Imperfect interface effect for nano-composites accounting for fiber section shape under antiplane shear

This paper investigates the elastic responses of fibrous nano-composites with imperfectly bonded interface under longitudinal shear. The proposed imperfect interface model is the shear lag (or the spring layer) model; the presented nano interfacial stress model is the Gurtin–Murdoch surface/interface model; and the three-phase confocal elliptical cylinder model is the geometry model accounting for the fiber section shape. By virtue of the complex variable method, a generalized self-consistent method is employed to derive the closed from solution of the effective antiplane shear modulus of the fibrous nano-composites with imperfect interface. Five existing solutions can be regarded as the limit form the present analytic expression. The influences of the interface elastic constant, the interfacial imperfection parameter, the size of the elliptic section fiber, the fiber section aspect ratio, the fiber volume fraction and the fiber elastic property on the effective antiplane shear modulus of the nano-composites are discussed. Particularly, numerical results demonstrate that the interfacial elastic imperfection will always cause a significant reduction in the effective antiplane shear modulus; and the fiber interface stress effect on the effective modulus of the fibrous nano-composites will weaken with the interfacial imperfection increases.     

Diffraction of pulse shear waves in a tunnel cavity of curvilinear section in an anisotropic massif

The problem of diffraction of longitudinal shear waves in the form of periodic triangular pulses by a cylindrical tunnel cavity in a rectilinearly orthotropic unbounded massif is reduced to a number of problems on the diffraction of harmonic shear waves of different relative length. Each of the problems on stationary diffraction is solved in series in the basis solutions of shear vibrations equations by using an affine coordinate transformation of the original domain. It is proposed to use a point method of least squares to seek the coefficients of the series. The influence of the degree of anisotropy of the elastic properties of the massif on the shear stress concentration at the boundary of a circular section is analyzed for a different position of the incident triangular pulse front relative to center of the cavity.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 70–76, 1989.     

The perturbed stress state of a closed cylindrical shell with longitudinal cracks on its inner surface

Using the Rice-Levy line spring model we find an approximate solution of the problem of elastic balance of a closed infinite circular cylindrical shell weakened by a periodic system of longitudinal interior cracks of identical length. The stress intensity factors are found at the center of a crack for various numbers of cracks with various parameters. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

周期裂纹削弱的无限长板条的应力分析

作出了周期裂纹削弱的无限长板条的应力分析．假设这些裂纹均在水平位置,又板条承受 y方向的拉伸力p．此时边值问题归结为一个复杂混合边值问题．发现,对此问题言,特征展开变分原理方法 ( eigenfunction expansion variational method,简称为EEVM)是非常有效的．研究了裂纹端的应力强度因子和T-应力．从拉伸力作用下的弹性变形考虑,开裂板条可等价于一不开裂的正交异性板条．还分析了等价正交异性板条的弹性性质．最后给出了算例和数值结果．     

Anti-plane analysis of an orthotropic strip weakened by several moving cracks

In this paper several finite cracks with constant length (Yoffe-type crack) propagating in an orthotropic strip were studied. The distributed dislocation technique is used to carry out stress analysis in an orthotropic strip containing moving cracks under anti-plane loading. The solution of a moving screw dislocation is obtained in an orthotropic strip by means of Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by moving cracks. Finally several examples are solved and the numerical results for the stress intensity factor are obtained. The influences of the geometric parameters, the thickness of the orthotropic strip, the crack size and speed have significant effects on the stress intensity factors of crack tips which are displayed graphically.     

Analytical solutions for the circular stress transducer embedded in rheological rock mass

The rheological stress recovery (RSR) method was proposed to measure the in-situ stress of rock mass with time-dependent property by drilling a hole and embedding transducers into it. To solve the stress distribution on the transducer, a viscoelastic axisymmetric plane model was firstly built considering an arbitrary stress boundary condition. The analytical solution was developed by dividing the stress boundary conditions into axisymmetric and anti-axisymmetric combining with Laplace transformation technique. The explicit stress expressions on interfaces of rock–grout and grout–transducer was obtained using Burgers and Boltzmann viscoelastic models, respectively. Furthermore, the variations of transducer surface final stress, which is related to rheological time, geometric and mechanical properties of rock mass, grout parameter, and transducer materials, was proposed for calculating the measured stress by RSR method. For both of Burgers and Boltzmann viscoelastic model, final stress increases as elastic modulus ratio increases when elastic modulus ratio under 20, and the final stress could be ignored when diameter ratio is over 1.4. The rheological time increases with increasing of viscosity coefficient and the modulus ratio, but decreases as the shear modulus increases. The results in here provide a simple method for stress analyzing and have great value for understanding the relationship between the initial stress of rock mass and the measured stress for the RSR method.     

HYPERSINGULAR INTEGRAL EQUATIONS FOR PERIODIC ARRAYS OF PLANAR CRACKS IN A PERIODICALLY-LAYERED ANISOTROPIC ELASTIC SPACE UNDER ANTIPLANE SHEAR STRESS

1991MRSubjectClassification75M25,45E991IntroductionDuringthelasttenyearsorsojmanyresearchersinappliedmathematicsandmechanicshaveshownasurginginterestinformulatinglinearcrackproblemsillterlllsofsystel-alsofHadamardfillite-part(hypersingular)integralequations,e.g.Ioakimidis['],Lin'kovandMogilevskaya[']andAnal'].Anadvantageofsuchaformulationisthatthe11nkllowllfllnctionsaredirectlyrelatedtothejlllxlpillthedisplacementsacrossoppositeera(:kfaces.Oncetheyaredeterlttillied,crackparaliietersofinter…     

功能梯度条硬币型裂纹扭转冲击响应

研究非均匀条中硬币型裂纹的扭转冲击问题．材料的剪切模量假定按特定的梯度变化．采用Laplace 和Hankel 变换将问题化为求解Fredholm积分方程,通过将Bessel函数渐进展开获得裂纹尖端动态应力场．考查非均匀参数和功能梯度条高度对裂尖动态断裂行为的影响．动应力强度因子和能量密度因子的清晰表达式表明,作为裂纹扩展力,对于这里所研究的问题,二者是等价的．动应力强度因子的数值结果显示,增加剪切模量的非均匀参数可以抑制动应力强度因子的幅度,而条形域的高度对动态断裂特性的影响较小．     

正交各向异性功能梯度材料Ⅲ型裂纹尖端动态应力场

研究了无限大正交各向异性功能梯度材料Griffith裂纹受反平面剪切冲击作用的问题.材料两个方向的剪切模量假定为成比例按特定梯度变化.通过采用积分变换-对偶积分方程方法,获得了裂纹尖端动态应力场.动态应力强度因子计算结果显示:增加剪切模量梯度或增加垂直于裂纹面方向的剪切模量可以抑制动态应力强度因子的幅值.     

双周期裂纹场平面弹性焊接的数学问题

本文讨论双周期胞腔中含任意形状裂纹的不同材料的弹性平面焊接(焊线为任意形状的封闭光滑曲线)的第二基本问题.运用Мусхелишвили复变函数方法,对这类弹性平面问题建立起了数学模型,将求解弹性平衡问题化归为寻求复应力函数满足一定边界条件的边值问题,然后构造其解的形式,再将其转化为正则型的奇异积分方程,数学上严格证明其解的存在与唯一.     

Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.     

Diffraction of a pulsed longitudinal shear wave by a cylindrical cavity in an anisotropic half-space

The problem of the stress concentration on a free tunnel-shaped cavity in an orthotropic half-space with a free flat boundary, on which shear waves in the form of periodic triangular pulses are incident, is solved. The initial problem is reduced to a series of problems of diffraction of harmonic shear waves. The effect of the degree of shear anisotropy on the tangential-stress concentration at the boundary of a cavity with a circular cross section is studied for different positions of the leading edge of the pulse relative to the boundary of the half-space and the cavity.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 62–66, 1990.     

Interaction between Griffith cracks in a sandwiched orthotropic layer

A study is made of the interaction between three coplanar Griffith cracks which are located symmetrically in the midplane of an orthotropic layer of finite thickness 2h sandwiched between two identical orthotropic half planes. The Fourier transform technique is used to reduce the elastostatic problem to a set of integral equations which have been solved by using the finite Hilbert transform and Cooke's results. Analytical expressions for the stress intensity factors at the tips of cracks are obtained for large values of h. Numerical results concerning the interaction effects are presented with physical significance. It is shown that interaction effects are either shielding or amplification depending on the location of cracks, spacing of crack-tips, and the thickness of the layer. The stress magnification factors at the crack-tips are also calculated.     

Fracture of an isotropic medium strengthened with a regular system of stringers

An isotropic medium containing a system of foreign transverse rectilinear inclusions is considered. Such a medium can be interpreted as an infinite plate strengthened with a regular system of ribs (stringers) whose cross section is a very narrow rectangle. The medium is weakened by a periodic system of rectilinear cracks. The action of the stringers is re placed by unknown equivalent concentrated forces at the points of their connection with the medium. The boundary-value problem on equilibrium of the periodic system of cracks under the action of external tensile forces is reduced to a singular integral equation, from the solution of which the stress in tensity factors are found. The condition of limiting state of equilibrium of the cracks is formulated based on a criterion of brittle fracture. The stress state in the case where crack faces come into a partial contact is also considered. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 59–72, January–February, 2007.     

Elastodynamic analysis of a cracked orthotropic half-plane

The solution of elastodynamic volterra-type dislocation in an orthotropic half-plane is obtained by means of the Fourier transforms. The distributed dislocation technique is used to construct integral equations for an orthotropic half-plane weakened by cracks where the domain is under time-harmonic anti plane traction. These equations are of Cauchy singular type at the location of dislocation which is solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determine stress intensity factors for multiple smooth cracks. Several examples are solved and the stress intensity factors for multiple cracks with different configuration are obtained.     

Doubly singular boundary value problems with derivative dependent source function: A fast‐converging iterative approach

In this paper, a fast‐converging recursive scheme is presented to approximate the solution of a class of derivative dependent doubly singular boundary value problems (DDSBVP). First, the original problem is reformulated as an equivalent integral equation. The resulting integral equation is then efficiently tackled by an improved homotopy analysis method (IHAM). This method contains a parameter, which greatly accelerates the convergence of the series solution. The convergence of the method is carried out. To illustrate the efficiency and accuracy of the proposed recursive approach, we consider three nonlinear examples, including one physical model problem, which describes stress distribution on a rotationally shallow membrane cap. Results show that our method excels over the existing methods.     

Non-local theory solution of two collinear mode-I cracks in piezoelectric materials

In this paper, the basic solution of two collinear cracks in a piezoelectric material plane subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem is solved with the help of two pairs of integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the integral equations, the jumps of displacements across the crack surfaces are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants and the lattice parameter on the stress field and the electric displacement field near crack tips. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion in piezoelectric materials.     

Investigation of the mechanical properties and internal stress field of directionally reinforced materials in transverse tension

The dependence of the transverse Young's modulus and Poisson's ration on the reinforcement ratio, type of structure, and internal fiber radius has been investigated. The microstress field in the structure of a composite in transverse tension is analyzed. Simple approximate expressions, which can be used to determine the corresponding mechanical characteristics without analyzing the state of stress, are obtained from the exact equations. A medium reinforced with doubly periodic groups is taken as a model. The exact solutions of the problem are obtained by means of elliptic functions.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 5, pp. 812–817, September–October, 1970.     

On diffraction of SH-waves by cuts in nonhomogeneous solids

The diffraction of SH-waves by an infinite periodic system of cuts in an infinite medium possessing nonhomogeneity has been studied. Assuming that shear modulus and density vary, the problem of diffraction of SH-waves by the periodic system of cuts depends on the solution of dual series equations which ultimately reduces to the solution of an infinite system of algebraic equations.     

考虑表面效应时孔边均布径向多裂纹Ⅲ型断裂力学分析

理论研究了纳米尺度孔边均布径向多裂纹的Ⅲ型断裂性能.基于Gurtin-Murdoch表面弹性理论和保角映射技术,获得了孔和裂纹应力场的解析解,给出了裂纹尖端应力强度因子的闭合解.基于解答分析了应力强度因子的尺寸效应,讨论了裂纹数量、裂纹/孔径比和缺陷表面性能对应力强度因子的影响.结果表明:当孔和裂纹尺寸在纳米量级时,无量纲应力强度因子具有显著的尺寸效应;应力强度因子随裂纹数量的变化规律受裂纹/孔径比的影响;裂纹/孔径比对应力强度因子的影响受到缺陷表面性能的制约,同时表面性能对应力强度因子的影响也受限于裂纹/孔径比;表面效应对应力强度因子的影响与裂纹数量无关.