Dynamic Propagation Problems Research Articles

Dynamic propagation problem of mode I semi‐infinite crack subjected to superimpose loads

ABSTRACTBy means of the theory of complex functions, dynamic propagation problems of a semi‐infinite crack subjected to the superimposed mode load were studied. Analytical solutions of stress, displacement and dynamic stress intensity factor are very readily obtained using the approaches of self‐similar functions. The corresponding closed solutions are obtained based on the Riemann–Hilbert problems.

Asymmetrical Dynamic Propagation Problems Concerning Mode III Interface Crack

By application of the theory of complex functions, asymmetrical dynamic propagation problems concerning mode III interface crack were studied. The general representations of analytical solutions are acquired by the approaches of self-similar functions. The problems considered can easily be transformed into Riemann–Hilbert problems and two examples under the action of moving increasing normal force Px/t and increasing loads Pt located at the origin of the coordinates of the crack are given, respectively. After those solutions are used by superposition theorem, the solutions of arbitrarily complex problems can be attained.

Damping efficiency of the Tchamwa–Wielgosz explicit dissipative scheme under instantaneous loading conditions

To deal with dynamic and wave propagation problems, dissipative methods are often used to reduce the effects of the spurious oscillations induced by the spatial and time discretization procedures. Among the many dissipative methods available, the Tchamwa–Wielgosz (TW) explicit scheme is particularly useful because it damps out the spurious oscillations occurring in the highest frequency domain. The theoretical study performed here shows that the TW scheme is decentered to the right, and that the damping can be attributed to a nodal displacement perturbation. The FEM study carried out using instantaneous 1-D and 3-D compression loads shows that it is useful to display the damping versus the number of time steps in order to obtain a constant damping efficiency whatever the size of element used for the regular meshing. A study on the responses obtained with irregular meshes shows that the TW scheme is only slightly sensitive to the spatial discretization procedure used. To cite this article: L. Mahéo et al., C. R. Mecanique 337 (2009).

Dynamic propagation problems concerning surfaces of asymmetrical mode III crack subjected to moving loads

With the theory of complex functions, dynamic propagation problems concerning surfaces of asymmetrical mode III crack subjected to moving loads are investigated. General representations of analytical solutions are obtained with self-similar functions. The problems can be easily converted into Riemann-Hilbert problems using this technique. Analytical solutions to stress, displacement and dynamic stress intensity factor under constant and unit-step moving loads on the surfaces of asymmetrical extension crack, respectively, are obtained. By applying these solutions, together with the superposition principle, solutions of discretionarily intricate problems can be found.

Dynamic stress intensity factor K III and dynamic crack propagation characteristics of anisotropic materials

Based on the mechanics of anisotropic materials, the dynamic propagation problem of a mode III crack in an infinite anisotropic body is investigated. Stress, strain and displacement around the crack tip are expressed as an analytical complex function, which can be represented in power series. Constant coefficients of series are determined by boundary conditions. Expressions of dynamic stress intensity factors for a mode III crack are obtained. Components of dynamic stress, dynamic strain and dynamic displacement around the crack tip are derived. Crack propagation characteristics are represented by the mechanical properties of the anisotropic materials, i.e., crack propagation velocity M and the parameter α. The faster the crack velocity is, the greater the maximums of stress components and dynamic displacement components around the crack tip are. In particular, the parameter α affects stress and dynamic displacement around the crack tip.

Dynamic Propagation Problems Concerning Asymmetrical Mode III Interface Crack

By application of the theory of complex functions, dynamic propagation problems concerning asymmetrical mode III interface crack were studied. The general expressions of analytical solutions are obtained by the methods of self-similar functions. The problems researched can be easily transformed into Riemann-Hilbert problems and analytical solutions of an asymmetrical propagation crack under the conditions of variable moving loads and unit-step moving loads, respectively, are attained. After those solutions are utilized by superposition theorem, the solutions of arbitrarily complex problems can be gained.

Dislocation Distribution Function of Two Fracture Dynamics Problems Concerning Aluminum Alloys

By the approaches of the theory of complex functions, dynamic propagation problems on the surfaces of mode I crack subjected to unit-step loads and instantaneous impulse loads located at the origin of the coordinates were studied for Aluminum alloys, respectively. Analytical solutions to stresses, displacements, dynamic stress intensity factors and dislocation distribution functions are gained by the methods of self-similar functions. The problems considered can be very facilely transformed into Riemann-Hilbert problem and their closed solutions are obtained rather straightforward by Muskhelishvili’s measure.

Dynamic interfacial crack propagation in elastic–piezoelectric bi-materials subjected to uniformly distributed loading

In transversely isotropic elastic solids, there is no surface wave for anti-plane deformation. However, for certain orientations of piezoelectric materials, a surface wave propagating along the free surface (interface) will occur and is called the Bleustein–Gulyaev (Maerfeld–Tournois) wave. The existence of the surface wave strongly influences the crack propagation event. The nature of anti-plane dynamic fracture in piezoelectric materials is fundamentally different from that in purely elastic solids. Piezoelectric surface wave phenomena are clearly seen to be critical to the behavior of the moving crack. In this paper, the problem of dynamic interfacial crack propagation in elastic–piezoelectric bi-materials subjected to uniformly distributed dynamic anti-plane loadings on crack faces is studied. Four situations for different combination of shear wave velocity and the existence of MT surface wave are discussed to completely analyze this problem. The mixed boundary value problem is solved by transform methods together with the Wiener–Hopf and Cagniard–de Hoop techniques. The analytical results of the transient full-field solutions and the dynamic stress intensity factor for the interfacial crack propagation problem are obtained in explicit forms. The numerical results based on analytical solutions are evaluated and are discussed in detail.

Asymmetrical dynamic propagation problems on mode III interface crack

By the application of the theory of complex functions, asymmetrical dynamic propagation problems on mode III interface crack are studied. The universal representations of analytical solutions are obtained by the approaches of self-similar function. The problems researched can be facilely transformed into Riemann-Hilbert problems and analytical solution to an asymmetrical propagation crack under the condition of point loads and unit-step loads, respectively, is acquired. After those solutions were used by superposition theorem, the solutions of arbitrarily complex problems could be attained.

The FEM Simulation and Full-Scale Blast Tests for Crack Deceleration in Gas Pipeline

Preventing pipeline from rapid crack propagation is a critical issue to avoid casualties and disasters. In this paper, by combining the energy balance theory with FEM simulation and arrest criteria, the numerical analysis is developed to solve the problem of crack dynamic propagation in gas pipeline. This simulation, in combination with the full-scale blast tests, provides a broad prediction of the dynamic fracture process. The crack tip opening angle (CTOA) criterion is consummated through the comparison between CTOA in FEM calculation and the critical value of (CTOA)C obtained by the experiment. The result of the simulation for the crack speed and location is consistent with data by Alliance and Japanese full-scale blast tests.

Application of the time discontinuous Galerkin finite element method to heat wave simulation

This paper deals with the numerical simulation of heat wave propagation in the medium subjected to different kinds of heat source, particularly heat impulse. The discontinuous Galerkin finite element method (DGFEM) proposed for the stress wave propagation in solids [X.K. Li, D.M. Yao, R.W. Lewis, A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non-linear solids and saturated porous media. Int. J. Numer. Meth. Eng. 57 (2003) 1775–1800] is extended to numerically solve for the non-Fourier heat transport equation constructed according to the CV model [C. Cattaneo, A form of heat-conduction equation which eliminates the paradox of instantaneous propagation, Compute Rendus 247 (1958) 431–433; P. Vernotte, Les paradoxes de la theorie continue de l’equation de la chaleur, Compute Rendus 246 (1958) 3154–3155]. Temperature and its time-derivative are chosen as primitive variables defined at each FE node. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite’s polynomial) and linear interpolations for both temperature and its time-derivative, respectively, in the time domain is particularly proposed. As a consequence the continuity of temperature at each discrete time instant is exactly ensured, whereas discontinuity of the time-derivative of temperature at discrete time levels remains. Numerical results illustrate good performance of the present method in the numerical simulation of heat wave propagation in eliminating spurious numerical oscillations and in providing more accurate solutions in the time domain.

Dynamic propagation problem on Dugdale model of mode III interface crack

By the theory of complex functions, the dynamic propagation problem on Dugdale model of mode III interface crack for nonlinear characters of materials was studied. The general expressions of analytical solutions are obtained by the methods of self-similar functions. The problems dealt, with can be easily transformed into Riemann-Hilbert problems and their closed solutions are attained rather simply by this approach. After those solutions were utilized by superposition theorem, the solutions of arbitrarily complex problems could be obtained.

Comments on ‘‘Dynamic crack propagation in piezoelectric materials—Part I. Electrode solution’’ by Shaofan Li, Peter A. Mataga [J. Mech. Phys. Solids 44 (1996) 1799–1830

In this comment, it is pointed out that the paper [Li and Mataga, 1996. J. Mech. Phys. Solids 44, 1799–1830], which presents original and valid solution strategy for an important problem of dynamic crack propagation in piezoelectric materials, contains ultimate quantitatively and qualitatively incorrect expressions, conclusions and plots due calculation errors. The correct calculations and corresponding correct conclusions and plots are presented.

A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non‐linear solids and saturated porous media

AbstractA time‐discontinuous Galerkin finite element method (DGFEM) for dynamics and wave propagation in non‐linear solids and saturated porous media is presented. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, in the time domain is particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non‐linear problems, as compared with that required for the existing DGFEM. Both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non‐linear problems. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. Copyright © 2003 John Wiley & Sons, Ltd.

3-D interpretation of reflected arrival times by finite‐difference techniques

Finite‐difference methods have generally been used to solve dynamic wave propagation problems over the last 25 years (Alterman and Karal, 1968; Boore, 1972; Kelly et al., 1976; and Levander, 1988). Recently, finite‐difference methods have been applied to the eikonal equation to calculate the kinematic solution to the wave equation (Vidale, 1988 and 1990; Podvin and Lecomte, 1991; Van Trier and Symes, 1991; Qin et al., 1992). The calculation of the first‐arrival times using this method has proven to be considerably faster than using classical ray tracing, and problems such as shadow zones, multipathing, and barrier penetration are easily handled. Podvin and Lecomte (1991) and Matsuoka and Ezaka (1992) extended and expanded upon Vidale’s (1988) algorithm to calculate traveltimes for reflected waves in two dimensions. Based on finite‐difference calculations for first‐arrival times, Hole et al. (1992) devised a scheme for inverting synthetic and real data to estimate the depth to refractors in the crust in three dimensions. The method of Hole et al. (1992) for inversion is computationally efficient since it avoids the matrix inversion of many of the published schemes for refraction and reflection traveltime data (Gjøystdal and Ursin, 1981).

Determination of the dynamic stress intensity factorsK I d andK II d , for a mixed-mode propagating crack

In this paper, the dynamic propagation problem of a mixed-mode crack was studied by means of the experimental method of caustics. The initial curve and caustic equations were derived under the mixed-mode dynamic condition. A multi-point measurement method for determining the dynamic stress intensity factors,K I d , andK II d , and the position of the crack tip was developed. Several other methods were adopted to check this method, and showed that it has a good precision. Finally, the dynamic propagating process of a mixed-mode crack in the three-point bending beam specimen was investigated with our method.

Semianalytic Hyperelement for Layered Strata

The paper presents a new numerical technique to solve dynamic wave propagation problems in layered systems. The method is based on the closed-form analytical solution in the direction parallel to the layering, and arbitrary displacement expansions in the direction perpendicular to it. Detailed information is given for the use of the theory for the particular case of a linear expansion. The main advantage of the theory is that a great reduction in the number of degrees-of-freedom necessary to model a system, as compared to conventional finite element models, can be attained. The advantage is achieved at the expense of having to solve the system in the frequency domain, and having to compute the stiffness matrix for each individual frequency. The method is, therefore, limited to linear systems. The relatively modest size of the problem allows a solution in fast memory, without having to resort to peripheral storage allocation.