Stress singularities in composite wedge-shaped materials

Calculations of the stress singularity in the vicinity of the tip of two bonded wedges are described. Plasticity in the near-tip region is accounted for within the framework of the Ramberg–Osgood relation. It is shown that the singularity can be larger for two bonded wedges than for a single wedge, and figures are presented to indicate how the stress singularity depends on the relevant parameters.     

Numerical analysis of stress field around the second-phase particle in polymeric composite materials

In this paper, the stress distribution around the spherical particle in polymeric based composite materials was analyzed using axisymmetric finite-element method. The effect of varying Young's modulus and volume fraction of second-phase particle was examined. Predicted values of the modulus of particle-filled composite were given as function of the volume fraction of filler. Based upon these calculated results, various damage mechanisms relevant to the stress conditions in loaded materials were discussed. It is shown that the numerical predictions are in qualitative agreement with the microscopical observations.     

Applications of thermoelastic stress analysis to composite materials

A number of applied thermoelastic stress analysis (TSA) studies on composite components and assemblies are described, for the purpose of illustrating the potential of the technique for use with composite materials.     

The stress analysis method for three-dimensional composite materials

This study proposes a stress analysis method for three-dimensionally fiber reinforced composite materials. In this method, the rule-of mixture for composites is successfully applied to 3-D space in which material properties would change 3-dimensionally. The fundamental formulas for Young's modulus, shear modulus, and Poisson's ratio are derived. Also, we discuss a strength estimation and an optimum material design technique for 3-D composite materials. The analysis is executed for a triaxial orthogonally woven fabric, and their results are compared to the experimental data in order to verify the accuracy of this method. The present methodology can be easily understood with basic material mechanics and elementary mathematics, so it enables us to write a computer program of this theory without difficulty. Furthermore, this method can be applied to various types of 3-D composites because of its general-purpose characteristics.     

Numerical thermo-elastic analysis of singularities in two-dimensions

Linear elastic two-dimensional problems with singular points subjected to steady-state temperature distribution are considered. The stress tensor in the vicinity of the singular points exhibits singular behavior characterized by the strength of the singularity and the associated thermal stress intensity factors (TSIFs). It is shown that the TSIFs and the strength of the stress singularity can be obtained using the principle of complementary energy together with the modified Steklov method and the p-version of the finite element method. Importantly, the proposed method is applicable not only to singularities associated with crack tips, but also to multi-material interfaces and non-homogeneous materials. Numerical results of crack-tip singularities in a rectangular plate and singular points associated with a two-material inclusion are presented.Research performed while the author served as a visiting assistant professor at the Center for Computational Mechanics, Washington University, St. Louis, Missouri 63130, USA.     

Numerical analysis of edge singularities in three-dimensional elasticity

A numerical method is described for the computation of eigenpairs which characterize the exact solution of linear elastostatic problems in three-dimensions in the vicinity of edge singularities. These may be caused by re-entrant corners, abrupt changes in boundary conditions or material properties. Such singularities are of great interest from the point of view of failure initiation: The eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigenpairs and their amplitudes. This paper addresses the problem of determining the edge eigenpairs numerically on the basis of the modified Steklov formulation (presented in Reference 1 in a 2-D framework) in conjunction with the p-version of the finite element method. Numerical results are presented for several cases including isotropic as well as anisotropic multi-material interfaces. © 1997 John Wiley & Sons, Ltd.     

Thermally induced logarithmic stress singularities in a composite wedge and other anomalies

Wedge paradoxes, which were first studied by Sternberg and Koiter (Sternberg E, Koiter WT. The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity. ASME Journal of Applied Mechanics 1958;4:575–81), occur due to multiple roots in the Williams (Williams ML. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. ASME Journal of Applied Mechanics 1952;19:526–28) eigenfunction expansion. The consequence of such a paradox is a change in behavior of the stresses from , to the ‘non-separable’ form,

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. The focus of this study is the problem of thermally induced logarithmic stress singularities in a composite wedge associated with ω=0. Both double and triple root examples are presented which lead to and behavior in the stresses, respectively. This behavior is primarily associated with incompressible materials for the clamped–clamped single material case, and for the full range of Poisson’s ratio for the clamped-free case. The study also includes non-separable eigenfunctions that occur when complex conjugate roots transition to double real roots. Perhaps the most interesting result is that for the clamped–clamped wedge with Poisson’s ratio equal to 1/2, the hydrostatic stress has a logarithmic singularity proportional to the thermal strain for all wedge angles. This result can be extended to conclude that for a confined, incompressible or nearly incompressible material with a relatively sharp corner, and subject to some expansion or contraction phenomena, high hydrostatic stresses can result.     

Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients

This paper proposed numerical methods to determine the multiple stress singularities (including the oscillatory stress singularities) and the related stress intensity coefficients, by the use of common numerical solutions (stresses or displacements) obtained by an ordinary numerical tool such as finite element method (FEM) or boundary element method (BEM). To verify the efficiency of the present methods, two models of bonded dissimilar materials under the plane strain state are analyzed by BEM, and the orders of the stress singularities and the related stress intensity coefficients are examined numerically. The results show that all the orders of the stress singularities at an interface edge can be determined precisely by the present method, and the related stress intensity coefficients can be determined by the extrapolation method with a very good linearity. It is found that the methods presented in this paper are very simple and efficient. Moreover, they can be easily extended to any singular problem.     

Stress singularities in composite materials with an arbitrarily oriented crack meeting an interface

The plane, bending and antiplane strain elastostatic problems for two bonded half planes containing an arbitrarily oriented semi-infinite crack meeting the interface, are examined. A simple analogy between the interfacial continuity conditions for the plane and bending problems is derived and its implications are discussed. Using the eigenfunction-expansion method the crack tip stress singularities for various crack orientations and material combinations are compared for the three problems. It is shown that, under certain conditions, singularities are no longer present in the stress field. In the case of the antiplane strain problem subsequent paths of crack propagation are predicted.

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Résumé On examine des problèmes élasto-statiques en déformation plane, de flexion, et antiplanaire rencontrés dans deux demi-plans solidaires comportant, dans leur interface, une fissure semi-infinie orientée arbitrairement. On tire une analogie simple entre les conditions de continuité et interfaciales qui se présentent dans le cas des problèmes plans et flexionnels; ses implications sont discutées. En utilisant une méthode d'expansion d'une eigenfonction, on peut comparer les singularités de contraintes aux extrémités de la fissure, pour diverses orientations de la fissure et de combinaisons de matériaux différents. On montre que, sous certaines conditions, le champ de contraintes peut ne plus présenter de singularités. On peut par ailleurs prédire les chemins de la propagation d'une fissure dans le cas d'une déformation antiplanaire.

     

Numerical analysis of fibre bridging and fatigue crack growth in metal matrix composite materials

The analysis of bridged crack configurations in unidirectional fibre-reinforced composites is relevant to a variety of crack growth problems, including the fatigue of metal matrix composites and the study of fibre failure in the wake of a bridged matrix crack. Details of numerical procedures for predicting fibre stresses and their effect on crack tip stress intensity factors are presented here to provide a useful overview of how standard bridging calculations are done. Results are presented and discussed in the context of predicting fatigue crack growth with fibre failure in metal matrix composites.     

Stress singularities in cracked composite full-planes

The nature of the singular stress field developed at the point where a crack intersects the bonded interface of a bimaterial full-plane was investigated by the complex variable method. The crack lips were assumed to be under homogeneous stress, displacement or mixed boundary conditions. The order of the singularity was shown to be dependent on both the geometry and the four elastic constants of the two materials of the composite. Graphs showing the variation of the stress singularity with the aforementioned parameters were given. Valuable results indicating how the stress singularity depends on the geometry, the elastic constants and the boundary conditions of the cracked composite full-plane were derived.

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Résumé On a étudié à l'aide de la méthode des variables complexes la nature du champ singulier de contraintes qui se développe à l'endroit où une fissure coupe l'interface d'un plan constitué de deux matériaux solidaires. On suppose que les lèvres de la fissure sont soumises à des contraintes et des déplacements homogènes et sont sous des conditions aux limites mixtes. On montre que l'ordre de la singularité dépend à la fois de la géométrie et des quatre constantes élastiques des matériaux qui constituent le composite. On donne des diagrammes établissant la variation de la singularité de la contrainte selon les paramètres mentionnés. Des résultats pertinents sont obtenus, qui indiquent comment la singularité de contrainte dépend de la géométrie, des constantes élastiques et des conditions aux limites du plan composite fissuré.

     

Free-volume dependent moisture diffusion under stress in composite materials

A model for the effect of external uniaxial loading on the equilibrium moisture content in uni-directional composite materials is proposed. The model attributes the effect to a change in the free volume of the resin matrix, which is equal to its volume strain.M _∞ of the stressed state is calculated as a function of the stress level, the volume fraction of the fibres and the angle between the applied stress and the fibre direction. Moisture absorption experiments were performed on pure epoxy matrix and on graphite-epoxy composites. The effect of the parameters mentioned above onM _∞ was examined, and excellent agreement was found between the experimental and the calculated values. For stresses which were applied in the direction of the fibres (θ = O°) the experiment showed thatM _∞ decreased with the increase in the stress. We attribute this anomaly to the generation of residual stresses in the composite due to swelling.     

Numerical analysis of singularities in two-dimensions part 1: Computation of eigenpairs

A numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second-order elliptic partial differential equations in two dimensions in the vicinity of singular points. The singularities may be caused by re-entrant corners and abrupt changes in boundary conditions or material properties. Such singularities are of great interest from the point of view of failure initiation: The eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigenpairs and their amplitudes. This paper addresses the problem of determining the eigenpairs numerically on the basis of the Steklov formulation. Numerical results are presented for several cases. Importantly, the method is applicable to three-dimensional cases.     

Frictional contact analysis of composite materials

In this paper, the highly non-linear frictional contact problems of composite materials are analysed. A proportional loading, the potential contact zone method and finite element analysis are used to solve the problems. A tree-like searching method is used to obtain the solution of the parametric linear complementary problem, which may overcome the anisotropic properties of contact equations caused by composite materials. In the frictional contact analysis of composite materials, the distributions of normal contact pressures, tangential contact stresses and relative tangential displacements are presented for different contact material systems and different coefficients of friction. The results show that the solutions in the paper have good agreement with Hertzian solutions. The influence of different contact material systems and different coefficients of friction on the contact stresses and displacements is large. As a numerical example, ball-indentation tests of composite materials are modelled by the three-dimensional finite element method.     

On stress singularities in linear elasticity

The stress field in many exact solutions of problems in linear elasticity is found to have singularities which occur at corners and or at points where the boundary conditions change in type. The dependence of the singularities on the local geometry and on the types of boundary conditions is investigated in this paper. It is found that the type of boundary conditions determines the type of singularity but that the geometry largely influences the magnitude of the singularity.     

Propagation of stress corrosion cracks in aligned glass fibre composite materials

A modified fracture toughness test has been used to measure the growth of cracks in unidirectional glass fibre/polyester resin composite materials in the presence of a dilute hydrochloric acid. Crack growth rates perpendicular to the fibre axis have been measured over a range of stress intensities. Scanning electron microscope studies of the fracture surfaces have shown that the micromechanisms of nucleation and propagation are dependent on stress intensity. The use of crack growth data to predict component lifetimes and the existence of inherent flaws in the material are discussed.     

Through-the-thickness stress predictions for laminated plates of advanced composite materials

A finite element formulation is developed with emphasis primarily focused on providing stress predictions for thin to moderately thick plate (shell) type structures. Plate element behaviour is specified by prescribing independently the neutral surface displacements and rotations, thus relaxing the Kirchhoff hypothesis. Numerical efficiency is achieved due to the simplicity of the element formulation, i.e. the approach yields a displacement dependent multi-layer model. In-plane layer stresses are determined via the constitutive equations, while the transverse shear and short-transverse normal stresses are determined via the equilibrium equations. Accurate transverse stress variations are obtained by appropriately selecting the displacement field for the element. A selective reduced integration technique is utilized in computing element stiffness matrices. Static and spectral (eigenvalue) tests are performed to demonstrate the element modelling capability.