A generally applicable approximate solution for mixed mode crack-inclusion interaction

Summary Based on the transformation toughening theory an approximate solution is developed for predicting the stress intensity factor for a crack interacting with an inclusion of arbitrary shape and size under I/II mixed mode loading conditions. The transformation strains in the inclusion induced by the crack tip field and the remotely applied stresses are evaluated based on the Eshelby equivalent inclusion theory. As validated by detailed finite element analyses, the solution is applicable with good accuracy for the inclusion of arbitrary shape and large size under mixed mode loadings.     

Crack-inclusion interaction for mode I crack analyzed by Eshelby equivalent inclusion method

Several simple formulas have been developed to predict the variations of stress intensity factors for mode I crack induced by the stiffness and geometry of the near crack-tip inclusion. The derivation of the fundamental formula is based on the transformation toughening theory. The unconstrained mismatch strains between matrix and inclusion, which induce the variation of the near crack-tip field, are estimated based on the Eshelby equivalent inclusion approach. As validated by numerical examples, the developed formulas have satisfactory accuracy for a wide range of the modulus ratio between inclusion and matrix as long as the inclusion is located in the K ₀-controlled field.     

Some simple formulas to predict the variation of stress intensity factors for mode I crack induced by near crack-tip inclusion

Several simple formulas have been developed to predict the variations of stress intensity factors (SIFs) for mode I crack induced by the inclusion within crack-tip field. The derivation of the fundamental formula is based on the transformation toughening theory. The unconstrained mismatch strains between matrix and inclusion, which induce the variation of the near crack-tip field, are estimated from the remote applied SIF K₀. As validated by numerical examples, the developed formulas have satisfactory accuracy for wide range of the modulus ratios between inclusion and matrix as long as the inclusion is located in the K₀-controlled field.     

Some simple formulas to predict the variation of stress intensity factors for mode I crack induced by near crack-tip inclusion

Several simple formulas have been developed to predict the variations of stress intensity factors (SIFs) for mode I crack induced by the inclusion within crack-tip field. The derivation of the fundamental formula is based on the transformation toughening theory. The unconstrained mismatch strains between matrix and inclusion, which induce the variation of the near crack-tip field, are estimated from the remote applied SIF K₀. As validated by numerical examples, the developed formulas have satisfactory accuracy for wide range of the modulus ratios between inclusion and matrix as long as the inclusion is located in the K₀-controlled field.     

Static and dynamic response of sector-shaped graphene sheets

Nonlocal elasticity theory is presented for the free vibration and bending analysis of nano-scaled graphene sheets having a sector shape. An eight-node curvilinear domain is used for transformation of the governing equation of motion of sector graphene from physical region to computational region in conjunction with the Kirchhoff plate theory. The discrete singular convolution method is employed for numerical solutions of resulting nonlocal governing differential equations and related boundary conditions. Then, the effects of nonlocal parameters, mode numbers, sector angles, and radius ratios on the static and vibration results of nano-scaled sector-shaped graphene sheets are discussed.     

The Effect of a Plastically Deformed Zone Near Crack Tip on the Stress Intensity Factors

A plastically deformed particle ahead of a crack-tip is treated as an equivalent transformation inclusion by means of Eshelby equivalent inclusion theory. A general solution to determine the effect of the plastically deformed particle on the stress intensity factor of mode I crack is obtained.     

Progressive debonding of aligned oblate inclusions and loss of stiffness in a brittle matrix composite

A micromechanical theory is developed to examine the progressive debonding process of a brittle matrix composite containing aligned oblate inclusions under a high state of triaxial tension. Complete debonding is taken to be the debonding mode under such a high triaxial loading and its debonding strength is assumed to be governed by a Weibull probability function in terms of the hydrostatic tension of the inclusions. The micromechanical theory provides the required hydrostatic tensile stress at a given stage of debonding for a given inclusion shape and concentration. This allows one to calculate the debonding process progressively as the applied stress increases. The resulting stress-strain relation of the progressively debonded system is found to start out with that of the perfectly bonded composite, then deviates from it and eventually approaches the stress-strain curve of the corresponding porous material. It is further revealed that debonding with spherical inclusions is completed faster than with thin discs and it also occurs faster at a lower volume concentration of inclusions. The loss of stiffness of the transversely isotropic composite is also established as a function of inclusion shape and concentration for all five independent moduli; these moduli are seen to decrease gradually in the initial stage, then drop sharply while progressively debonding and finally level off again as all inclusions become debonded.     

Modelling the evolution of particle size distribution during nucleation,growth and coarsening

Abstract

A model, based on the numerical framework of Kampmann and Wagner, has been developed to predict the evolution of particle size distribution (PSD) during the decomposition of supersaturated solid solutions by nucleation, growth, and coarsening. During the early stages of transformation, where nucleation and growth are dominant, the PSD shape is predicted to be constantly evolving. Only during the latter stages of transformation, when coarsening becomes dominant, does the PSD tend towards a steady state shape, which closely matches that expected from classical coarsening theory. It is also predicted that as the PSD evolves, a transient double peak forms and then decays. The physical basis of this double peak has been investigated and the effect of supersaturation on its formation has been predicted.     

The shielding effects of the crack-tip plastic zone

In this paper we demonstrate that a plastically deformed zone around a stressed crack tip can be, mechanically, identified with an inclusion of transformation strain by means of Eshelby equivalent inclusion method. Thus, the shielding effect of the plastic zone can be quantitatively evaluated by the present transformation toughening theory. A closed-form solution to determine the change in the stress intensity factor induced by the plastic zone is given both for plane stress and plane strain mode I cracks under small-scale yielding conditions. By using the present solution, the effects of the strain-hardening behavior of the material, the plane stress and plane strain states and the T-stress on the crack-tip shielding effects are identified.     

Free vibration analysis of rotating Timoshenko beams with multiple delaminations

Analytical solutions are developed to study the free vibrations of rotating Timoshenko beams with multiple delaminations. The Timoshenko beam theory and both the ‘free mode’ and ‘constrained mode’ assumptions in delamination vibration are adopted. Parametric studies are performed to study the influences of Timoshenko effect and rotating speed on delamination vibration. Results show that the effect of delamination on both modes 1 and 2 natural frequencies is significantly influenced by Timoshenko effect and the rotating speed. Also, the comparison between ‘free mode’ assumption and ‘constrained assumption’ are conducted. Furthermore, the effect of delamination on mode shapes is also influenced by rotating speed and Timoshenko effect. For both Timoshenko effect and rotating speed, the influences on the second vibration mode shape are more significant.     

Voltammetry at micropipet electrodes

The use of micropipet electrodes for quantitative voltammetric measurements of ion-transfer (IT) and electron-transfer (ET) reactions at the interface between two immiscible electrolyte solutions (ITIES) requires knowledge of geometry of the liquid interface. The shape of the meniscus formed at the pipet tip was studied in situ by video microscopy under controlled pressure. The shape of the interface can be changed from a complete sphere to a concave spherical cap by varying the pressure applied to the pipet, and the diffusion current to the pipet changes accordingly. With no external pressure applied, the water/organic interface turned out to be flat, and the voltammetric response of a pipet must follow the well-known theory for a microdisk electrode. The large deviations from this theory observed previously can be attributed to a small amount of the filling aqueous solution which escapes from the pipet and forms a thin layer on its outer wall. This effect can be eliminated by making the outer pipet wall hydrophobic. Procedures have been developed for independent silanization of the inner and outer walls of the pipet. Pipets with a silanized inner wall can be filled with an organic solvent (e.g., 1,2-dichloroethane) and be used for voltammetric measurements in aqueous solutions. Another mode of voltammetry is based on trapping of a thin layer of organic solvent in the narrow shaft of a pipet between the filling solution and the aqueous outer phase. This arrangement is potentially useful for electrochemical catalysis and sensor applications.     

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.     

Elastic behaviour of an orthotropic beam/one-dimensional plate of uniform and variable thickness

In this paper, the analysis of the title problem is based on mixed first-order thick-beam one-dimensional plate theory, and on using a small-parameter approach towards its numerical solution. The boundary conditions at the edges of the beam may be quite general, and between these two edges the beam may have varying thickness. Closed-form solutions have been developed for the static response of orthotropic beams with nonlinear thickness variation subjected to uniform loading. The accuracy of the present model is demonstrated by problems for which exact solutions and numerical results are available, and the results are also presented for a variety of problems whose solutions are not available in the literature.     

任意分布多个椭圆形刚性夹杂的反平面问题

对于硬夹杂与软基体的复合材料,考虑夹杂间的相互影响,采用坐标变换和复变函数的依次保角映射方法,构造任意分布且相互影响的多个椭圆形刚性夹杂模型的复应力函数,同时满足各个夹杂的边界条件,利用围线积分将求解方程化为线性代数方程,推导出了在无穷远双向均匀剪切,椭圆形刚性夹杂任意分布的界面应力解析表达式,算例分析给出了单夹杂模型与多夹杂模型的夹杂形状对界面应力最大值的影响规律,并进行了对比,描绘出了曲线.     

剪切型基础平动与转动结构随机地震响应分析的复模态法

对多层剪切型结构考虑基础平动与转动的随机地震响应问题进行了系统研究。首先建立了结构运动方程,并根据结构的振动反应以第一阶振型为主的特点,将结构按第一振型展开;然后针对所得运动方程为非对称质量与非经典阻尼矩阵的情况,用复模态法解耦,获得了体系位移、速度和加速度随机响应的解析表达式,并给出了算例。该方法可推广到非平稳地震响应分析     

一种模态振型的直接扩阶方法

传统的模型缩阶及模态扩阶方法是以计算传递矩阵为手段.该文提出一种实测模态振型空间不完备问题的直接处理方法,即未测试自由度振型值的直接估计方法.通过引入一复合向量,该向量由对应于主自由度的实测振型值与对应于从自由度的待估算值组成.依靠实测模态及有限元模型信息对复合向量中未测试自由度的振型值进行重新估算,进而获得空间完备的模态振型.该文采用5 自由度的弹簧-质量体系验证方法的正确性;借助平面框架结构探讨新方法较传统方法在低阶模态振型估算精度上的优势.数值结果表明:所提出方法不需质量归一化条件以及实测模态与有限元振型的比例条件;同时,该方法较传统方法在低阶模态振型扩阶上具有更好的精度,并允许有限元模型具有一定的模型误差.     

混凝土结构I型裂纹裂尖塑性区研究

应用双剪统一强度理论,研究了I型裂纹的塑性变形问题。给出了包含反映材料拉压性能差异的参数拉压比及反映中间主应力效应的参数b的I型裂纹裂尖塑性区形状和大小的统一解。已有的Tresca准则、Mises准则和Mohr-Coulomb准则解均是本文的特例或线性逼近。针对混凝土结构,画出了不同参数b情况下的裂尖塑性区半径变化图。得出了材料拉压比对I型裂纹裂尖塑性区影响很大。b对I型裂纹裂尖塑性区影响随拉压比的不同而不同,拉压比较大时,b对塑性区影响大,拉压比较小时,b对塑性区影响小的结论。该统一解可以适应于各种不同材料,能充分发挥材料潜力,具有普遍性和广泛的适应性,有一定的工程应用价值。结论对于研究各种材料的断裂问题有参考作用。     

On the transient elastic field for an expanding spherical inclusion in 3-D solid

Summary. The three-dimensional transient elastic field of an infinite isotropic elastic medium is investigated when a phase transformation is nucleated from a point and proceeds through the crystal dynamically. The phase transformation keeps the spherical shape and expands at a speed of arbitrary time profile. This process is modeled by an expanding spherical inclusion with a spatially uniform eigenstrain. The objective of this paper is to present a general method to determine the transient displacement field for points either covered or not covered by the transformation area. This method can be applied to investigate the nucleation and expanding mechanism of phase transformation. Using a Green's function approach, an explicit procedure is presented to evaluate the 3-D displacement field when the expanding history of the spherical inclusion is given. As numerical examples, the explicit formulations are given for the transient elastic fields, when the spherical inclusion expands at a constant or an exponent damping speed with a pure dilatational eigenstrain or pure shear eigenstrain. It is found that the elastic field inside the expanding inclusion remains constant with respect to time, which is consistent with the well-known Eshelby solution for a static inclusion case.     

Geometrical and physical models of martensitic transformations in ferrous alloys

The classical theory of the crystallography of martensitic transformations developed in the 1950s is based on the notion that the interface between the parent and product phases is an invariant plane of the shape deformation. Underlying this hypothesis is the expectation that such interfaces do not exhibit long-range strain, and the geometric theory is an algorithm for finding invariant planes, the orientation relationship and transformation displacement. In the context of ferrous alloys, the classical theory has been applied successfully to transformations with {295} habit planes, but is less satisfactory for {575} for example. A new model of martensitic transformations has been presented recently based on dislocation theory, incorporating developments in the understanding of the topological properties of interfacial defects. Topological arguments show that glissile motion of transformation dislocations, or disconnections, can only occur in coherent interphase interfaces. Hence, the interface in the model comprises coherent terraces with a superimposed network of disconnections and crystal dislocations. It is demonstrated explicitly that this defect network accommodates the coherency strains, and that lateral motion of the disconnections across the interface effects transformation in a diffusionless manner. Moreover, it is shown that a broader range of habit planes is predicted on the basis of the semi-coherent interface model than the invariant plane notion. In the case of ferrous alloys, it will be shown that a range of viable solutions arise which include {575}.