Infinite Elastic Plane Research Articles

Action of Nonaxisymmetric Dynamic Loads on a Circular Hole in the Infinite Elastic Plane

A modified finite-difference method with respect to time and the method of Fourier series with respect to the angular variable are used to solve a plane dynamic problem of the elasticity theory on the action of nonaxisymmetric loads on the boundary of a circular hole in the infinite plane. The distribution of stress is determined. The time dependence of the stress concentration on the boundary of a circular hole is numerically found for various Poisson’s ratios and the load distributed according to a certain law.

Equivalence of the notch stress intensity factor, tip opening displacement and energy release rate for a sharp V-notch

For an infinite elastic plane with a sharp V-notch under the action of symmetrically loading at infinity, the length of crack initiation ahead of the V-notch’s tip is estimated according to a modified Griffith approach. Applying a new conservation integral to the perfectly plastic strip (Dugdale model) ahead of the V-notch’s tip, the relationship between notch stress intensity factor (NSIF) and notch tip opening displacement (NTOD) is presented. Also, the relationship between NSIF and perfectly plastic strip size (PPSS) is found. Since there are three fracture parameters (NSIF, NTOD, and PPSS) with changeable notch opening angle in two basic relationships, it is necessary to select one critical parameter with changeable notch opening angle or two independent critical parameters, respectively. With the help of a characteristic length, it is found by this new conservation integral that the NSIF, NTOD and energy release rate are equivalent in the case of small-scale yielding. Especially, the characteristic length possesses clear physical meaning and, for example, depends on both the critical NSIF and SIF or both the NTOD and CTOD, respectively, in which SIF and CTOD are from the tip of a crack degenerated from the sharp V-notch. The dependence of NSIF on NTOD and PPSS is presented according to the equivalence, and the critical NSIF depending on the critical NTOD with a notch opening angle is also predicted.

On the load transfer from elastic inclusions to an infinite elastic orthotropic plane with cuts

The present paper deals with the problem of load transfer from elastic inclusions to an infinite elastic orthotropic plane with cuts located on one of the principal orthotropy directions. The constitutive system of equations of this problem is derived under the assumption that the inclusions are in a uniaxial stress state. The obtained system consists of a singular integro-differential equation and a singular integral equation for the jumps of the tangential stresses acting on the inclusion shores and for the derivative of the the cut opening function. The behavior of solutions of the system of constitutive equations at the endpoints of the inclusions and cuts is studied, and the solution of this system is constructed by the numerical-analytic discrete singularity method.

Fundamental Formulation for Transformation Toughening in Anisotropic Solids

In this paper the problem of transformation toughening in anisotropic solids is addressed in the framework of Stroh formalism. The fundamental solutions for a transformed strain nucleus located in an infinite anisotropic elastic plane are derived first. Furthermore, the solution for the interaction of a crack tip with a residual strain nucleus is obtained. On the basis of these expressions, fundamental formulations are presented for the toughening arising from transformations using the Green's function method. Finally, a representative example is studied to demonstrate the relevance of the fundamental formulation.

Analytical Solutions of Dynamic Crack Models of Bridging Fiber Pull-Out in Unidirectional Composite Materials

An elastic analysis of an internal central crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane was analyzed, and the crack extension should occur in the format of self-similarity. When the fiber strength is over its maximum tensile stress, the fiber breaks. By means of complex variable functions, the problem considered can be easily translated into Reimann-Hilbert mixed boundary value problem. Utilizing the built dynamic model of bridging fiber pull-out in unidirectional composite materials, analytical solutions of the displacements, stresses and stress intensity factors under the action of increasing loads Pt5/x5, Px5/t4 are obtained, respectively. After those analytical solutions were used by superposition theorem, the solutions to arbitrary complex problems were acquired.

On the Solution of an Elliptical Inhomogeneity in Plane Elasticity by the Equivalent Inclusion Method

Stress analysis of an elliptical inhomogeneity in an infinite isotropic elastic plane is a classical elasticity problem, which is usually solved by means of the complex variable formulation. In this work, we demonstrate that an alternative method of solution for such a problem, via the equivalent inclusion method, may be more convenient and straightforward without recourse to complex potentials or curvilinear coordinates. The explicit analytical solution can be derived through simple algebraic manipulation, although the longitudinal eigenstrain component should be handled with care in the case of plane strain. Since the ex- terior Eshelby tensor for an elliptical inclusion is available in closed-form, the present study provides a full field stress solution expressed in Cartesian coordinates. Furthermore, the in-plane stress components are represented in terms of Dundurs' parameters. The solution methodology and the convenient formulae of the stress concentration may be of practical use to the engineers in developing benchmarks for design evaluation.

Application of the multipole expansion method to evaluation of the stress state of an infinite elastic plane containing several circular holes

The 2D-elasticity problem of several closely-spaced circular holes in an infinite plane under biaxial loading at infinity is considered. The solution of the problem is found using the method of multipole decomposition, which is based on the representation of the jump displacement function as a sum of multipoles. Different arrangements of holes and their configurations are considered. The stress state in the region of mutual interaction of holes and near their contours is obtained. Special attention is given to the stress concentration around the holes. The points and magnitudes of stress concentration are determined and compared with those obtained around a single hole under the same loading conditions.

An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials

An elastic analysis of an internal crack with bridging fibers parallel to the free surface in an infinite orthotropic elastic plane is studied. An asymmetrical dynamic model for bridging fiber pull-out of unidirectional composite materials is presented for analyzing the distributions of stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-homogenous stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is determined by maximum tensile stress, resulting in fiber rupture and hence the crack propagation would occur in a self-similarity manner. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of unidirectional composite materials under the conditions of two moving loads given is obtained, respectively. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be obtained.

Asymmetrical dynamic fracture model of bridging fiber pull-out of unidirectional composite materials

An elastic analysis of an internal crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane is studied, and asymmetrical dynamic fracture model of bridging fiber pull-out of unidirectional composite materials is presented for analyzing the distributions of stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-homogenous stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is ascertained by maximum tensile stress, the fiber ruptures and hence the crack propagation should also appear in the modality of self-similarity. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of unidirectional composite materials under the conditions of two increasing loads given is obtained, respectively. In terms of correlative material properties, the variable rule of dynamic stress intensity factor was depicted very well. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be gained.

First fundamental problems of anisotropic elastic plane weakened by periodic collinear cracks

Using the method of complex functions, we discuss the first fundamental problems of an anisotropic infinite elastic plane weakened by periodic collinear cracks and with periodic boundary loads on both sides of the cracks. This problem was considered by Cai [Engineering Fracture Mechanics 46(1), 133–142 (1993)]. However, the previous method is imperfect. Therefore, the results are incorrect. Here, we revise the method and give a correct solution.

Determining the material constant in the equilibrium problem for an infinite elastic plane weakened by a circular hole

The averaging method is used to determine the material functions of the linear Cosserat theory of elasticity. The proposed method is used to find the material constant in the equilibrium problem for an infinite plane weakened by a circular hole.

Dynamic T-Stress for a Mode-I Crack in an Infinite Elastic Plane

An integral equation method is presented to determine dynamic elastic T-stress. Special attention is paid to a single crack in an infinite elastic plane subjected to impact loading. By using the Laplace and Fourier transforms, the associated initial-boundary value problem is transformed to a Fredholm integral equation. The dynamic T-stress in the Laplace transform domain can be expressed in terms of its solution. Moreover, an explicit expression for initial T-stress is derived in closed form. Numerically solving the resulting equation and performing the inverse Laplace transform, the transient response of T-stress is determined in the time space, and the response history of the T-stress is shown graphically. Results indicate that T-stress exhibits apparent transient characteristic.

Direct boundary integral procedure for a Boltzmann viscoelastic plane with circular holes and elastic inclusions

A direct boundary integral method in the time domain is presented to solve the problem of an infinite, isotropic Boltzmann viscoelastic plane containing a large number of randomly distributed, non-overlapping circular holes and perfectly bonded elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, in general, be different. The method is based on a direct boundary integral approach for the problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described by Crouch and Mogilevskaya [1], and a time marching strategy for viscoelastic analysis described by Mesquita and Coda [2–8]. Benchmark problems and numerical examples are included to demonstrate the accuracy and efficiency of the method.

A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions

This paper considers the problem of an infinite, isotropic viscoelastic plane containing an arbitrary number of randomly distributed, non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size. All inclusions are assumed to be perfectly bonded to the material matrix but the elastic properties of the inclusions can be different from one another. The Kelvin model is employed to simulate the viscoelastic plane. The numerical approach combines a direct boundary integral method for a similar problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described in [Crouch SL, Mogilevskaya SG. On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries. Int J Numer Methods Eng 2003;58:537–578], and a time-marching strategy for viscoelastic material analysis described in [Mesquita AD, Coda HB, Boundary integral equation method for general viscoelastic analysis. Int J Solids Struct 2002;39:2643–2664]. Several numerical examples are given to verify the approach. For benchmark problems with one inclusion, results are compared with the analytical solution obtained using the correspondence principle and analytical Laplace transform inversion. For an example with two holes and two inclusions, results are compared with numerical solutions obtained by commercial finite element software—ANSYS. Benchmark results for a more complicated example with 25 inclusions are also given.

Loosening of elastic inclusions

A numerical method is presented for simulating the occurrence of localized slip and separation along the interfaces of multiple, randomly distributed, circular elastic inclusions in an infinite elastic plane. The method is an extension of a direct boundary integral approach previously described elsewhere for solving problems involving perfectly bonded circular inclusions. Here, we allow displacement discontinuities to develop along the inclusion/matrix interfaces in accordance with a linear Mohr–Coulomb yield condition combined with a tensile strength cut-off. The displacements, tractions, and displacement discontinuities on the inclusion boundaries are all represented by truncated Fourier series, and an explicit iterative algorithm is adopted to determine zones of slip and separation under the prevailing loading conditions. Several examples are given to demonstrate the accuracy and generality of the approach.

Dynamic crack models on problem of bridging fiber pull-out of composite materials

An elastic analysis of an internal central crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane was performed. A dynamic model of bridging fiber pull-out of composite materials was presented. Resultingly the fiber failure is governed by maximum tensile stress, the fiber breaks and hence the crack extension should occur in self-similar fashion. By the methods of complex functions, the problem studied can be transformed into the dynamic model to the Reimann-Hilbert mixed boundary value problem, and a straightforward and easy analytical solution is presented. Analytical study on the crack propagation subjected to a ladder load and an instantaneous pulse loading is obtained respectively for orthotropic anisotropic body. By utilizing the solution, the concrete solutions of this model are attained by ways of superposition.

A dynamic model of bridging fiber pull-out of composite materials

An elastic analysis of an internal central crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane is carried out. In this paper a dynamic model of bridging fiber pull-out is presented for analyzing the distributions stress and displacement of composite materials with the internal central crack under the loading conditions of an applied non-uniform stress and the traction forces on crack faces yielded by the fiber pull-out model. Thus the fiber failure is determined by maximum tensile stress, the fiber breaks and hence the crack propagation should occur in self-similar fashion. By reducing the dynamic model to the Keldysh–Sedov mixed boundary value problem, a straightforward and easy analytical solution can be attained. When the crack extends, its fibers continue to break. Analytical study on the crack extension under the action of an inhomogeneous point force Px/ t, Pt is obtained for orthotropic anisotropic body, respectively; and it can be utilized to attain the concrete solutions of the model by the ways of superposition.

On the path dependence of the J-integral in notch problems

This study attempts to clarify the conditions under which the J-integral is path-independent in U- and V-shaped notch problems. The key is to determine the contribution to the J-integral evaluated in the global coordinate system from the second component of the J k -vector evaluated in the local coordinate system along the traction-free surfaces that form part of the integration path. It is found that the global J-integral is path-independent only if the projected contribution from the J 2-integral to J vanishes. The J-integral for a V-shaped notch is, strictly speaking, path-dependent even under remote symmetrical loading. This is due to the fact that, unlike in the case of a line- or plane-crack, the value of the J-integral calculated along a closed contour surrounding the V-shaped notch is dependent on the selection of the starting and ending points on the notch surface. In other words, the traction-free surface of the V-shaped notch does contribute to the J-integral due to the non-zero projected values induced from the J 2-integral. For a U-shaped notch, the path-independence of the J-integral is established if the integration path completely encloses the notch root. This is because both the upper and lower notch surfaces of the U-shaped notch are parallel to the geometric symmetrical line (the x 1-axis) and hence the projected values from the J 2-integral vanish. Furthermore, it is found that the small arc at the root of the notch (whether U- or V-shaped) also contributes to the J-integral even if the remote loading is symmetrical. These conclusions are derived by detailed analytical manipulations and by numerical examples; the analytical solution obtained by Lazzarin and Tovo [Int. J. Fracture 78 (1996) 3] and Lazzarin et al. [Int. J. Fracture 91 (1998) 269] for stress field in the vicinity of a notch root in an infinite elastic plane is used to calculate the contribution induced from the arc with different radii. Some useful results for studying the fracture and fatigue of notches are discussed.

Dynamic stress intensity factors around three cracks in an infinite elastic plane subjected to time-harmonic stress waves

The time-harmonic problem for an infinite elastic plane weakened by three parallel cracks has been solved. In this problem, two cracks are situated symmetrically on either side of a central crack and incident stresses impinge perpendicular to the cracks. Using the Fourier transform technique, the boundary conditions are reduced to four simultaneous integral equations. To solve the equations, the differences of displacements inside the cracks are expanded in a series. The unknown coefficients in the series are solved by the Schmidt method. The dynamic stress intensity factors are calculated numerically for several crack configurations.

On the fundamental problem for an infinite elastic plane bonded by different anisotropic materials with cracks

In this paper, the first fundamental problem for an infinite elastic plane bonded by different anisotropic materials with cracks of arbitrary shape is discussed. The problem is reduced to a certain system of singular integral equations with several undetermined constants, which is proved to be uniquely solvable when these constants are suitably and uniquely chosen.