Plastic instability studied experimentally on a semi-crystalline polymer through thermomechanical heat source identification: the flow stress concept revisited

Micromechanical deformation phenomena such as those leading to macroscopic viscoelastic and plastic behavior must be studied from a thermodynamic viewpoint, as they induce complex and partly irreversible heat effects. Calorimetric measurements of the intrinsic volumetric thermomechanical heat sources (THS) activated in the material bulk during mechanical loads can produce valuable information with respect to that aim. They can be based on infrared imaging if submitted to inverse algorithms that allow a correct reconstruction of THS to be produced. Here, an inverse method relying on a diffusion-advection heat transfer model is applied to experimental temperature maps recorded during tensile tests. These are made on a semi-crystalline polymer that shows a strong development of plastic instabilities. Along with simultaneous kinematic observables produced with a digital image correlation system, the competition between advection and diffusion phenomena may be clearly established. 1-D profiles of the reconstructed THS and measured strain rates illustrate clearly that thermomechanical effects associated with necking onset and propagation follow the kinematic variable in a rather direct manner. Finally, we show for tensile experiments that THS estimations lead to analyze plasticity as a rheological behavior controlled by the flow stress, responsible of necking development and propagation.     

Thermomechanical formulation of ductile damage coupled to nonlinear isotropic hardening and multiplicative viscoplasticity

In this paper, we present a thermomechanical framework which makes use of the internal variable theory of thermodynamics for damage-coupled finite viscoplasticity with nonlinear isotropic hardening. Damage evolution, being an irreversible process, generates heat. In addition to its direct effect on material's strength and stiffness, it causes deterioration of the heat conduction. The formulation, following the footsteps of Simó and Miehe (1992), introduces inelastic entropy as an additional state variable. Given a temperature dependent damage dissipation potential, we show that the evolution of inelastic entropy assumes a split form relating to plastic and damage parts, respectively. The solution of the thermomechanical problem is based on the so-called isothermal split. This allows the use of the model in 2D and 3D example problems involving geometrical imperfection triggered necking in an axisymmetric bar and thermally triggered necking of a 3D rectangular bar.     

Heat transfer and entropy generation analysis of non-Newtonian fluid flow through vertical microchannel with convective boundary condition

The entropy generation and heat transfer characteristics of magnetohydrodynamic(MHD) third-grade fluid flow through a vertical porous microchannel with a convective boundary condition are analyzed. Entropy generation due to flow of MHD non-Newtonian third-grade fluid within a microchannel and temperature-dependent viscosity is studied using the entropy generation rate and Vogel's model. The equations describing flow and heat transport along with boundary conditions are first made dimensionless using proper non-dimensional transformations and then solved numerically via the finite element method(FEM). An appropriate comparison is made with the previously published results in the literature as a limiting case of the considered problem.The comparison confirms excellent agreement. The effects of the Grashof number, the Hartmann number, the Biot number, the exponential space-and thermal-dependent heat source(ESHS/THS) parameters, and the viscous dissipation parameter on the temperature and velocity are studied and presented graphically. The entropy generation and the Bejan number are also calculated. From the comprehensive parametric study, it is recognized that the production of entropy can be improved with convective heating and viscous dissipation aspects. It is also found that the ESHS aspect dominates the THS aspect.     

Constraints, Constitutive Limits, and Instability in Finite Thermoelasticity

This paper examines all the possible types of thermomechanical constraints in finite-deformational elasticity. By a thermomechanical constraint we mean a functional relationship between a mechanical variable, either the deformation gradient or the stress, and a thermal variable, temperature, entropy or one of the energy potentials; internal energy, Helmholtz free energy, Gibbs free energy or enthalpy. It is shown that for the temperature-deformation, entropy-stress, enthalpy-deformation, and Helmholtz free energy-stress constraints equilibrium states are unstable, in the sense that certain perturbations of the equilibrium state grow exponentially. By considering the constrained materials as constitutive limits of unconstrained materials, it is shown that the instability is associated with the violation of the Legendre–Hadamard condition on the internal energy. The entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints do not exhibit this instability. It is proposed that stability of the rest state (or equivalently convexity of internal energy) is a necessary requirement for a model to be physically valid, and hence entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints are physical, whereas temperature-deformation constraints (including the customary notion of thermal expansion that density is a function of temperature only), entropy-stress constraints, enthalpy-deformation constraints, and Helmholtz free energy-stress constraints are not.     

Two-temperature continuum thermomechanics of deforming amorphous solids

There is an ever-growing need for predictive models for the elasto-viscoplastic deformation of solids. Our goal in this paper is to incorporate recently developed out-of-equilibrium statistical concepts into a thermodynamically consistent, finite-deformation, continuum framework for deforming amorphous solids. The basic premise is that the configurational degrees of freedom of the material – the part of the internal energy/entropy that corresponds to mechanically stable microscopic configurations – are characterized by a configurational temperature that might differ from that of the vibrational degrees of freedom, which equilibrate rapidly with an external heat bath. This results in an approximate internal energy decomposition into weakly interacting configurational and vibrational subsystems, which exchange energy following a Fourier-like law, leading to a thermomechanical framework permitting two well-defined temperatures. In this framework, internal variables, that carry information about the state of the material equilibrate with the configurational subsystem, are explicitly associated with energy and entropy of their own, and couple to a viscoplastic flow rule. The coefficients that determine the rate of flow of entropy and heat between different internal systems are proposed to explicitly depend on the rate of irreversible deformation. As an application of this framework, we discuss two constitutive models for the response of glassy materials, a simple phenomenological model and a model related to the concept of Shear-Transformation-Zones as the basis for internal variables. The models account for several salient features of glassy deformation phenomenology. Directions for future investigation are briefly discussed.     

On the formulation of coupled thermoplastic problems with phase-change

This paper deals with a numerical formulation for coupled thermoplastic problems including phase-change phenomena. The final goal is to get an accurate, efficient and robust numerical model, allowing the numerical simulation of solidification processes in the metal casting industry. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing differential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain fixed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. The constitutive behavior has been defined by a thermoelastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. A brief summary of the thermomechanical frictional contact model is included. The numerical model has been implemented into the computational Finite Element code COMET developed by the authors. A numerical assessment of the isentropic and isothermal operator splits, regarding the nonlinear stability behavior, has been performed for weakly and strongly coupled thermomechanical problems. Numerical simulations of solidification processes show the performance of the computational model developed.     

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.     

Thermal and mechanical analysis of material response to non-steady ramp and steady shock wave loading

Ramp wave experiments on the Sandia Z accelerator provide a new approach to study the rapid compression response of materials at pressures, temperatures and stress or strain rates not attainable in conventional shock experiments. Due to its shockless nature, the ramp wave experiment is often termed as an isentropic (or quasi-isentropic) compression experiment (ICE). However, in reality there is always some entropy produced when materials are subjected to large amplitude compression even under shockless loading. The entropy production mechanisms that cause deformation to deviate from the isentropic process can be attributed to mechanical and thermal dissipations. The former is due to inelasticity associated with various deformation mechanisms and the rate effect that is inherent in all the deformation processes and the latter is due to irreversible heat conduction. The main purpose of the current study is to gain insights into the effects of ramp and shock loading on the entropy production and thermomechanical responses of materials. Another purpose is to investigate the role of heat conduction in the material response to both the non-steady ramp wave and steady shock.Numerical simulations are used to address the aforementioned research objectives. The thermomechanical response associated with a steady shock wave is investigated first by solving a set of nonlinear ordinary differential equations. Using the steady wave solutions as the reference, the material responses under non-steady ramp waves are then studied with numerical wave propagation simulation. It is demonstrated that the material response to ramp and shock loading is essentially a manifestation of the interaction between the time scale associated with the loading and the intrinsic time scales associated with mechanical deformation and heat transfer. At lower loading rates as encountered in ramp loading, the loading path is closer to an isentrope and results in lower entropy production. The reasonable ramp rate to obtain a quasi-isentropic state depends on the intrinsic time scales of the dissipation mechanisms which are strongly material dependent. Thus shockless loading does not necessarily produce an isentropic response. Between two equilibrium states, heat conduction was shown to have significant effect on the temperature history but it contributes little to the overall temperature change if the specific heat remains constant. It also affects the history of entropy, but only the irreversible part of heat conduction contributes to the net entropy change. The various types of thermomechanical responses of materials would manifest themselves more significantly in terms of the thermal history than the mechanical history. Thus temperature measurement appears to be an important experimental tool in distinguishing the various mechanisms for the thermomechancial responses of the materials.     

Critical cooling rates to avoid transient-driven cracking in thermal barrier coating (TBC) systems

This paper presents critical cooling rates to avoid cracking in thermal barrier coatings (TBCs) driven by thermal transients. A complete thermomechanical model is presented for multilayers; it determines the history of temperature, deformation and stress distributions in the layers, as well as the steady-state energy release rate (ERR) for delamination for all possible crack locations. The model is used to analyze bilayers over a broad range of properties and cooling rates; critical cooling rates are identified that distinguish scenarios in which the transient delamination driving force is higher than that associated with the fully cooled state. Implications for coating the durability are discussed.     

Prigogine theorem of minimum entropy production applied to the average-atom model

Thermodynamics of irreversible processes is applied to study the interaction of matter and a radiation field in non-local thermodynamic equilibrium within the framework of the average-atom model. The rate of entropy production of matter and the radiation field, in contact with a free electron reservoir in local thermodynamic equilibrium, is obtained using conjugates of the state variables. The average-atom one-electron populations are determined by minimizing the rate of entropy production at fixed electronic density, electronic temperature, and radiation field. Numerical results and comparisons with experiment for a photoionized iron plasma are presented and discussed. Our approach, which is based on first principles, can be used in a large variety of non-local thermodynamic equilibrium situations.     

Variational asymptotic homogenization of temperature-dependent heterogeneous materials under finite temperature changes

The variational asymptotic method is used to construct a thermomechanical model for homogenizing heterogeneous materials made of temperature-dependent constituents subject to finite temperature changes with the restriction that the strain is small. First, we presented the derivation for a Helmholtz free energy suitable for finite temperature changes using basic thermodynamics concepts. Then we used this energy to construct a thermomechanical micromechanics model, extending our previous work which was restricted to small temperature changes. The new model is implemented in the computer code VAMUCH using the finite element method for the purpose of handling real heterogeneous materials with arbitrary periodic microstructures. A few examples including binary composites, fiber reinforced composites, and particle reinforced composites are used to demonstrate the application of this model and the errors introduced by assuming small temperature changes when they are not necessarily small.     

Predicting thermal shape memory of crosslinked polymer networks from linear viscoelasticity

The viscoelastic behavior of an amorphous shape memory polymer network and its dependence on time and temperature were measured by dynamic mechanical analysis. The resulting thermo-mechanical behavior was modeled and implemented in a commercial finite element code. The ability of the resulting thermomechanical model to simulate and, eventually, predict the shape storage and shape recovery of the material was evaluated against experimental shape memory thermomechanical torsion data in a large deformation regimen. The simulations showed excellent agreement with experimental shape memory thermomechanical cycle data. This demonstrates the dependence of the shape recovery on time and temperature. The results suggest that accurate predictions of the shape recovery of any amorphous polymer networks under any thermomechanical conditions combination solely depends on considering the material viscoelasticity and its time–temperature dependence.     

联合极限状态下基于最大熵可靠度的易损性分析

考虑工程需求参数(EDP)的前四阶矩,提出基于最大熵可靠度理论的地震易损性分析方法.基于SAP2000建立钢筋混凝土框剪模型,选择最大层间位移角和最大层加速度衡量结构的联合性能极限状态,建立极限状态方程.不对EDP的分布进行人为假定,在不同峰值加速度(PGA)下计算两种EDP的前四阶矩,并作为约束条件,建立极限状态方程...     

Polyatomic gases with dynamic pressure: Kinetic non-linear closure and the shock structure

This paper is concerned with the analysis of polyatomic gases within the framework of kinetic theory. Internal degrees of freedom are modelled using a single continuous variable corresponding to the molecular internal energy. The state of the gas is determined by the 6 fields—5 standard fields (mass density, velocity and temperature) and the dynamic pressure. Using the maximum entropy principle and the non-equilibrium entropy density, it is shown that dynamic pressure appears as a natural measure for deviation from equilibrium state. A proper collision cross section is constructed which obeys the micro-reversibility requirement. The non-linear source term in the balance law for dynamic pressure, and the entropy production rate, are determined using collision operator in the form which generalizes the known results obtained within the framework of extended thermodynamics. They are also compared with the results obtained using BGK approximation. For the proposed model the shock structure problem is thoroughly analyzed and discussed for different values of the parameters in the source term.     

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.     

Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling

In this paper, we examine the influence of thermomechanical coupling on the behavior of superelastic shape memory alloys subjected to cyclic loading at different loading rates. Special focus is given to the determination of the area of the stress-strain hysteresis loop once the material has achieved a stabilized state. It is found that this area does not evolve monotonically with the loading rate for either transient or asymptotic states. In order to reproduce this observation analytically, a new model is developed based on the ZM model for shape memory alloys which was modified to account for thermomechanical coupling. The model is shown to predict the non-monotonic variation in hysteresis area to good accord. Experimentally observed variations in the temperature of SMA test samples are also correctly reproduced for lower strain rates.     

On the constitutive modeling of coupled thermomechanical phase-change problems

This paper deals with a thermodynamically consistent numerical formulation for coupled thermoplastic problems including phase-change phenomena and frictional contact. The final goal is to get an accurate, efficient and robust numerical model, able for the numerical simulation of industrial solidification processes. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing differential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain fixed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. All the material properties have been considered to be temperature dependent. The constitutive behavior has been defined by a thermoviscous/elastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. The constitutive model proposed accounts for a continuous transition between the initial liquid state, the intermediate mushy state and the final solid state taking place in a solidification process. In particular, a pure viscous deviatoric model has been used at the initial fluid-like state. A thermomecanical contact model, including a frictional hardening and temperature dependent coupled potential, is derived within a fully consistent thermodinamical theory. The numerical model has been implemented into the computational finite element code COMET developed by the authors. Numerical simulations of solidification processes show the good performance of the computational model developed.     

Thermomechanical modeling of metals at finite strains: First and mixed order finite elements

The aim of this paper is to describe an updated EAS (Enhanced Assumed Strain) finite element formalism developed to model the thermomechanical behavior of metals submitted to large strains. We will also expose the use of mixed order elements (first order mechanical elements strongly coupled with quadratic thermal elements) which, as we will show, is of particular interest for modeling fast processes inducing important temperature gradients. The features of this formalism, used jointly with an Updated Lagrangian approach and an hypoelastic anisothermal constitutive formulation, will be described. Three applications involving finite strains and important thermomechanical couplings will be studied. The results obtained will be compared with the results given by the now classical SRI (Selective Reduced Integration) formalism.     

Application of the scale entropy diffusion model to describe a liquid atomization process

Whatever the situation, liquid atomization processes show a continuous evolution of the liquid system shape. However, such a system is a multiscale object, i.e., its shape cannot be fully described by a single geometrical parameter. The present work makes use of the scale entropy function to describe this multiscale object. This function is found similar to the scale distribution previously introduced to take into account the droplet shape in liquid spray characterization. Time-averaged scale entropy is locally measured on images of atomizing liquid flows issuing from a low injection pressure single-hole triple-disk nozzle. The advantage in using this nozzle is that the atomization process and the spray are inscribed in a plane and can be fully described by 2-D visualizations. The measurements are performed from the nozzle exit down to the spray region. The operating conditions consider varying injection pressure and liquid physical properties. The temporal evolution of the scale entropy is described by the scale entropy diffusion model. Initially developed in turbulence, this model introduces new parameters such as the scale diffusivity and the local scale entropy flux sink, which characterize the diffusion dynamic of the scale entropy in the scale space. For the first time, these parameters are measured and strong correlations between them and the working conditions are evidenced. Furthermore, new parameters are introduced such as a scale viscosity and the total scale entropy flux lose. These results demonstrate the relevance of the scale entropy diffusion model to describe a liquid atomization process. This application is the first of its kind.