Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Mathematica--A System for Doing Mathematics by Computer.

Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates

Composites science and technology

THE appearance of a treatise in English upon the mathematical theory of elasticity is an event the potential importance of which may be judged by the that the author, in his frequent suggestions for collateral reading, refers to only three such, those of Southwell, Timoshenko, and Love. In spirit and content Sokolnikoff}s book differs greatly from each and all of these. It may be described by a possible sub-title: “A pure mathematician surveys topics related to certain problems in the mathematical theory of elasticity”. It is symptomatic of the change in outlook of American mathematics over the past few decades. Mathematical Theory Of Elasticity Prof. I. S. Sokolnikoff with the collaboration of Asst. Prof. R. D. Speche. Pp. xi + 373. (New York and London: McGraw-Hill Book Co., Inc., 1946.) 22s. 6d.

Mathematical Theory Of Elasticity

The ultrasonic compaction might be a suitable technique to obtain well compacted composite laminates in order to implement them in out-of-autoclave automated curing processes. This procedure generates an ultrasonic vibration over the laminate that makes the resin heat and allows the composite plies to be compacted. The numerical study of this heat generation and its distribution within the laminate requires the modelling of the resin and the fibre layers separately, requiring a very fine mesh. Furthermore, the process needs a discretization along the time with very short time steps, in order to model the ultrasonic vibration properly. For these reasons, a fine FEM solution is very costly to compute. Space-time separated representations as the ones considered in Proper Generalized Decomposition (PGD) techniques seem to be an appealing choice for addressing the solution of ultrasonic compaction thermomechanical models. In this work, the formulation and implementation of a PGD solution of a model for the viscous heating and temperature distribution during the compaction of composite layers is presented. The compaction will be studied in two operating modes: it will be studied as a transient problem, assuming that the compactor is still over the laminate and as a steady-state problem, assuming that the compactor is moving along the laminate. Finally, the solution obtained with the transient model will be compared with experimental measurements, showing that the model predicts the behavior of the temperature inside the laminate properly.

Study of the ultrasonic compaction process of composite laminates—part II: advanced numerical simulation

One of the reasons for the application of composites to mobile structures is their capacity to absorb energy under impact As their components have a very reduced interval of deformation till breakage, the mechanism of absorption of energy cannot be based on plasticity but on the capacity of generating free surfaces inside the material, the debonding between fibers and matrix being the most direct materialization of the mechanism of absortion of energy in composites In this chapter the onset and propagation of damage in between fiber and matrix in a fibrous composite, particularly under transverse loading is studied and characterized The tools to apply are interfacial fracture mechanics (IFM), linear elastic brittle interface model, finite fracture mechanics (FFM), and cohesive zone model (CZM), combined with a numerical tool, in this case the boundary element method (BEM) Different loading situations are analyzed, single tension and compression, bi-dimensional loading as well as curing effects Finally single fiber, two fibers, and multi-fiber configurations are considered in this chapter, the key being the evolution of the energy released by the debonding crack representing damage in between fiber and matrix Numerical predictions are compared with experimental evidence to support the adequacy of the approaches followed

2.16 Micromechanics of Interfacial Damage in Composites

This paper establishes that the Stroh orthogonality relations for an anisotropic body are a direct consequence of the fact that the system of equations of equilibrium is self-adjoint and positive definite. It is demonstrated that, assuming a complex representation of displacements and boundary tractions, the Betti theorem of reciprocity implies the ‘orthogonality’, and positive definiteness of strain energy implies the full rank of the normalization matrix, in the Stroh orthogonality relations. The presented proof is applicable to both the Lekhnitskii and Eshelby theories of an anisotropic body.

On Stroh orthogonality relations: An alternative proof applicable to Lekhnitskii and Eshelby theories of an anisotropic body

Explicit closed-form real-variable expressions of a fundamental solution and its derivatives for three-dimensional problems in transversely linear elastic isotropic solids are presented. The expressions of the fundamental solution in displacements U ik and its derivatives, originated by a unit point force, are valid for any combination of material properties and for any orientation of the radius vector between the source and field points. An ex- pression of U ik in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector is used as starting point. Working from this expression of U ik , a new approach (based on the application of the rotational symmetry of the material) for deducing the first and second order derivative kernels, U ik,j and U ik,jl respectively, has been developed. The expressions of the fundamental solution and its derivatives do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational symmetry axis. The expressions of U ik , U ik,j and U ik,jl are presented in a form suitable for an efficient computational implementation in BEM codes.

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Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

The objective of this chapter is to show the modelling of composite materials at the micro-meso level, involving the nonlinearities associated with the damage that may appear in a composite laminate. The potential damage includes debondings between the fibres and matrix where the initial damage occurs in the weakest lamina of a laminate, and the cracking of the matrix that appears when a debonding abandons the fibre–matrix interface, kinking and penetrating into the matrix. Integrated multiscale models based on the boundary element method are developed, allowing us to analyze the whole laminate in a model while also having the possibility of modelling the damage object of study at the micromechanical level in this chapter. The incidence of the features of the model in the representativity of the results is first analyzed, correlating the predictions with the experimental observations from testing laminates. The models carried out will follow a line of association with particular problems regarding the use of composites, with a special emphasis on the study of the scale effect in composite laminates, emphasizing the correspondence between the model performed to simulate the damage and the actual mechanism of damage that a laminate experiences.

Federico París

Papers

Study of the ultrasonic compaction process of composite laminates—part II: advanced numerical simulation

2.16 Micromechanics of Interfacial Damage in Composites

On Stroh orthogonality relations: An alternative proof applicable to Lekhnitskii and Eshelby theories of an anisotropic body

Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

Modelling fibre–matrix interface debonding and matrix cracking in composite laminates