Dynamic stresses created by the sudden appearance of a transverse crack in periodically layered composites

The time-dependent stress field generated by the sudden appearance of a transverse crack in a periodically layered composite that is subjected to a remote loading is determined. The resulting two-dimensional elastodynamic problem is solved by combining two approaches. In the first one, the representative cell method, which has been presently generalized to dynamic problems, is employed for the construction of the time-dependent Green’s functions generated by the displacement jumps along the crack line. This is performed in conjunction with the application of the double finite discrete Fourier transform. Thus the original problem for the cracked periodic composite is reduced to the problem of a domain with a single period in the transform space. The second approach is based on a wave propagation in composites theory which has been presently generalized to admit arbitrary types of loading. This theory is based on the elastodynamic continuum equations where the transformed time-dependent displacement vector is expressed by a second-order expansion, and the equations of motion and the various interfacial and boundary conditions are imposed in the average (integral) sense. The time-dependent field in any observation point in the plane can be obtained by the application of the inverse transform. This field is valid as long as no reflected waves from external boundaries have been arrived. Results along the crack line as well as the full field are given for cracks of various lengths for Mode I, II and III deformations. In particular the dynamic magnification with respect to the static case is determined at the interface within the first unbroken stiff layer.     

On the intersonic crack propagation in an orthotropic medium

Motivaded by recent theoretical studies the elastodynamic response of an orthotropic material with a semi-infinite line crack, which propagates intersonically. is revisited through an approach which differs from those used in previous studies. The near tip stress and displacement fields are obtained for Mode I and Mode II of steady state crack propagation. The strain energy release rate analysis confirms that the Mode I is physically impossible due to the order of stress singularity, which is larger then one half. For Model II the order of stress is less than one half and it is shown that a steady state intersonic propagation is allowed only for a particular crack tip velocity which is a function of the material orthotropy.     

Dynamic response for non-homogeneous piezoelectric medium with multiple cracks

A general procedure to analyze the dynamic response of non-homogeneous piezoelectric medium containing some non-collinear cracks is developed. It is assumed that all the material properties only depend on the coordinates y (along the thickness direction). The assumption is made that the non-homogeneous medium is composed of numerous laminae with their surfaces perpendicular to the thick direction. The solution method is based upon the Fourier and Laplace transforms to reduce the boundary value problem to a system of generalized singularity equations in the Laplace transform domain. The singular integral equations for the problem are derived and numerically solved by weight residual value method. The time-dependent full field solutions are obtained in the time domain. As numerical illustration, the stress and electric displacement intensity factors for a three-layer plate specimen with two cracks are presented. It is found that the stress and electric fields are coupled in the crack plane ahead of the crack tip for non-homogenous piezoelectric materials.     

热弹性动力学耦合问题的插值型移动最小二乘无网格法研究

该文基于插值型移动最小二乘法,将无网格局部Petrov-Galerkin（MLPG）法用于二维耦合热弹性动力学问题的求解。修正的Fourier热传导方程和弹性动力控制方程通过加权余量法来离散,Heaviside分段函数作为局部弱形式的权函数,从而得到描述热耦合问题的二阶常微分方程组。然后利用微分代数方法,温度和位移作为辅助变量,将上述二阶常微分方程组转换成常微分代数系统,采用Newmark逐步积分法进行求解。该方法无需Laplace变换可直接得到温度场和位移场数值结果,同时插值型移动最小二乘法构造的形函数由于满足Kroneckerdelta特性,因此能直接施加本质边界条件。最后通过两个数值算例来验证该方法的有效性。     

Scattering of SH waves by an arbitrarily orientated closed crack

The 2D problem of a time-harmonic plane shear horizontal (SH) wave scattered by a finite closed crack in an isotropic material is presented in the paper. The crack is arbitrarily orientated with regard to the incident wave. A spring model based on the assumption that the traction components on the crack surfaces are linearly related to the crack opening displacement (COD) is used to model the closed crack. The problem is formulated in a set of boundary integral equations which contains the CODs as unknowns. Numerical examples are presented for the CODs, elastodynamic stress intensity factors, and the scattered displacement field for various parameters, such as spring stiffness, crack sizes and crack orientations. The results show that both the crack closure and orientation have significant effects on the scattered displacement field for the closed crack.     

Time-domain BEM for three-dimensional fracture mechanics

The present paper deals with time-domain analysis of three-dimensional transient dynamic crack problems. The time-domain formulation of the boundary element method for 3-D elastodynamic problems is used. Quarter-point and singular quarter-point elements represent displacements and tractions, respectively, near the crack front. Special attention is paid to integration and algorithms to preserve stability. Cracks in finite and unbounded regions under single and mixed mode dynamic loading conditions are studied. To the authors’ knowledge, no previous BE approach for 3-D elastodynamic crack problems based on the time-domain displacement representation exists.     

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

A hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.     

Study of elastic wave propagation induced by piezoelectric actuators for crack identification

The current paper provides a theoretical study of elastic wave propagation in a cracked elastic medium induced by an embedded piezoelectric actuator. The formulation of the problem is based on a newly developed actuator model and the solution of coupled crack and actuator problems. The interaction between an actuator and a crack is studied to develop basic understanding of the effect of the crack upon the wave field generated. The resulting wave field is used to simulate 'sensor signals'. Scattered wave from the crack is then regenerated by solving a reverse elastodynamic problem and used in an imaging technique to identify the crack.     

Determination of Stress Intensity Factor for Cracks in Orthotropic Composite Materials using Digital Image Correlation

Abstract: This paper focuses on the application of the digital image correlation (DIC) technique to determine the stress intensity factor (SIF) for cracks in orthotropic composites. DIC is a full‐field technique for measuring the surface displacements of a deforming object and can be applied to any type of material. To determine the SIF from full‐field displacement data, the asymptotic expansion of the crack‐tip displacement field is required. In this paper the expansion of the crack tip displacement field is derived from an existing solution for strain fields. Unidirectional fibre composite panels with an edge crack aligned along the fibre were tested under remote tensile loading and the displacements were recorded using DIC. The SIF was calculated from the experimental data by fitting the theoretical displacement field using the least squares method. The SIF thus determined was in good agreement with theoretical results and therefore demonstrates the applicability of the derived displacement field and DIC technique for studying fracture in composites.     

Electro-elastic stress analysis for a Zener-Stroh crack interacting with a coated inclusion in a piezoelectric solid

Summary. The problem of a Zener-Stroh crack initiated near a coated circular inclusion in a piezoelectric medium is investigated in this paper. By using the solution of a single piezoelectric screw dislocation near a coating inclusion as the Greens function, a Mode III displacement loaded crack is investigated. The proposed problem is formulated as a set of singular integral equations which are solved by numerical techniques. The influence of various parameters, such as the material constants of the inclusion, the coating, the matrix, the coating layer thickness, etc., on the crack behavior is studied. The stress and electric displacement intensity factors of the crack are derived. Several numerical examples are given and the results obtained are discussed in detail.     

Elastic and plastic fracture analysis of a crack perpendicular to an interface between dissimilar materials

Elastic and plastic fracture analysis of a Mode I crack perpendicular to an interface between dissimilar materials is carried out. Continuously distributed dislocations are used to simulate the crack. The simulation will cause singular integral equations with Cauchy kernel. By solving the singular integral equations numerically, the effects of crack depth (distance from the interface to the crack middle point) and Dundurs’ parameters on the Mode I stress intensity factor are investigated systematically. Then, based on the Dugdale model, the plastic zone size, and the crack tip opening displacement of the crack under uniform loadings are investigated. The effects of uniform loadings, crack depth, and Dundurs’ parameters on the plastic zone size and the crack tip opening displacement are examined. Numerical results show that when the crack is embedded in a stiffer material, the values of both the normalized plastic zone size and the normalized crack tip opening displacement are larger than 1. On the contrary, if the crack is embedded in a softer material, the values of both the normalized plastic zone size and the crack tip opening displacement are less than 1.     

两端用圆板封口的环肋柱壳振动声辐射性能分析

研究两端用圆板封口的环肋圆柱壳结构在流场中的振动声辐射特性,着重分析了由环肋圆柱壳和端部圆板所围空腔的内部声场对整个系统结构振动和声辐射的影响.基于扩展的Rayleigh-Ritz法,利用Hamilton变分原理推导出圆柱壳与圆板的耦合振动方程.考虑了静水压力的影响,环肋圆柱壳与端部圆板之间采用二自由度弹簧模拟弹性连接的等效刚度.环肋圆柱壳声辐射的研究中,用Helmholtz波动方程和流固交界面上的速度相容条件求得壳体表面辐射声压的表达式,该表达式对有限长圆柱壳进行Fourier积分变换得到壳体外表面辐射声压的解;用Green函数法来求解环肋圆柱壳和端板所围空腔的内部压力场.     

Calculation and Measurement of Scattering Coefficients for Surface-Breaking Cracks Sonified by a Focused Array

Abstract

A simulation of a focused beam of surface wave motion generated by a line array of surface wave transducers is used to calculate the field scattered back from a surface-breaking crack. The backscattered field is expressed in terms of a scattering coefficient, which is derived from the reciprocal identity for elastodynamic states, and which is calculated numerically by using the Kirchhoff approximation. In an earlier work the validity of the focused beam simulation has been verified. In this paper, the theoretical results for the backscattered field are compared with measurements for two crack sizes, one of which is used for calibration.     

均匀热流下导热裂纹Ⅱ型温度应力强度因子的解析解

研究含中心裂纹无限大板受远场均匀热流作用,热流密度方向与裂纹有一夹角的情况。当垂直于裂纹面方向有定量热流穿过裂纹时,采用复变函数理论,得出了温度、应力与位移场解析解。利用位移单值条件,确定出温度应力强度因子的解析表达式。针对铝合金LY12材料进行了数值计算,研究了裂纹导热情况与热流方向对温度场及温度应力强度因子的影响。研究表明:该文给定的温度边界条件下,只产生Ⅱ型温度应力强度因子,不产生Ⅰ型温度应力强度因子。热荷载可等效为一个远场均匀作用的剪应力。Ⅱ型温度应力场取决于热流密度沿垂直裂纹面方向的分量,平行于裂纹方向的热流分量对温度应力场没有影响。     

Calculation and Measurement of Scattering Coefficients for Surface-Breaking Cracks Sonified by a Focused Array

A simulation of a focused beam of surface wave motion generated by a line array of surface wave transducers is used to calculate the field scattered back from a surface-breaking crack. The backscattered field is expressed in terms of a scattering coefficient, which is derived from the reciprocal identity for elastodynamic states, and which is calculated numerically by using the Kirchhoff approximation. In an earlier work the validity of the focused beam simulation has been verified. In this paper, the theoretical results for the backscattered field are compared with measurements for two crack sizes, one of which is used for calibration.     

Comparison of two viscoplastic flow laws as applied to a problem of crack growth

The finite element method is incorporated in the study of the mechanical response of an advanced nickel-based superalloy IN-100 with inelastic time-dependent material behavior at 1350°F (732°C). The material's time-dependent plastic deformation (viscoplasticity) is analytically modeled by both the Bodner-Partom equations and the Malvern overstress flow law. The Bodner-Partom constitutive equations involve the use of nine material parameters which are determined from a complex procedure. Various forms of the Malvern overstress flow law are studied since this flow law uses only a few material parameters which are easily determined from uniaxial experimental data. Both the Malvern and Bodner-Partom constitutive equations are formulated in multiaxial from in a two-dimensional finite element program incorporating the constant strain triangle. The program is applied to the analysis of a center crack plate of IN-100 under a monotonic tension load. The residual force method is utilized to handle variations in material stiffness due to time-dependent plastic deformation. A Hybrid Experimental Numerical (HEN) procedure is used in order to track crack opening displacement near the crack tip. This HEN procedure controls the crack growth rate in the finite element model by following the experimental displacement rates accurately. Thus, crack growth predictions become a byproduct of both the rate sensitive model and the near field experimental displacement rates. The predictions of the two constitutive models are compared with respect to total plastic work and effective stress profiles in addition to crack growth rate.     

Cylindrical interface cracks in 1-3 piezocomposites

1-3 Piezocomposites are made by embedding piezoelectric fibers/rods in polymer matrix materials. Fiber–matrix interface fracture can affect the performance of piezocomposites. In this paper, axisymmetric interfacial cracks in piezocomposites are studied by considering an idealized model of a single piezoelectric fiber in a matrix material. The displacement discontinuity method is used to formulate the Mode I and II crack problems. The fundamental solutions required for DDM are derived explicitly by using the electroelastic field equations and Fourier integral transforms. The dependence of Mode I and II stress intensity factors of single and multiple interface cracks on fiber and matrix material properties, crack length and distance between cracks are investigated.     

Transmission of anti-plane shear waves past an interface crack in dissimilar media

The scattered field generated by horizontally polarized incident shear waves in a system of layers having a flaw at the interface is studied. The physical model is idealized to the case of a crack sandwiched between two dissimilar media of infinite height. The local intensification of the dynamic stresses due to the crack is analyzed for incident waves directed at an arbitrary angle. Results of numerical computations are obtained by solving pairs of coupled integral equations and reveal the variations of the stress and displacement fields with ratios of densities and shear moduli of the two adjoining materials.     

Impact response of a strip composite with a finite crack

The dynamic response of a central crack in a strip composite under normal impact is analyzed. The crack is oriented normally to the interfaces. Laplace and Fourier transform techniques are used to reduce the elastodynamic problem to a pair of dual integral equations. The integral equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometrical parameters is discussed.     

Non-local theory solution of a mode-I crack in a piezoelectric/piezomagnetic composite material plane

The non-local theory solution of a mode-I permeable crack in a piezoelectric/piezomagnetic composite material plane was given by using the generalized Almansi’s theorem and the Schmidt method in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement and magnetic flux singularities are present at the crack tips in piezoelectric/piezomagnetic composite materials. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.