Qualitative analysis of 3D elastostatic contact problems with orthotropic Coulomb friction and solution-dependent coefficients of friction

This paper analyzes a discrete form of 3D contact problems with local orthotropic Coulomb friction and coefficients of friction which may depend on the solution itself. The analysis is based on the fixed-point reformulation of the original problem. Conditions guaranteeing the existence and uniqueness of discrete solutions are established. Finally, numerical results of a model example are presented.     

A class of dynamic contact problems with Coulomb friction in viscoelasticity

The aim of this work is to study a class of nonsmooth dynamic contact problems which model several surface interactions, including relaxed unilateral contact conditions, adhesion and Coulomb friction laws, between two viscoelastic bodies of Kelvin–Voigt type. An abstract formulation which generalizes these problems is considered and the existence of a solution is proved by using Ky Fan’s fixed point theorem, suitable approximation properties, several estimates and compactness arguments.     

Sufficient conditions of non‐uniqueness for the Coulomb friction problem

We consider the Signorini problem with Coulomb friction in elasticity. Sufficient conditions of non‐uniqueness are obtained for the continuous model. These conditions are linked to the existence of real eigenvalues of an operator in a Hilbert space. We prove that, under appropriate conditions, real eigenvalues exist for a non‐local Coulomb friction model. Finite element approximation of the eigenvalue problem is considered and numerical experiments are performed. Copyright © 2003 John Wiley & Sons, Ltd.     

The Effects of Poisson Contraction on Matrix Cracking in Brittle-Matrix Composites

The effects of Poisson contraction on matrix cracking in unidirectional fiber-reinforced brittle-matrix composites are studied in this paper. The fibers, initially held in the matrix by a compressive pressure due to the thermal expansion mismatch, are subjected to frictional slipping over the matrix as soon as a fiber-bridged crack is formed. The friction between the fibers and the matrix is assumed to follow the Coulomb friction law. A shear-lag model, which includes the Poisson contraction and the friction due to the relative fiber/matrix slipping, is adopted to calculate the stress and strain fields in the fibers and matrix. Using the energy balance approach, a relation for the critical matrix cracking stress for propagating of a semi-infinite fiber-bridged crack is derived. The results obtained show that the Poisson contraction has a strong effect on the predicted matrix cracking stress in brittle-matrix composites, especially in composites with a stiff matrix.     

Raman spectroscopy investigation of stiffness change and residual strains due to matrix cracking

The matrix cracking models developed for cross-ply composite laminates have been poorly extended in the past to more complex geometries used in practice, and they are still under development. In this paper, a new detailed analysis of the effect of matrix cracking on the behaviour of cross-ply and [0/45]_s laminates under uniaxial tension is attempted. The model used in this work is applicable both to cross-ply laminates and unbalanced systems. It gives exact closed-form expressions for all thermomechanical properties of a general symmetric laminate with cracks in arbitrary layers. The theoretical approach is backed by experimental data obtained by microscopic strain-state variation measurements within a specimen, with using the technique of laser Raman spectroscopy. Glass-fibre-reinforced epoxy systems were investigated. Embedded aramid fibres-sensors within the 0° ply and near the 0°/θ ° interface were necessary due to the poor Raman signal of glass. Using experimental Raman data, the residual strain and the stiffness reduction are determined as functions of increase in crack density. The stiffness reduction is predicted with a high accuracy, whereas the measured residual strains are larger than predicted. The good results for the reduction in the elastic modulus show that the basic assumption of the model is accurate. The difference is explained by the viscoelastic-viscoplastic behaviour of the off-axis layer in shear, which in creases the “apparent” residual strain. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 771–786, November–December, 2006.     

Strong influence of a small fiber on shear stress in fiber-reinforced composites

In stiff fiber-reinforced composites, it has been known that the shear stress increases at the rate of as the distance ? between adjacent fibers approaches 0. This paper reveals a strong influence of a combination of a triple fiber, as well as the distance between a pair of fibers, on the blow-up so that the stress concentration can be significantly accelerated by adding a small fiber in-between fibers. Specifically, if a fiber F₂ with a small diameter δ is located in-between fibers F₁ and F₃, ?₁=dist(F₁,F₂) and ?₂=dist(F₂,F₃), then the stress blows up at the exact rates of and between F₁ and F₂ and between F₂ and F₃, respectively. This estimate still holds even when a part of F₂ overlaps with F₃. The magnification factor yields the enormous increase in the stress that greatly surpasses the expectancy by previous methods.     

Effect of fiber curvature on the modulus of elasticity for unidirectional glass-reinforced plastics in tension

The effect of curvature of the fibers on the modulus of elasticity of unidirectional glass-reinforced plastics (GRP) in tension has been investigated on the basis of the theory of layered reinforced media with random initial irregularities [1,2]. It is shown experimentally that relatively minor distortion of the fibers during manufacture may result in important changes in the modulus of elasticity. A model is proposed for determining the effect of fiber curvature on Young's modulus; experiments on specimens with a given regular fiber curvature indicate good agreement with the theory. The effect of prestressing the fibers on the modulus of elasticity has been studied using commercial AG-4S material and the optimum prestress has been established.Mekhanika Polimerov, Vol. 3, No. 2, pp. 243–249, 1967     

Change in the structure of polymer matrix of particulate-filled composites: The fractal treatment

An obligatory change (disturbance) in the structure of a polymer matrix is shown to arise upon injecting a disperse filler in it, which is determined by the fractal structure of the skeleton formed by filler particles. The characteristic of this disturbance is obtained in terms of change in the fractal dimension d _f of polymer matrix structure. Tentative estimates show a good correspondence between the values of d _f obtained and experimental data.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 827–834, November–December, 2004.     

Effect of fiber orientation on the structure of shock waves in carbon fiber-epoxy composites

Using an analytical relation between the Hugoniot (anisotropic and isotropic) states and other thermodynamic (anisotropic and isotropic) states at high pressures, the effect of fiber orientation on the structure of shock waves in carbon fiber-epoxy composites of various symmetry is investigated. A correct nonlinear model of propagation of shock waves in anisotropic materials is proposed, which employs the conception of total generalized pressure and the pressure corresponding to the thermodynamic response, i.e., to the equation of state. The equation generalizes the nonlinear Hugoniot equation to anisotropic materials and is reduced to the classical variant in the case of isotropy. Invoking the relations of nonlinear anisotropic solids and the generalized decomposition of stress tensor, the double structure of shock waves, consisting of nonlinear anisotropic and isotropic elastic parts, is examined. The numerical calculations of Hugoniot levels of stress agree well with experimental data for a carbon fiber-epoxy composite selected.     

Effect of structural and processing parameters of the reinforcement on the strength of unidirectional metalfiber composites under different types of loading

Conclusion The above studies show that the strength of unidirectional FCP's in uniaxial and biaxial tension is largely determined by the strength of the bond between the fibers and matrix and the internal geometry of the reinforcement. This dependence is particularly significant when the FCP product is loaded across as well as along the fibers. Impulsive loading as a method of making FCP's makes it possible to broadly vary the strength of the fiber-matrix interfaces. The unidirectional FCP's and tubular products of FCP's obtained by this method have a high strength under different types of loading, and this strength can be predicted by calculation.Presented at the Sixth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, November, 1986).Translated from Mekhanika Kompozitnykh Materiaiov, No. 6, pp. 1033–1038, November–December, 1986.     

Primitive Ideals in the Algebra of Regular Functions on Quantum m × n Matrices

Mathematical Notes -     

Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss–Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems.     

Effect of technological factors on the strength of unidirectional fiberglass plastics in compression and shear

The effect of variables in the technological process on the strength properties of fiberglass plastics has been studied. An investigation is made by the methods of regression analysis.     

A refined stress-strain analysis in the load transfer zone of flat specimens of high-strength unidirectional composites in uniaxial tension 3. Effect of grip misalignment

The influence of a preliminary bending induced by misalignment of the grips of a testing machine on the stress-strain state of a flat specimen made of a high-strength unidirectional carbon fiber composite under uniaxial tension is estimated. An analysis of the analytical solution obtained is carried out, and the results of finite-element calculations and modeling experiments are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 745–760, November–December, 2007.