Effect of random irregularities of the creep of reinforced layered plastics

The authors investigate the creep of inhomogeneous materials consisting of a large number of stiff orthotropic elastic layers alternating with layers of linear isotropic viscoelastic material. The elastic layers are assumed to be almost plane; the functions describing the irregularities (curvature) form a random field. The averaged characteristics of the medium are found together with the variation of the averaged displacements and strains in time. An analogous problem was previously considered in [1, 6] on the assumption that the binder layers are elastic. The present paper is based on the equations of [1] and the elastic-viscoelastic correspondence principle [4]. When the correlation scales of the irregularities are small as compared with the dimensions of the body and the characteristic distances over which the averaged parameters of the stress-strain state vary appreciably is considered in detail. A relation is established between the creep functions for simple cases of the state of stress and the parameters characterizing the properties of the components, the properties of the random field of initial irregularities, etc. The development of perturbations with different wave numbers is investigated. The theory is used to describe the creep of reinforced layered plastics.Mekhanika Polimerov, Vol. 2, No. 5, pp. 755–762, 1966     

Basic equations of the theory of reinforced media

The author derives the basic equations of the theory of composite elastic media obtained by reinforcing some elastic medium with a large number of linear or planar elastic elements with high strength and deformation resistance. The argument is based on macrostructural considerations. The stress-strain state of each of the reinforcing elements is considered with allowance for interaction with the matrix material. In addition, the "smoothing" principle introduced in [1–3] is applied. This corresponds to approximating the reinforced medium with some equivalent quasi-homogeneous anisotropic medium.The case of a fibrous medium in which the reinforcing elements are rods or filaments [4] is discussed in detail. Allowance for moment effects leads to equations analogous to the equations of the Voight-Cosserat moment theory and its later generalizations. Similar equations are obtained for the case of laminated media, where the reinforcing elements are membranes or plates. On the basis of the viscoelastic analogy [7], the equations of the theory of reinforced media are extended to include the case in which the matrix and/or reinforcing materials are linear viscoelastic.Mekhanika Polimerov, Vol 1, No. 2, pp. 27–37, 1965     

A generalized self-consistent method for composites with random elastic properties of inclusions

The generalized self-consistent method is extended to the problems of statistical mechanics of composites with random elastic properties of inclusions. This approach makes it possible to reduce the problem of predicting the effective elastic properties of composites with random structures to a sequence of simpler homogenized boundary-value problems for solitary inclusions with inhomogeneous elastic transition layers in a homogeneous effective elastic medium and with the corresponding boundary conditions. The elastic properties of a solitary inclusion for the gth homogenized problem are found from the solutions of the gth and (g+1)th homogenized problems. The elastic properties and sizes of the transition layers account for the random distribution, random sizes, and random elastic properties of inclusions in the composite. A test problem of predicting the effective elastic properties of a transversely isotropic layer composite with random elastic properties of some layers is solved by using the method proposed. The solution obtained coincides with the known exact solution [1].Perm State Technical University, Perm, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 785–796, November–December, 1999.     

Thermal deformation of a composite reinforced with hybrid cloth strips

Conclusion We tested (for mechanical and thermal effects) composites reinforced with hybrid cloth COS and VAI strips; five alternate schemes of material, which differred in terms of the content of VAI layers and layers reinforced with COS, were tested. The elasticity characteristics, tension diagrams, and CLTE of the composites were determined. It was established experimentally that variation in the relative content of the above-indicated layers makes it possible to regulate the thermal expansion of the composite in the longitudinal direction of the reinforcing strips Objectively over significant ranges; in this case, the elastic modulus varies negligibly, while the specific elastic modulus remains virtually unchanged,An alternate scheme for determining the elasticity characteristics and CLTE of laminar polymeric materials reinforced with hybrid cloth strips on the basis of component properties is developed. The model according to which the structural organization of the composite is subdivided into several levels is primarily a computational model. The stress-strain state of the repeating structural elements is evaluated by methods of the strength of materials. The proposed algorithm for computing the physicomechanical characteristics of laminar composites is implemented in the form of a computer program. The experimental elasticity characteristics and CLTE obtained for composites with a different content of COS and VAI layers are compared with those calculated in accordance with the method developed (the computed values correspond to the experimental with an accuracy acceptable for engineering applications).Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 392–401, May–June, 1988.     

Elastic constants of glass-reinforced plastics with anisotropic structure

The macroscopic elastic constants of plastics reinforced in one direction are calculated by solving the statistical boundary value problem of the theory of elasticity. The results obtained and the method proposed in [15] are used to compute the macromoduli of GRP with an anisotropic layered structure created by an orthogonal arrangement of unidirectional tapes and to compute the moduli of elasticity of a unidirectional layered material with random deviations of the layers from the direction of preferred orientation.V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, Sverdlovsk Branch. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 631–637, July–August, 1968.     

Averaging of a finely laminated elastic medium with roughness or adhesion on the contact surfaces of the layers

The governing relations of a laminated elastic medium with non-ideal contact conditions in the interlayer boundaries are obtained by an asymptotic averaging method. The interaction of rough surfaces is described by a non-linear contact condition which simulates the local deformation of the microroughnesses using a certain penetration of the nominal surfaces of the elastic layers. The cohesive forces, caused by the thin adhesive layer, are described within the limits of the Frémond model which includes a differential equation characterizing the change in the cohesion function. A piecewise-linear approximation of the initial positive segment of the Lennard–Jones potential curve is proposed to describe of the adhesive forces between smooth dry surfaces. A comparison is made with the solution obtained within the limits of the Maugis–Dugdale model based on a piecewise-constant approximation. Solutions of the above problems are constructed taking account of the possible opening of interlayer boundaries.     

Wave propagation in,layered, transversally isotropic,elastic media

Wave propagation in a transversally isotropic, elastic medium consisting of plane-parallel layers and half spaces is considered. A generalized matrix method is used to derive the dispersion equation of this medium and to find the coefficients of reflection and refraction. This method makes it possible to consider dispersion curves and the coeffients of reflection and refraction in a broader domain than with Haskell's method. The results obtained generalize to layers in which the elastic characteristics vary with depth according to an arbitrary law. For such layers it is possible to find matrices in the form of series which converge rapidly for low and high frequencies. Moreover, a rule is formulated which makes it possible on the basis of a known field in an isotropic medium to find the field in the corresponding transversally isotropic medium.     

Plane strain deformation of an initially unstressed elastic medium

Selim and Ahmed [1] used the eigenvalue approach by assuming distinct eigenvalues to calculate the elastic deformation due to an inclined load at any point as a result of an inclined line load of initially stressed orthotropic elastic medium. They studied the plane strain problem and obtained the corresponding results for an unstressed orthotropic medium as a particular case. In the present paper, it is shown that all the eigenvalues do not remain distinct, but become repeated when the elastic medium is free from the initial compressive stresses. Further, the displacements and stresses for an unstressed elastic medium have been independently obtained. The variation of the displacements and stresses due to normal and tangential line load are also shown graphically.     

Analysis of the stress-strain state of a ring-reinforced cylindrical section of an underground pipe

A method is presented for solving the problem of determining the stress-strain state of closed circular cylindrical shells in an elastic medium. The problem relates to the design of underground pipelines. The work of cylindrical shells is examined from the viewpoint of the theory of thin-walled three-dimensional systems, with allowance being made for the unilateral character of the interaction with the elastic medium. The stress-strain state of a cylindrical section of an underground pipe reinforced in the middle by a ring is investigated. It is shown that different factors influence the stress-strain state of the shell of the pipe.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 66–72, 1987.     

Compliance tensor for an elastic material reinforced in two directions

The relations for all the independent components of the elastic compliance and stiffness tensors of an elastic material reinforced in two directions are examined on the assumption that it is a continuous, macroscopically homogeneous, anisotropic (orthotropic) medium. The results obtained are used to construct illustrative curves and are compared with certain experimental data.Mekhanika Polimerov, Vol. 2, No. 3, pp. 372–379, 1966     

Effect of interphase layers on the elastic properties of a carbon-nanotube-reinforced composite

A variant of a stepwise analysis of the elastic properties of a carbon-nanotube-reinforced composite with account of the effect of interphase layers between the nanotubes and the polymer matrix is reported. The preliminary calculation of the elastic constants of a structural element incorporating a nanotube and an interphase layer and the subsequent calculation of independent elastic constants of a composite with such transversely isotropic structural elements oriented in one direction are both performed by using the Mori–Tanaka theory of an equivalent medium. The calculations are carried out for a wide range of ratios between the elastic moduli of the interphase layer and matrix. The elastic constants of a composite with randomly oriented nanotubes are obtained by using the method of orientational averaging.     

Elastic constants of randomly reinforced plastics

A previously developed [4] general method of solving the nonlinear (in the statistical sense) boundary value problem of the theory of elasticity is used to determine the macroscopic moduli of elasticity of randomly reinforced plastics whose structure is simulated by a two-phase microinhomogeneous medium. The macroscopic modulus is represented in the form of a series composed of the sum of the mean value of the modulus and a sequence of corrections that take into account the central moments of the distribution of elastic constants. The limits of convergence of this series are established. The values of the macroscopic moduli for a glass-reinforced plastic obtained by calculation are compared with the experimental data.Mekhanika Polimerov, Vol. 3, No. 2, pp. 259–265, 1967     

Stress state in multilayer cylindrical tubes made of elastically compressible linear viscoelastic material

Through the use of Abel's operators and э operators, the authors obtained an implicit solution for N-layered cylindrical tubes with alternating elastic and viscoelastic layers under conditions of elastic compressibility of the viscoelastic medium. Explicit expressions were written for the reactive pressures in 3- and 2-layered cylindrical tubes with alternating elastic and viscoelastic layers; these are analyzed in detail. Based on the solution obtained earlier for a condition where the viscoelastic operator corresponding to Poisson's coefficient is taken as a constant, it has been shown that the hypotheses discussed will lead in time to qualitatively different stress states. The overall results obtained are illustrated on an example.     

Problems of the concentration of elastic stresses near defects in spherically multilayered media

A spherically multilayered medium, whose elastic parameters change abruptly on the spherical surfaces, with defects in the form of cracks or thin rigid inclusions, is considered. The method of solving problems of the stress concentration near such defects is based on the introduction of linear combinations of the displacements and stresses as the fundamental unknowns. This enables the difficulties related to the presence of an arbitrary number of layers to be effectively overcome. The method is described initially for an unbounded elastic medium and defects of spherical form, situated on the surfaces where the elastic parameters change (interphase defects) and a way of extending this to the case of an elastic medium of finite dimensions, defects of other forms and not situated on these surfaces, is indicated. The method is described in detail as it applies to the case of a two-layer medium with an interphase crack when a torsion centre at the origin of coordinates acts on the medium. The problem is reduced to an integral equation, an effective method of solving it is given, and a formula is obtained for the stress intensity factor.     

非均匀变截面弹性圆环在任意载荷下的弯曲问题

本文在等刚度弹性圆环的初参数公式的基础上,利用[2]提出的阶梯折算法,进一步研究非均匀变截面弹性圆环的弯曲,得到了这类问题的通解,应当指出,这组通解对非均匀变截面圆柱拱的相应问题也是适用的.为验证所得的公式并说明这种方法的应用,文末给出了示例并进行了求解,圆环、圆拱是工程上经常采用的结构,它们的弯曲,Timoshenko,S.[5],Barber,J.R.[3],Roark,R J[4],津村利光[6]等曾作过很多研究.然而,迄今只求得了均匀材料、等截面圆环的通解。对变截面问题,仅仅求得了抗弯刚度是坐标的线性函数这一特殊情况的解.由于非均匀变截面问题常常导出变系数微分方程,它们的求解遇到很大的数学困难.本文通过阶梯折算法把非均匀变截面弹性圆环弯曲问题的变系数微分方程转化成一等效的等刚度圆环弯曲的常系数微分方程.为保证内力连续,引入虚拟内力,并以[1]导出的初参数公式为影响函数,通过积分构造出了非齐次解,从而求得了非均匀变截面弹性圆环弯曲问题的通解.     

Determination of the elastic moduli of reinforced materials

A method is proposed for determining the elastic moduli of reinforced materials at arbitrary concentrations of the components. The method is based on the successive symmetrization of the frequency function of the moduli of the component elements of a two-component material with subsequent application of the correlation theory [1]. Algorithms are constructed for determining the elastic moduli of layered, fibrous, and granular materials and typical calculations are presented.Mekhanika Polimerov, Vol. 4, No. 1, pp. 78–85, 1968     

A theory of elasticity with spatial distribution of matter. Three-dimensional complex structure

References [1 and 2] consider a theory of elasticity with spatial distribution of matter for a medium having simple structure and for a one-dimensional medium having complex structure. In the present article the general case of a three-dimensional medium with complex structure is examined. The general scheme of the one-dimensional case [2] is retained; chief attention is directed toward the specific character of the three-dimensional problem. The original micro-model is a complex crystal lattice [3]. In Section 1 this model is generalized to the case of a continuous distribution of matter. The displacements of the mass centers of the unit cells and the micro-strains of the cells are introduced as the kinematic variables. The force variables are the micro-moments. The transition to an exact continuous representation is carried out, and the equations of an elastic medium of complex structure with spatial distribution of matter are derived. The operators corresponding to the continuous theory are expressed in terms of the original microparameters. It is shown that the well known conditions of symmetry of the tensor of elastic constants, which are usually interpreted as the condition of absence of initial stresses [3 and 4], are consequences of the invariance of the elastic energy under translation and rotation. In Section 2 some special models are examined, and the equations of a medium are obtained for the approximation of weak dispersion of matter. These equations contain as a special case the equations of linear nonsymmetric elasticity (couple-stress theory) [5 to 7]. However, in the latter it turns out that the orders of approximation are inconsistent in the various equations from the point of view of the theory of spatial distribution.

In Section 3 the equations of a medium having complex structure are transformed in the acoustic range into equations, one of which contains only a single kinematic variable (the displacement of the mass centers) and the others of which are explicitly solvable for the remaining kinematic variables. The first equation of this set coincides in form with the equation for a medium with simple structure, but differs from it by the presence of a timewise dispersion which is unrelated to energy dissipation. Expressions are written for the energy density, and it is shown that it is possible to introduce a symmetric stress tensor, as in the case of a simple structure.     

The limiting state of a rigidly fixed nonlinearly elastic multilayer rod

The loss of the load-carrying capacity of a nonlinearly elastic multilayer rod is investigated. The rod, whose layers have various thickness and are made of different materials, is rigidly fixed at both its ends. Rigid contact conditions between the layers are assumed. The problem posed is solved by using the variational method of mixed type in combination with the Rayleigh-Ritz method. The initial analysis is reduced to the solution of the Cauchy problem for a nonlinear ordinary differential equation solved for the first derivative. As the initial condition, the maximum initial eccentricity of the rod is assumed. In the case of zero eccentricity, the Shanley critical force for an axially compressed rod is determined. For a three-layer rod whose outer layers have equal thickness and are made of the same material, numerically, for various degrees of nonlinearity, the effect of physicomechanical and geometric parameters on the critical load of buckling instability is determined. It is found that, by matching the heterogeneity of the rod, it is possible to raise its load-carrying capacity. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 347–360, May–June, 2006.     

Generalized self-adjustment method for statistical mechanics of composite materials

A new method is developed for the statistical mechanics of composite materials — the generalized selfadjustment method — which makes it possible to reduce the problem of predicting effective elastic properties of composites with random structures to the solution of two simpler “averaged” problems of an inclusion with transitional layers in a medium with the desired effective elastic properties. The inhomogeneous elastic properties and dimensions of the transitional layers take into account both the “approximate” order of mutual positioning, and also the variation in the dimensions and elastics properties of inclusions through appropriate special averaged indicator functions of the random structure of the composite. A numerical calculation of averaged indicator functions and effective elastic characteristics is performed by the generalized self-adjustment method for a unidirectional fiberglass on the basis of various models of actual random structures in the plane of isotropy.     

States of stress in multilayer spherical vessels,cylindrical tubes,and circular disks of linear viscoelastic material

By using the properties of Abel operators and -operators [3], unique implicit solutions for N-layer structures with alternating elastic and viscoelastic layers in the case where the viscoelastic operator corresponding to the Poisson ratio reduces to a constant. For an N-layer spherical vessel an analogous implicit solution is also obtained on the assumption that the viscoelastic medium is elastically compressible. Explicit expressions for the reaction pressures are written down for the case of all possible two- and three-layer structures with alternating elastic and viscoelastic layers and analyzed in detail. It is shown that the hypotheses in question lead to qualitatively different states of stress. The general results obtained are illustrated by an example.Institute of Hydrodynamics, Siberian Division, Academy of Sciences of the USSR, Novosibirsk. Translated from Mekhanika Polimerov, No. 1, pp. 110–116, January–February, 1973.