Constitutive relation for rheological processes: Properties of theoretic creep curves and simulation of memory decay

We study the nonlinear stress-strain constitutive relation proposed earlier for describing one-dimensional isothermal rheological processes in the case of monotonous variation of the strain (in particular, viscoplasticity, creep, relaxation, plasticity, and superplasticity). This relation contains integral time operators of the strain and strain rate, which are the norms in the Lebesgue and Sobolev spaces equipped with special weight factors, one material function, and nine material parameters determined by the results of tests of the material for relaxation, creep, long-term strength, and constant-rate strain.We analytically inverse the constitutive relation and study the properties of the inverse operator. We derive the equation of creep curves corresponding to an arbitrary law of loading at the stage of passing from the zero stress to a given constant level. We study their dependence on the material parameters and the loading stage characteristics and find restrictions on the material parameters which ensure that the asymptotic behavior of the creep curves for large times is independent of the length of the loading stage and the specific law of stress variation during this stage, i.e., we find the conditions of the model memory decay in creep. Thus we have proved that the constitutive relation proposed above can adequately model both creep and the effect of the material memory decay.     

Fracture criteria under creep with strain history taken into account,and long-term strength modelling

In the present paper, we continue to study the nonlinear constitutive relation (CR) between the stress and strain proposed in [1] to describe one-dimensional isothermal rheological processes in the case of monotone variation of the strain (in particular, relaxation, creep, plasticity, and superplasticity). We show that this CR together with the strain fracture criterion (FC) leads to theoretical long-term strength curves (LSC) with the same qualitative properties as the typical experimental LSC of viscoelastoplastic materials. We propose two parametric families of fracture criteria in the case of monotone uniaxial strain, which are related to the strain fracture criterion (SFC) but take into account the strain increase history and the dependence of the critical strain on the stress. Instead of the current strain, they use other measures of damage related to the strain history by time-dependent integral operators. For any values of the material parameters, analytic studies of these criteria allowed us to find several useful properties, which confirm that they can be used to describe the creep fracture of different materials. In particular, we prove that, together with the proposed constitutive relations, these FC lead to theoretical long-term strength curves (TLSC) with the same qualitative properties as the experimental LSC. It is important that each of the constructed families of FC forms a monotone and continuous scale of criteria (monotonously and continuously depending on a real parameter) that contains the SFC as the limit case. Moreover, the criteria in the first family always provide the fracture time greater than that given by the SFC, the criteria in the second family always provide a smaller fracture time, and the difference can be made arbitrarily small by choosing the values of the control parameter near the scale end. This property is very useful in finding a more accurate adjustment of the model to the existing experimental data describing the fracture time dependence on the stress, temperature, radiation, and other factors: if these data are poorly described by the SFC, then one can choose a more appropriate criterion from the constructed families by varying the value of the control parameter smoothly and monotonously.     

Sn60Pb40钎料合金的具有蠕变和塑性边界的粘塑性本构方程

采用具有蠕变和塑性边界的粘塑性本构方程对Sn-Pb共晶合金的基本力学行为进行了模拟和预测。结果表明：在较宽的外部变量范围（应变率为10^-5-10^-2S^-1,温度为－55－125℃）内,模拟结果与实验结果吻合良好。证明了该方程用于描述Sn-Pb共晶合金的力学行为具有良好的预测能力。     

Coupled processes of deformation and long-term damage of layered materials under thermal loading

The failure criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle fracture on the difference between the equivalent stress and its limit, which is the ultimate strength, according to the Schleicher–Nadai failure criterion, and assumed to be a random function of coordinates. An equation of damage (porosity) balance in the layers at an arbitrary time is formulated taking into account the thermal component. Algorithms of calculating the time dependence of microdamage and macrostresses are developed. Corresponding curves are plotted. The effect of temperature on the deformation and microdamage of the layers is studied     

Coupled processes of deformation and long-term damage of fibrous materials under thermal loading

A theory of long-term damage of fibrous composites under thermal loading is set up. The damage of the matrix is modeled by randomly dispersed micropores. The failure criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle fracture on the difference between the equivalent stress and its limit, which is the ultimate strength, according to the Schleicher–Nadai failure criterion, and assumed to be a random function of coordinates. An equation of damage (porosity) balance in the matrix at an arbitrary time is formulated taking into account the thermal component. Algorithms of calculating the time dependence of microdamage and macrostresses are developed. Corresponding curves are plotted. The effect of temperature on the deformation and microdamage of the material is studied     

Constitutive relation of weakly anisotropic polycrystal with microstructure and initial stress

Man (Nondestr Test Eval 15:191–214, 1999) derived the constitutive relation of a weakly-textured orthorhombic aggregate of cubic crystallites with effects of microstructure and initial stress. In this paper, a computational expression on the integration is given. Then, by means of the computational expression, the general constitutive relation of a weakly-textured anisotropic polycrystal with the consideration of microstructure and initial stress is derived. As special cases of our general constitutive relation, two constitutive relations are given for an isotropic polycrystal and a weakly-textured anisotropic aggregate of cubic crystallites. The acoustoelastic tensor of the reference cubic crystal is derived to determine the material constants of the polycrystal. Two examples are given for understanding the physical meaning of the texture coefficients and the constitutive relations. The project supported by the National Natural Science Foundation of China (10562004, 10662004), the Natural Science Foundation of Jiangxi of China (0512021), the Science Foundation of Jiangxi Educational Department of China([2006]3), and the Foundation of Training Academic and Technical Header for Main Majors of Jiangxi of China.     

Constitutive and damage model for 63Sn37Pb solder under uniaxial and torsional cyclic loading

A constitutive model with Ohno–Wang kinematic hardening rule is developed and employed to simulate the isothermal cyclic behavior of Sn–Pb solder under uniaxial and torsional loading. An implicit constitutive integration scheme is presented for inelastic flow of solder. Then a modified low cycle fatigue life prediction model is put forward in which the sum of maximum shear strain range and normal strain range based on the critical plane concept is adopted to replace the uniaxial strain range used by Stolkarts et al. [Stolkarts, V., Keer, L.M., Fine, M.E., 1999. Damage evolution governed by microcrack nucleation with application to the fatigue of 63Sn–37Pb solder. J. Mech. Phys. Solids 47, 2451–2468]. Comparison of the experimental results and simulation verifies that the stress strain hysteresis loops and peak stress decline curve of solder can be reasonably modeled over a wide range of loading conditions with implement of damage coupled constitutive model, and the lifetime estimations of 63Sn37Pb solder based on the assumption of microcrack nucleation governed damage is effective to provide a conservative prediction.     

平面应变各向异性本构关系及在应力波传播模拟中的应用

为了研究平面应变条件下各向异性材料中应力波传播的特点,利用各向异性弹性Hooke定律、 Tsai-Hill屈服准则、经典塑性流动理论,引入修正的物态方程计及高压下的体积压缩非线性,建立了平面应 变条件下正交各向异性复合材料的弹塑性本构关系,并且分析了二维问题中材料变形引起的主轴旋转及客 观应力率修正问题。最后采用动态显式有限元方法自行编写程序模拟某种纤维增强复合材料碰撞过程中平 面应力波的传播,模拟结果显示,在平面应变条件下应力波在该材料的传播过程中表现出明显的二维效应、各 向异性特点及弹塑性特点。     

Phase field model for coupled displacive and diffusive microstructural processes under thermal loading

A non-isothermal phase field model that captures both displacive and diffusive phase transformations in a unified framework is presented. The model is developed in a formal thermodynamic setting, which provides guidance on admissible constitutive relationships and on the coupling of the numerous physical processes that are active. Phase changes are driven by temperature-dependent free-energy functions that become non-convex below a transition temperature. Higher-order spatial gradients are present in the model to account for phase boundary energy, and these terms necessitate the introduction of non-standard terms in the energy balance equation in order to satisfy the classical entropy inequality point-wise. To solve the resulting balance equations, a Galerkin finite element scheme is elaborated. To deal rigorously with the presence of high-order spatial derivatives associated with surface energies at phase boundaries in both the momentum and mass balance equations, some novel numerical approaches are used. Numerical examples are presented that consider boundary cooling of a domain at different rates, and these results demonstrate that the model can qualitatively reproduce the evolution of microstructural features that are observed in some alloys, especially steels. The proposed model opens a number of interesting possibilities for simulating and controlling microstructure pattern development under combinations of thermal and mechanical loading.     

Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials

Following the modelling of Zener, we establish a connection between the fractional Fokker-Planck equation and the anomalous relaxation dynamics of a class of viscoelastic materials which exhibit scale-free memory. On the basis of fractional relaxation, generalisations of the classical rheological model analogues are introduced, and applications to stress–strain relaxation in filled and unfilled polymeric materials are discussed. A possible generalisation of Reiner's Deborah number

is proposed for systems which exhibit a diverging characteristic relaxation time.     

A micromechanics constitutive theory for forward transformation plasticity with shear and dilatation effect: I,nonproportional loading history

A micromechanics constitutive theory which takes into account both the dilatation and shear effects of the transformation is proposed to describe the macroscopic plastic behavior of structure ceramics during forward transformation under different temperatures. Under some basic assumptions, the analytic expressions of the Helmholtz and complementary free energy of the constitutive element are derived in a self-consistent manner by using the Mori-Tanaka's method which takes into account the interaction between the transformed inclusions. In the framework of Hill-Rice's internal variable constitutive theory, the forward transformation yield function and incremental stress strain relations, in analogy to the theory of metal plasticity, for non-proportional loading histories are obtained. The project supported by National Natural Science Foundation of China     

Polymer Networks with Slip-links: 2. Constitutive Equations for a Cross-linked Network

Stress–strain relations are derived for the mechanical response of elastomers at arbitrary three-dimensional deformations with finite strains. An elastomer is treated as an incompressible network of chains bridged by permanent (chemical cross-links and physical cross-links whose lifetime exceeds the characteristic time of deformation) and temporary (entanglements modeled as slip-links) junctions. Two types of chains are introduced in the network to distinguish between permanent and temporary nodes. Type-I chains have free ends, and their motion at the micro-level is constrained by a random number of slip-links. Type-II chains are Gaussian chains permanently connected to the network. Concentration of type-I chains is fixed, while the number of type-II chains per unit volume can change under deformation. The governing equations involve two (networks with constant concentrations of type-II chains) or three (networks where the content of type-II chains is affected by mechanical factors) material parameters. These parameters are found by fitting observations on rubbers, thermoplastic–elastomers, and thermoplastic-elastomer composites. Good agreement is demonstrated between the experimental data in uniaxial tensile tests and the results of numerical simulation at elongations up to 1,000%. It is shown that the adjustable parameters are affected by chemical composition and molecular architecture of polymers in a physically plausible way.     

Polymer Networks with Slip-links: 1. Constitutive Equations for an Uncross-linked Network

Constitutive equations are derived for the mechanical response of polymers at three-dimensional deformations with finite strains. A polymer is treated as an incompressible network of flexible chains with free ends whose motion at the micro-level is constrained by a random number of slip-links. The slip-links move affinely with macro-deformation, whereas chains can slide with respect to slip-links. When a free end of a chain slides through a slip-link, the slip-link disappears. Stress–strain relations are developed by using the laws of thermodynamics. They involve only one material constant with a transparent physical meaning.     

Constitutive theories for nonlocal micropolar continua with implicity and with multiple interactions

In this paper the constitutive theory for nonlocal micropolar continua which was proposed by A. C. Eringen is extended to the cases for nonlocal micropolar continua with implicity and with multiple interactions. Here nonlocal micropolar thermoelastic solids with implicity and with multiple interactions are cited as instances to illustrate the procedure for the establishment of their constitutive theories as well as two relevant theorems concerning the constitutive theories for those solids are given.     

Technique for selecting the functions of the constitutive equations of creep and long-term strength with one scalar damage parameter

The strain-strength characteristics of aerostructures made of hardening materials under uniaxial tension in creep conditions are determined. The problem is reduced to a system of ordinary differential equations of the kinetic theory of creep with one scalar damage parameter. The approximate solutions of the problem are obtained with the help of the implicit Euler method and of the arc length method in combination with the explicit methods of the Runge–Kutta family for cylindrical St.45 steel samples and 3V titanium alloy plates.