Stress intensity factors for two offset parallel circular cracks: Part II — Semi-infinite solid

With the aid of the formulation in [1] (R. Muki, Progress in Solid Mechanics (North-Holland, 1961)) for general three-dimensional asymmetric problems and the superposition principle, Part II of this work makes use of the method in Part I (G.A.C. Graham and Q. Lan, J. Theor. Appl. Fract. Mech. 20, 207–225 (1994) [2]) to examine the interaction of arbitrarily located penny-shaped cracks in an infinite elastic solid to the case of a semi-infinite solid. As in Part I for the infinite body, the problem of a semi-infinite solid containing two penny-shaped cracks is reduced to a system of Fredholm integral equations of the second kind. These integral equations are then solved for some special cases when cracks are far apart and far away from the boundary. Some asymptotic solutions are presented and comparisons are made with the results for the special case where there is only one crack under axisymmetric loading.     

Stress intensity factors for two offset parallel circular cracks: Part I — Infinite elastic solid

The method (W.D. Collins, Proc. R. Soc. London A274, 507–528 (1963); W.S. Fu and L.M. Keer, Int. J. Eng. Sci. 7, 361–372 (1969) [1,2]) used to solve co-planar penny-shaped cracks is generalized to investigate interaction of arbitrarily located penny-shaped cracks. The solution (M.K. Kassir and G.C. Sih, Three-dimensional Crack Problems (Noordhoff International, 1975) [3]) for the problem of an isolated crack in an infinite solid is applied together with the superposition principle to reduce the problem to a system of Fredholm integral equations of the second kind. These integral equations are then solved iteratively when the cracks are far apart. Some asymptotic solutions for the stress intensity factors are presented and comparisons are made whenever possible. Numerical solutions reveal some interesting phenomena.     

A model of high speed penetration into ductile targets

A model developed by Mileiko et al. [J. Appl. Mech. Tech. Phys. 5 (1981) 711–713; Theor. Appl. Fracture Mech. 21 (1994) 9–16] describing a high speed penetration of an impactor into a ductile target is generalized.     

Acoustic field in a borehole within fluid formations with a horizontal interface

The acoustic field in a cylindrical borehole embedded in a horizontally layered infinite medium is studied. The modes in each layer are found to consist of continuous as well as discrete ones, the orthogonality and completeness of which are proven. The relevant weights associated with the two kinds of modes are determined by solving a set of integral equations deduced from the boundary conditions at the layer interface. This technique is not limited to low frequencies, but applies well to frequencies prevailing in typical logging environment. Reflection, transmission and coupling coefficients of different modes are evaluated. The numerical results are verified by comparison with the method of real axis integration (RAI) and the hybrid mehtod proposed by Tsang [“Transient acoustic waves in a fluid-filled borehole with a horizontal bed boundary separating two solid formations”, J. Acoust. Soc. Am. 81, 844–853 (1987)]. The scaled laboratory experiments are carried out and the corresponding results are presented for comparison with the numerical computation.     

On a moderate rotation theory of laminated anisotropic shells—Part 1. Theory

A moderate rotation theory of laminated anisotropic shells, proposed by Schmidt and Reddy [J. appl. Mech. 55, 611–617.1988], is developed and its application is presented. All aspects of the derivations are explicitly developed and specific forms of the equations are derived in this part. The finite-element formulation and its applications are presented in Part 2 of the paper.     

Analysis of three-dimensional edge cracks under tensile loading

Three-dimensional edge cracks are analyzed using the Self-Similar Crack Expansion (SSCE) method with a boundary integral equation technique. The boundary integral equations for surface cracks in a half space are presented based on a half space Green's function (Mindlin, 1936). By using the SSCE method, the stress intensity factors are determined by the crack-opening displacement over the crack surface. In discrete boundary integral equations, the regular and singular integrals on the crack surface elements are evaluated by an analytical method, and the closed form expressions of the integrals are given for subsurface cracks and edge crakcs. This globally numerical and locally analytical method improves the solution accuracy and computational effort. Numerical results for edge cracks under tensile loading with various geometries, such as rectangular cracks, elliptical cracks, and semi-circular cracks, are presented using the SSCE method. Results for stress intensity factors of those surface breaking cracks are in good agreement with other numerical and analytical solutions.     

Stress intensity factors for anisotropic layered composites: Part II—isolated double periodic cracks

The formulation in Part I (Theoret. Appl. Fracture Mech. 17, 205–219 (1992)) of this work for collinear cracks in alternate layers of an anisotropic laminate is extended to a system where each of the cracked layer contains a periodic array of parallel cracks. These cracks are also collinear such that they are periodic in two mutually perpendicular directions. Finite Fourier transform is applied at discrete points for one of the space variables reducing the problem to a singular integral equation. The unknown is expressible in terms of the crack opening displacement as in Part I. Displayed graphically are the normalized stress intensity factor, effective stiffness of the laminate, and the interlayer stresses. The local stress intensity factor for the double array crack system is always less than that for a single isolated crack depending on the periodicity ratio. The interlayer stresses directly ahead of the crack are elevated, the intensity of which increases with decreasing distance between the crack tip and interface. Increase in the thickness of the adjoining layer tends to decrease the interlayer stresses nearest to the crack tip, a result that is to be expected.     

Estimation of the largest Lyapunov exponent-like (LLEL) stability measure parameter from the perturbation vector and its derivative dot product (part 2) experiment simulation

Controlling system dynamics with use of the Largest Lyapunov Exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. This article is the second part of the one presented in Dabrowski (Nonlinear Dyn 67:283–291, 2012). It develops method LLEDP of the LLE estimation and shows that from the time series of two identical systems, one can simply extract value of the stability parameter which value can be treated as largest LLE. Unlike the method presented in part, one developed method (LLEDPT) can be applied to the dynamical systems of any type, continuous, with discontinuities, with time delay and others. The theoretical improvement shows simplicity of the method and its obvious physical background. The proofs for the method effectiveness are based on results of the simulations of the experiments for Duffing and Van der Pole oscillators. These results were compared with ones obtained with use of the Stefanski method (Stefanski in Chaos Soliton Fract 11(15):2443–2451, 2000; Chaos Soliton Fract 15:233–244, 2003; Chaos Soliton Fract 23:1651–1659, 2005; J Theor Appl Mech 46(3):665–678, 2008) and LLEDP method. LLEDPT can be used also as the criterion of stability of the control system, where desired behavior of controlled system is explicitly known (Balcerzak et al. in Mech Mech Eng 17(4):325–339, 2013). The next step of development of the method can be considered in direction that allows estimation of LLE from the real time series, systems with discontinuities, with time delay and others.     

Numerical simulations of chaotic dynamics in a model of an elastic cable

The finite motions of a suspended elastic cable subjected to a planar harmonic excitation can be studied accurately enough through a single ordinary-differential equation with quadratic and cubic nonlinearities.The possible onset of chaotic motion for the cable in the region between the one-half subharmonic resonance condition and the primary one is analysed via numerical simulations. Chaotic charts in the parameter space of the excitation are obtained and the transition from periodic to chaotic regimes is analysed in detail by using phase-plane portraits, Poincaré maps, frequency-power spectra, Lyapunov exponents and fractal dimensions as chaotic measures. Period-doubling, sudden changes and intermittency bifurcations are observed.Part of this work was presented at the XVIIth Int. Congr. of Theor. and Appl. Mech., Grenoble, August 1988.     

Scattering of elastic waves by a small inclined surface-breaking crack

I we examine the scattering of Rayleigh waves by an inclined two-dimensional plane surface-breaking crack in an isotropic elastic half-plane. We follow the method already introduced by the authors (A and W , 1992a, J. Mech. Phys. Solids 40, 1683) to obtain an analytical solution when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The procedure employed is the method of matched asymptotic expansions, which involves determining the scattered wave field both away from and near the crack. The outer solution is specialized from the general expansion in the first part of this work (A and W , 1992a, J. Mech. Phys. Solids 40, 1683), and the inner problem is exactly solved by the Wiener-Hopf technique. The displacement field and scattered Rayleigh waves are found uniformly to third order in , and concluding remarks are made about the general method as well as the results presented here.     

Mathematical modelling of transformation plasticity in steels II: Coupling with strain hardening phenomena

The models for the plastic behaviour of steels during phase transformations proposed in Part I and in a previous paper ( et al. [1986b]) for the case of ideal-plastic phases are extended to include strain-hardening effects (isotropic or kinematic hardening). An expression for the transformation plastic strain rate is obtained by modifying the treatment of Part I in a suitable manner. The classical plastic strain rate is also studied in a similar way. Complementary evolution equations for the hardening parameters are finally given, taking into account the possible “recovery” of strain hardening during transformations (i.e., the fact that the newly formed phase can “forget,” partially or totally, the previous hardening).     

On the dynamic behaviour of a piezoelectric laminate with multiple interfacial collinear cracks

This paper investigates the dynamic behaviour of a piezoelectric laminate containing multiple interfacial collinear cracks subjected to steady-state electro-mechanical loads. Both the permeable and impermeable boundary conditions are examined and discussed. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which can be solved using Chebyshev polynomial expansions. Numerical results are provided to show the effect of the geometry of interacting collinear cracks, the applied electric fields, the electric boundary conditions along the crack faces and the loading frequency on the resulting dynamic stress intensity and electric displacement intensity factors.     

Damage analysis of thermopiezoelectric properties: Part I - crack tip singularities

This is Part I of the work on a two-dimensional analysis of thermal and electric fields of a thermopiezoelectric solid damaged by cracks. It deals with finding the singular crack tip behavior for the temperature, heat flow, displacements, electric potential, stresses and electric displacements. By application of Fourier transformations and the extended Stroh formalism, the problem is reduced to a pair of dual integral equations for the temperature field with the aid of an auxiliary function. The electroelastic field is governed by another pair of dual integral equations. The inverse square root singularity is found for the heat flow field while the logarithmic singularity prevailed for the electroelastic field regardless of whether the crack lies in a homogeneous piezoelectric solid or at an interface of two dissimilar piezoelectric materials. Results are given for the energy release rate and a finite length crack oriented at an arbitrarily angle with reference to the external disturbances. Part II of this paper considers the modelling of a piezoelectric material containing microcracks. A representative cracked area element is used to obtain the effective conductivity and electroelastic modulus. Numerical results are given for a peizoelectric Bati O₃ ceramic with cracks.     

State-space approach of two-temperature generalized thermoelasticity of infinite body with a spherical cavity subjected to different types of thermal loading

In this work, we will consider an infinite elastic body with a spherical cavity and constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory (Youssef in J Appl Math Mech 26(4):470–475 2005a, IMA J Appl Math, pp 1–8, 2005). The medium is assumed initially quiescent. Laplace transform and state space techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem when the bounding plane of the cavity is subjected to thermal loading (thermal shock and ramp-type heating). The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the two-temperature and the ramping parameters.     

Analysis of multiple axisymmetric annular cracks in a piezoelectric medium

An axisymmetric annular electric dislocation is defined. The solution of axisymmetric electric and Volterra climb and glide dislocations in an infinite transversely isotropic piezoelectric domain is obtained by means of Hankel transforms. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks with so-called permeable and impermeable electric boundary conditions on the crack faces where the domain is under axisymmetric electromechanical loading. These equations are solved numerically to obtain dislocation densities on the crack surfaces. The dislocation densities are employed to determine field intensity factors for a system of interacting annular and/or penny-shaped cracks.     

On the overall elastic moduli of composites with spherical coated fillers

A micromechanical approach is presented to estimate the overall linear elastic moduli of three phase composites consisting of two phase coated spherical particles randomly dispersed in a homogeneous isotropic matrix. The theoretical method is based on Eshelby’s equivalent inclusion method and its recent extension by Shodja and Sarvestani [J. Appl. Mech. 68 (2001) 3] to evaluate the local field variables in case of double (multi) inhomogeneities. Using Tanaka–Mori theorem [J. Elasticity 2 (1972) 199] and a decomposition of Green’s function integral equation, the pair-wise average phase values of stress and strain in two interacting coated particles are estimated. Following Ju and Chen [Acta Mech. 103 (1994) 103; Acta Mech. 103 (1994) 123] the ensemble phase volume average of stress and strain fields can be evaluated within a representative volume element containing a finite number of coated particles. Comparisons with classical bounds are presented to illustrate the accuracy of the proposed method.     

THE BEHAVIOR OF TWO COLLINEAR CRACKS IN MAGNETO-ELECTRO-ELASTIC COMPOSITES UNDER ANTI-PLANE SHEAR STRESS LOADING

In this paper, the behavior of two collinear cracks in magneto-electro-elastic composite material under anti-plane shear stress loading is studied by the Schmidt method for permeable electric boundary conditions. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of displacements across the crack surfaces. In solving the triple integral equations, the unknown variable is expanded in a series of Jacobi polynomials. Numerical solutions are obtained. It is shown that the stress field is independent of the electric field and the magnetic flux.     

Cracks’ closure in 3-D fracture dynamics: the effect of relative location of two coplanar cracks

In this paper, the problem of two equal coplanar cracks with allowance for the crack faces contact interaction was investigated. The problem of the cracks located in homogeneous, isotropic, and linearly elastic solid subjected to normally incident tension–compression wave is solved by the boundary integral equations method. The influence of the distance between two cracks on the stress intensity factors (opening mode and transverse shear mode) is studied for a range of wave numbers. The results are compared with those obtained neglecting cracks’ closure.     

Two parallel penny-shaped cracks under the action of dynamic impacts

Summary The three-dimensional elastodynamic response of two parallel penny-shaped cracks embedded in an infinite elastic solid under the action of impact loading is investigated. A time-domain boundary integral equation method is used for calculating the time-dependent crack opening displacements and subsequently the dynamic stress intensity factors. Numerical computations are carried out for various geometry parameters. The results are presented in graphical form and discussed. The effect of the locations of the cracks on the dynamic stress intensity factors is presented. Received 8 May 1996; accepted for publication 25 September 1996     

Closed-form solution for two collinear mode-III cracks in an orthotropic elastic strip of finite width

The problem of an orthotropic strip containing two collinear cracks normal to the strip boundaries is considered. The Fourier series method is used to reduce the associated boundary value problem to triple series equations, then to a singular integral equation, which can be solved analytically. Under remote uniform antiplane shear loading, the stress field and the crack sliding displacement are determined analytically and stress intensity factors are also given in a closed form.