Elastic constants of monotropic plastic foams. 2. Deformation parallel to foam rise direction. Numerical calculations

In order to perform numerical calculations of the elastic constants of monotropic plastic foams with a pronounced strut-like structure, a package of computing programs has been developed. Numerical values of the elastic constants were determined using the Simpson method for the calculation of triple averaging integrals and stepwise unconditional minimization of the one-argument potential energy function. The parameters of the numerical calculation process providing acceptable accuracy of the results were evaluated. Dimensions of the structural elements in the model of isotropic plastic foams were determined and analyzed.Institute of Polymer Mechanics, Riga, LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 163–172, March–April, 1998.     

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.     

A Deformation Model of Monotropic Plastic Foams in Shear Perpendicularly to the Isotropy Plane

Based on an ellipsoidal model for the cellular structure of monotropic plastic foams, their deformational properties in shear perpendicularly to the isotropy plane are predicted. The theoretical results are obtained for the case of pure shear. The deformation energy is calculated by the method of coordinates assuming that the polymer struts of the structure can change both their length and orientation. Numerical values of shear moduli are found for monotropic and isotropic plastic foams, which agree satisfactorily with the experimental data for polyurethane and polyvinylchloride foams available in the literature.     

A Deformation Model for Monotropic Plastic Foams under Compression/Tension Perpendicularly to the Foam Rise Direction

The deformation properties of monotropic plastic foams under uniaxial compression or tension perpendicularly to the foam rise direction are considered. Theoretical results are obtained for the case where the volume- deformation hypothesis is assumed. In order to perform numerical calculations, the foams are devided into three groups with different degrees of monotropy. For these groups, different methods of calculating the numerical value of the relative volume strain are used. The Young's modulus in the plane of isotropy is determined using both the mixed Reuss-Voigt and Voigt averaging. The Poisson ratios are expressed using the Reuss averaging. If only the axial deformation of the load-bearing elements (polymer struts) is allowed for, the results obtained do not agree with experimental data. Therefore, we also take into account the changes in their orientation. The values of Young's modulus and Poisson ratios are obtained for a wide range of degrees of extension of the model cell. A comparison of the theoretical results with the experimental data available shows a satisfactory agreement.     

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.     

Model of Deformation of Monotropic Plastic Foams Parallel to the Foam Rise Direction Based on the Volume-Deformation Hypothesis

The deformation properties of monotropic plastic foams under uniaxial deformation (compression or tension) parallel to the foam rise direction is considered. The theoretical results are obtained in the case where the volume-deformation hypothesis is assumed. The validity of neglecting the fluctuation component in calculations of the effective volume strains of foams is substantiated. A tie condition permitting the sought-for semiaxes to vary not obligatorily equally is derived. The values of Young's modulus and Poisson coefficients are obtained for a wide range of model cell stretch ratios and foam space-filling coefficients. A comparison of the theoretical results with the experimental data available is performed.     

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.     

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.     

Experimental Investigations on Polymeric Foams

The knowledge of the material behaviour of polymeric foams and their experimental investigation is the starting point for the structure of the chosen constitutive equations and for the following identification of the material constants therein. Especially for the parameter identification, it is necessary to make an adequate set of experimental data available. In this regard, it is important that the experiments make the different kinds of material behaviour visible like elastic, plastic or viscous material properties. For this reason, the foam is observed under uniaxial tension and compression and under simple shear tests combined with different deformation states in axial direction. Unfortunately, due to different reasons, e.g., the foam must be sticked on the fastener to realize the tests mentioned above, it is very difficult to initialize a homogenous deformation state in the specimen. Therefore, the experiments are recorded with a standard digital camcorder to get local information of the deformation state by tracking single points with algorithms of the digital image processing. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of cellular structure on the mechanical properties of plastic foams

A generalized equation for the compressive stress-strain diagrams of plastic foams is derived on the basis of a 14-faced model of the cell. The results obtained make it possible to predict the polymer base and type of cellular structure required to obtain a foam with predetermined mechanical properties in compression. The calculated values are shown to be in satisfactory agreement with the experimental data.Vladimir Scientific-Research Institute of Synthetic Resins. Translated from Mekhanika Polimerov, No. 4, pp. 594–602, July–August, 1970.     

Effective moduli of elasticity of a composite composed of anisotropic layers

Relations are obtained for the effective moduli of elasticity and Poisson's ratios of a laminated fiber-reinforced composite, each layer of which has at least orthorhombic symmetry. The elastic properties of the composite in terms of the elastic constants of the layer are expressed exactly, and the elastic constants of the individual layer in terms of the values for the fiber and the matrix are expressed approximately. Two approximations are considered: one corresponds to the Hashin-Shtrikman variational approach, while in the second the comparison material is assigned elastic properties equal to the Voigt or Reuss means of the values for each layer. A numerical example is worked for the combination boron fibers-epoxy resin. The results of the calculation are compared with the exact solution of the problem for a composite composed of alternating layers of boron and epoxy resin.     

Incompatibility Induced Nonlocal Crystal Plasticity

Motivated by the fact that locally inhomogeneous elastic or plastic deformations may result in incompatibilities of the fictitious intermediate configuration a strain gradient crystal plasticity model is developed. Thereby incompatibilities can be accounted for and scale dependent material behavior, as also observed experimentally, is predictable. A nonlocal extension of existing local formulations is proposed which does not require additional boundary conditions and thus maintains the classical BVP structure. On the numerical side key developments are an extended FE-formulation for rate-(in)dependent strain gradient plasticity and a local FE-formulation which bases the gradient computation on an operator split combined with a smoothing algorithm. Comparative numerical studies for classical examples proove the superior efficiency of the second approach. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of the Rounding Radius of Supports on the Accuracy of Determining the Interlayer Shear Modulus of Reinforced Plastics from Short-Beam Bending Tests

The accuracy of determining the elastic constants of reinforced plastics is estimated based on bending tests of short beams using the results of a numerical experiment with known elastic constants of the specimen material. Several combination variants for the initial values of the shear modulus and support radius are considered. It is shown that the calculation error of the shear modulus is considerably higher than that of the elastic modulus A decrease in the shear modulus increases the accuracy of its determination. The radius of supports affects this accuracy insignificantly.     

Computational modeling of elastic and plastic multiple cracks by the fundamental solutions

The fundamental solutions of elasticity are used to establish a numerical method for elastic and plastic multiple crack problems in two dimensions. The continuous distributions of the point forces, dislocations, and the plastic sources are used systematically to model the crack, non-crack boundary, and the plastic deformation. Use of these singularities are guided strictly by the physical interpretation of the problem. We adopt Muskhelishvili's complex variable formalism that facilitate the analytical evaluation of the integrals representing the continuous distributions of the singularities. The resulting numerical method is concise and accurate enough to be used for elastic and plastic multiple crack problems.     

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.     

An extended beam theory for smart materials applications part I: Extended beam models,duality theory,and finite element simulations

An extended theory for elastic and plastic beam problems is studied. By introducing new dependent and independent variables, the standard Timoshenko beam model is extended to take account of shear variation in the lateral direction. The dynamic governing equations are established via Hamilton's principle, and existence and uniqueness results for the solution of the static problem are proved. Using the theory of convex analysis, the duality theory for the extended beam model is developed. Moreover, the extended theory for rigid-perfectly plastic beams is also established. Based on the extended model, a finite-element method is proposed and numerical results are obtained indicating the usefulness of the extended theory in applications.The work of the first author was supported in part by National Science Foundation under Grant DMS9400565.     

Evaluation of two methods for the determination of the elastic constants of trigonal crystals

Two procedures for the computation of the elastic constants (trigonal system) from measured values of elastic wave velocities are discussed. The simplified procedure first used by Bhimasenachar is incorrect in principle although, in very rare cases, it may lead to only small numerical errors. A new procedure is mentioned which simplifies the use of the Christoffel equations in an exact way.