The penny-shaped crack in three-dimensional electrostriction

Zusammenfassung Zur Untersuchung der Spannungs- und Verschiebungsfelder in einem unendlichen elastischen Dielektrikum verwenden wir die dreidimensionale lineare ungekoppelte Theorie der Elektrostriktion.Von besonderem Interesse ist das Wesen der Spannungs-Singularität am Rande eines flachen kreisförmigen Spaltes innerhalb der begrenzenden Geometrie. Ein zweifacher Sachverhalt wurde festgestellt: 1. diese Spannungs-Singularität kann von höherer Ordnung sein als diejenige, welche mit den klassischen Umgebungen mechanischer und thermischer Art verknüpft ist;2. diese Singularität neigt nicht dazu, den Spalt in bezug auf eine spannungslose Spaltoberfläche auszubreiten. Die obigen Ergebnisse stimmen mit unseren früheren zweidimensionalen Analysen überein.

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This work was supported by the United States Atomic Energy Commission.     

Partial hybrid finite elements for composite laminates

Three types of partial hybrid finite elements are presented in order to set up a global/local finite element model for analysis of composite laminates. In the global/local model, a composite laminate is divided into three different regions: global, local, and transition regions. These are modeled using three different elements. In the global region, a 4-node degenerated plate/shell element is used to model the overall response of the composite laminate. In the local region, a multilayer element is used to predict detailed stress distribution. In the transition region, a multilayer transition element is used to smoothly connect the two previous elements. The global/local finite element model satisfies the compatibility of displacement at the boundary between the global region and the local region. It also satisfies the continuity of transverse stresses at interlaminar surfaces and traction conditions on the top and bottom surfaces of composite laminates. The global/local finite element model has high accuracy and efficiency for stress analysis of composite laminates. A numerical example of analysis of a laminated strip with free edge is presented to illustrate the accuracy and efficiency of the model.     

Theoretical analysis of three-dimensional interface crack

Using the fundamental solution of interface crack and the method of finite-part integral, the problem of three-dimensional interface crack is reduced to solve a set of two-dimensional hypersingular integrodifferential equations with unknown displacement discontinuities of crack surface. Then a systematically theoretical analysis for solving these equations is presented.     

Stress analysis of cross-ply composite laminates with transverse cracks

A boundary-discontinuous Fourier expansion method to analyze the displacements and stresses in cross-ply composite laminates with transverse cracks is presented. The governing equations of the problem are derived on the basis of the generalized plane strain assumption and two-dimensional equations of elasticity. By employing the boundary-discontinuous Fourier expansion method, the governing equations in the form of coupled high-order ODEs are transformed to a set of systems of linear algebraic equations. The method is used to obtain solutions for which published results can be found for comparisons. Compared with the conventional numerical methods for solving coupled high-order ODEs, the method presented is more efficient. Further parametric studies are carried out for cracked laminates with various geometric and material properties. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 795–826, November–December, 2008.     

Optimum stacking sequence of composite laminates for maximum strength

A method is presented for maximum strength optimum design of symmetric composite laminates subjected to in-plane and transverse loadings. The finite element method based on shear deformation theory is used for the analysis of composite laminates. Ply orientation angles are chosen as design variables. The quadratic failure criterion which is meant to predict fracture, is used as an object function for optimum stacking sequence design of a laminated plate. The Broydon-Fletcher-Goldfarb-Shanno optimization technique is employed to solve the optimization problem effectively. Numerical results are given for various loading conditions, boundary conditions, and aspect ratios. The results show that the quadratic failure criterion such as Tsai-Hill theory is effective for the optimum structural design of composite laminates.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 3, pp. 393–404, May–June, 1995.     

Isogeometric vibration and buckling analyses of curvilinearly stiffened composite laminates

Isogeometric analysis (IGA) with the polynomial splines over hierarchical T-meshes (PHT-splines) is used to provide an efficient tool capable of carrying out the vibration and buckling analyses of the stiffened laminates. IGA offers increased accuracy and efficiency using the PHT-splines, which represent exact geometry of the stiffeners and make the refinement more flexible near the areas where the stiffeners and composite plate are connected. Numerical examples are given to validate the correctness and superiority of the present method, comparing with the results from existing literatures and commercial softwares. Besides, the influences of the orientation, curvature, location and cross-section size of the stiffeners on the natural frequencies and buckling loads are also studied. The results show that the optimization of the shape and size of the stiffeners has an important effect on the vibration and buckling characteristics of stiffened laminates.     

Plane harmonic wave propagation in three-dimensional composite media

An important characteristic of waves propagating through periodic materials is the existence of stop bands. A stop band implies the range of frequencies over which a medium completely reflects all incident waves and there is no transmission. Predicting stop band phenomena in periodic materials is regarded as the first step toward designing composite microstructures capable of propagating energy in a predetermined manner. In this paper a global–local modeling methodology previously proposed by the authors is used to successively predict stop bands in three-dimensional composite media. Numerical results reveal that the first stop band of the considered microstructures occurs where an acoustic shear mode veers with the lowest optical branch of the same symmetry class.     

Modeling the nonlinear deformation of composite laminates based on plasticity theory

The plasticity theory has been successfully used for describing the nonlinear deformation of laminated composite materials under a monotonically increasing loading. Generally, several tests are needed to determine the parameters of the plastic potential for a laminate. We explore an alternative approach and obtain the plastic potential by using theoretical considerations based on a laminate analysis. The model is shown to provide an accurate prediction for the response of a cross-ply glass/epoxy laminate under uniaxial tensile loading at different angles to the material orthotropy axes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 309–318, May–June, 2007.     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

Detection of matrix cracks in composite laminates by using the modal strain energy method

As the foremost mechanism of damage development, matrix cracking is the critical damage found in the early stage of structural failure of composites. This study aims to nondestructively detect matrix cracks in composite laminates by using an experimental modal analysis (EMA). An AS4/PEEK composite was used to fabricate cross-ply [0₂/90₁₂/0₂] and quasi-isotropic [(±45/0/90)₄] _s laminates. The damage in the form of a matrix crack in the laminates was created by using a tensile load. The EMA was conducted on the laminates to obtain the modal displacements before and after damage. The displacements were then employed to compute the modal strain energy and to define the damage index, which is used for detecting matrix cracks. Limited by the mesh points of measurements, we used the differential quadrature method to calculate the partial differentials in the strain energy formula. The results obtained were validated by using the X-ray radiography method and three-point bending tests. The experimental results showed that the damage index well identified the location of breadthwise matrix cracks inside the laminates. However, the resolution of the damage index became poor if the spans of matrix cracks were short or the matrix cracks were scattered over the laminates.     

Analysis of the free-edge effect in composite laminates by the boundary finite element method

Because of the risk of delamination due to high interlaminar stresses in the vicinity of free edges of composite laminates, there is a strong interest in efficient methods for the analysis of this free-edge effect. By the example of a symmetric [0°/90°]_s cross-ply laminate, the Boundary Finite Element Method is presented as a very efficient numerical method, which combines the advantages of the finite element method and the boundary element method. Analogously to the boundary element method, only the boundary is discretized, while the element formulation is finite element based. The resultant stress field is shown to be in very good agreement qualitatively and quantitatively with the comparative finite element analysis. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 355–366, March–April, 2000.     

Substructuring FE–XFE approaches applied to three-dimensional crack propagation

Two substructuring methods are investigated in order to allow for the use of the eXtended Finite Element Method (X-FEM) within commercial finite element (FE) codes without need for modifying their kernel. The global FE problem is decomposed into two subdomains, the safe domain and the cracked domain based on the value of the level sets representing the crack. The safe domain is treated by the host FE software while the cracked domain is treated by an independent XFE code. The first substructuring method consists of calculating the Schur matrix of a cracked super-element with the XFE code. The second technique introduces the finite element tearing and interconnecting method (FETI) which ensures the compatibility of the displacements at the interface between the cracked and safe subdomains. The stiffness matrices and nodal forces are provided by the XFE and FE codes for the cracked and safe subdomains, respectively. The solutions obtained with these two techniques are rigorously equivalent to those computed with the stand-alone XFE code. First, the computational efficiency of the two approaches is demonstrated. Second, a validation is proposed towards comparison with reference values of the stress intensity factors in simple 3D cracked geometries. Finally, this contribution presents an application of the FE–XFE–FETI method to the computation of the stress intensity factor induced by a crack inside a hydraulic cylinder under internal pressure.