Analysis method of planar interface cracks of arbitrary shape in three-dimensional transversely isotropic magnetoelectroelastic bimaterials

Using the fundamental solutions for three-dimensional transversely isotropic magnetoelectroelastic bimaterials, the extended displacements at any point for an internal crack parallel to the interface in a magnetoelectroelastic bimaterial are expressed in terms of the extended displacement discontinuities across the crack surfaces. The hyper-singular boundary integral–differential equations of the extended displacement discontinuities are obtained for planar interface cracks of arbitrary shape under impermeable and permeable boundary conditions in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. An analysis method is proposed based on the analogy between the obtained boundary integral–differential equations and those for interface cracks in purely elastic media. The singular indexes and the singular behaviors of near crack-tip fields are studied. Three new extended stress intensity factors at crack tip related to the extended stresses are defined for interface cracks in three-dimensional transversely isotropic magnetoelectroelastic bimaterials. A penny-shaped interface crack in magnetoelectroelastic bimaterials is studied by using the proposed method.The results show that the extended stresses near the border of an impermeable interface crack possess the well-known oscillating singularity r^?1/2±iε or the non-oscillating singularity r^?1/2±κ. Three-dimensional transversely isotropic magnetoelectroelastic bimaterials are categorized into two groups, i.e., ε-group with non-zero value of ε and κ-group with non-zero value of κ. The two indexes ε and κ do not coexist for one bimaterial. However, the extended stresses near the border of a permeable interface crack have only oscillating singularity and depend only on the mechanical loadings.     

Corrigendum to “Solution of a crack in an electrostrictive solid” [International Journal of Solids and Structures 47 (2010) 444–453]

In our previous work [Gao, C.F., Mai, Y.W., Zhang, N., 2010. Solution of a crack in an electrostrictive solid. International Journal of Solids and Structures 47, 444–453.] the intensity factor of the total stress for an impermeable crack is directly written by using the corresponding result of a permeable crack. This is based on the fact that an impermeable crack can be considered as a special case of a permeable crack where the electric field is not zero. However, the singularity of total stresses for the impermeable crack can also be analyzed directly from the complex potentials. In this Corrigendum, the singularity of the total stresses is further studied for the impermeable crack, and the intensity factors are re-derived by using the obtained complex potentials. It is shown that for an impermeable crack, the total stresses still have an inverse square-root singularity but their intensity factor is different from that obtained by the solution of a permeable crack. Therefore, it is concluded that solutions for impermeable cracks cannot be obtained directly from those of permeable cracks, since the assumption of the electric boundary condition has not only influenced the electric fields on the crack-faces but also on the electric body force inside the material.     

A magnetoelectroelastic medium with an elliptical cavity under combined mechanical–electric–magnetic loading

The solution for an elliptical cavity in an infinite two-dimensional magnetoelectroelastic medium subject to remotely uniformly applied combined mechanical–electric–magnetic loadings is obtained by using the Stroh formalism and the exact boundary conditions along the surface of the cavity. By letting the minor-axis of the cavity to zero the solution for a crack is deduced. A self-consistent method is proposed to calculate the real crack opening under the combined mechanical–electric–magnetic loadings. The method requires that the crack opening is the minor-axis of the elliptical opening profile. Beside the real crack solution, four different extreme models, i.e., the impermeable crack, permeable crack, electrically impermeable and magnetically permeable crack and electrically permeable and magnetically impermeable crack, are discussed. An expression of the strain energy density factor is derived. Numerical results of the strain energy density at the crack tip are given for a BaTiO₃–CoFe₂O₄ composite with the piezoelectric BaTiO₃ material being the inclusion and the magnetostrictive CoFe₂O₄ material being the matrix. The effects of the proportion of the two phases, permeability of the crack to electric and magnetic fields, the electric and magnetic loadings on the strain energy density factor are discussed.     

Anomalies associated with energy release parameters for cracks in piezoelectric materials

For a central crack in a piezoelectric plate, the mode-I stress intensity factor (K_I), electric displacement intensity factor (K_D), energy release rates (G, G_M) and energy density factor (S) are obtained from the finite element results. For the impermeable crack, the numerical results of K_I and K_D are coupled; this error is contrary to the uncoupled analytical solutions. The error has little effect on the total energy release rate G and energy density factor S, but in some cases, large errors in the mechanical energy release rate G_M are observed. G is global while SED is local. Also G is negative which defies physics where energy cannot be created while crack attempts to extend as implied by G. Computations should be made for the J-integral and also show that J becomes negative. What this shows is that the global fracture energy criterion is not suitable to address the local release of energy because it includes the overall energy which are irrelevant to fracture initiation being a local behavior. In addition, the case study shows that the energy density theory is the better fracture criterion for the piezoelectric material. According to the results of S, it retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees with analytical and experimental evidence in the past references.     

The determination of dynamic fracture toughness of AISI 4340 steel by the shadow spot method

Dynamic crack propagation experiments have been performed using wedge loaded double cantilever beam specimens of an austenitized, quenched and tempered 4340 steel. Measurements of the dynamic stress intensity factor have been made by means of the optical method of caustics. The interpretation of experimental data, obtained from the shadow spot patterns photographed with a Cranz-Schardin high speed camera, is based on an elastodynamic analysis. The instantaneous value of the dynamic stress intensity factor K^d_I is obtained as a function of crack tip velocity. Finally, the interaction of reflected shear and Rayleigh waves with the moving crack tip stress field is considered.     

Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading

Analytical solutions for an anti-plane Griffith moving crack inside an infinite magnetoelectroelastic medium under the conditions of permeable crack faces are formulated using integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic material. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities. The moving speed of the crack have influence on the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF), while the dynamic stress intensity factor (DSIF) does not depend on the velocity of the moving crack. When the crack is moving at very lower or very higher speeds, the crack will propagate along its original plane; while in the range of M_c1 < M < M_c2, the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

Dynamic stress-intensity factors for unsymmetric dynamic isochromatics

The mixed mode, near-field state of stresses sourrounding a crack propagating at constant velocity is used to derive a relation between the dynamic stress-intensity factorsK _I,K _II, the remote stress component σ _ox and the dynamic isochromatics. This relation, together with an over-deterministic least-square method, form the basis of a datareduction procedure for extracting dynamic,K _I,K _II and σ _ox from the recorded dynamic photoelastic pattern surrounding a running crack. The overdeterministic least-square method is also used to fit static isochromatics to the numerically generated dynamic isochromatics. The resultant staticK _I,K _II and σ _ox are compared with the corresponding dynamic values and estimats of errors involved in using static analysis to process dynamic isochromatic data are obtained. The data-reduction procedure is then used to evaluate the branching stress-intensity factor associated with crack branching and the mixed-mode stress-intensity factors associated with crack curving.     

Propagation of an anti-plane moving crack in a functionally graded piezoelectric strip

Summary The propagation of an anti-plane moving crack in a functionally graded piezoelectric strip (FGPS) is studied in this paper. The governing equations for the proposed analysis are solved using Fourier cosine transform. The mixed boundary value problems of the anti-plane moving crack, which is assumed to be either impermeable or permeable, are formulated as dual integral equations. By appropriate transformations, the dual integral equations are reduced to Fredholm integral equations of the second kind. For the impermeable crack, the stress intensity factor (SIF) of the crack in the FGPS depends on both the mechanical and electric loading, whereas, the SIF for the permeable crack depends only on the mechanical loading. The results obtained show that the gradient parameter of the FGPS and the velocity of the crack have significant influence on the dynamic SIF.Support from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7081/00E) is acknowledged. Support from the National Natural Science Foundation of China (Project No. 10072041) is also acknowledged.     

Dynamic analysis of a penny-shaped dielectric crack in a magnetoelectroelastic solid under impacts

The transient response of a magneto-electro-elastic material with a penny-shaped dielectric crack subjected to in-plane magneto-electro-mechanical impacts is made. To simulate an opening crack with a dielectric interior, the crack-face electromagnetic boundary conditions are supposed to depend on the crack opening displacement and the jumps of electric and magnetic potentials across the crack. Four ideal crack-face electromagnetic boundary conditions involving a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions can be reduced. The Laplace and Hankel transform techniques are further utilized to solve the mixed initial-boundary-value problem. Three coupling Fredholm integral equations are obtained and solved by the composite Simpson's rule. Dynamic field intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD), electric potential and magnetic potential are given in the Laplace transform domain. By means of a numerical inversion of the Laplace transform, numerical results are calculated to show the variations of the physical parameters of concern versus the normalized time in graphics. The effects of applied electric and magnetic loads on the dynamic intensity factors of stress and COD, and the dynamic energy release rate for a BaTiO₃-CoFe₂O₄ composite with a penny-shaped vacuum crack are discussed in detail.     

Crack-tip dynamic isochromatics in the presence of small-scale yielding

A plasticity correction factor for the dynamic stress-intensity factor,K _I ^dyn , associated with a propagating crack tip in the presence of small-scale yielding, is derived from Kanninen's solution for a constant-velocity Yoffe crack with a Dugdale-strip yield zone. Distortions in the otherwise elastic isochromatics surrounding the constant-velocity crack tip are also studied by the use of this model. This plasticity correction factor is then used to evaluateK _I ^dyn from the dynamic isochromatics of a propagating crack in a 3.2-mm-thick polycarbonate wedge-loaded rectangular double-cantilever-beam specimen. The correctedK _I ^dyn is in good agreement with the corresponding values computed by a dynamic, elastic-plastic finite-element code executed in its generation mode.     

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.     

An interface crack with contact zones in a piezoelectric/piezomagnetic bimaterial

An interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric/piezomagnetic spaces under the action of a remote mechanical loading, magnetic and electric fluxes as well as concentrated forces at the crack faces is considered. Assuming that all fields are independent on the coordinate x ₂ co-directed with the crack front, the stresses, the electrical and the magnetic fluxes as well as the derivatives of the jumps of the displacements, the electrical and magnetic potentials are presented via a set of analytic functions in the (x ₁, x ₃)-plane with a cut along the crack region. Two cases of magneto-electric conditions at the crack faces are considered. The first case assumes that the crack is electrically and magnetically permeable, and in the second case the crack is assumed electrically permeable while the open part of the crack is magnetically impermeable. For both these cases due to the above-mentioned representation the combined Dirichlet–Riemann boundary value problems have been formulated and solved exactly. Stress, electric and magnetic induction intensity factors are found in a simple analytical form. Transcendental equations and a closed form analytical formula for the determination of the real contact zone length have been derived for both cases of magnetic conditions in the crack region. For a numerical illustration of the obtained results a bimaterial BaTiO₃–CoFe₂O₄ with different volume fractions of BaTiO₃ has been used, and the influence of the mechanical loading and the intensity of the magnetic flux upon the contact zone length and the associated intensity factors as well as the energy release rate has been shown.     

Effect of residual surface stress on the fracture of nanoscale materials

The problem of a mode I crack in nanomaterials under a remote mechanical load is investigated. The effect of the residual surface stress on the crack surface is considered and the solutions to the crack opening displacement (COD) and the stress intensity factor (K_I) are obtained. The results show that the surface effect on the crack deformation and crack tip field are prominent at nanoscale. Moreover, COD and K_I are influenced by the residual surface stress not only on the surface near the crack tip region but also on the entire crack surface.     

A mixed electric boundary value problem for a two-dimensional piezoelectric crack

In this paper, a mixed electric boundary value problem for a two-dimensional piezoelectric crack problem is presented, in the sense that the crack face is partly conducting and partly impermeable. By the analytical continuation method, the unknown electric charge distributions on the upper and lower conducting crack faces are reduced to two decoupled singular integral equations and then these two equations are converted into algebraic equations to find the full field solution. Though the results suggest that the stress intensity factors at the crack tip are identical to those of conventional piezoelectric materials, but the electric field and electric displacement are related to the electric boundary conditions on the crack faces. The electric field and electric displacement are singular not only at crack tips but also at the junctures between the impermeable part and conducting parts. Numerical results for the variations of the electric field, electric displacement field and J-integral with respect to the normalized impermeable crack length are shown. Some discussions for the energy release rate and the J-integral are made.     

压电压磁材料中正n边形孔边裂纹分析

基于电磁复合材料力学,运用Stroh型公式和复变函数方法,针对压电压磁材料中含正n边形孔边裂纹反平面问题进行了研究。利用Schwarz-Christoffel变换技术,结合Cauchy积分公式和留数定理,导出了磁电全非渗透型边界条件下任意正n边形裂纹尖端场强度因子和能量释放率的解析解。当缺失磁场时,所得解退化为已有结果,以此验证方法的有效性。通过数值算例,对比分析了n=3,n=4和n=5三种特殊情形对应的孔口边长、裂纹长度和受到的力、电和磁载荷对等效场强度因子和无量纲能量释放率的影响规律。研究结果发现,正n边形孔洞的尺寸和裂纹长度均会促进裂纹扩展,且前者的影响更显著一些;正n边形边的数量增加会阻止裂纹的扩展;在磁电全非渗透型边界条件下,机械载荷始终促进裂纹的扩展,电位移载荷可以促进或抑制裂纹的扩展,磁载荷对裂纹的扩展贡献较少。本研究结果适用于任意正n边形孔边裂纹求解问题,为压电压磁材料元器件的优化设计和断裂特性分析提供了新思路。     

Analytical and experimental evaluations of the influence of fracture surface roughness,its sliding actions and the associated plasticity on fatigue crack propagation

An investigation of fatigue crack propagation in rectangular AM60B magnesium alloy plates containing an inclined through crack is presented in this paper. The behavior of fatigue crack growth in the alloy is influenced by the fracture surface roughness. Therefore, in the present investigation, a new model is developed for estimating the magnitude of the frictional stress intensity factor, k_f, arising from the mismatch of fracture surface roughness during in-plane shear. Based on the concept of k_f, the rate of fatigue crack propagation, db/dN, is postulated to be a function of the effective stress intensity factor range, Δk_eff. Subsequently, the proposed model is applied to predict crack growth due to fatigue loads. Experiments for verifying the theoretical predictions were also conducted. The results obtained are compared with those predicted using other employed mixed mode fracture criteria and the experimental data.     

Stress intensity factor and crack velocity relationship for polyester/TiO2 nanocomposites

The dynamic fracture behavior of polyester/TiO₂ nanocomposites has been characterized and compared with that of the matrix material. A relationship between the dynamic stress intensity factor,K _I and the crack tip velocity,å, has been established. Dynamic photoelasticity coupled with high-speed photography has been used to obtain crack tip velocities and dynamic stress fields around the propagating cracks. Birefringent coatings were used to conduct the photoelastic study due to the opaqueness of the nanocomposites. Single-edge notch tension and modified compact tension specimens were used to obtain a broad range of crack velocities. Fractographic analysis was conducted to understand the fracture process. The results showed that crack arrest toughness in nanocomposites was 60% greater than in the matrix material. Crack propagation velocities prior to branching in nanocomposites were found to be 50% greater than those in polyester.     

Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material

Based on mechanics of anisotropic material, the dynamic crack propagation problem of I/II mixed mode crack in an infinite anisotropic body is investigated. Expressions of dynamic stress intensity factors for modes I and II crack are obtained. Components of dynamic stress and dynamic displacements around the crack tip are derived. The strain energy density theory is used to predict the dynamic crack extension angle. The critical strain energy density is determined by the strength parameters of anisotropic materials. The obtained dynamic crack tip fields are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. The obtained results show Crack propagation characteristics are represented by the mechanical properties of anisotropic material, i.e., crack propagation velocity M and fiber direction α. In particular, the fiber direction α and the crack propagation velocity M give greater influence on the variations of the stress fields and displacement fields. Fracture angle is found to depend not only on the crack propagation but also on the anisotropic character of the material.     

Fracture analysis of circular-arc interface cracks in piezoelectric materials

The anti-plane problem of N arc-shaped interfacial cracks between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix is investigated by means of the complex variable method. Cracks are assumed to be permeable and then explicit expressions are presented, respectively, for the electric field on the crack faces, the complex potentials in media and the intensity factors near the crack-tips. As examples, the corresponding solutions are obtained for a piezoelectric bimaterial system with one or two permeable arc-shaped interfacial cracks, respectively. Additionally, the solutions for the cases of impermeable cracks also are given by treating an impermeable crack as a particular case of a permeable crack. It is shown that for the case of permeable interfacial cracks, the electric field is jumpy ahead of the crack tips, and its intensity factor is always dependent on that of stress. Moreover all the field singularities are dependent not only on the applied mechanical load, but also on the applied electric load. However, for the case of a homogeneous material with permeable cracks, all the singular factors are related only to the applied stresses and material constants.     

A novel method in determination of dynamic fracture toughness under mixed mode I/II impact loading

The objective of this paper is to propose a novel methodology for determining dynamic fracture toughness (DFT) of materials under mixed mode I/II impact loading. Previous experimental investigations on mixed mode fracture have been largely limited to qusi-static conditions, due to difficulties in the generation of mixed mode dynamic loading and the precise control of mode mixity at crack tip, in absence of sophisticated experimental techniques. In this study, a hybrid experimental–numerical approach is employed to measure mixed mode DFT of 40Cr high strength steel, with the aid of the split Hopkinson tension bar (SHTB) apparatus and finite element analysis (FEA). A fixture device and a series of tensile specimens with an inclined center crack are designed for the tests to generate the components of mode I and mode II dynamic stress intensity factors (DSIF). Through the change of the crack inclination angle β (=90°, 60°, 45°, and 30°), the K_II/K_I ratio is successfully controlled in the range from 0 to 1.14. A mixed mode I/II dynamic fracture plane, which can also exhibit the information of crack inclination angle and loading rate at the same time, is obtained based on the experimental results. A safety zone is determined in this plane according to the characteristic line. Through observation of the fracture surfaces, different fracture mechanisms are found for pure mode I and mixed mode fractures.