack problem of functionally gradient materials (FGMs) with the gradient direction parallel to the crack is discussed, and the differences of stress distribution between the gradient materials and the homogeneous materials are analyzed. It is shown that a mode I crack problem of FGMs with the gradient direction parallel to the crack direction can become a mixed‐mode crack problem. In FGMs, the crack initiation angles are determined by the fracture toughness gradient, elastic modulus and crack mode. If the gradient coefficients are small, the crack initiation angles in FGMs are the same as those in homogeneous materials. If the elastic modulus gradient is large, the principal stress terms without the gradient coefficients can be ignored in obtaining the crack initiation angle. In this study, all the above results are generalized to the mixed‐mode crack problems with arbitrary angle between the gradient direction and the crack direction.     

Determination of stress intensity factor solutions for cracks in finite-width functionally graded materials

A generalized method to determine the stress intensity factor equations for cracks in finite-width specimens of functionally graded materials (FGMs), based on force balance in regions ahead of the crack tip is provided. The method uses the Westergaard's stress distribution ahead of the crack in an infinite plate and is based on the requirement of isostrain deformation of layers of varying moduli ahead of the crack tip. It is shown that the modified Westergaard equation describes the normal stress distribution and the singular stress state ahead of the crack tip in a reasonably accurate manner. Based on this, closed-form analytical equations for the stress intensity factors of cracks in finite-width center cracked specimens were derived. Comparisons of the K values from the analytical equations with that obtained from FEM simulations indicate that the derived stress intensity factor equations for FGMs are reasonably accurate. For the finite-width center-cracked-tension (CCT) specimen, the errors are less than 10% for most of the crack lengths for materials with the outer layer modulus ratios varying from 0.2 to 5. The stress intensity factors were found to be sensitive to the absolute values of moduli of the layers, the modulus ratio of the outer layers as well as the nature of gradation including the increasing and the decreasing functional forms. The stress intensity factor equations are convenient for engineering estimates of stress intensity factors as well as in the experimental determinations of fracture toughness of FGMs.     

Einfluss eines Gefügegradienten auf die rauhigkeitsinduzierte Rissschließung bei TIMETAL 1100

Influence of a Grain Size Gradient on Roughness Induced Crack Closure of TIMETAL 1100 Crack closure effects, which affect the crack propagation behaviour at low R‐values, become crack length dependent in gradient materials. It is not known, if the crack closure at a given ΔK is locally identical to that in a homogeneous microstructure or if closure effects are crack direction dependent. To get some insights to these questions, coarse lamellar microstructures (showing only roughness induced crack closure) with a gradient in the lamellae package sizes where produced on TIMETAL 1100 by recrystallisation. Their crack closure behaviour was compared to that of two different homogenous microstructures. The results clearly show, that crack closure effects show identical dependencies on the stress intensity and the local fracture surface roughness for all microstructures. The roughness, in turn, is uniquely determined by the sizes of the lamellae. Thus no crack direction effects where observed. It is shown that a model for roughness induced crack closure, which was developed for homogeneous microstructures, can be used to calculate the crack closure stress intensity for any gradient and loading condition.     

Analysis of a propagating crack in functionally graded materials with property variation angled to crack direction

The stress and displacement fields for a crack propagating in functionally graded materials (FGMs) with property variation angled to crack direction are obtained. The FGMs have a linear variation of shear modulus with a constant density and Poisson’s ratio. The solutions for higher order terms in the dynamic equilibriums are obtained by transforming the general differential equations to Laplace’s equations. Using the stress fields, the effects of the nonhomogeneity and the angled properties on stress components are investigated. In addition to, the contours of the constant maximum shear stress around the static and propagating crack tip are generated. The contours of the constant maximum shear stress around the static and propagating crack tip tilt toward the property gradation direction.     

Modelling dynamic crack propagation using the scaled boundary finite element method

This study presents a novel application of the scaled boundary finite element method (SBFEM) to model dynamic crack propagation problems. Accurate dynamic stress intensity factors are extracted directly from the semi‐analytical solutions of SBFEM. They are then used in the dynamic fracture criteria to determine the crack‐tip position, velocity and propagation direction. A simple, yet flexible remeshing algorithm is used to accommodate crack propagation. Three dynamic crack propagation problems that include mode‐I and mix‐mode fracture are modelled. The results show good agreement with experimental and numerical results available in the literature. It is found that the developed method offers some advantages over conventional FEM in terms of accuracy, efficiency and ease of implementation. Copyright © 2011 John Wiley & Sons, Ltd.     

Dominance of asymptotic crack tip fields in elastic functionally graded materials

The stress field surrounding an edge crack in an elastic functionally graded plate is calculated using two dimensional finite element analysis. The property gradient direction is parallel to the crack line and loading is constrained to be symmetric such that a pure mode I situation is achieved. The extent of dominance of asymptotic fields is evaluated by comparing the stress field calculated from the finite element analysis to that calculated by asymptotic equations. Two separate forms of the asymptotic stress fields, one for homogeneous materials and another for continuously nonhomogeneous materials are used. The shape and extent of the dominance regions of each asymptotic field and their dependence on crack length and material nonhomogeneity is also presented. Under the pure mode I conditions considered here, it is seen that both asymptotic fields exist around the crack tip with the one for homogeneous materials in general being embedded in the one for continuously nonhomogeneous materials. The ligament length is seen to primarily control the extent of development of the asymptotic stress field for nonhomogeneous materials. The steepness of the material gradient affects the relationship between the two asymptotic stress fields and therefore the extent of their dominance.     

Constraint effects on the elastic–plastic fracture behaviour in strain gradient solids

ABSTRACT Crack‐tip constraint effects (or T‐stress effects) on the elastic–plastic fracture behaviour in strain gradient materials are analysed in the present study. The T‐stress effects on the stress distributions along the plane ahead of the stationary and growing crack tip, respectively, are analysed by using the Fleck and Hutchinson strain gradient plasticity formation. For a steadily growing crack, the T‐stress effects on the steady‐state fracture toughness are analysed by adopting the embedded fracture process zone model. In addition, the analysis for the growing crack is applied to an interfacial cracking experiment for a metal/ceramic system, and the material length‐scale parameter appearing in the strain gradient plasticity theory is predicted. In the present analyses, a new finite element method specially designed for strain gradient problems by Wei and Hutchinson is adopted.     

The mode III full-field solution in elastic materials with strain gradient effects

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical K_III field; stresses have r ^-3/2 singularity near the crack tip, and are significantly larger than the classical K _III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K _III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K _III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral. This revised version was published online in August 2006 with corrections to the Cover Date.     

Study on crack-inclusion interaction using digital gradient sensing method

The interaction between matrix crack and a round inclusion was studied by the method of digital gradient sensing. First, the stress fields at the matrix crack tip in the neighbor of a round inclusion were derived based on transformation toughening theory and Eshelby inclusion method, and the effect of the inclusion on the stress intensity factor of the matrix crack was analyzed. Then, the non-contact optical measurement system of digital gradient sensing was built up, and a three-point bending test was carried out using a single-edge cracked specimen. The mode I stress intensity factor was extracted from the angular deflection of the light rays. Finally, the effect of the inclusion on the angular deflection fields and the stress intensity factor at the crack tip was analyzed experimentally. These results will play an important role for evaluating the fracture mechanism of crack-inclusion interaction in composites.     

Measurement of interfacial fracture parameters using coherent gradient sensing (CGS)

An optical interferometry called coherent gradient sensing (CGS) has been extended for mapping interface crack tip fields and for evaluating fracture parameters. The optical technique is a double grating shearing interferometer with an on line spatial filtering arrangement. The method offers real time full field measurements and can be used both in transmission made and reflection mode. The interferometer measures small angular deflections of light rays which can be further related to in plane gradients of _x + _y in transmission through elasto-optic relations. Direct interfacial crack tip measurements in a high stiffness mismatch PMMA-aluminium bimaterial system are performed. A variety of crack tip mode mixities are studied using asymmetric four point bend specimens subjected to different far field mechanical loads. The. complex stress intensity factors and the associated phase angles are measured from CGS patterns using an asymptotic expansion field. The measurements are compared with finite element results.     

Shear mode threshold for a small fatigue crack in a bearing steel

The fatigue behaviour of small, semi‐elliptical surface cracks in a bearing steel was investigated under cyclic shear‐mode loading in ambient air. Fully reversed torsion was combined with a static axial compressive stress to obtain a stable shear‐mode crack growth in the longitudinal direction of cylindrical specimens. Non‐propagating cracks less than 1 mm in size were obtained (i) by decreasing the stress amplitude in tests using notched specimens and (ii) by using smooth specimens in constant stress amplitude tests. The threshold stress intensity factor ranges, ΔK_IIth and ΔK_IIIth, were estimated from the shape and dimensions of non‐propagating cracks. Wear on the crack faces was inferred by debris and also by changes in microstructure in the wake of crack tip. These effects resulted in a significant increase in the threshold value. The threshold value decreased with a decrease in crack size. No significant difference was observed between the values of ΔK_IIth and ΔK_IIIth.     

The extended finite element method with new crack‐tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X‐FEM) with new crack‐tip enrichment functions. In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and crack‐tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ? class and κ class, two classes of crack‐tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J‐integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.     

Experimental Investigations on Deformation and Fracture Behavior of Glass Sphere Filled Epoxy Functionally Graded Materials

In this paper, deformation and fracture behavior of glass sphere filled epoxy functionally graded materials (FGM) are numerically evaluated and experimentally studied. The fabrication of the FGM is described in detail, and the spatial gradation of elastic modulus and the microscopic structure in FGM are measured and analyzed. The deformation and fracture characterization of the FGM specimen with a crack oriented along the direction of the elastic gradient under three point bend are studied by the experimental and the finite element method. The influences of crack location at both the stiff and the compliant sides of the FGM specimen on crack initiation, deformation field and stress intensity factor are analyzed. The results are: (a) The neutral-axis in the FGM specimen under three-point-bending will shift toward the stiffer side; (b) The initial fracture load increases with the increase of elastic modulus at the crack tip; (c) The elastic gradients shield a crack on the compliant side and lower the stress intensity factor when compared to the one with crack on the stiff side. These results will be useful for better design and reliable evaluation of FGM.     

An implicit gradient application to fatigue of sharp notches and weldments

This paper addresses the problem of stress singularities at the tip of sharp V-notches by means of a non-local implicit gradient approach. A non-local equivalent stress is defined as a weighted average of a local stress scalar quantity computed on the assumption of linear elastic material behaviour. In the case of a crack, we propose an analytical solution for the non-local equivalent stress at the crack tip when the local equivalent stress assumes the analytical form proposed by Irwin. For open notches, several numerical procedures are possible.For welded joints, we assume that the material obeys a linear elastic constitutive law. In this case, the non-local equivalent stress obtained from the implicit gradient approach is assumed as the effective stress for assessments of joint fatigue. Using the principal stress as local equivalent stress and a notch tip or weld toe radius equal to zero, we analyse many series of arc welded joints made of steel and subjected to either tensile or bending loading, and we propose a unifying fatigue scatter band. If the welded joints are subjected only to mode I loading, an analytical relationship between the relevant Notch Stress Intensity Factors (NSIF) of mode I and the effective stress is established; otherwise, the effective stress is evaluated by means of a simplified numerical analysis. For complex welded structures, however, a completely numerical solution is proposed; when different crack initiation sites are present (i.e. either weld toes or roots), the proposed approach correctly estimates the actual critical point.     

On the slope of the fatigue resistance curve for laser stake‐welded T‐joints

The paper investigates the influence of highly localised stress distribution around the notch tips of the laser stake‐welded T‐joints to the slope of the fatigue resistance curve. The study considers experimental data of eight series involving joints under tension or bending loads. Various boundary conditions and plate thicknesses are considered. The stress distribution in the singularity‐dominated zone ahead of the notch tips is investigated by means of the finite element analysis. The aim is to relatively distinguish the stress distribution from one case to another. The growth rate of the elastic singular stress with respect to the distance from the tip is described by the dimensionless gradient. This gradient is equal to the slope of the linear stress‐distance function when presented in double‐logarithmic scale. The slope of the fatigue resistance curve varies approximately from 4 to 8. It is observed that the change of the slope can be closely associated with the gradient of the maximum principal stress evaluated in the plane that is orthogonal to the crack path. The orthogonal plane corresponds to the maximum principal stress direction. In contrast, there is a large scatter in the relation between the slope and the gradient evaluated in the commonly assumed crack plane. The study shows that the dimensionless gradient exhibits sensitivity towards plate thicknesses, local weld geometry and the loading condition.     

Complex fracture parameters for an interface crack between twohardening materials: a photoelastic study

In this paper, complex stress intensity factors (SIFs) at an interface crack are determined for a range of applied loads, crack lengths and remote mode mixes using automated photoelasticity. The specimen geometries comprise epoxy resin and aluminium alloy halves bonded together, and are loaded in either compact tension in mixed‐mode conditions or in three‐point bend under mode I conditions. In the experiments, full‐field isochromatic data were obtained from the epoxy half using an established phase‐stepping technique. A reworked approach to the determination of the SIFs was developed by combining a least‐squares over‐deterministic method for fitting crack‐tip stress equations to the data and a weighting factor that ensures that only data in the singularity zone are used. For comparison, some of the specimens were tested using a linear‐elastic finite element (FE) analysis and/or by experiment using homogeneous test specimens. Excellent agreement between the experimental and numerical SIF moduli was achieved for remote mode I loadings. However, for good agreement to be made between the phase angle results requires an additional phase term to be added to the FE solution at each load to account for the development of a crack‐tip plastic zone. Further, results for the SIFs from remote mixed‐mode loadings of the compact tension specimen only have a meaningful interpretation in light of small‐scale yielding conditions. It is shown, qualitatively, that the experiments verify some of the predictions made in the literature of asymptotic behaviour at interface crack tips from results of elasto‐plastic FE analyses.     

承受双向等拉应力的平面应变I型裂纹线弹性断裂的I1准则与K准则一致性分析

采用线弹性有限元方法计算了承受双向等拉应力的平面应变I型裂纹的应力场,分析了裂纹尖端各应力分量间的关系,拟合了各非零应力分量关于裂纹半长度a和裂纹尖端最小网格尺寸l₁的函数,分析了应力第一不变量I₁与应力场强度因子K_I的相关性。结果表明,裂纹尖端各非零应力分量间存在稳定的比例关系;各非零应力分量值和加载应力的比值与裂纹半长度a的1/2次幂呈正比例关系、与裂纹尖端最小网格尺寸l₁的1/2次幂呈反比例关系;相同最小网格尺寸条件下,裂纹尖端的应力第一不变量与应力场强度因子的比值l₁/K_I为与加载应力和裂纹长度无关的常数,证明了承受双向等拉应力的平面应变I型裂纹线弹性断裂的I₁准则与K准则具有一致性。