A constitutive model of cold drawing in polycarbonates

This paper presents a set of constitutive equations to model cold-drawing (necking) in polycarbonates (PC). The model is based on a representation of cold drawing as a double glass transition, i.e., a transition from a glass into a rubbery state, when a certain yield surface in the stress space is reached, and a transition back to the glassy state upon unloading or when a certain molecular orientation (draw ratio) is achieved. The stretching process in the rubbery state is modeled by a hyperelastic extension of the J₂-flow theory to the finite strain range. An appropriate yield surface and an associative flow rule (defined via the Kuhn–Tucker optimality conditions) are presented to simulate this process in polycarbonates. The isochoric constraint during double glass transition is treated via an exact multiplicative decomposition of the deformation gradient into volume preserving and spherical parts. Numerical constitutive integration algorithm is based on an operator splitting technique where constraint/consistency during inelastic deformation is enforced via return mapping algorithm. Numerical results are presented to demonstrate the correspondence with the experimental data.     

Segmental orientation in plastically deformed glassy PMMA

In this study, spatial orientational distribution functions of labeled chain segments of cross-linked and linear PMMA were obtained by solid-state NMR as a function of finite deformation (far) below and (far) above the glass transition temperature T_g. The applied data analysis allows comparison of theoretical predictions and experimental data, both in terms of the orientational probability distributions as a function of two polar angles, as well as in terms of moments of the distribution. Orientation-strain relationships of chain segments agreed above and below T_g with predictions from the rubber-elastic affine network model, but suggests a much denser network below T_g than given by the cross-link density or the entanglement density in the melt. This suggested network structure is believed to be the generator of segmental orientation during plastic deformation in the glassy state, independent of the range of applied cross-link densities and deformation rates used in this study.     

A network theory for the thermal conductivity of an amorphous polymeric material

A model to relate the thermal conductivity tensor to the deformation of an amorphous polymeric material above the glass transition temperature is presented. The basis of the model is formed by the transient network theory for polymer melts. With this theory it is possible to calculate the average orientation of the macromolecular segments as a function of the history of the deformation. Combined with an expression which relates the thermal conductivity to the orientation of the molecules, this provides us with the information needed to calculate the heat conduction tensor. Despite the fact that the simplest possible network model is chosen, there is good agreement with the sparse, experimental results.     

Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling

Shape memory polymers (SMPs) can retain a temporary shape after pre-deformation at an elevated temperature and subsequent cooling to a lower temperature. When reheated, the original shape can be recovered. Relatively little work in the literature has addressed the constitutive modeling of the unique thermomechanical coupling in SMPs. Constitutive models are critical for predicting the deformation and recovery of SMPs under a range of different constraints. In this study, the thermomechanics of shape storage and recovery of an epoxy resin is systematically investigated for small strains (within ±10%) in uniaxial tension and uniaxial compression. After initial pre-deformation at a high temperature, the strain is held constant for shape storage while the stress evolution is monitored. Three cases of heated recovery are selected: unconstrained free strain recovery, stress recovery under full constraint at the pre-deformation strain level (no low temperature unloading), and stress recovery under full constraint at a strain level fixed at a low temperature (low temperature unloading). The free strain recovery results indicate that the polymer can fully recover the original shape when reheated above its glass transition temperature (T_g). Due to the high stiffness in the glassy state (T < T_g), the evolution of the stress under strain constraint is strongly influenced by thermal expansion of the polymer. The relationship between the final recoverable stress and strain is governed by the stress–strain response of the polymer above T_g. Based on the experimental results and the molecular mechanism of shape memory, a three-dimensional small-strain internal state variable constitutive model is developed. The model quantifies the storage and release of the entropic deformation during thermomechanical processes. The fraction of the material freezing a temporary entropy state is a function of temperature, which can be determined by fitting the free strain recovery response. A free energy function for the model is formulated and thermodynamic consistency is ensured. The model can predict the stress evolution of the uniaxial experimental results. The model captures differences in the tensile and compressive recovery responses caused by thermal expansion. The model is used to explore strain and stress recovery responses under various flexible external constraints that would be encountered in applications of SMPs.     

A finite deformation fractional viscoplastic model for the glass transition behavior of amorphous polymers

The stress response of amorphous polymers exhibits tremendous change during the glass transition region, from soft viscoelastic response to stiff viscoplastic response. In order to describe the temperature-dependent and rate-dependent stress response of amorphous polymers, we extend the one-dimensional small strain fractional Zener model to the three-dimensional finite deformation model. The Eyring model is adopted to represent the stress-activated viscous flow. A phenomenological evolution equation of yield strength is used to describe the strain softening behaviors. We demonstrate that the stress response predicted by the three-dimensional model is consistent with that of one-dimensional model under uniaxial deformation, which confirms the validity of the extension. The model is then applied to describe the stress response of an amorphous thermoset at various temperatures and strain rates, which shows good agreement between experiments and simulation. We further perform a parameter study to investigate the influence of the model parameters on the stress response. The results show that a smaller fractional order results in a larger yield strain while has little effect on the yield stress when the temperature is below the glass transition temperature. For the stress relaxation tests, a smaller fractional order leads to a slower relaxation rate.     

An effective temperature theory for the nonequilibrium behavior of amorphous polymers

Amorphous polymers lack an organized microstructure, yet they exhibit structural evolution, where physical properties change with time, temperature, and inelastic deformation. To describe the influence of structural evolution on the mechanical behavior of amorphous polymers, we developed a thermomechanical theory that introduces the effective temperature as a thermodynamic state variable representing the nonequilibrium configurational structure. The theory couples the evolution of the effective temperature and internal state variables to describe the temperature-dependent and rate-dependent inelastic response through the glass transition. We applied the theory to model the effect of temperature, strain rate, aging time, and plastic pre-deformation on the uniaxial compression response and enthalpy change with temperature of an acrylate network. The results showed excellent agreement with experiments and demonstrate the ability of the effective temperature theory to explain the complex thermomechanical behavior of amorphous polymers.     

Evolution of plastic anisotropy in amorphous polymers during finite straining

The large strain deformation response of amorphous polymers results primarily from orientation of the molecular chains within the polymeric material during plastic straining. Molecular network orientation is a highly anisotropic process, thus the observed mechanical response is strongly a function of the anisotropic state of these materials. Through mechanical testing and material characterization, the nature of the evolution of molecular orientation under different conditions of state of strain is developed. The role of developing anisotropy on the mechanical response of these materials is discussed in the context of assessing the capabilities of several models to predict the state of deformation-dependent response. A three-dimensional rubber elasticity spring system that is capable of capturing the state of deformation dependence of strain hardening is used to develop a tensorial internal state variable model of the evolving anisotropic polymer response. This fully three-dimensional constitutive model is shown to be successfully predictive of the true stress vs. true strain data obtained in our isothermal uniaxial compression and plane strain compression experiments on amorphous polycarbonate (PC) and polymethylmethacrylate (PMMA) at moderate strain rates. A basis is established for providing the polymer designer with the ability to predict the flow strengths and deformation patterns of highly anisotropic materials. A companion paper by Arruda, Boyce, and Quintus-Bosz [in press] shows how the model developed herein is used to predict various anisotropic aspects of the large strain mechanical response of preoriented materials. Additional work has been done to extend the model to include the effects of strain rate and temperature in Arruda, Jayachandran, and Boyce [in press].     

Testing thermodynamic constitutive equations for polymers by adiabatic deformation experiments

During adiabatic deformation experiments on polyisobutylene of various molecular weights and on polyvinylacetate, the temperature change was measured. The thermal effects occurring during the subsequent stress relaxation were also recorded. From all data, the conclusion was drawn that the entropic elasticity theory is obeyed for temperatures sufficiently above the glass transition temperature. When the value of T_g is approached, some interesting energy effects become appreciable.     

Mechanics and chemical thermodynamics of phase transition in temperature-sensitive hydrogels

This paper uses the thermodynamic data of aqueous solutions of uncrosslinked poly(N-isopropylacrylamide) (PNIPAM) to study the phase transition of PNIPAM hydrogels. At a low temperature, uncrosslinked PNIPAM can be dissolved in water and form a homogenous liquid solution. When the temperature is increased, the solution separates into two liquid phases with different concentrations of the polymer. Covalently crosslinked PNIPAM, however, does not dissolve in water, but can imbibe water and form a hydrogel. When the temperature is changed, the hydrogel undergoes a phase transition: the amount of water in the hydrogel in equilibrium changes with temperature discontinuously. While the aqueous solution is a liquid and cannot sustain any nonhydrostatic stress in equilibrium, the hydrogel is a solid and can sustain nonhydrostatic stress in equilibrium. The nonhydrostatic stress can markedly affect various aspects of the phase transition in the hydrogel. We adopt the Flory-Rehner model, and show that the interaction parameter as a function of temperature and concentration obtained from the PNIPAM-water solution can be used to analyze diverse phenomena associated with the phase transition of the PNIPAM hydrogel. We analyze free swelling, uniaxially and biaxially constrained swelling of a hydrogel, swelling of a core-shell structure, and coexistent phases in a rod. The analysis is related to available experimental observations. Also outlined is a general theory of coexistent phases undergoing inhomogeneous deformation.     

Modeling thermomechanical processes in shape memory polymers under finite deformations

A model taking into account finite deformations is constructed for the behavior of a shape memory polymer which undergoes a transition from the highly elastic to the vitreous state and back during deformation and temperature change. The obtained relations are tested on problems which have experimental support.     

A two characteristic model for cohesionless soil and its calibration using optimization

An elastro-plastic constitutive model is developed to characterize the stress-strain behavior of cohesionless soils. The model utilizes the concept of two characteristics states represented by two characteristic state lines in the stress space. The two characteristic state lines represent the state of the soil at failure and the state at which the soil passes from compressive to dilative mode of deformation during shearing. The proposed approach provides a better control in predicting volumetric behavior of cohesionless soils. To increase the accessibility and usefulness of the model and reduce the difficulty of the model calibration process, an optimization procedure is used to evaluate the material parameters associated with the model. The model is verified with respect to a test that is used for finding the parameters, and a number of other tests that are not used for finding the parameters. A good correlation between predictions and experimental data is observed.     

水蒸汽驱动高聚物形状记忆效应的力学模拟

部分形状记忆高聚物在相对湿度较高的环境中会从其临时形状恢复到永久形状,这种效应被称之为水蒸汽驱动形状记忆效应。由于不需要升高温度就可实现形状恢复,水蒸汽驱动的形状记忆效应在多个领域都有着潜在的应用价值。本文拟建立一个热-力-化学多场耦合的理论模型来模拟非晶态高聚物的水蒸汽驱动形状记忆行为。该理论模型采用自由体积的概念来模拟玻璃态转变,采用Fick定律来模拟水蒸汽在高聚物基体中的扩散行为。相关有限元模拟结果表明,该模型能定性地描述文献中观察到的恢复温度、相对湿度以及溶剂分子扩散速度对形状恢复行为的影响,也能模拟复杂变形条件下水蒸汽驱动的形状记忆效应。     

水蒸汽驱动高聚物形状记忆效应的力学模拟

部分形状记忆高聚物在相对湿度较高的环境中会从其临时形状恢复到永久形状，这种效应被称之为水蒸汽驱动形状记忆效应。由于不需要升高温度就可实现形状恢复，水蒸汽驱动的形状记忆效应在多个领域都有着潜在的应用价值。本文拟建立一个热-力-化学多场耦合的理论模型来模拟非晶态高聚物的水蒸汽驱动形状记忆行为。该理论模型采用自由体积的概念来模拟玻璃态转变，采用Fick定律来模拟水蒸汽在高聚物基体中的扩散行为。相关有限元模拟结果表明，该模型能定性地描述文献中观察到的恢复温度、相对湿度以及溶剂分子扩散速度对形状恢复行为的影响，也能模拟复杂变形条件下水蒸汽驱动的形状记忆效应。     

Thermoplastic Polymer Shrinkage in Emerging Molding Processes

In this paper, in situ experiments have been designed using the full-field deformation technique of Digital Image Correlation (DIC) to characterize non-uniform shrinkage in thermoplastic polymers commonly used in traditional and emerging molding processes. These experiments are capable of characterizing the differences in strains that develop due to thermal gradients and stiction as the polymer shrinks from the molten to the solid state during molding processes. The experimental set-up consists of simulated open molds, a heating stage, thermocouples for temperature measurements, and a video imaging system for DIC. From these experiments, it has been shown that there is a large increase in shrinkage strain associated with the transition of the polymer from the molten to the solid state in a mold with reduced side rigidity, and as it is cooled below the Vicat softening point. Changing the cooling rate from air-cooled to quasi-steady state can eliminate the transition at the Vicat softening point. Furthermore, substantial decreases in shrinkage strain are observed when the polymer is melted in an open mold without mold release, while using mold release produces results similar to that observed with reduced side rigidity. A simple 1D model reasonably explains and predicts the observed trends in the shrinkage behavior due to temperature differences through the thickness of the polymer melt when using high conductivity molds as well as constraint in the polymer melt near the mold resulting from stiction.     

Temperature evolution associated with phase transition from quasi static to dynamic loading

Thermo-mechanical coupling is an intrinsic property of first order martensitic transformation. In this paper, we study the temperature evolution during phase transition at a wider strain rates from quasi static to impact loading to reveal the thermodynamic nature of the strain rate effect of phase transition materials. Based on the laws of thermodynamics and the principle of maximum dissipated energy, a thermal-mechanically coupled model was proposed. The model shows that, in the quasi static case, the temperature profile grades around the moving phase boundary, while for the dynamic case, thermal response of the specimen can be reached homogeneously due to random nucleation. The predicted results of the model are in good agreement with the experimental results, suggesting that the interaction between the self-heating effect and the temperature dependence of phase transition behavior plays a leading role in the process of the transformation deformation mechanism associated with the loading rate.

     

Prediction of nonlinear viscoelastic behavior of polymeric composites using an artificial neural network

Creep tests at constant stress are performed for the carbon-fiber reinforced epoxy composite at various temperatures and initial stresses. A nonlinear viscoelastic constitutive model is developed, and its material parameters are determined by fitting it to creep test data. Model results are found to agree very well with the experimental data at low temperature and low stress conditions. However, the agreement deteriorates at high temperatures, particularly in the vicinity of the glass transition temperature.An alternative model based on an artificial neural network (ANN) is developed to predict the stress relaxation of the polymer matrix composite. The ANN model is trained and validated with 9000 experimental data sets obtained from stress relaxation tests performed at various constant strain (initial stress) and constant temperature conditions. Training of the ANN employs a scaled conjugate gradient method. The optimal brain surgeon algorithm is employed to optimize the topology. The optimal ANN configuration has 88 processing elements (3 in the input layer, 45 in the first hidden layer, 39 in the second hidden layer, and 1 in the output layer) and 410 links. The predictions of the ANN model are found to be more accurate over a wider range of stress and temperature conditions than those of the explicit nonlinear viscoelastic model, in particular near the glass transition temperature.     

锆基非晶合金的动态弛豫机制和高温流变行为

非晶合金的动态弛豫机制对于理解其塑性变形,玻璃转变行为,扩散机制以及晶化行为都至关重要.非晶合金的力学性能与动态弛豫机制的本征关联是该领域当前重要科学问题之一.本文借助于动态力学分析(DMA),探索了Zr₅₀Cu₄₀Al₁₀块体非晶合金从室温到过冷液相区宽温度范围内的动态力学行为.通过单轴拉伸实验,研究了玻璃转变温度附近的高温流变行为.基于准点缺陷理论(quasi-point defects theory),对两种力学行为的适用性以及宏观力学行为变化过程中微观结构的演化规律进行描述.研究结果表明,准点缺陷理论可以很好地描述非晶合金损耗模量α弛豫的主曲线.基于非晶合金的内耗行为,玻璃转变温度以下原子运动的激活能U_β为0.63 eV.与准点缺陷浓度对应的关联因子χ在玻璃转变温度以下约为0.38,而在玻璃转变温度以上则线性增大.Zr₅₀Cu₄₀Al₁₀块体非晶合金在玻璃转变温度附近,随温度和应变速率的不同而在拉伸实验中显示出均匀的或不均匀的...     

Creep performance of PVC aged at temperature relatively close to glass transition temperature

In order to predict the mechanical performance of the polyvinyl chloride(PVC) at a high operating temperature,a series of short-term tensile creep tests(onetenth of the physical aging time) of the PVC are carried out at 63 C with a small constant stress by a dynamic mechanical analyzer(DMA).The Struik-Kohlrausch(SK) formula and Struik shifting methods are used to describe these creep data for various physical aging time.A new phenomenological model based on the multiple relaxation mechanisms of an amorphous polymer is developed to quantitatively characterize the SK parameters(the initial creep compliance,the characteristic retardation time,and the shape factor) determined by the aging time.It is shown that the momentary creep compliance curve of the PVC at 63 C can be very well fitted by the SK formula for each aging time.However,the SK parameters for the creep curves are not constant during the aging process at the elevated temperatures,and the evolution of these parameters and the creep rate versus aging time curves at the double logarithmic coordinates have shown a nonlinear phenomenon.Moreover,the creep master curves obtained by the superposition with the Struik shifting methods are unsatisfactory in such a case.Finally,the predicted results calculated from the present model incorporating with the SK formula are in excellent agreement with the creep experimental data for the PVC isothermally aged at the temperature relatively close to the glass transition temperature.     

Self-consistent modeling of large plastic deformation, texture and morphology evolution in semi-crystalline polymers

A self-consistent model for semi-crystalline polymers is proposed to study their constitutive behavior, texture and morphology evolution during large plastic deformation. The material is considered as an aggregate of composite inclusions, each representing a stack of crystalline lamellae with their adjacent amorphous layers. The deformation within the inclusions is volume-averaged over the phases. The interlamellar shear is modeled as an additional slip system with a slip direction depending on the inclusion's stress. Hardening of the amorphous phase due to molecular orientation and, eventually, coarse slip, is introduced via Arruda-Boyce hardening law for the corresponding plastic resistance. The morphology evolution is accounted for through the change of shape of the inclusions under the applied deformation gradient. The overall behavior is obtained via a viscoplastic tangent self-consistent scheme. The model is applied to high density polyethylene (HDPE). The stress-strain response, texture and morphology changes are simulated under different modes of straining and compared to experimental data as well as to the predictions of other models.     

Modelling large deformation behaviour under loading–unloading of semicrystalline polymers: Application to a high density polyethylene

In this work, the large deformation behaviour under monotonic loading and unloading of a high density polyethylene (HDPE) is studied. To analyze the nonlinear time-dependent response of the material, mechanical tests were conducted at room temperature under constant true strain rates and stress relaxation conditions. A physically-based inelastic model written under finite strain formulation is proposed to describe the mechanical behaviour of HDPE. In the model, the inelastic mechanisms involve two parallel elements: a visco-hyperelastic network resistance acting in parallel with a viscoelastic–viscoplastic intermolecular resistance where the amorphous and crystalline phases are explicitly taken into consideration. The semicrystalline polymer is considered as a two-phase composite. The influence of the crystallinity on the loading and unloading behaviour is investigated. Numerical results are compared to experimental data. It is shown that the model is able to accurately reproduce the experimental observations corresponding to monotonic loading, unloading and stress relaxation behaviours at different strain levels. Finally, the model capabilities to capture cyclic loading–unloading behaviour up to large strains are discussed. To demonstrate the improved modelling capabilities, simulations are also performed using the original model of Boyce et al. [Boyce, M.C., Socrate, S., Llana, P.G., 2000. Constitutive model for the finite deformation stress–strain behavior of poly(ethylene terephthalate) above the glass transition. Polymer 41, 2183–2201] modified by Ahzi et al. [Ahzi, S., Makradi, A., Gregory, R.V., Edie, D.D., 2003. Modeling of deformation behavior and strain-induced crystallization in poly(ethylene terephthalate) above the glass transition temperature. Mechanics of Materials 35, 1139–1148].