Constitutive Modelling of Resins in the Stiffness Domain

An analytic method for inverting the constitutive compliance equations of viscoelasticity for resins is developed. These equations describe the HWKK/H rheological model, which makes it possible to simulate, with a good accuracy, short-, medium- and long-term viscoelastic processes in epoxy and polyester resins. These processes are of first-rank reversible isothermal type. The time histories of deviatoric stresses are simulated with three independent strain history functions of fractional and normal exponential types. The stiffness equations are described by two elastic and six viscoelastic constants having a clear physic meaning (three long-term relaxation coefficients and three relaxation times). The time histories of axiatoric stresses are simulated as perfectly elastic.The inversion method utilizes approximate constitutive stiffness equations of viscoelasticity for the HWKK/H model. The constitutive compliance equations for the model are a basis for determining the exact complex shear stiffness, whereas the approximate constitutive stiffness equations are used for determining the approximate complex shear stiffness. The viscoelastic constants in the stiffness domain are derived by equating the exact and approximate complex shear stiffnesses. The viscoelastic constants are obtained for Epidian 53 epoxy and Polimal 109 polyester resins. The accuracy of the approximate constitutive stiffness equations are assessed by comparing the approximate and exact complex shear stiffnesses. The constitutive stiffness equations for the HWKK/H model are presented in uncoupled (shear/bulk) and coupled forms. Formulae for converting the constants of shear viscoelasticity into the constants of coupled viscoelasticity are given as well.     

Modeling of nonlinear viscoelastic contact problems with large deformations

Contact problems are one of the most important engineering problems. These problems become much more tedious when one of the contacting bodies behaves nonlinear viscoelasticity and large deformations. This paper presents an incremental-iterative finite element model for the analysis of two dimensional quasistatic frictionless contact problems. Nonlinear viscoelastic behavior and large deformations are considered. The Schapery’s single-integral creep model with stress-dependent properties is used for nonlinear viscoelasticity. The constitutive equations are transformed into an incremental form resulting in a recursive relationship. Thereby, the need to store the entire strain histories is eliminated, except that from the previous time increment. The updated Lagrangian formulation is used to model the material and geometrical nonlinearities. Also, the Lagrange multiplier method is adopted to enforce the contact constraints. The converged solution is obtained using the Newton–Raphson iterative technique. The developed model has been verified with the previously published works and found a good agreement with them. To demonstrate the efficient capability of the developed computational model, three contact problems with different nature are analyzed.     

Creep of Poly(Vinyl Chloride)/ Chlorinated Polyethylene Blends

The results of experimental investigations of the creep behavior of blends of poly(vinyl chloride) (PVC) with chlorinated polyethylene (CPE) are presented. Eight types of specimens with the PVC/CPE weight ratios of 100/0, 90/10, 80/20, 60/40, 40/60, 20/80, 10/90, and 0/100 are examined. The creep tests were continued for 1000 h. It is discussed how the blend composition affects the elastic and inelastic behavior of the material. The elastic compliance of the blend can be determined from the properties of its components by using the Kerner and Budiansky equations for heterogeneous systems with a phase structure of statistic-dispersion type. The creep compliance (the total current compliance minus the elastic compliance) obeys the power law of creep with coefficients depending on the blend composition.     

Partial three dimensional deformations for the perfectly elastic Mooney material

Summary The governing equations for finite elastic deformations are highly nonlinear and there is still only a limited number of known exact solutions. In general for large elastic fully three dimensional deformations of the isotropic incompressible perfectly elastic neo-Hookean and Mooney materials, a non-trivial deformation for say the neo-Hookean strain-energy function, is frequently not well-defined for the general Mooney strain-energy function because the additional coupling imposes extra constraints on the deformation which are generally inconsistent with one another. Here we note two fully three dimensional deformations for which this is not the case. In both cases the resulting coupled systems of ordinary differential equations need to be integrated numerically but the deformations are nevertheless well-defined for the general Mooney material. The first deformation is simply noted because the details are given elsewhere. For the second deformation, the coupled system is derived and some new simple special solutions are given. Such deformations are important and noteworthy because of the scarcity of exact solutions in finite elasticity.     

Wave propagation in thermo-chiral elastic medium

The propagation of time harmonic waves through an infinite thermo-chiral elastic material has been investigated. The elastic field of thermo-chiral medium has been described by extending the governing equations and constitutive relations of hemitropic micropolar material to include temperature field. Seven basic waves consisting of three coupled dilatational waves and four coupled shear waves traveling with distinct speeds may exist in the medium. All the waves are found to be dispersive, however the coupled dilatational waves are attenuating and temperature dependent, while the coupled shear waves are independent of temperature field. The phase speeds and corresponding attenuation quality factors of all the coupled dilatational waves have been computed numerically for a specific model. The effect of chirality and temperature field have been shown graphically.     

Method of determining the volume and shear characteristics of elastico-viscous hereditary media from uniaxial-tension (compression) experiments

A method is proposed for determining the elastic constants — instantaneous modulus of elasticity, Poisson's ratio, shear modulus, bulk modulus, and the shear and volume influence functions — the shear creep kernel, the shear creep rate kernel, and the corresponding relaxation kernels from the data of creep or relaxation tests.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 754–758, July–August, 1969.     

Nonlinear physical stress - strain relationship of cellular polymers and glass plastics under conditions of attenuating creep

The coupling equations of hard cellular polymers and glass plastics based on these are analyzed. The material is considered as an elastically relaxing medium for the case of small deformations. The physical relationship between the rubber-elastic stresses and strains is derived in explicit form for the case of attenuating creep and a uniform three-dimensional stressed state. The total deformations are described by the one-dimensional rheological model of a "typical body" with a reduced stress and an instantaneous elastic modulus, having the appropriately chosen viscosity of a non-Newtonian liquid.Institute of Technical Mechanics, Bulgarian Academy of Sciences, Sofia. Translated from Mekhanika Polimerov, No. 2, pp. 231–235, March–April, 1972.     

Calculation of deformative and thermal properties of multiphase composite materials filled with composite or hollow spherical inclusions by the effective medium method

A theoretical investigation was carried out to examine the possibilities of a structural approach to prediction of elastic constants, creep functions, and thermal properties of multiphase polymer composite materials filled with composite or hollow spherical Inclusions of several types. The problem of determining effective properties of the composite was solved by generalizing the effective medium method, a variant of the self-consistent method, for the case of a four-phase kernel-shell-matrix-equivalent homogeneous medium model. Exact analytical expressions for the bulk modulus thermal expansion coefficient, thermal conductivity coefficient, and specific heat were obtained. The solution for the shear modulus is given in the form of a nonlinear equation whose coefficients are the solution of a system of 12 linear equations.To be presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 462–472, July–August, 1995.     

Application of the Green Tensor to the Modeling of Solution-Precipitation Creep

Our aim is to present a mechanical model for solution-precipitation creep, a diffusive deformation process occurring in polycrystalline and granular materials. Within the scope of the model, a Lagrangian consisting of the elastic power and dissipation is proposed. The former is kept in a standard form typical for linear material behavior while the latter is assumed as a surface integral depending on the velocity of the material transport and the velocity of the motion of the boundary. The minimization with respect to the total deformations leads to the equilibrium equation which is solved analytically, by using Green's function. The evolution equations are obtained as the result of the minimization with respect to the internal variables and are solved via finite element method. The contribution closes with a numerical example showing the deformation of a polycrystal consisting of hexagonal crystals. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Creep of plastics with spherical inclusions

A new variant of the theory of creep of plastics with spherical inclusions or pores is proposed on the basis of approximate equations for the integral parameters and the Volterra principle. Rabotnov's theory of viscoelasticity is used to describe linear creep of the matrix. The remaining components of the composite are assumed to be elastic. The complete system of operator equations of the linear viscoelasticity of plastics with spherical inclusions is obtained on the basis of the hypothesis of elastic deformation of the composite and hydrostatic pressure. Sample calculations are performed. A. A. Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences, Moscow. Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 668–675, September–October, 1996.     

On the direct connection of rheological elements in nonlinear continuum mechanics

The structure of complicated phenomenological material models at finite strains is often exemplified with the help of rheological elements. Thereby, simple material behaviour, i.e. elasticity or viscous and plastic flow, are composes by components. In our approach, we directly apply this concept to obtain material models at finite strains. Towards this end, the thermodynamically consistent material behaviour of single elements is defined first. Subsequently, the elements are connected by evaluation of stress equilibria equations formulated on interconnecting configurations. The basic equations of this concept are presented using the example of nonlinear viscoelasticity of Maxwell type. The model results from a series connection of an elastic and a viscous element, whereas both are formulated in a thermodynamically consistent way within the framework on nonlinear continuum mechanics. Furthermore, an approach of numerical implementation using the stress equilibria is suggested. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One of the simplest means of accounting for nonlinearity in viscoelastic media

The author examines a simple extension, to the nonlinear case, of memory-type theory based on the Boltzmann-Volterra superposition principle. It is shown that given certain assumptions the quasi-linear theory of viscoelasticity reduces to introduction into the equations of linear memory theory of a single stress- or strain-intensity function. This function is determined from creep or relaxation tests. A successive-approximation method is presented for solving problems of nonlinear viscoelasticity with the aid of the equations introduced. It is shown that in the case of simple loading the equations of the theory of small elastic-plastic deformations are an analog of the equations considered.Mekhanika Polimerov, Vol. 3, No. 2, pp. 207–212, 1967     

Numerical simulation of rheological processes in hardening plastics under stress control

The paper concerns the simulation of rheological processes in hardening plastics (resins) under stress control. It is assumed that the resins work in the glassy state, under normal conditions, and the rheological processes are quasi-static and isothermal. The reduced stress levels do not exceed 30% of the instantaneous tensile strength. A resin is modelled as a homogeneous, isotropic, linearly viscoelastic material. The HWKK/H rheological model, developed recently by the author, is used. Short-term, medium-term, and long-term shear strain components are considered and described by one fractional and two normal exponential functions as the stress history (memory) functions. A novel algorithm for the numerical simulation of rheological processes in resins has been developed, which is unified for all stress history functions in the HWKK/H model. The algorithm employs the Boltzmann superposition principle, a virtual table for the classic creep process, and a high-rank Gaussian quadrature. The stress function is approximated with a stair case function. The constitutive equations governing the HWKK/H model are trans formed into an algebraic form suitable for algorithmization. The problem of quasi-exact calculation of the double-improper integral resulting from the fractional exponential function is solved effectively. The algorithm has been tested successfully on selected loading programs of unidirectional tension of epoxide. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 201–212, March–April, 2007.     

Representation of the linear viscoelastic behaviour of wood with a constitutive theory based on fractional time derivatives

Like many other materials used in mechanical and civil engineering, wood shows a pronounced history-dependent mechanical material behaviour. Due to its anisotropy its rheological behaviour is strongly dependent on the direction. In this research project, the material behaviour is represented with a phenomenological theory of anisotropic fractional viscoelasticity. In order to identify the material functions and parameters, the time-dependent creep compliances are measured in three orthogonal directions under tension and shear. As a result of the developed constitutive approach, the experimentally observed creep data is described by several power functions. In the second part of the presentation, some differences between classical models of viscoelasticity which are based on Kelvin-Voigt or Maxwell elements and the fractional approach are presented. The assets and drawbacks with respect to wood are discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai Equations

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.     

Isotropic continuum damage/repair model for alveolar bone remodeling

Several authors have proposed mechanical models to predict long term tooth movement, considering both the tooth and its surrounding bone tissue as isotropic linear elastic materials coupled to either an adaptative elasticity behavior or an update of the elasticity constants with density evolution. However, tooth movements obtained through orthodontic appliances result from a complex biochemical process of bone structure and density adaptation to its mechanical environment, called bone remodeling. This process is far from linear reversible elasticity. It leads to permanent deformations due to biochemical actions. The proposed biomechanical constitutive law, inspired from Doblaré and García (2002) [30], is based on a elasto-viscoplastic material coupled with Continuum isotropic Damage Mechanics (Doblaré and García (2002) [30] considered only the case of a linear elastic material coupled with damage). The considered damage variable is not actual damage of the tissue but a measure of bone density. The damage evolution law therefore implies a density evolution. It is here formulated as to be used explicitly for alveolar bone, whose remodeling cells are considered to be triggered by the pressure state applied to the bone matrix. A 2D model of a tooth submitted to a tipping movement, is presented. Results show a reliable qualitative prediction of bone density variation around a tooth submitted to orthodontic forces.     

Structural elastic and creep models of a UD composite in longitudinal shear

The transient creep of a UD composite with a quadratic arrangement of elastic fibers of quadratic cross section is investigated. The deformational properties of the composite are determined from the known properties of its constituents. A structural model of the UD composite is developed, whose minimal elementary cell contains four elements. The stress-strain state of the elements is assumed homogeneous. Two types of basic and resolving governing equations of transient creep are deduced, which are based on static or kinematic assumptions. In each of the cases, a formula for the longitudinal elastic shear modulus of the composite is found. The stationary solutions of creep equations allow one to obtain formulas of the steady-state creep of the composite in a form similar to Norton’s law. Numerical calculations are also performed, and a comparison of the results with data given in the literature bears witness to the efficiency of the models developed and the solutions obtained. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 437–448, July–August, 2007.     

Dynamic modeling for rotating composite Timoshenko beam and analysis on its bending-torsion coupled vibration

A formulation is presented for steady-state dynamic responses of rotating bending-torsion coupled composite Timoshenko beams (CTBs) subjected to distributed and/or concentrated harmonic loadings. The separation of cross section's mass center from its shear center and the introduced coupled rigidity of composite material lead to the bending-torsion coupled vibration of the beams. Considering those two coupling factors and based on Hamilton's principle, three partial differential non-homogeneous governing equations of vibration with arbitrary boundary conditions are formulated in terms of the flexural translation, torsional rotation and angle rotation of cross section of the beams. The parameters for the damping, axial load, shear deformation, rotation speed, hub radius and so forth are incorporated into those equations of motion. Subsequently, the Green's function element method (GFEM) is developed to solve these equations in matrix form, and the analytical Green's functions of the beams are given in terms of piecewise functions. Using the superposition principle, the explicit expressions of dynamic responses of the beams under various harmonic loadings are obtained. The present solving procedure for Timoshenko beams can be degenerated to deal with for Rayleigh and Euler beams by specifying the values of shear rigidity and rotational inertia. Cantilevers with bending-torsion coupled vibration are given as examples to verify the present theory and to illustrate the use of the present formulation. The influences of rotation speed, bending-torsion couplings and damping on the natural frequencies and/or shape functions of the beams are performed. The steady-state responses of the beam subjected to external harmonic excitation are given through numerical simulations. Remarkably, the symmetric property of the Green's functions is maintained for rotating bending-torsion coupled CTBs, but there will be a slight deviation in the numerical calculations.