Influence of initial stresses on fracture of composite materials containing interacting cracks

Considered in this study are the axially-symmetric problems of fracture of composite materials with interacting cracks, which are subjected to initial (residual) stresses acting along the cracks planes. An analytical approach within the framework of three-dimensional linearized mechanics of solids is used. Two geometric schemes of cracks location are studied: a circular crack is located parallel to the surface of a semi-infinite composite with initial stresses, and two parallel co-axial penny-shaped cracks are contained in an infinite composite material with initial stresses. The cracks are assumed to be under a normal or a radial shear load. Analysis involves reducing the problems to systems of second-kind Fredholm integral equations, where the solutions are identified with harmonic potential functions. Representations of the stress intensity factors near the cracks edges are obtained. These stress intensity factors are influenced by the initial stresses. The presence of the free boundary and the interaction between cracks has a significant effect on the stress intensity factors as well. The parameters of fracture for two types of composites (a laminar composite made of aluminum/boron/silicate glass with epoxy-maleic resin and a carbon/plastic composite with stochastic reinforcement by short ellipsoidal carbon fibers) are analyzed numerically. The dependence of the stress intensity factors on the initial stresses, physical-mechanical parameters of the composites, and the geometric parameters of the problem are investigated.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

Thermomechanical behavior of nonmetallic electrical conductors during high-temperature treatment

We give a mathematical statement and initial relations for the problem of determining the temperature fields and stresses in nonmetallic electrical conductors under induction heating to high temperatures. We study the influence of the temperature dependence of the characteristics of the material and the radiant heat transfer with the surrounding medium on the thermomechanical behavior of a cylinder. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 74–79.     

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

Planar deformation of a plate with a one- or two-sided cover

We solve the problem of planar deformation of an isotropic plate fastened at one surface or symmetrically in its two surfaces to cover plates of a different material. As initial data we use the approximate relations of the two-dimensional formulation, which take account of the shear and the transverse compression of the plate. We study the dependence of the interlayer stresses on the ratio of the elastic moduli of the materials of the plate and the cover.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 99–102.     

On the single fibre pullout and pushout problem: Effect of fibre anisotropy

Based on our recent theoretical model of the interfacial debonding and fibre pullout/pushout stresses, an improved analysis including the effect of fibre anisotropy has been performed for single fibre pullout and pushout. During fibre-matrix debonding the debonded region is considered as an interfacial crack and its extension is dependent on a fracture energy criterion being satisfied. The elastic properties of the fibre are taken to be transversely isotropic about the fibre axis and the matrix is assumed to be isotropic. The results show that the maximum debond stress for fibre pullout and the initial frictional pullout stress increase as the ratio of the Young's moduli of the fibre between the axial and radial directionsE _f /E _f ^r increases, and this is particularly so at large embedded fibre lengths. However, in the case of fibre pushout these debond and pushout stresses decrease asE _f /E _f ^r increases.     

Limit equilibrium of a closed transversally isotropic cylindrical shell with a nonthrough longitudinal crack

In the context of an analog of the Leonov-Panasyuk-Dagdeil model we consider the problem of limit equilibrium of a nonshallow transversally isotropic cylindrical shell weakened by a nonthrough surface longitudinal crack. Based on the equations that take account of the initial stresses, we reduce the problem to a system of two singular integral equations with unknown limits of integration. We carry out a numerical analysis of the dependence of the opening of the edges of the crack on the load and the geometric and physico-mechanical parameters of the shell. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 31–36.     

Elastic and Sorption Characteristics of an Epoxy Binder in a Composite During Its Moistening

Results of an experimental investigation into the elastic and sorption characteristics of a model composite material (CM) — epoxy resin filled with LiF crystals — during its moistening are presented. Properties of the binder in the CM with different filler contents (v _f = 0, 0.05, 0.11, 0.23, 0.28, 0.33, 0.38, and 0.46) were evaluated indirectly by using known micromechanical models of CMs. It was revealed that, for the CM in a conditionally initial state, the elastic modulus of the binder in it and the filler microstrain (change in the interplanar distance in the crystals, measured by the X-ray method) as functions of filler content had the same character. The elastic modulus of the binder in the CM with a low filler content was equal to that for the binder in a block; the elastic modulus of the binder in the CM decreased with increasing filler content. The maximum (corresponding to water saturation of the CM) stresses in the binder and the filler microstresses as functions of filler content were of the same character. Moreover, the absolute values of maximum stresses in the binder and of filler microstresses coincided for high and low contents of the filler. At v _f = 0.2–0. 3, the filler microstrains exceeded the stresses in the binder. The effect of moisture on the epoxy binder in the CM with a high filler content was not entirely reversible: the elastic characteristics of the binder increased, the diffusivity decreased, and the ultimate water content increased after a moistening-drying cycle.__________Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 4, pp. 499–511, July–August, 2005.     

Initial stresses in filament-wound reinforced-plastic rings

Methods of experimentally determining the initial stresses in filament-wound GRP rings are examined and the effect of winding tension is investigated. It is shown that by varying the winding tension it is possible to exercise a considerable degree of control over the magnitude and distribution of the initial stresses.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 505–511, May–June, 1969.     

On the use of jump conditions at a thin inclusion in solving problems of elasticity theory for a half-space with protuberances

We propose a method of approximate solution of problems of elasticity theory for a half-space with protuberances based on the use of jump conditions in the stresses and displacements at a thin elastic element. The problem of determining the stresses reduces to a system of two-dimensional integral equations of Newtonian potential type for determining the contact stresses between the protuberances and the half-space. We consider the case when the elastic characteristics of the material of the protuberances are different from the material of the half-space.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 156–160.     

On the extremal value of the thermoelastic energy of a material with an inclusion subject to mechanical and thermal action

Under prescribed thermoelastic stresses and known properties of the matrix and the inclusion in an elastic medium with an inhomogeneity, we find the shape of inhomogeneity that leads to an extremal value of the thermoelastic energy. From the necessary conditions for an extremum of the thermoelastic energy functional we find a condition for seeking the interface. For the case of isotropic cornponents and under loads of stretching (compression) type and uniform heating of the medium the shape of the inclusion can be found explicitly within a certain range of initial parameter values. The results of numerical study are presented and analyzed. One table. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 26–34.     

Moduli of elasticity of polymer melts

The physical significance of the so-called initial modulus, determined in measuring the rheological properties of thermoplastic melts from the initial slope of the time dependence of the stresses at given constant shear rate, is considered. It is shown that this quantity is related with the ratio of the shear and normal stresses.Topchiev Institute of Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 534–538, May–June, 1969.     

Effect of a uniaxial load on moisture absorption by an epoxy binder

Conclusion The above-developed method of studying the effect of uniaxial compressive and tensile loads on moisture absorption in polymers was tested on epoxy binder EDT-10. It was established experimentally for this material that tensile stresses up to 50 MPa significantly increase the equilibrium moisture content (up to 20% of the initial value). There is a corresponding increase in the rate of moisture absorption during the initial stage of the sorption process — up to 80% of the initial value. No changes in the diffusion coefficient were seen in this case. The application of compressive stresses of up to 40 MPa also had no effect on the kinetics of moisture absorption.Deceased.Translated from Mekhanika Kompozitnykh Materialov, Vol. 29, No. 1, pp. 110–115, January–February, 1993.     

Geometrically nonlinear dilation of an orthotropic plane with an elliptical hole

In a geometrically nonlinear formulation we obtain expressions for the stresses and displacements on the edge of an elliptical hole in an orthotropic plane dilated by a strain at infinity in the direction of the principal axes of anisotropy. We exhibit the dependence of the concentration of stresses on the magnitude of the load acting on the plane. We prove that the stresses are finite at the ends of the crack and we compute their values. We confirm the need to take account of the geometric nonlinearity of the problem for low-modulus orthotropic materials.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 90–96.     

Linear Viscoelasticity of Liquid-Crystalline Polymers

A linear (small-amplitude) periodic shear deformation of anisotropic viscoelastic liquids obeying the Akay–Leslie rheological model is considered. The frequency dependences of the real and imaginary components of the complex shear modulus and complex normal-stress coefficient are determined. A comparison between calculation results and test data on the shear flow of poly(-benzylglutamate) in m-cresol is carried out. It is stated that, if the material is characterized by some initial orientation, both components of the complex shear modulus contain a multiplier which depends on the degree of the initial orientation and increases the values of the components compared with those for an initially isotropic material. The model predicts that, in a periodic shear flow, the components of shear and normal stresses are constant and, like the components of shear modulus, are independent of deformation frequency. If the parameter d ₀ of the Akay–Leslie model is equal to zero, the values of its other parameters can be determined from experimental results on periodic shear flow.     

Normal stresses in non-Newtonian polymer flows

The calculation of the normal stresses from the flow curve is considered on the basis of the method of correlation of the frequency and stationary characteristics of flowing polymer systems. Simple expressions are given for finding the initial normal stress coefficient and the high-elastic modulus from the point corresponding to the beginning of non-Newtonian flow. A necessary condition for the appearance of normal stresses is a viscosity anomaly. It is shown that for a bounded value of the initial normal stress coefficient to exist the derivative of the effective viscosity with respect to shear rate must be zero at the initial point.Topchiev Institute of Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 506–514, May–June, 1971.     

Axisymmetric contact problems for a prestressed incompressible elastic layer

Two axisymmetric problems of the indentation without friction of an elastic punch into the upper face of a layer when there is a uniform field of initial stresses in the layer are considered. The model of an isotropic incompressible non-linearly elastic material, specified by a Mooney potential, is used. Two cases are investigated: when the lower face of the layer is rigidly clamped after it is prestressed, and when the lower face of the layer is supported on a rigid base without friction after it is prestressed. It is assumed that the additional stresses due to the action of the punch on the layer are small compared with the initial stresses; this enables the problem of determining the additional stresses to be linearized. The problem is reduced to solving integral equations of the first kind with symmetrical irregular kernels relative to the pressure in the contact area. Approximate solutions of the integral equations are constructed by the method of orthogonal polynomials for large values of the parameter characterizing the relative layer thickness. The case of a punch with a plane base is considered as an example.     

Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

Determination of residual stresses in glass shells by a computational-experimental method

A method is proposed for determining the technological residual stresses in glass shells of revolution based on the solution of inverse problems of shell theory with proper stresses and the use of experimental data obtained by the photoelastic method. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 100–105.     

Residual stresses in a closed cylindrical shell with a longitudinal crack

We study the influence of the residual stresses on the size and character of the variations in the strain intensity and moment intensity coefficients in a closed cylindrical shell welded together from two parts and under the action of axisymmetric loads. We perform a numerical analysis in the case when the shell with residual stresses is acted on by internal pressure.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 40–45.