Elastic constants of monotropic plastic foams. 1. Deformation parallel to foam rise direction. A mathematical model

A mathematical model of the deformative properties and structure of lightweight, monotropic (or isotropic in the limiting case) plastic foams with a pronounced strut-like structure has been elaborated in the linear theory of deformation. A selection of five independent elastic constants is described. For the integral characterization of the deformative properties of plastic foams as micrononhomogeneous composite materials, the elastic constants are introduced as the effective constants. In order to describe the plastic foam structure, a local model consisting of two parts is proposed, i.e., a model of a continuous medium for the calculation of stresses and a local structure model. Considering deformation parallel to the foam rise direction when the semiaxes hypothesis is assumed, the Young modulus and Poisson's ratio are determined.Institute of Polymer Mechanics. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 719–733, November–December, 1997.     

Calculation of Dependent Elastic Constants of Plastic Foams with a Pronounced Strut-Like Structure

Seven dependent elastic constants of monotropic plastic foams with an expressed strut-like structure are calculated. For this purpose, the basic results of the previously elaborated mathematical model for light-weight plastic foams is used. The model includes a model cell of local structure for monotropic/isotropic plastic foams and an ensemble of structural elements, which allows one to calculate the seven dependent elastic constants, taking into account the pronounced polydispersity of the structure of plastic foams. The numerical values of the constants are compared with the available experimental data, and a satisfactory agreement is found to exist. As a final result, a full set of general expressions and numerical values are obtained for all 12 elastic constants of monotropic plastic foams.     

Elastic constants of monotropic plastic foams. 3. Deformation parallel to foam rise direction. Analysis of the results

Based on an elaborate mathematical model shaped like an ellipsoidal cell, the Poisson's ratio v ₃₁ ^* and Young's modulusE ₃ ^* are calculated for monotropic (isotropic in the limiting case) plastic foams when loading parallel to the foam rise direction is considered and the hypothesis of half-axis is assumed. The effect of the state of the strut system on the calculation results is studied. The dependence of the calculated elastic constants on the characteristics of plastic foams such as the space filling coefficient, degree of anisotropy and knot parameter is analyzed. The theoretical results are compared with the experimental results as well as the results of other authors.Institute of Polymer Mechanics, University of Latvia, 23 Aizkraukles St., Riga, LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 823–838, November–December, 1998.     

Elastic and strength characteristics of a glass-reinforced plastic in compression

The elastic constant and strength characteristics in compression have been experimentally determined for a cold-cured glass-reinforced plastic in the direction of all three principal axes of anisotropy. It is shown that the elastic constants obtained satisfy the constraints imposed on orthotropic materials by the thermodynamic relations.Translated from Mekhanika Polimerov, No. 5, pp. 866–869, September–October, 1970.     

Deformative characteristics of plastics reinforced with high-modulus anisotropic fibers

The independent elastic constants of plastics unidirectionally reinforced with transversely isotropic fibers have been determined. It has been assumed that the distribution of reinforcement in a transverse section of the plastic is regularly rectangular or hexagonal. To determine the transverse elastic modulus and the shear modulus in the plane of reinforcement, a constancy-of-plane-sections hypothesis was used. Values of deformative characteristics determined by the assumed calculational dependences have been compared with the experimental ones for plastics reinforced with graphite fibers.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 631–639, July–August, 1972.     

Numerical model of one possibility of determining the tensor of elastic constants of an anisotropic body

An algorithm for a digital computer is presented which permits calculating the tensor of the elastic constants of a material with any given symmetry on the basis of experimentally measured Young moduli and Poisson ratios for several specimens oriented in unequivalent crystallographic directions. The results of determining the components of the tensor of elastic constants by means of it for a modification of orthotropic reinforced plastic are also given.Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 436–442, May–June, 1973.     

Tensile behavior of plastic foams

The tensile stress-strain diagram of plastic foams is calculated on the basis of a previously proposed 14-faced cell model. An S-shaped inflection due to the flexural deformations of the ties is detected on the initial section of the diagram. It is shown that for closed-celled foams this inflection may degenerate. The results of testing PVC and polyurethane foams (volume content of polymer base about 3–6%) prove to be in satisfactory agreement with the calculations.Vladimir Scientific-Research Institute of Synthetic Resins. Translated from Mekhanika Polimerov, No. 4, pp. 670–675, July–August, 1971.     

Large Elastic Strains of Plastic Foams

The elastic deformation of plastic foams with a low (< 6%) volume fraction of solid phase is described based on a 4-rod equivalent element. A criterion is proposed which allows one to determine the parameter of structure of this element. Based on an analysis of the equivalent element, a procedure is developed for constructing the compression diagram of plastic foams in the region of large (> 70%) strains. The calculation results are compared with data found in the literature and experimental results for polyurethane foams obtained by the present authors. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 5, pp. 619–632, September–October, 2005.     

Structural Modeling in the Mechanics of Porous Materials

A new method for analyzing the deformation behavior of rigid and elastic foams with a small volume content of solid phase ( < 0.2) is developed. Various structural models for describing the elastic behavior of rigid and elastic plastic foams are used and compared. The results of structural simulation of anisotropic auxetic (i.e., having a negative Poisson ratio) foams with concave cells are presented. For cyclic uniaxial compression of rigid foams and volumetric deformation of elastic foams, the stress-strain curves are obtained. The general shape of the curves agrees well with the nonlinearly elastic behavior of plastic foams observed in experiments.     

Mechanical behavior of cylindrical laminated plastic shells

The static and dynamic behavior of a cylindrical glass-reinforced plastic shell with an open cross section under long-term loading and unloading has been experimentally and theoretically investigated. The experimental results are compared with the theoretical data obtained using a geometrically nonlinear theory based on the Kirchhoff hypothesis. For calculation purposes the glass-reinforced plastic is assumed to be a homogeneous orthotropic linearly elastic material.Building Research Institute, Prague. Translated from Mekhanika Polimerov, No. 1, pp. 110–115, January–February, 1976.     

Temperature-frequency dependence of the elastic constants of certain resins

The authors present the results of an experimental investigation of the elastic constants of certain crosslinked and linear polymers in the temperature range from –50 to +110° C and the frequency range from 10¹ to 10⁵ Hz. The elastic constants were determined by both quasi-static and dynamic methods. The range of application of the different methods are characterized with respect to a time scale. It is shown that when certain experimental conditions are observed (strain rate, frequency), the values of the constants obtained by the different methods almost coincide. On the basis of temperature measurements it is postulated that rigid epoxies have a secondary low-temperature dispersion region.Mekhanika Polimerov, Vol. 3, No. 1, pp. 60–65, 1967     

Shear effects during bending of oriented glass-reinforced plastics

An oriented glass-reinforced plastic may be treated as a regular multilayer material. Given large number of layers we can replace the discrete model with a continuous medium whose elastic constants are determined by the properties of the components. Using this approach and assuming that the reinforcing and the resin obey Hooke's law, we examine the problem of the bending of oriented glass-reinforced plastics and by means of two examples demonstrate the importance of shear effects in the bending of beams. The experimental results are in good agreement with theory.Mekhanika Polimerov, Vol 1, No. 2, pp. 38–46, 1965     

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     

Elastoplastic deformation during projectile–wall collision

The Smoothed Particle Hydrodynamics method for elastic solid deformation is modified to include von Mises plasticity with linear isotropic hardening and is then used to investigate high speed collisions of elastic and elastoplastic bodies. The Lagrangian mesh-free nature of SPH makes is very well suited to these extreme deformation problems eliminating issues relating to poor element quality at high strains that limits finite element usage for these types of problems. It demonstrates excellent numerical stability at very high strains (of more than 200%). SPH can naturally track history dependent material properties such as the cumulative plastic strain and the degree of work hardening produced by its strain history. The high speed collisions modelled here demonstrate that the method can cope easily with collisions of multiple bodies and can also naturally resolve self-collisions of bodies undergoing high levels of plastic strain. The nature and the extent of the elastic and plastic deformation of a rectangular body impacting on an elastic wall and of an elastic projectile impacting on a thin elastic wall are investigated. The final plastically deformed shapes of the projectile and wall are compared for a range of material properties and the evolution of the maximum plastic strain throughout each collision and the coefficient of restitution are used to make quantitative comparisons. Both the elastoplastic projectile–elastic wall and the elastic projectile–elastoplastic wall type collisions have two distinct plastic flow regimes that create complex relationships between the yield stress and the responses of the solid bodies.     

Thermal deformation of unidirectional hybrid composites. 2

Conclusions A variant of the solution of the problem of the thermorheologically complex temperature strain of a hybrid composite containing viscoelastic thermorheologically simple components with differing functions for temperature-time reduction in addition to elastic components, is proposed. An experimental study is conducted on unidirectional specimens of organic- and glass-fiber-reinforced plastic, organic- and carbon-fiber-reinforced plastic, and carbon- and glass-fiber-reinforced plastic at a constant rate of temperature change in the 20–150 °C range. Satisfactory correspondence is obtained between predicted and experimental data.For previous communication, see [1].Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 969–979, November–December, 1989.     

Deformative properties of flexible plastic foams in compression

A calculation of the compression diagrams of flexible plastic foams during their long-time testing under a load is given on the basis of the proposed cell model, on the assumption that permanent set is related to buckling of the strands. The character of change of the nominal compressive strength and corresponding deformation as a function of the magnitude of permanent set is established. A marked change of the second section of the compression diagram (plateau), even to its degeneration, in the presence of considerable permanent set was found. A comparison of the experimental and calculated data showed their satisfactory correspondence.Vladimir Research Institute of Synthetic Resins. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 443–449, May–June, 1973.     

Analysis of effective elastic properties of unidirectional boron fiber-reinforced plastic by a generalized self-consistent method

A new generalized self-consisrtent method is developed for the statistical mechanics of composites which makes it possible to reduce the problem of predicting the effective elastic properties of composites with random structures to solution of a simpler averaged problem of an inclusion with a transitional layer in a material with the effective elastic properties sought. The typical size of the transition layer is determined by the correlation radius of the random structure, and its elastic properties are considered as both the close order of the mutual position and the variation of inclusion dimensions in terms of a special averaged indicator function of the structure. A numerical calculation is presented by the generalized self-consistent method for the average indicator function and the transversely-isotropic tensor for effective elastic properties of unidirectional boron fiber-reinforced plastic based on different models for actual random structure in the plane of isotropy. Analysis of the numerical results compared with experimental data and known solutions of other authors demonstrates the high accuracy of the generalized self-consistent method for a broad class of random composite structures.Perm State Mechanical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 747–758, November–December, 1996.     

Determination of rational structure for spherofibrous plastics

A complete set of elastic constants for various structures of triorthogonally reinforced plastics with a matrix dispersely reinforced by spherical particles has been determined on the basis of the proposed algorithm. Approximate equations are obtained for determination of all integral parameters from data on the components and structure geometry. It is established that for the reinforcement schemes investigated, the shear moduli of the composites have the lowest possible values and are calculated with a higher accuracy than the other elastic constants.A. A. Blagonravov Institute of Machine Science, Russian Academy of Sciences. Moscow, Russian. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 155–160, March–April, 1997.     

Fundamentals of the theory of non-fabric glass-reinforced plastics

Starting from the known mechanical properties of the components (glass reinforcement and resin) the stress-strain relations for an anisotropic viscoelastic material (glass-reinforced plastic) are determined. Using Volterra's principle [2] of replacing the elastic constants in the solution of the problem of the theory of elasticity [5] by integral operators, relations for the given material are obtained and found to be in fairly good agreement with the experimental data. In a separate study the high-temperature behavior of GRP is investigated by calculating the integral operators, which are functions of the viscous properties of the resin and the temperature.Mekhanika polimerov, Vol. 1, No. 1, pp. 151–158, 1965     

Contribution to the theory of the elastic properties of amorphous-crystalline polymers

The elastic constants of an amorphous-crystalline polymer are calculated from the known values of the elastic constants of its amorphous and crystalline components.Moscow Lenin Pedagogical Institute, Problem Laboratory of Polymer Physics. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 213–218, March–April, 1969.