Fundamental formulation for transformation toughening

In this paper, the transformation toughening problem is addressed in the framework of plane strain. The fundamental solution for a transformed strain nucleus located in an infinite plane is derived first. With this solution, the transformed inclusion problems are formulated by a Green’s function method, and the interaction of a crack tip with a single transformation source is found. On the basis of this solution, the fundamental formulations for toughening arising from martensitic and ferroelastic transformation are formulated also using the Green’s function method. Finally, some examples are provided to demonstrate the validity and relevance of the fundamental formulations proposed in the paper.     

On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium

In this study, a bilinear cohesive zone model is employed to describe the transformation toughening behavior of a slowly propagating crack along an interface between a shape memory alloy and a linear elastic or elasto-plastic isotropic material. Small scale transformation zones and plane strain conditions are assumed. The crack growth is numerically simulated within a finite element scheme and its transformation toughening is obtained by means of resistance curves. It is found that the choice of the cohesive strength t₀ and the stress intensity factor phase angle φ greatly influence the toughening behavior of the bimaterial. The presented methodology is generalized for the case of an interface crack between a fiber reinforced shape memory alloy composite and a linear elastic, isotropic material. The effect of the cohesive strength t₀, as well as the fiber volume fraction are examined.     

Dynamic steady crack growth in elastic-plastic solids—propagation of strong discontinuities

The near-tip field of a mode I crack growing steadily under plane strain conditions is studied. A key issue is whether strong discontinuities can propagate under dynamic conditions. Theories which impose rather restrictive assumptions on the structure of an admissible deformation path through a dynamically propagating discontinuity have been proposed recently. Asymptotic solutions for dynamic crack growth, based on such theories, do not contain any discontinuities. In the present work a broader family of deformation paths is considered and we show that a discontinuity can propagate dynamically without violating any of the mechanical constitutive relations of the material. The proposed theory for the propagation of strong discontinuities is corroborated by very detailed finite element calculations. The latter shows a plane of strong discontinuity emanating from the crack tip (with its normal pointing in the direction of crack advance) and moving with the tip. Elastic unloading ahead of and/or behind the plane of discontinuity and behind the crack tip have also been observed.The numerical investigation is performed within the framework of a boundary layer formulation whereby the remote loading is fully specified by the first two terms in the asymptotic solution of the elasto-dynamic crack tip field, characterized by K₁, and T. It is shown that the family of near-tip fields, associated with a given crack speed, can be arranged into a one-parameter field based on a characteristic length, L_g, which scales with the smallest dimension of the plastic zone. This extends a previous result for quasi-static crack growth.     

Crack growth resistance of shape memory alloys by means of a cohesive zone model

Crack growth resistance of shape memory alloys (SMAs) is dominated by the transformation zone in the vicinity of the crack tip. In this study, the transformation toughening behavior of a slowly propagating crack in an SMA under plane strain conditions and mode I deformation is numerically investigated. A small-scale transformation zone is assumed. A cohesive zone model is implemented to simulate crack growth within a finite element scheme. Resistance curves are obtained for a range of parameters that specify the cohesive traction-separation constitutive law. It is found that the choice of the cohesive strength t₀ has a great influence on the toughening behavior of the material. Moreover, the reversibility of the transformation can significantly reduce the toughening of the alloy. The shape of the initial transformation zone, as well as that of a growing crack is determined. The effect of the Young's moduli ratio of the martensite and austenite phases is examined.     

A boundary element modeling of fatigue crack growth in a plane elastic plate

This paper presents an extension of a boundary element method to fatigue growth analysis of mixed-mode cracked plane elastic bodies. The method consists of the non-singular displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity element due to the author. In the boundary element implementation the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the modified maximum strain energy density criterion. In numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the boundary element method. Crack growth is simulated by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. Some numerical results of fatigue growth in a plane elastic plate with a center-inclined crack under uniaxial cyclic loading are given.     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

On the uses of special crack tip elements in numerical rock fracture mechanics

Numerical methods such as boundary element methods are widely used for the stress analysis in solid mechanics. These methods are also used for crack analysis in rock fracture mechanics. There are singularities for the stresses and displacements at the crack tips in fracture mechanics problem, which decrease the accuracy of the numerical results in areas very close to the crack ends. To overcome this, higher order elements and isoperimetric higher order elements have been used. Recently, special crack tip elements have been proposed and used in most of the numerical fracture mechanics models. These elements can drastically increase the accuracy of the results near the crack tips, but in most of the models only one special crack tip element has been used for each crack end. In this study the uses of higher order crack tip elements are discussed and a higher order displacement discontinuity method is used to investigate the effect of these elements on the accuracy of the results in some crack problems. The useful shape functions for two special crack tip elements, are derived and given in the text and appendix for both infinite and semi-infinite plane problems. In this analysis both Mode I and Mode II stress intensity factors are computed . Some example problems are solved and the computed results are compared with the results given in the literature. The numerical results obtained here are in good agreement with those cited in the literature. For the curved crack problem, the strain energy release rate, G can be calculated accurately in the vicinity of the crack tips by using the higher order displacement discontinuity method with a quadratic variation of displacement discontinuity elements and with two special crack tip elements at each crack end.     

A numerical analysis of perpendicular cracks under general in-plane loading with a hybrid displacement discontinuity method

In this paper, a numerical analysis of perpendicular cracks under general in-plane loading is performed by using a hybrid displacement discontinuity method which consists of the non-singular displacement discontinuity element presented by Crouch and Starfied and the crack tip displacement discontinuity elements by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the ordinary non-singular displacement discontinuity elements that cover the entire crack surface and the other boundary. The present numerical results show that the numerical approach is simple, yet very accurate for calculating numerically stress intensity factors for perpendicular cracks under general in-plane loading.     

FATIGUE GROWTH MODELING OF MIXED-MODE CRACK IN PLANE ELASTIC MEDIA

This paper presents an extension of a displacement discontinuity method with cracktip elements （a boundary element method） proposed by the author for fatigue crack growth analysis in plane elastic media under mixed-mode conditions. The boundary element method consists of the non-singular displacement discontinuity elements presented by Crouch and Starfield and the crack-tip displacement discontinuity elements due to the author. In the boundary element implementation the left or right crack-tip element is placed locally at the corresponding left or right crack tip on top of the non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the maximum circumferential stress criterion. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the numerical approach. Crack growth is modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characteristics of some related elements are adjusted according to the manner in which the boundary element method is implemented. As an example, the fatigue growth process of cracks emanating from a circular hole in a plane elastic plate is simulated using the numerical simulation approach.     

内部压力作用下矩形板中源于椭圆孔的分支裂纹应力强度因子的一种数值分析

应用一种边界元方法来研究内部压力作用下矩形板中源于椭圆孔的分支裂纹。该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者最近提出的裂尖位移不连续单元构成。在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界。本数值结果进一步证实这种数值方法对计算有限大板中复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示裂纹体几何对应力强度因子的影响。     

双轴载荷作用下源于椭圆孔的分支裂纹的一种边界元分析

利用一种边界元方法来研究双轴载荷作用下无限大板中源于椭圆孔的分支裂纹．该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者提出的裂尖位移不连续单元构成．在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界,文中算例说明本数值方法对计算平面弹性裂纹的应力强度因子是非常有效的。该文对双轴载荷作用下无限大板中源于椭圆孔的分支裂纹的数值结果进一步证实本数值方法对计算复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示双轴载荷及裂纹体几何对应力强度因子的影响。     

The application of the Eshelby equivalent inclusion method for unifying modulus and transformation toughening

When a crack is lodged in an inclusion, both difference between the modulus of the inclusion and matrix material and stress-free transformation strain of the inclusion will cause the near-tip stress intensity factor to be greater (amplification effect) or less (shielding or toughening effect) than that prevailing in a homogeneous material. In this paper, the inclusion may represent a second phase particle in composites and a transformation or microcracked process zone in brittle materials, which may undergo a stress-free transformation strain induced by phase transformation, microcracking, thermal expansion mismatch and so forth. A close form of solution is derived for predicting the toughening (or amplification) effect. The derivation is based on Eshelby equivalent inclusion approach that provides rigorous theoretical basis to unify the modulus and transformation contributions to the near-tip field. As validated by numerical examples, the developed formula has excellent accuracy for different application cases.     

Isolated crack in three-dimensional piezoelectric solid. Part II: Stress intensity factors for circular crack

This is part II of the work concerned with finding the stress intensity factors for a circular crack in a solid with piezoelectric behavior. The method of solution involves reducing the problem to a system of hypersingular integral equations by application of the unit concentrated displacement discontinuity and the unit concentrated electric potential discontinuity derived in part I [1]. The near crack border elastic displacement, electric potential, stress and electric displacement are obtained. Stress and electric displacement intensity factors can be expressed in terms of the displacement and the potential discontinuity on the crack surface. Analogy is established between the boundary integral equations for arbitrary shaped cracks in a piezoelectric and elastic medium such that once the stress intensity factors in the piezoelectric medium can be determined directly from that of the elastic medium. Results for the penny-shaped crack are obtained as an example.     

The analysis of crack problems with non-local elasticity

IntroductionTheclassicalconhnuummechanicshasbeenusedtosolvemanyproblemsinmacrofracturemechanics,butencountersdifficulheswhentheeffectofITilcrocharacteristicdimensionshouldbetakenintoaccount.Thestressfieldverynearthecracktipisstillnotclear.Somephenomenaofshortcrackscannotbeexplained["']andsomemechanismoffracturehasnotbeensolvedyet.Thenon-localelashcitytheoryseemsattractivetotheseproblems.Thetheoryofnon-localelasticity,establishedanddevelopedbyEringenetal[3),connectstheclassicalcontinuummechan…     

A special crack tip displacement discontinuity element

Based on the analytical solution to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite elastic solid and the note of the crack tip element by Crouch, in the present paper, the special crack tip displacement discontinuity element is developed. Further the analytical formulas for the stress intensity factors of crack problems in general plane elasticity are given. In the boundary element implementation the special crack tip displacement discontinuity element is placed locally at each crack tip on top of the non-singular constant displacement discontinuity elements that cover the entire crack surface. Numerical results show that the displacement discontinuity modeling technique of a crack presented in this paper is very effective.     

Finite deformation analysis of COD,J-integral and crack tip intense strain region in plane strain large-scale yielding

A finite element analysis was performed to simulate crack tip blunting and the development of the intense strain region in a small compact tension specimen (0.4 T CT) of SA533B-1 under plane strain large-scale yielding, with the condition of large-geometry change around the crack tip taken into consideration. The region where the equivalent plastic strain $\text{}\text{?}$g3_p is greater than 0.15 was defined as the intense strain region, which corresponded to the recrystallized-etched zone delineated experimentally around the blunting crack tip. The development of the intense strain region was discussed as a function of the J-integral and the crack opening displacement. A linear relationship was obtained between the plastic work W_p dissipated within the intense strain region and (J/σ_y)² or b², where b is the crack opening displacement, defined as the separation of the two points at which the boundary of the intense strain region surrounding the crack tip intersects with the free surfaces of the crack.     

Theoretical and experimental studies in fracture mechanics of crack-like defects under biaxial loading

It is a common point of view in fracture mechanics that, for any geometry of the body with a crack and any boundary conditions for the loading acting in the body plane, the stress and displacement components near the crack tip can be approximated in the framework of the theory of elasticity by a one-parameter or one-term representation, i.e., strictly in terms of the stress intensity coefficients K _I and K _II for an arbitrary failure crack [1, 2]. The authors of [2] specified the Westergaard function of the singular solution for a central crack under the biaxial loading of a plate. This approximate two-component solution has satisfactory accuracy. It is clear from [2] that this method cannot be admitted as a general statement [1], although it has long been assumed to be correct. The cause is that one cannot reasonably justify neglecting the second term in the Williams representation of the stress components in the plane case in the form of eigenfunction series; the contribution of this term in the rectangular coordinate system x, y is independent of the distance from the crack tip. This method may result in a serious mistake, from both the qualitative and quantitative viewpoints, in the prediction of local stresses, displacements, and related variables that are of interest. Apparently, this can best be demonstrated by an example of biaxial loading of a plate with a crack [1]. The unfounded neglect of the second term (whose contribution is independent of the distance from the crack tip) in the series representing the stress components is the source of the above-mentioned difficulties. In this problem, the influence of the load applied in the direction parallel to the crack plane manifests itself only in the second term of the series [3]. Therefore, this term should be clearly determined and studied in detail in the case of technological welding defects (faulty fusions, incomplete fusions, undercuts, and slag inclusions) and crack-like defects (scratches and cuts) in the base metal. The influence of the stress σ _OX along the crack axis on the stress tensor σ _x , σ _y , τ _xy and on the displacements u _x and u _y is confirmed by experimental studies of cracks by the photoelasticity method [4].     

Nonsteady motion of a transverse-shear crack across the interface between elastic media

This article examines the motion of a crack along the line joining two different elastic half-planes under the influence of variable shear stresses. Analogous to the case of a homogeneous medium [1–3], the law of motion of the edge is assumed to be known. Among the features of the physical situation being examined are the nonsymmetrical character of the solution with a symmetrical load distribution and the dependence of the number of Rayleigh wave which can be generated (two, one, none) on the ratios of the elastic parameters. The problem decomposes in the image space into a scalar problem of conjugating two functions reflecting the connection between the displacement discontinuity on the crack and the shear stress on the crack extension. The formula must then be inverted to represent the normal stress. The solution is constructed by the method of factorization, which was used in [2, 3] for a problem with a movable separation point for the boundary conditions. The properties of the Rayleigh boundary function for contacting elastic bodies are also studied. It is shown that the Holder continuity condition for the input functions is sufficient to determine the asymptotes at the edge of the crack, analogous to the case of steady crack movement [4]. With transformations of the convolutions, we used the methods of contour integration and applied the residue theorem. This made it possible to somewhat simplify the results [2]. The subject of crack starting is addressed in an examination of special types of loading. The solution of a similarity problem was given in [5].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 129–138, November–December, 1986.     

Kinked crack analysis by a hybridized boundary element/boundary collocation method

In this research a two dimensional displacement discontinuity method (which is a kind of indirect boundary element method) using higher order elements (i.e. a source element with a cubic variation of displacement discontinuities having four sub-elements) is used to obtain the displacement discontinuities along each boundary element. In this paper, three kinds of the higher order boundary elements are used: the ordinary elements, the kink elements and the special crack tip elements.The boundary collocation technique is used for the calculation of the displacement discontinuities at the center of each sub-elements. Again a special boundary collocation technique is used to treat the kinked source elements occur in the crack analysis. Considering the two source elements (each having four sub-elements) joined at a corner (kink point). The collocation points in the cubic element model which are outside of the kink point are moved to the crack kink then the displacement discontinuities on the left and right sides of the kink are calculated. The displacement discontinuities of the kink point are obtained by averaging the corresponding values of its left and right sides. The special crack tip elements are also treated by the boundary displacement collocation technique considering the singularity variation of the displacements and stresses near the crack tip. Some simple example problems are solved numerically by the proposed method. The numerical results are compared with the corresponding results obtained by the previous methods cited in the literature. This comparison shows a very good agreement between the results and verify the accuracy and validity of the proposed method.     

An analytical approach for the calculation of stress-intensity factors in transformation-toughened ceramics

Stress-induced transformation toughening in Zirconia-containing ceramics is described analytically by means of a quantitative model: A Griffith crack which interacts with a transformed, circular Zirconia inclusion. Due to its volume expansion, a ZrO₂-particle compresses its flanks, whereas a particle in front of the crack opens the flanks such that the crack will be attracted and finally absorbed. Erdogan's integral equation technique is applied to calculate the dislocation functions and the stress-intensity-factors which correspond to these situations. In order to derive analytical expressions, the elastic constants of the inclusion and the matrix are assumed to be equal.