Time-dependent magnetothermoelastic creep modeling of FGM spheres using method of successive elastic solution

Magnetothermoelastic creep behavior of thick-walled spheres made of functionally graded materials (FGM) placed in uniform magnetic and distributed temperature fields and subjected to an internal pressure is investigated using method of successive elastic solution. The material creep, magnetic and mechanical properties through the radial graded direction are assumed to obey the simple power law variation. Using equations of equilibrium, stress-strain and strain-displacement a differential equation, containing creep strains, for displacement is obtained. A semi-analytical method in conjunction with the Mendelson’s method of successive elastic solution has been developed to obtain history of stresses and strains. History of stresses, strains and effective creep strain rate from their initial elastic distribution at zero time up to 55 years are presented in this paper. Stresses, strains and effective creep strain rate are changing in time with a decreasing rate so that after almost 50 years the time-dependent solution approaches the steady state condition when there is no distinction between stresses and strains at 50 and 55 years.     

The limit equilibrium of an anisotropic medium under the general plasticity condition

A theory of the limit equilibrium of an anisotropic medium under the general plasticity condition in the plane strain state is developed. The proposed yield criterion (the limit equilibrium condition) is obtained by combining the von Mises–Hill yield criterion of an ideally plastic anisotropic material and Prandtl's limit equilibrium condition for a medium under the general plasticity law. It is shown that the problem is statically determinate, i.e., if the boundary conditions are specified in stresses, the stress state in plastic region can only be obtained using equilibrium equations. It is established that the equations describing the stress state are hyperbolic and have two families of characteristic curves that intersect at variable angles. In deriving the equations describing the velocity field, the material is assumed to be rigid plastic, and the associated law of flow is applied. It is shown that the equations for the velocities are also hyperbolic, and their characteristic curves are identical with those of the equations for stresses. However, the directions of the principal values of the stress and strain rate tensors are different due to the anisotropy of the material. The characteristic directions differ from the isotropic case in that the normal and tangential components of the stress tensor do not satisfy the limit conditions. It is established that the equations obtained allow of partial solutions, and in this case, at least one family of characteristic curves consists of straight lines. The conditions along the lines of discontinuity of the velocity are investigated, and it is shown that, as in the isotropic case, these are characteristic curves of the system of governing equations. In the anisotropic formulation, the well-known Rankine problem of the limit state of a ponderable layer is solved. From an analysis of the velocity field it is shown that plastic flow of the entire layer is possible only for a slope angle equal to the angle of internal friction. For slope angles less than the angle of internal friction, the solutions obtained are solutions of problems of the pressure of the medium on the retaining walls. The change in this pressure as a function of the parameters of anisotropy is investigated, and turns out to be significant.     

Simple loading of a polymer material with nonlinear creep

Nonlinear tensor relations between strain, stress, and time are examined for a memory-type medium using degenerate kernels. The material parameters are determined from creep tests in a simple state of stress. Expressions for the strain associated with a complex state of stress and simple loading, found on the basis of the local strains theory, are in satisfactory agreement with the experimental data obtained for specimens of high-density polyethylene.Mekhanika Polimerov, Vol. 3, No. 2, pp. 236–242, 1967     

Creep of polymer materials under periodically varying loads

A method is proposed for constructing the creep curves of a material whose nonlinear memory properties are described by Rozovskii's nonlinear integral equation [2] (with allowance for the stress dependence of the relaxation time) under given periodic loading from known creep curves recorded at constant stress. In deriving the theoretical relation certain simplifying assumptions are made (the creep strain accumulated in 1–2 cycles is small, no vibration [4–6]). An experimental check shows that the proposed method can be used to predict the behavior of a material under periodic loading with an accuracy sufficient for practical purposes.Mekhanika Polimerov, Vol. 2, No. 3, pp. 330–336, 1966     

Modeling of primary and secondary creep for a wide stress range

Many materials exhibit a stress range dependent creep behavior. The power–law creep observed for a certain stress range changes to the viscous type creep if the stress value decreases. Recently published experimental data for advanced heat resistant steels indicate that the high creep exponent (in the range 5–12 for power–law behaviour) may decrease to the low value of approximately 1 within the stress range relevant for engineering structures. The aim of this paper is to confirm the necessity of the assumption of the stress range dependent power–law–viscous creep transition for the solution of stress relaxation problems affected by creep behavior at elevated temperatures. A constitutive model for the minimum creep rate is introduced to describe both the linear and the power law creep depending upon the stress level. The proposed constitutive model includes a strain hardening function to describe the primary creep stage. To demonstrate the existence of the linear creep behaviour in the low stress range of application area and the influence of the primary creep behaviour on relaxation, several solutions of a uniaxial stress relaxation problem are presented for the loading values relevant to engineering applications. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

State of stress of flexible plastic shells with an opening

The state of stress of flat flexible shells with an opening is investigated with allowance for the viscoelastic properties of the material. The equilibrium equations and boundary conditions are written in finite-difference form. A nonlinear system of algebraic equations is solved by successive approximations. A method of accelerating the convergence of slowly converging iteration processes is proposed. The effect of the viscoelastic properties of the shell material on its state of stress is investigated with reference to the example of a polymethyl methacrylate shell. The variations of the ring moment and ring forces at the free edge of the shell are plotted for various moments of time, load values, and flatness parameters. It is shown that as soon as the viscosity factor begins to take effect, the state of stress and strain of the shell changes sharply; the concentration of forces and moments increases in the flexible viscoelastic (as compared with the elastic) shell.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1071–1075, November–December, 1973.     

高体积百分比颗粒增强聚合物材料的有效粘弹性性质

聚合物材料通常表现为粘弹性性质．为了改进聚合物材料的力学性能,通常将某种无机材料以颗粒或纤维的形式填充到聚合物中,从而得到增强、增韧的聚合物基复合材料．提出了一个新的细观力学模型,用于预测颗粒增强聚合物复合材料的有效粘弹性性质,尤其针对高体积百分比的颗粒夹杂复合材料,该方法基于Laplace变换和双夹杂相互作用的弹性模型．计算了玻璃微珠/ED-6复合材料的有效松弛模量以及恒应变率下的应力应变关系．计算结果表明在高体积百分比下该文方法比基于Mori Tanaka方法预测的粘弹性效应明显减弱．     

A non-linear thermo-viscoelastic rheological model based on fractional derivatives for high temperature creep in concrete

In this paper, a novel non-linear thermo-viscoelastic rheological model based on fractional derivatives for high temperature creep in concrete is proposed. The rheological model consists of a linear springpot unit placed in series with a second springpot used for non-linear creep which activates under high stress and temperature. The model parameters which include the dynamic viscosities of the springpots and the fractional exponent are calibrated using existing experimental data of basic creep strain in concrete under constant stress and temperatures for various aggregate types. The power law form of the naturally resulting creep compliance allows an accurate representation of experimental data with the use of only a few model parameters. Furthermore, the variable-order fractional differential stress-strain equation provides a compact method for analytical and numerical modelling of basic creep under conditions of time-varying stress and temperature. In addition, applications of the proposed model to determine axial deformations in columns and transverse deflections in beams under constant and varying temperatures are demonstrated.     

Numerical calculation of the tangent stiffness matrix in materials modeling

The Finite Element Method in the field of materials modeling is closely connected to the tangent stiffness matrix of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness matrix can be derived analytically only with great effort, if at all. Numerical methods are therefore widely used for its calculation. We present our method to calculate the tangent stiffness matrix for the logarithmic strain measure. The approach is compared with other commonly used procedures. A significant increase in accuracy can be achieved with the proposed method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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     

A Variational Approach to Dynamic Recrystallization Using Probability Distributions

We investigate a model of dynamic recrystallization in polycrystalline materials. A probability distribution function is introduced to characterize the state of individual grains by grain size and dislocation density. Specifying free energy and dissipation within the polycrystalline aggregate we are able to derive an evolution equation for the probability density function via a thermodynamic extremum principle. Once the distribution function is known macroscopic quantities like average strain and stress can be calculated. For distribution functions which are constant in time, describing a state of dynamic equilibrium, we obtain a partial differential equation in parameter space which we solve using a marching algorithm. Numerical results are presented and their physical interpretation is given. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limiting states of rigid polyvinyl chloride in creep

Creep tests at constant stresses have been carried out on crystalline polyvinyl chloride (PVC) at 16, 40, and 60° to determine its limiting states (i.e., states defining its creep strength, complete cessation of creep strain, and the boundary of the range of invariance of the mechanical properties), which are shown to be described by exponential functions. An approximate method of determining creep is developed; in this method the creep strength is defined as the stress corresponding to the point of intersection of the limiting equilibrium curves and the stress/time-to-rupture curves.Mekhanika Polimerov, Vol. 1, No. 3, pp. 81–86, 1965     

Numerical Prediction of the Strain State of Thermoset Matrix/Mineral Filler Composite Plates

The numerical prediction of the fields of inelastic strains (the linear invariant of the tensor of inelastic strains) in thermoset polyester/marble filler composite plates is discussed. A uniformly distributed load is applied to the plates, which lie on a steel base. The strain fields are predicted by means of the boundary element method by using creep test data for the composites and the polyester matrix itself. Identical creep tests were performed for two ages of the materials (1 month and 13 years), which allowed evaluating the aging effect. The study is carried out in two stages. At the first stage, the application of the generalized Maxwell-Gurevich equation to the thermoset matrix/mineral filler composite is demonstrated. The model parameters determined from the experimental creep data is used for the second stage, where the state of inelastic strains in the plates is predicted by applying the boundary element method. The influence of composite formulation (filler content) and physical aging of the polyester matrix on the state of inelastic strains in the plates is shown.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 145–156, March–April, 2005.     

Quasistatic anti-plane motion in the simplest theory of nonlinear viscoelasticity

We consider a very simple model in the framework of differential viscoelastic materials which are isotropic and incompressible. In this model the Cauchy stress tensor is split in an elastic part and a dissipative part. The elastic part is derived from a strain-energy density function only of the first invariant of the Cauchy–Green strain tensor. The dissipative part is like the Navier–Stokes equations: linear in the stretching tensor with a constant viscosity parameter. For this model we provide some time and spatial estimates in the quasistatic approximations for the equations governing anti-plane shear motions. Several explicit examples for specific form of the strain energy are produced. Our results impose analytical restrictions on the mathematical properties of the strain energy to ensure a physical behavior in the creep and recovery experiments. Moreover, we show polynomial decay for the spatial behavior in the class of stress-hardening (or strain-stiffening) materials. For stress-softening materials a Phragmen–Lindelof alternative is proved.     

Estimation of temperature in rubber-like materials using non-linear finite element analysis based on strain history

A finite element procedure for hyper-elastic materials such as rubber has been developed to estimate the temperature rise during cyclic loading. The irreversible mechanical work developed in rubber has been used to determine the heat generation rate for carrying out thermal analysis. The evaluation of the heat energy is dependent on the strains. The finite element analysis assumes Green–Lagrangian strain displacement relations, Mooney–Rivlin strain energy density function for constitutive relationship, incremental equilibrium equations, and Total Lagrangian approach and the stress and strain of the rubber-like materials are evaluated using a degenerated shell element with assumed strain field technique, considering both material and geometric non-linearities. A transient heat conduction analysis has been carried out to estimate the temperature rise for different time steps in rubber-like materials using Galerkin's formulations. A numerical example is presented and the computed temperature values for various load steps agree closely with the experimental results reported in the literature.     

Simulation of thick adhesive layers at finite strains utilizing an associated flow rule in a thermodynamically consistent framework

The application of elastoplastic material models is commonly used for the modelling of adhesive layers with high strength adhesives as realized with polyurethane or epoxy resin. To fulfill thermodynamic consistency often restrictions on the choice of material parameters are requested. One of them is the introduction of a non-associated flow rule, which always ensures positive dissipation. Nevertheless, this assumption is a non-essential criterion, which will be addressed in this work. Continuing along this argumentation, the constitutive relations for the material is modified based on an associated flow rule. The applied model for the simulation of the adhesives is based on a small strain theory. A yield surface including two stress invariants, the hydrostatic pressure as well as the deviator stress state, set the elastic limit of the material response. Linear as well as exponential hardening is incorporated and material softening that arises subsequently is also included by substituting effective invariants in the yield function. This material model as proposed from literature was extended to finite strain application with the concept of generalized stress-strain-measure, which was realized in a previous work. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A study of creep of plastic (polymethyl methacrylate) simultaneously stressed in tension and torsion

It was established that isotropic polymethyl methacrylate in the vitreous states is, in the physical sense, a tensorially nonlinear material in creep. The tensorial nonlinearity is not very substantial, and the hypothesis of the similarity of stress and strain deviators gives sufficiently accurate results. The creep strain under relatively simple programmed loading conditions is dependent on the form of the stress state.Mekhanika Polimerov, Vol. 2, No. 5, pp. 671–677, 1966     

Three-dimensional model of the deformation of twisted multilayered rods of composite materials

Conclusion An analytical approach based on certain assumptions regarding the character of the kinematic relations and equilibrium conditions was developed to calculate the stress-strain state of naturally twisted rods of composite materials. A computer program for calculating the stress and strain parameters in each layer was also developed. The results of calculations performed in accordance with the linearized variant (Eq. (5)) of general relations (4) agree satisfactorily with experimental data within a broad range of acting longitudinal loads. The method that was developed has been widely used in the design and optimization of several types of structures.Translated from Mekhanika Kompozitnykh Materialov, No. 4, pp. 610–614, July–August, 1990.