A constitutive equation for nonlinear stress–strain curves of crystalline polymers

We present a new method for the analysis of stress–strain curves for crystalline polymers such as polypropylene and polyethylene. A nonlinear constitutive equation that includes terms that cover the plastic deformation and anharmonicity of the spring is developed. In order to quantitatively characterize the nonlinear viscoelasticity using this equation, data on the transient moduli during elongation at a constant rate of strain are required. Hence, the simultaneous measurements of linear oscillatory viscoelastic moduli during a constant rate of elongation were investigated. It was found that the present method makes possible the evaluation of the plastic deformation fraction and the Gruneisen constant for crystalline polymers. © 1998 Chapman & Hall     

A constitutive model for the viscoplastic behavior of rubbery polymers at finite strains

Summary. A constitutive model is derived for the viscoplastic behavior of rubbery polymers at finite strains. A polymer is treated as an equivalent network of chains bridged by permanent junctions. The elastic response of the network is attributed to the elongation of strands, whereas its plastic behavior is associated with the sliding of nodes with respect to their initial positions. Unlike conventional stress–strain relations in finite viscoplasticity, the rate-of-strain tensor for the sliding of junctions is expressed in terms of the rate-of-strain tensor for macro-deformation. Constitutive equations are developed by using the laws of thermodynamics. These relations are simplified for simple shear of an incompressible medium with finite strains. The governing equations are determined by 3 material constants. To verify the model, a series of shear tests is performed on polycarbonate melts reinforced with short glass fibers. Adjustable parameters in the stress–strain relations are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is shown that the material constants change with the filler content in a physically plausible way.Department of Chemical Engineering, Kuwait University, Safat 13060, Kuwait     

A finite deformation constitutive model for shape memory polymers based on Hencky strain

In many engineering applications, shape memory polymers (SMPs) usually undergo arbitrary thermomechanical loadings at finite deformation. Thus, development of 3D constitutive models for SMPs within the finite deformation regime has attracted a great deal of interest. In this paper, based on the classical framework of thermodynamics of irreversible processes, employing the logarithmic (or Hencky) strain as a more physical measure of strain, a 3D large-strain macromechanical model is presented. In the constitutive model development, we adopt a multiplicative decomposition of the deformation gradient into elastic and stored parts. In addition, employing the averaging scheme, the logarithmic elastic strain tensor is decomposed into the rubbery and glassy parts. The evolution equations for internal variables are introduced for both cooling and heating processes. The time-discrete form of the proposed model in the implicit form is also presented. Comparing the predicted results with experimental data reported in the literature, the model is validated. Finally, using the finite element method, two boundary value problems e.g., a 3D beam and a medical stent made of SMPs are numerically simulated.     

A constitutive equation for concrete

A more rational approach to strength criterion development for concrete is proposed to cover the composite nature and complex failure mechanism of concrete materials. The use of scalar valued function theory as applied to concrete failure prediction is demonstrated. The results are applicable to general brittle materials     

A constitutive equation for hot deformation range of 304 stainless steel considering grain sizes

A general constitutive equation based on the framework of invariant theory by consideration of hot deformation key variables and also the properties of the material such as initial grain size is presented in the current work. Soundness of the considered parameters to be used in the developed formula was initially verified based on the important axioms such as objectivity, entropy principle, and thermodynamics stability. To access the prediction ability of the method, the formula was simplified for the simple hot compression test. To evaluate the simplified formula, single-hit hot compression tests were carried out at the temperature range of 900–1100 °C under true strain rate of 0.01–1 s⁻¹ on a AISI 304 stainless steel. The capability of proposed formula for reproducing the variation of flow stress with strain and the strain hardening rate with stress for the resultant flow stress data was examined. The good agreement between model predictions and actual results signified the applicability of this method as a general constitutive equation in hot deformation studies.     

A constitutive relation for the deformation of snow

In this paper a constitutive equation which describes the uniaxial deformation of snow is developed. The basic assumption underlying this work is that the stress-strain response can be derived by considering the structure of the material. The equation which describes the plastic portion of the deformation is developed by considering the relationship between three fundamental variables: the mean spacing between ice grains, the relative velocity between grains, and the fraction of the total number of grains which participate in the deformation process.The mean distance between ice grains is determined by a stereological investigation of the snow structure, and the velocity component is found by empirically characterizing the relaxation of the snow. To determine the mobility of the ice grains acoustic emission data are used. An equation describing the pattern of acoustic emissions for constant rates of deformation is derived and applied to a number of tests. Combining the above variables produces a compressive and tensile constitutive equation which reflects the behavior of the snow under both uniaxial deformations.     

Mg-9Al-3Si-0.375Sr-0.78Y合金的热变形行为及本构模型

利用Gleeble-3500热模拟试验机对Mg-9Al-3Si-0.375Sr-0.78Y合金试样进行等温恒应变速率压缩实验,研究其在温度250~400℃、应变速率0.001~10s~(-1)条件下的热变形行为。结果表明:在热变形过程中,峰值应力随着应变速率的降低和温度的升高而减小,且峰值应力对应变速率的敏感性随着变形温度的下降而增强。建立了考虑应变的热变形Arrhenius本构模型,模型精度良好,在300,350℃及0.001~10s~(-1)范围内,模型的平均绝对误差分别为1.57%和1.76%;合金的平均变形激活能为183.58k J/mol,平均应变速率敏感指数为0.1616。热变形过程中,α-Mg相呈现明显的动态再结晶特征,β-Mg₁₇Al₁₂相尺寸减小且分布均匀,初生Mg_2Si相较小。在低温(250~300℃)变形时,动态再结晶仅发生在晶界处。在高温(350~400℃)变形时,初生α-Mg晶粒发生了明显的动态再结晶。随着温度的增加和应变速率的降低,再结晶程度提高,再结晶晶粒逐渐长大。     

A constitutive equation for non-linear electro-active solids

Summary Electro-active solids are solids that are either infused with electrorheological fluids or embedded with electrically conducting particles, the body as a whole however conducting negligible current. In this paper, we provide a mathematical framework, within the context of continuum mechanics, for the study of electro-active solids. The theory assumes that the body can be considered as a continuum, in the sense of homogenization, which is isotropic, incompressible, elastic and is capable of responding to an electric field. Appealing to standard techniques in continuum mechanics, we obtain a constitutive relation for the stresses in terms of the deformation and electric field. This is used in a study of triaxial extension, simple shear and anisotropy induced by the electric field.     

A simple novel constitutive equation for concrete

A simple novel constitutive equation, three parameter strength criterion for concrete is proposed to represent the composite nature and complex failure mechanism of material of concrete. In this paper, the study is to demonstrate the use of scalar valued function, invariant theory, as applied to concrete failure prediction. Without the loss of accuracy of prediction, a three parameter strength criterion is developed.     

A thermo-viscoelastic constitutive model for compressible amorphous polymers

We propose a compressible thermo-viscoelastic constitutive model at finite deformation for amorphous polymers. Firstly, the compressible hyperelastic deformation energy is extended to include thermal effect. Secondly, in order to describe the viscous property of the materials, internal variables are introduced into the above thermo-hyperelastic constitutive relation. As the Maxwell model is equivalent to the transient network model under certain condition, the rationality of introducing the internal variables for a general deformation mode is proved. The model only contains a few material parameters, just necessary for a complicated thermal-mechanical coupling system. Besides, this model provides an explanation for the deformation energy function from the microscopic mechanism. Based on this model, the influences of the loading rate, compressibility and the thermal expansion coefficient on the coupled thermo-mechanical behavior of polymers are discussed.     

A rate-dependent constitutive equation for 5052 aluminum diaphragms

In this article, both experimental and numerical approaches are conducted to present a constitutive equation for 5052 aluminum diaphragms under quasi-static strain rate loadings. For this purpose the stress–strain curves at different strain rates are obtained using tensile tests. Brittle behavior during tensile tests is observed due to samples thin thicknesses. Employing Johnson–Cook constitutive equation no yields in reasonable agreement with these tensile tests results. Therefore, developing a more suitable constitutive equation for aluminum diaphragms is taken into consideration. This equation is then implemented into the commercial finite element software, ABAQUS, via a developed user material (UMAT) subroutine utilizing von Mises plasticity theory and an own solution algorithm. A single-element pathological test method is adopted to show the well-development of the UMAT subroutine. In order to verify the proposed constitutive equation for precision predicting of mechanical behavior, a bulge test is performed in which demonstrates a good agreement between experimental and numerical results.     

A viscoplastic constitutive model for dynamic fracture

The use of a viscoplastic approach to dynamic fracture prediction is proposed with the aim of producing a model which is applicable to ductile situations. A numerical (finite-element) approach is adopted with specially developed joint elements being used to simulate behaviour within the fracture-process zone. Results are presented for an expanding edge-crack problem. Experimental results are used to calibrate the numerical model by determining the material-specimen parameters under dynamic-fracturing conditions. Detailed results are presented for both LEFM and nonlinear fracture-mechanics approaches.     

A 3D finite deformation constitutive model for amorphous shape memory polymers: A multi-branch modeling approach for nonequilibrium relaxation processes

Shape memory polymers (SMPs) are materials that can recover a large pre-deformed shape in response to environmental stimuli. For a thermally activated amorphous SMP, the pre-deformation and recovery of the shape require the SMP to traverse its glass transition temperature (T_g) to complete the shape memory (SM) cycle. As a result, the recovery behavior of SMPs shows strong dependency on both the pre-deforming temperature and recovery temperature. Generally, to capture the multitude of relaxation processes, multi-branch models (similar to the 1D generalized viscoelastic model or Prony series) are used to model the time-dependent behaviors of polymers. This approach often requires an arbitrary (usually numerous) number of branches to capture the material behavior, which results in a substantial number of material parameters. In this paper, a multi-branch model is developed to capture the SM effect by considering the complex thermomechanical properties of amorphous SMPs as the temperature crosses T_g. The model utilizes two sets of nonequilibrium branches for fundamentally different modes of relaxation: the glassy mode and Rouse modes. This leads to a significant reduction in the number of material parameters. Model simulation comparisons with a range of thermomechanical experiments conducted on a tert-butyl acrylate-based SMP show very good agreement. The model is further utilized to explore the intrinsic recovery behavior of an SMP and the size effects on the free recovery characteristics of a magneto-sensitive SMP composite.     

A constitutive theory for the inelastic behavior of concrete

A general theory for the inelasticity of concrete is proposed, the main constituents being a new, rate independent model of distributed damage for mortar and the application of mixture theories to account for the composite nature of concrete. The proposed theory of damage is capable of accommodating fully anisotropic elastic degradation, both in tension and in compression, in a manner which is ideally suited for computation. Mixture theories, on the other hand, are found to provide a simple yet effective tool for characterizing the values of the phase stresses that act on mortar and aggregate and which drive damage and plastic flow. This uneven distribution of stresses between mortar and aggregate is seen to lie at the foundation of effects such as the characteristic splitting failure modes in uniaxial compression and the unloading hysteretic loops that arise during cyclic loading. Further to furnishing useful insights into the physical mechanisms underlying the inelastic behavior of concrete, the proposed model provides a simple means of quantifying such behavior in a way which can be readily implemented in any standard finite element code. Possible generalizations of the theory are suggested. In particular, it is noted how rate and rheological effects can be incorporated into the proposed framework by extending it into the viscoplastic range and through the use of Eyring's theory of thermal activation.     

Zirlo锆合金高温变形行为及本构关系EI北大核心CSCD

采用Gleeble-3800型热模拟试验机,对Zirlo合金进行等温恒应变速率压缩实验,研究其在变形温度550~700℃,应变速率0.01~10 s^(-1)范围内的热变形行为;并在Arrhenius型双曲正弦函数方程基础上引入应变量,构建了基于应变补偿的Arrhenius本构模型,同时构建了基于位错密度演化加工硬化模型和基于唯象型的软化模型的分段唯象型本构模型。结果表明:Zirlo合金的流变应力随着温度的降低和应变速率的提高而升高,低应变速率下流变应力呈现更高的温度敏感性,流变应力曲线在不同变形条件下分别呈现加工硬化、动态回复、动态再结晶特征。经过误差分析可知,基于应变补偿的Arrhenius本构模型大部分预测值的误差均在15%以内,具有较高的准确性,而分段唯象型本构模型相对平均绝对误差最大值不超过3%,具有97%以上的准确率,可以很好地预测合金的应力-应变曲线,具有良好的拓展性,并且可初步判断曲线类型,具有良好的实用性。     

A phenomenological constitutive model for the nonlinear viscoelastic responses of biodegradable polymers

We formulate a constitutive framework for biodegradable polymers that accounts for nonlinear viscous behavior under regimes with large deformation. The generalized Maxwell model is used to represent the degraded viscoelastic response of a polymer. The large-deformation, time-dependent behavior of viscoelastic solids is described using an Ogden-type hyperviscoelastic model. A deformation-induced degradation mechanism is assumed in which a scalar field depicts the local state of the degradation, which is responsible for the changes in the material’s properties. The degradation process introduces another timescale (the intrinsic material clock) and an entropy production mechanism. Examples of the degradation of a polymer under various loading conditions, including creep, relaxation and cyclic loading, are presented. Results from parametric studies to determine the effects of various parameters on the process of degradation are reported. Finally, degradation of an annular cylinder subjected to pressure is also presented to mimic the effects of viscoelastic arterial walls (the outer cylinder) on the degradation response of a biodegradable stent (the inner cylinder). A general contact analysis is performed. As the stiffness of the biodegradable stent decreases, stress reduction in the stented viscoelastic arterial wall is observed. The integration of the proposed constitutive model with finite element software could help a designer to predict the time-dependent response of a biodegradable stent exhibiting finite deformation and under complex mechanical loading conditions.     

A non linear integral constitutive equation for viscoelastic fluids

An intégral constitutive equation is written using a particular reference frame, built with unit vectors along the principal axes of the rate-of-deformation tensor, and using the associated intrinsic rate-of-rotation. This equation is easier to handle in calculations than corotational or codeformational models. The material functions for a rheological model including the first six terms of the constitutive equation have been studied in steady and unsteady shear flows, as well as in elongational flows. Material functions are readily written from six memory functions and no inconsistency comes out.