Screw dislocations interacting with a partially debonded interface in cylindrically anisotropic composites

Interaction between screw dislocations and a partially debonded interface in cylindrically anisotropic composites subjected to uniform stress at infinity is investigated in this paper. Using Muskhelishvili’s complex variable method, the closed forms of complex potentials are obtained for a screw dislocation and a screw dislocation dipole located inside either matrix or inhomogeneity. Explicit expressions of stress intensity factors at the crack tips, image forces and image torques acting on dislocation or the center of dipole are provided. The results show that the crack and dislocation geometry combination plays an important role in the interaction between screw dislocations and interface crack. Furthermore, it is found that the anisotropy of solids may change the shielding and anti-shielding effects arising from screw dislocations and the equilibrium position of screw dislocations. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some novel or previously known results.     

Crack nucleation from a wedge disclination dipole with shift of rotation axes

This paper analyzes crack nucleation from a wedge disclination dipole in the presence of a remote stress, accounting for the shift of the rotation axes within the dipole arm of the disclinations. A Zener–Griffith crack is assumed and an energy method is employed for the analysis. A single energy equation determines the equilibrium crack lengths and the crack head opening. Uniaxial and biaxial dipoles are compared: in the former the disclinations share a common rotation axis while in the latter the axes are separate. The results show that stable and unstable cracks can nucleate from the positive disclination of the dipole, but some of them are energetically unfavorable. A uniaxial dipole is stable against crack nucleation when the axis is located away from the positive disclination. Biaxial dipoles are more stable when the rotation axis of each disclination approaches the defect line of the other disclination. If the negative disclinations of a uniaxial dipole and a biaxial dipole have the same axis shift, the critical nucleation stress of the biaxial dipole is larger if its positive disclination shift is more than that of the uniaxial dipole. Stable crack lengths generally increase, while the crack head openings decrease, with the axis shift of the positive disclination. The crack head opening to crack length ratio is of the order of 0.001–0.01, and can be higher if an applied stress is present.     

Wedge disclinations in the shell of a core–shell nanowire within the surface/interface elasticity

The elastic behaviors of a two-axes dipole of wedge disclinations and an individual wedge disclination located inside the shell of a free standing core–shell nanowire is studied within the surface/interface elasticity theory. The corresponding boundary value problem is solved using complex potential functions, defined through modeling the disclination dipole by two finite walls of infinitesimal edge dislocations. The stress field, disclination strain energies and image forces acting on the disclinations, are calculated and studied in detail. It is shown that the stresses are rather inhomogeneous across the nanowire cross section, change their signs and reach local maxima and minima far from the disclination lines in the bulk or on the surface of the nanowire. For negative values of the surface/interface modulus and relatively small values of the ratio of the shell and core shear moduli, the surface/interface effect manifests itself through non-classical stress oscillations along the shell free surface in the case of a disclination dipole and core–shell interface in both the cases of a disclination dipole and an individual disclination. The non-classical solution for the strain energy deviates from the classical solution with different effects caused by the surface/interface moduli on the wedge disclination dipole and an individual disclination. When the core is softer than the shell, the dipole with radial orientation of its arm has an unstable equilibrium position in the shell. In general, if the surface/interface modulus is positive, the surface/interface effects are rather weak; however, if it is negative, the effect can be very strong, especially near the shell surface.     

螺型位错偶极子与界面刚性线的弹性干涉

研究了晶体材料中螺型位错偶极子和界面刚性线夹杂的弹性干涉作用。利用复变函数方法,得到了该问题的复势函数以及应力场的封闭形式解答。求出了作用在螺型位错偶极子中心的像力和力偶矩,并分析了界面刚性线几何条件和不同材料特征组合对位错偶极子平衡位置的影响规律。研究结果表明:当位错偶极子不断靠近刚性线时,刚性线对螺型位错偶极子的运动有很强的排斥作用。当刚性线的长度和材料剪切模量比达到临界值时,可以改变偶极子和界面之间的干涉机理。同时,偶极子偶臂的方向对其自身的平衡也有很大的影响。     

Stress and strain energy of a periodic array of interfacial wedge disclination dipoles in a transversely isotropic bicrystal

Analytical closed-form solutions are obtained for the elastic stress and strain energy density fields of a periodic array of interfacial wedge disclination dipoles in a bicrystal. The adjoining crystals are transversely isotropic with maximum dissimilar in-plane crystallographic orientations (0 and π/2). The solutions are obtained by the method of image dislocations. The strain energy per unit area of the bicrystal interface is also obtained numerically. The results show that significant discrepancies can exist between the bicrystal and the isotropic homogeneous solutions. The rates of decrease/increase of the strain energy density and stresses from the interface are smaller in the crystal whose larger stiffness direction is perpendicular to the interface. Also, the strain energy of the bicrystal boundary is a function of the dipole arm length (2a) and period (L). The maximum strain energy occurs at a/L=0.25 and is estimated to be ∼8.9 J/m² if the dipole period is 10 nm and the disclination strength is π/2.     

The interaction between a screw dislocation and a nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface

The interaction between a screw dislocation and a circular nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface is investigated. By using Riemann-Schwartz’s symmetry principle integrated with the analysis of singularity of complex functions and the conformal mapping technique, the analytical expressions of the stress field in both the circular nano-inhomogeneity and the infinite matrix, the image force acting on the screw dislocation and the stress intensity factor at the crack tip are obtained. The influence of elastic mismatch of materials, inhomogeneity size, interface stress, wedge crack opening angle and the relative location of dislocation on the image force and on the equilibrium position of the screw dislocation and the shielding effect of the screw dislocation are discussed in detail. The results show that interface stress has a significant impact on the movement of dislocations near the interface, and the effect of interface stress enhances when the inhomogeneity radius decreases. With the decrease in the wedge crack opening angle, the influence of interface stress on the movement of the screw dislocation and on the SIF enhances. With the increment of the relative shear modulus, the influence of interface stress weakens with the screw dislocation locating in the inhomogeneity and strengthens with the screw dislocation locating in the matrix. When the screw dislocation is located in the inhomogeneity, the positive (negative) interface stress increases (decreases) the shielding effect, while this phenomenon is opposite when the screw dislocation locates in the matrix.     

Stress analysis on a Zener crack nucleation from an eccentric wedge disclination in a cylinder

The stress relaxation of an eccentric (off-center) negative wedge disclination in an isotropic homogeneous cylinder by nucleation of a Zener crack has been investigated with a continuum model. The nucleated Zener crack is simulated with distributed edge dislocations. The stress intensity factor (SIF) at the sharp tip of the Zener crack is computed through solving the singular integral equations formulated. By enforcing the fracture criterion at the sharp tip, the critical disclination power to nucleate a Zener crack is determined. The equilibrium crack lengths of the crack are then calculated when the disclination power is above the critical value. It is found that there is a special position at which the critical disclination power reaches the minimum value. As the disclination deviates from this position, the critical disclination power increases. Two or four equilibrium crack lengths could be found for the Zener crack, dependent upon the power and off-center position of the disclination. The influence of the off-center distance on the equilibrium crack lengths and the dependence of the critical disclination power and stable equilibrium crack lengths on cylinder radius are also discussed.     

Stress investigation on a Griffith crack initiated from an eccentric disclination in a cylinder

Disclinations are rotational line defects which may be introduced in metal wires during the manufacturing process. In this work, the relaxation of an eccentric (off-center) negative wedge disclination in a cylinder by nucleation of a Griffith crack is investigated. The nucleated crack is simulated with distributed edge dislocations. The stress intensity factors (SIFs) of the crack are evaluated by solving a set of singular integral equations. By enforcing the condition that the SIFs at the two crack tips should keep the same value in the nucleation process, the crack length growth on each side of the wedge disclination is determined. The critical disclination power and equilibrium crack lengths are then numerically determined. Some important characteristics of the Griffith crack nucleation are revealed. (1) The two tips of the nucleated Griffith crack grow asymmetrically when the disclination locates eccentrically. The tip closer to the cylinder edge travels a shorter length. This asymmetry is getting more severe as the normalized off-center distance increases. (2) The critical disclination power increases monotonically with the normalized off-center distance. (3) The normalized stable equilibrium crack length decreases as the normalized off-center distance increases while the normalized unstable equilibrium crack length shows an opposite dependence. The dependence of the critical disclination power and the equilibrium crack lengths on the disclination power and cylinder radius is also discussed in this work. It is believed that this work helps to predict the strength of disclinated metal wires at various length scales.     

双金属层合板垂直界面裂纹疲劳的扩展过程

采用四点弯曲加载方式进行奥氏体不锈钢／低碳锅炉钢双金属层合板垂直界面裂纹的疲劳扩展实验,研究了组元强度配合、爆炸焊接影响的区域性能（晶粒大小、形变强化、界面脱粘、独立塑性区等）对裂纹扩展行为的影响,以及垂直界面裂纹疲劳扩展的不同过程及其所对应的扩展机制．结果表明：由于强度错配,裂纹起始于高强度材料一倜时其疲劳扩展速率提高,而起始于低强度材料一倜时其疲劳扩展速率降低;当裂纹尖端接近界面时,界面区域的存在对上述两种情况下疲劳裂纹的扩展均起到了一定的屏蔽减速作用．     

压电材料中螺型位错偶极子与圆弧形界面裂纹的电弹干涉效应

研究了在无穷远力电荷载作用下广义螺型位错偶极子与圆弧形界面裂纹的电弹干涉作用。运用复变函数方法,导出了该问题的一般解答,并获得了界面上只有一条裂纹时的封闭形式解,求得了基体及夹杂区域复势函数、广义应力场、裂纹尖端的广义应力强度因子以及作用在螺型位错偶极子上的位错力和力偶矩。讨论了裂纹长度、压电材料电弹常数以及位错偶极子的位置对裂纹尖端应力强度因子、偶极子中心的位错力和像力偶矩的影响。     

螺型位错偶极子与圆形夹杂界面裂纹的弹性干涉

该文研究了螺型位错偶极子和圆形夹杂界面裂纹的弹性干涉作用。利用复变函数方法,得到了该问题的复势函数以及应力场的封闭形式解答。求出了裂纹尖端的应力强度因子以及作用在螺型位错偶极子中心的像力和像力偶矩,并分析了位错偶极子对应力强度因子的影响及界面裂纹几何条件和不同材料特征组合对位错偶极子平衡位置的影响规律。研究结果表明:位错偶极子对应力强度因子具有很强的屏蔽或反屏蔽效应;硬夹杂排斥位错偶极子,而裂纹吸引位错偶极子,在一定条件下,位错偶极子在裂纹附近出现一个平衡位置;当裂纹的长度和材料剪切模量比达到临界值时,可以改变偶极子和界面之间的干涉机理。同时,裂纹长度对位错偶极子中心像力偶矩也有很大的影响。     

A hybrid finite element analysis of interface cracks

A versatile hybrid finite element scheme consisting of special crack-tip elements and crack face contact elements is developed to analyse a partially closed interface crack between two dissimilar anisotropic elastic materials. The crack-tip element incorporates higher-order asymptotic solutions for an interfacial crack tip. These solutions are obtained from complex variable methods in Stroh formalism. For a closed interfacial crack tip, a generalized contact model in which the crack-tip oscillation is eliminated is adopted in the calculation. The hybrid finite element modelling allows the stress singularity at an open and closed crack tip to be accurately treated. The accuracy and convergence of the developed scheme are tested with respect to the known interface crack solutions. Utilizing this numerical scheme, the stress intensity factors and contact zone are calculated for a finite interface crack between a laminated composite material.     

Effect of cooperative grain boundary sliding and migration on dislocation emitting from a semi-elliptical blunt crack tip in nanocrystalline solids

A theoretical model is established to investigate the interaction between the cooperative grain boundary (GB) sliding and migration and a semi-elliptical blunt crack in deformed nanocrystalline materials. By using the complex variable method, the effect of two disclination dipoles produced by the cooperative GB sliding and migration process on the emission of lattice dislocations from a semi-elliptical blunt crack tip is explored. Closed-form solutions for the stress field and the force acting on the dislocation are obtained in complex form, and the critical stress intensity factors for the first dislocation emission from a blunt crack under mode I and mode II loadings are calculated. Then, the influence of disclination strength, curvature radius of blunt crack tip, crack length, locations and geometry of disclination dipoles, and grain size on the critical stress intensity factors is presented detailedly. It is shown that the cooperative GB sliding and migration and the grain size have significant influence on the dislocation emission from a blunt crack tip.     

Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media

A complete form of stress and electric displacement fields in the vicinity of the tip of an interfacial crack, between two dissimilar anisotropic piezoelectric media, is derived by using the complex function theory. New definitions of real-valued stress and electric displacement intensity factors for the interfacial crack are proposed. These definitions are extensions of those for cracks in homogeneous piezoelectric media. Closed form solutions of the stress and electric displacement intensity factors for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media are also obtained by using the mutual integral.     

Stress intensity factors for penny-shaped cracks perpendicular to graded interfacial zone of bonded bi-materials

This paper examines the stress intensity factors that are associated with a penny-shaped crack perpendicular to the interface of a bi-material bonded with a graded interfacial zone. Elastic modulus of the graded interfacial zone is assumed to be an exponential function of the depth. The stress intensity factors are calculated numerically using a so-called generalized Kelvin solution based boundary element method. Three cases of normal or shear tractions acting on the crack surfaces are examined. Values of the stress intensity factors are examined by taking into account the effects of the following four parameters: (a) the crack front position; (b) the non-homogeneity parameter of the graded interfacial zone; (c) the crack distance to the graded interfacial zone; and (d) the graded interfacial zone thickness. The numerical results are compared well with existing solutions under some degenerated conditions. These results are useful to furthering our knowledge on fracture behavior of bi-material systems with or without a graded interfacial zone.     

Analysis of Bimaterial Wedges Using a New Singular Finite Element

This paper is concerned with the singular stress field at the vertex of a bimaterial wedge under in-plane loading. The boundary value problem is initially formulated in terms of the complex function method. The eigenequations are obtained using the continuity conditions along the interface and the traction-free conditions along the free edges, leading to the development of explicit expressions for the singular stress and displacement fields for a general bimaterial wedge. These expressions are then used to develop a new singular finite element. This element enables the determination of the singular stress field and the associated stress intensity factors reliably and efficiently. To establish the validity of the method, test cases are examined and compared with existing solutions. The method is then applied to evaluate the effect of the wedge geometry and the elastic mismatch upon the resulting stress intensity factors.     

Stress intensity factor analysis at an interfacial corner between anisotropic bimaterials under thermal stress

A numerical method using a path-independent H-integral based on the Betti reciprocal principle was developed to analyze the stress intensity factors of an interfacial corner between anisotropic bimaterials under thermal stress. According to the theory of linear elasticity, asymptotic stress near the tip of a sharp interfacial corner is generally singular as a result of a mismatch of the materials’ elastic constants. The eigenvalues and the eigenfunctions are obtained using the Williams eigenfunction method, which depends on the materials’ properties and the geometry of an interfacial corner. The order of the singularity related to the eigenvalue is real, complex or power-logarithmic. The amplitudes of the singular stress terms can be calculated using the H-integral. The stress and displacement fields around an interfacial corner for the H-integral are obtained using finite element analysis. A proposed definition of the stress intensity factors of an interfacial corner involves a smooth expansion of the stress intensity factors of an interfacial crack between dissimilar materials. The asymptotic solutions of stress and displacement around an interfacial corner are uniquely obtained using these stress intensity factors.     

Conducting cracks in dissimilar piezoelectric media

Complete stress and electric fields near the tip of a conducting crack between two dissimilar anisotropic piezoelectric media, are obtained in terms of two generalized bimaterial matrices proposed in this paper. It is shown that the general interfacial crack-tip field consists of two pairs of oscillatory singularities. New definitions of real-valued stress and electric field intensity factors are proposed. Exact solutions of the stress and electric fields for basic interface crack problems are obtained. An alternate form of the J integral is derived, and the mutual integral associated with the J integral is proposed. Closed form solutions of the stress and electric field intensity factors due to electromechanical loading and the singularities for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media, are also obtained by using the mutual integral.     

坝基界面在非线性水压力驱动下的非线性断裂过程模拟

基于多边形比例边界有限元法和粘聚裂缝模型提出了混凝土坝坝基界面在随缝宽非线性变化的水压力驱动下的非线性断裂数值模型。混凝土和基岩采用多边形比例边界单元模拟,界面裂缝的断裂过程区采用粘性界面单元模拟。因为界面裂缝总是处于复合断裂模态,故同时引入了法向和切向的界面单元,且考虑了裂纹面作用有法向和切向任意荷载时的应力强度因子求解。以裂尖为原点,裂尖附近的位移场和应力场在径向上解析求解,在环向具有有限元精度。因此无需在裂尖附近加密网格或采用富集技术即可求得高精度的解。对于界面断裂,可模拟出与两种材料差异性相关的非1/2奇异性。断裂过程区的水压力随缝面宽度变化,采用指数函数的形式进行表征,通过参数调整可实现不同分布的水压力的模拟。水压力与粘聚力考虑为与裂缝宽度相关的组合函数,便于非线性迭代的实现。结合多边形网格生成和重剖分技术,可方便地模拟界面裂缝在水力驱动下的扩展过程。算例研究表明了该文模型的有效性,从中也可看出考虑缝内水压及其具体分布形式对研究坝的稳定性具有重要影响。     

A penny-shaped crack above the pole of a spherical inhomogeneity

Stress investigation for the problem of a penny-shaped crack located above the pole of a spherical particle (inhomogeneity) in 3D elastic solid under tension has been carried out. Both the inhomogeneity and the solid are isotropic but have different elastic moduli. The analysis is based on Eshelby's equivalent inclusion method and superposition theory of elasticity. An approximation according to the Saint-Venant principle is made in order to decouple the interaction between the crack and the inhomogeneity. An analytical solution for the stress intensity factors on the boundary of the crack is thus evaluated. It is found that both Mode I and Mode II intensity factors exist, even the loading applied at infinity is uniform tension. Results obtained show that shielding and anti-shielding (amplifying) effects of the inhomogeneity to the crack are solely determined by the modulus ratios of the inhomogeneity to the matrix. Numerical examples also indicate the interaction between the crack and the inhomogeneity is strongly influenced by the distance between the centers.