Impact response of a cracked orthotropic medium-revisited

Elastodynamics response of an infinite orthotropic medium containing a central crack under impact loading has been investigated. Laplace and Fourier transforms have been employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform domain which has finally been solved by the method of iteration in the low frequency case. Analytic expressions for the stress intensity factors and crack opening displacement are also obtained for low frequency.     

Asymptotic stress field in a cracked orthotropic strip

Christensen's theory of viscoelastic fracture allows the crack propagation velocity to be determined in terms of dissipation whose calculation requires the knowledge of the stress field in the vicinity of the crack tip: the simplest configuration leading to a constant velocity is that of a straight semi-infinite crack contained in an infinitely long strip whose clamped edges are displaced normal to the crack; although experimental data pertaining to this problem have been obtained for a number of materials, no analytical solution is available. When the material is highly anisotropic, an asymptotic solution involving a small parameter related to the ratio of shear modulus to the larger Young's modulus can be attempted. As the corresponding perturbation problem is singular, a matched asymptotic expansion has to be used: it is the sum of outer and inner approximations; both of these are solutions to simple boundary-value problems which can be solved in closed form. The so-constructed asymptotic solution is shown to agree with finite element results, even when the small parameter is as large as 0.2.     

Nonlinear adhesive behavior effects in a cracked orthotropic sheet stiffened by a semi-infinite orthotropic sheet

The stress-intensity factors are determined for a cracked orthotropic sheet adhesively bonded to an orthotropic stringer where the adhesive layer is modeled with a nonlinear stress-strain curve. Since the stringer is modeled as a semi-infinite sheet, the solution is most appropriate for a crack tip located near a stringer edge. By the use of Green's functions and the complex variable theory of orthotropic elasticity developed by Lekhnitskii, a set of integral equations is obtained. The integral equations are replaced by an equivalent set of algebraic equations, which are solved to obtain the shear stress distribution in the adhesive layer. With these adhesive stresses, the crack-tip stress-intensity factors are found.When the adhesive was modeled with a nonlinear stress-strain curve, the peak shear stresses in the adhesive were considerably reduced in comparison to the solution for the linear elastic adhesive. This resulted in increases in the stress-intensity factors for the nonlinear adhesive solution compared to the linear adhesive solution. The nonlinear adhesive did not have a significant effect on the stress-intensity factor unless the near crack tip was beneath the stringer. The present investigation assumes that the adhesive bond remains intact. Onset of adhesive failure is predicted to occur at decreasing levels of applied stress as the crack propagates beneath the stringer.     

Out of plane shear of a cracked rectangular orthotropic block

The problem under consideration is the out of plane shear of a cracked rectangular orthotropic block. The exact solution is obtained by stating the problem in terms of a triple trigonometric series relation, which in turn can be shown to be equivalent to a singular integral equation whose solution is known. For the case of constant shear, the solution simplifies greatly and numerical results are given.     

A crack within a half-space of orthotropic elastic material

Summary The dislocation layer method combined with a technique of images is used to study a mode III loaded Griffith-type elastic strip crack situated parallel to the free-surface of a semi-infinite orthotropic crystal. The density function of the proposed distribution of dislocations is shown to satisfy a complex singular integral equation. Its closed-form solution provides a compact expression for an appropriate combination of the resulting stress field components. Some representative numerical results are presented in tabular form and discussed.     

Transient response of a cracked magnetoelectric material under the action of in-plane sudden impacts

Dynamic analysis of a crack embedded in a magnetoelectric material is made when subjected to in-plane mechanical, electric and magnetic impacts. The Laplace and Fourier transforms are applied to reduce the associated initial- and mixed-boundary value problem to dual integral equations, and then to singular integral equations with Cauchy kernel. By numerically solving the resulting equation, the dynamic field intensity factors as well as CODs, and energy release rates near the crack tip are evaluated and presented graphically. The effects of applied magnetic and electric impacts on crack growth are discussed. Obtained results show that, different from the static results, applied magnetic and electric impacts can strongly affect dynamic stress intensity factors.     

Elastostatic analysis of edge cracked orthotropic strips

Summary. The elastostatic problem of an edge cracked orthotropic strip is considered. The crack possesses a semi-infinite length. The crack surfaces are subjected to opening mode I fracture, by a concentrated force action, while the strip surfaces are traction free. Fourier transforms and asymptotic analyses are employed to reduce the problem to a first kind singular integral equation. The stress intensity factor is determined in a closed form expression. The effects of geometric and elastic characteristics of the strip on the values of the stress intensity factor are explained.     

Structural model of a microinhomogeneous cracked material

We develop a mathematical model of microscopically inhomogencous but macroscopically isotropic materials with statistically distributed components of tensors of stiffness and strength. In this model, the material is represented as the continuous set of “characteristic” (i.e., typical of a given material) disjoint microscopic domains (microvolumes). The microinhomogeneous material is identified with an “effectively homogeneous” material in such a way that, at the same points, the components of the displacement vector determined for these materials are equal. It is assumed that, for each “characteristic” microvolume the parameters of stiffness and strength of the material are constant and can be obtained as values of an arbitrary random variable distributed according to the Weibull law and averaged over a certain random interval of any length. The components of the tensor averaged as indicated above are also regarded as random variables distributed according to the normal law with the same probability of hitting any arbitrarily located “characteristic” microvolume. The model is based on the assumption that the material is isotropic both macroscopically and in any “characteristic” microvolume. The stress-strain state of the microinhomogeneous material is described by the “effective” (averaged over its volume) components of the stress tensor. The model takes into account cracks in the material if their length exceeds the size of the relevant “characteristic” volume. The model is justified for the case of an infinite microinhomogeneous cracked plane under uniaxial tension. It is shown that the parameters determining the stressed state of this plane are not independent in the vicinity of the crack tip. The relevant constraints are given by equations of the model. The choice of these parameters which ignores the indicated constraints leads to results contradicting well-known physical facts. By using the symmetry properties of the system under consideration and physical reasoning, we obtain equations for the determination of the size of “characteristic” domains and physically reasonable dependences of the maximal “effective” tensile stresses and their direction on the parameter of inhomogeneity of the material and average volume of defects. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, Lviv. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 4, pp. 5–16, July–August, 1996.     

Determination of SIF in a cracked plane orthotropic strip by the Wiener-Hopf technique

The stress intensity factor for a long cracked strip was determined within the context of the linear orthotropic clasticity. The body had the form of an infinite strip containing a semi-infinite crack at the middle distance of the strip faces. Fourier transforms in combination with the Wiener-Hopf technique were employed to evaluate asymptotically the cleavage stress and its intensity at the crack tip.

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Résumé On détermine le facteur d'intensité d'entaille relatif à une longue bande fissurée, dans un contexte d'élasticité linéaire et orthotrope. On considère un corps ayant la forme d'une bande infinie et comportant une fissure semi-infinie à mi-distance des faces de la bande. On recourt à une transformée de Fourier en combinaison avec la technique de Wiener-Hopf pour évaluer par voie asymptotique la tension de clivage et son intensité à l'extrémité de la fissure.

     

Dynamic response of top-down cracked asphalt concrete pavement under a half-sinusoidal impact load

A 3D finite element analysis model of cracked asphalt pavement is established by the FEM software ABAQUS. Based on dynamics mechanics, fracture mechanics and finite element theory, this paper studies the influence of various vehicle speeds, crack location, crack depth, damping ratio etc. on the dynamic response. The results show that the surface deflection, the maximum tensile strain at the bottom of the asphalt layer, and the maximum shear stress of the asphalt layer decreased with the increase in vehicle velocity when there is no crack in the pavement. No matter where the transverse position of the crack is the stress intensity factors increase with the increase in crack depth and decrease exponentially with the increase in longitudinal distance between the vehicle center and the crack. In the case of the crack locating in the center of wheel clearance, the surface deflection increases with the crack depth increasing. But if the crack is at the edge of the wheel track, there will be a critical vehicle velocity where the surface deflection is smaller than the asphalt pavement without crack if the vehicle velocity is above it. The maximum tensile strain at the bottom of the asphalt layer and the maximum shear stress of the asphalt layer are also smaller than the asphalt pavement without crack. The maximum tensile strain and the maximum shear stress decrease with the damping ratio increasing. So the increase in damping ratio can help to alleviate the propagation of cracks.     

A contour integral computation of stress intensity factors in the cracked orthotropic elastic plates

Stress intensity factors in the cracked orthotropic elastic plates are obtained numerically using a reciprocal work contour integral technique. The integral representation for calculating the intensities of elastic singular stresses at cracks, notches and corners can be extended to treat the rectilinearly anisotropic crack problems. An extension of the method can be done immediately with identification of the characteristic singular solutions and construction of the corresponding complementary elastic states. A centrally cracked tension plate with various degree of orthotropy is analyzed to assess accuracy of the computational procedures newly adopted. Two cracked orthotropic plates modelling bidirectional fiber-reinforced composites are treated: symmetric Mode I type of double edge crack tension plate and mixed mode type of cantilever plate with a single edge crack.     

A note on bogie impact loads

Simple formulae are presented for the linear elastic and elasto-plastic response of a line of carriages or freight cars that are arrested suddenly. Starting with impact at the leading car, a wave of compression flows along the train during which, ideally, a quarter of a period passes until rest is momentarily achieved. Kinetic energy of the vehicles is changed into strain energy and heat in the couplers. The formulae are approximate but are, nonetheless, useful for design purposes, it being assumed that de-railing does not occur during this collision or buffer incident.     

Dynamics of cracked composite material structures

As the result of this work models of the finite cracked beam element, delaminated beam element, and also delaminated plate element have been developed. These models have been used for the analysis of the influence of the fatigue cracks and delaminations on the dynamic characteristics of the constructions made of unidirectional composite materials. The method of modelling cracks and delaminations presented in this work enables its easy modification according to specific cases of damage (i.e. oblique crack, two-side crack, inside crack, multiple delaminations etc.). The results obtained from numerical calculations of the presented models are in good agreement with the known influence of damage parameters like: the position and the depth of the crack or the length and the position of the delamination on the natural frequencies. Simultaneously, a strong influence of the material parameters on these changes has been observed, which does not exist in the case of isotropic materials.     

Complex-variable and integral-transform methods for elastodynamic solutions of cracked orthotropic strips

Problems concerning cracked bodies in the form of an infinite strip subjected to antiplane stresses or displacements were solved within the context of the linear and anisotropic theory of elasticity. Two basic approaches were utilized, namely a technique based on the complex-variable theory and another based on the exponential Fourier transform and the Wiener-Hopf method. Two geometric configurations were also considered. In the first case the long strip was internally weakened by a constant-length crack, whereas in the second case by a semi-infinite crack. The problems were analyzed under the assumption of a steady-state elastodynamic crack motion.     

Shakedown of a cracked body consisting of kinematic hardening material

In this paper, the shakedown behaviour of a cracked body is studied. The main idea is to consider the crack as a notch. Then no singular stresses appear at crack tip. Due to the local character of the problem, Melan's shakedown theorem is used. By solving shakedown as an optimization problem, the limited stress intensity factor (SIF) for shakedown K_sh is obtained. It is found that the shakedown limit SIF of a cracked body is proportional to the initial yield stress σ_y of the material times the square root of the effective crack tip radius π, i.e. K_sh ∝ σ_y√ϱ. Comparison of shakedown limit SIFs with fatigue thresholds for certain materials, so far as can be found in literature, shows that these two quantities agree well with each other. This agreement indicates that shakedown of the cracked body is one of the reasons for arrest of the crack under cyclic loads. Shakedown investigation is then a new method for predicting the fatigue threshold of a cracked body. Thus, a transition from shakedown to cyclic fracture mechanics has been achieved.