A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: Application to thermally responsive gels

An elastomeric gel is a cross-linked polymer network swollen with a solvent, and certain gels can undergo large reversible volume changes as they are cycled about a critical temperature. We have developed a continuum-level theory to describe the coupled mechanical deformation, fluid permeation, and heat transfer of such thermally responsive gels. In discussing special constitutive equations we limit our attention to isotropic materials, and consider a model based on a Flory–Huggins model for the free energy change due to mixing of the fluid with the polymer network, coupled with a non-Gaussian statistical–mechanical model for the change in configurational entropy—a model which accounts for the limited extensibility of polymer chains. We have numerically implemented our theory in a finite element program. We show that our theory is capable of simulating swelling, squeezing of fluid by applied mechanical forces, and thermally responsive swelling/de-swelling of such materials.     

Thermally actuated shape-memory polymers: Experiments, theory, and numerical simulations

With the aim of developing a thermo-mechanically coupled large-deformation constitutive theory and a numerical-simulation capability for modeling the response of thermally actuated shape-memory polymers, we have (i) conducted large strain compression experiments on a representative shape-memory polymer to strains of approximately unity at strain rates of 10⁻³ and 10⁻¹ s⁻¹, and at temperatures ranging from room temperature to approximately 30 °C above the glass transition temperature of the polymer; (ii) formulated a thermo-mechanically coupled large-deformation constitutive theory; (iii) calibrated the material parameters appearing in the theory using the stress-strain data from the compression experiments; (iv) numerically implemented the theory by writing a user-material subroutine for a widely used finite element program; and (v) conducted representative experiments to validate the predictive capability of our theory and its numerical implementation in complex three-dimensional geometries. By comparing the numerically predicted response in these validation simulations against measurements from corresponding experiments, we show that our theory is capable of reasonably accurately reproducing the experimental results. As a demonstration of the robustness of the three-dimensional numerical capability, we also show results from a simulation of the shape-recovery response of a stent made from the polymer when it is inserted in an artery modeled as a compliant elastomeric tube.     

A high-resolution finite-difference method for simulating two-fluid,viscoelastic gel dynamics

An important class of gels are those composed of a polymer network and fluid solvent. The mechanical and rheological properties of these two-fluid gels can change dramatically in response to temperature, stress, and chemical stimulus. Because of their adaptivity, these gels are important in many biological systems, e.g. gels make up the cytoplasm of cells and the mucus in the respiratory and digestive systems, and they are involved in the formation of blood clots. In this study we consider a mathematical model for gels that treats the network phase as a viscoelastic fluid with spatially and temporally varying material parameters and treats the solvent phase as a viscous Newtonian fluid. The dynamics are governed by a coupled system of time-dependent partial differential equations which consist of transport equations for the two phases, constitutive equations for the viscoelastic stresses, two coupled momentum equations for the velocity fields of the two fluids, and a volume-averaged incompressibility constraint. We present a numerical method based on a staggered grid, second order finite-difference discretization of the momentum equations and a high-resolution unsplit Godunov method for the transport equations. The momentum and incompressibility equations are solved in a coupled manner with the Generalized Minimum Residual (GMRES) method using a multigrid preconditioner based on box-relaxation. We present results on the accuracy and robustness of the method together with an illustration of the interesting behavior of this gel model for the four-roll mill problem.     

Modeling of the pH-sensitive behavior of an ionic gel in the presence of diffusion

In this paper, we develop a model for swelling of an ionic gel in a solvent of varying pH and diffusion of the solvent through the swollen gel by applying a variational method originally presented by Baek and Srinivasa (Int. J. Nonlinear Mech. 39 (2004) 201). The approach presented here, based on the balance laws of a single continuum with mass diffusion and ionic chemical reactions, delivers system equations and boundary conditions by assuming two constitutive scalar functions for the free energy of the system and the rate of dissipation, instead of assuming osmotic pressure, electrostatic repulsive force, etc. In the equilibrium case, the model describes pH-dependent behavior and the effect of other counterions on the swelling of ionic gels. In the non-equilibrium case, it accounts for the pH-dependent mass flux through ionic gels. Moreover, the model shows that the mass flux can be induced by the gradient of chemical potential, the concentration of the mobile species and ionic charges. This model is applied to a typical carboxylated copolymer gel, and compared with an experiment for equilibrium swelling, and predicts the pH-dependence for a pressure-induced mass flux.     

A theory of coupled diffusion and large deformation in polymeric gels

A large quantity of small molecules may migrate into a network of long polymers, causing the network to swell, forming an aggregate known as a polymeric gel. This paper formulates a theory of the coupled mass transport and large deformation. The free energy of the gel results from two molecular processes: stretching the network and mixing the network with the small molecules. Both the small molecules and the long polymers are taken to be incompressible, a constraint that we enforce by using a Lagrange multiplier, which coincides with the osmosis pressure or the swelling stress. The gel can undergo large deformation of two modes. The first mode results from the fast process of local rearrangement of molecules, allowing the gel to change shape but not volume. The second mode results from the slow process of long-range migration of the small molecules, allowing the gel to change both shape and volume. We assume that the local rearrangement is instantaneous, and model the long-range migration by assuming that the small molecules diffuse inside the gel. The theory is illustrated with a layer of a gel constrained in its plane and subject to a weight in the normal direction. We also predict the scaling behavior of a gel under a conical indenter.     

On a framework for small-deformation viscoplasticity: free energy,microforces, strain gradients

This study develops a general theory for small-deformation viscoplasticity based on a system of microforces consistent with its own balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that allows for dependences on plastic strain-gradients. The microforce balance and the constitutive equations—suitably restricted by the second law—are shown to be together equivalent to a flow rule that accounts for variations in free energy due to flow. When this energy is the sum of an elastic strain energy and a defect energy quadratic, isotropic, and positive definite in the plastic-strain gradients, the flow rule takes the form of a second-order parabolic PDE for the plastic strain coupled to the usual PDE arising from the standard macroscopic force balance and the elastic stress-strain relation. The classical macroscopic boundary conditions are supplemented by nonstandard boundary conditions associated with viscoplastic flow. As an aid to solution, a weak (virtual power) formulation of the nonlocal flow rule is derived.     

Molecular simulation guided constitutive modeling on finite strain viscoelasticity of elastomers

Viscoelasticity characterizes the most important mechanical behavior of elastomers. Understanding the viscoelasticity, especially finite strain viscoelasticity, of elastomers is the key for continuation of their dedicated use in industrial applications. In this work, we present a mechanistic and physics-based constitutive model to describe and design the finite strain viscoelastic behavior of elastomers. Mathematically, the viscoelasticity of elastomers has been decomposed into hyperelastic and viscous parts, which are attributed to the nonlinear deformation of the cross-linked polymer network and the diffusion of free chains, respectively. The hyperelastic deformation of a cross-linked polymer network is governed by the cross-linking density, the molecular weight of the polymer strands between cross-linkages, and the amount of entanglements between different chains, which we observe through large scale molecular dynamics (MD) simulations. Moreover, a recently developed non-affine network model (Davidson and Goulbourne, 2013) is confirmed in the current work to be able to capture these key physical mechanisms using MD simulation. The energy dissipation during a loading and unloading process of elastomers is governed by the diffusion of free chains, which can be understood through their reptation dynamics. The viscous stress can be formulated using the classical tube model (Doi and Edwards, 1986); however, it cannot be used to capture the energy dissipation during finite deformation. By considering the tube deformation during this process, as observed from the MD simulations, we propose a modified tube model to account for the finite deformation behavior of free chains. Combing the non-affine network model for hyperelasticity and modified tube model for viscosity, both understood by molecular simulations, we develop a mechanism-based constitutive model for finite strain viscoelasticity of elastomers. All the parameters in the proposed constitutive model have physical meanings, which are signatures of polymer chemistry, physics or dynamics. Therefore, parametric materials design concepts can be easily gleaned from the model, which is also demonstrated in this study. The finite strain viscoelasticity obtained from our simulations agrees qualitatively with experimental data on both un-vulcanized and vulcanized rubbers, which captures the effects of cross-linking density, the molecular weight of the polymer chain and the strain rate.     

VISCOELASTICITY AND POROELASTICITY IN ELASTOMERIC GELS

An elastomeric gel is a mixture of a polymer network and a solvent.In response to changes in mechanical forces and in the chemical potential of the solvent in the environment,the gel evolves by two concurrent molecular processes:the conformational change of the network,and the migration of the solvent.The two processes result in viscoelasticity and poroelasticity,and are characterized by two material-specific properties:the time of viscoelastic relaxation and the effective diffusivity of the solvent through the network.The two properties define a materialspecific length.The material-specific time and length enable us to discuss macroscopic observations made over different lengths and times,and identify limiting conditions in which viscoelastic and poroelastic relaxations have either completed or yet started.We formulate a model of homogeneous deformation,and use several examples to illustrate viscoelasticity-limited solvent migration,where the migration of the solvent is pronounced,but the size of the gel is so small that the rate of change is limited by viscoelasticity.We further describe a theory that evolves a gel through inhomogeneous states.Both infinitesimal and finite deformation are considered.     

Kinetics of thermally induced swelling of hydrogels

We present a continuum model for thermally induced volume transitions in stimulus–responsive hydrogels (SRHs). The framework views the transition as proceeding via the motion of a sharp interface separating swollen and collapsed phases of the underlying polymer network. In addition to bulk and interfacial force and energy balances, our model imposes an interfacial normal configurational force balance. To account for the large volume changes exhibited by SRHs during actuation, the governing equations are developed in the setting of finite-strain kinematics. The numerical approximations to the coupled thermomechanical equations are obtained with an extended finite element/level-set method. The solution strategy involves a non-standard operator split and a simplified version of the level-set update. A number of representative problems are considered to investigate the model and compare its predictions to experimental observations. In particular, we consider the thermally induced swelling of spherical and cylindrical specimens. The stability of the interface evolution is also examined.     

Flow-induced nonequilibrium thermodynamics of lamellar semicrystalline polymers

In this work we present a first attempt to quantify the effect of flow deformation on the microstructure of semicrystalline polymers. This necessitates bridging the macroscopic flow length scale with the microscopic (segment) length scale of the semicrystalline structure. To achieve this connection we developed a hierarchical approach where a thermodynamically consistent macroscopic constitutive equation is interfaced with a microscopic lattice-based Monte Carlo (MC) simulation of the polymer chain conformation. We first illustrate this approach in a two-dimensional (2D) “toy” application where the 2D equivalent of a macroscopic constitutive equation based on reptation theory is applied to describe the chain deformation and extended free energy in the amorphous bulk phase. The values for the derivative of the free energy with respect to the mean segment orientation tensor, calculated for a planar extensional flow, are then used as an extended nonequilibrium thermodynamic forcing term. This is added in a traditional Metropolis Monte Carlo scheme, developed for a 2D lattice representation of a lamellar semicrystalline polymer, to drive the flow-induced microstructure. Significant flow-induced changes are calculated, steadily increasing as the Weissenberg number increases.We subsequently extend these ideas further in a much more realistic three-dimensional (3D) application where the information for the thermodynamics of the bulk amorphous phase under a uniaxial extensional flow is extracted from a macroscopic network model, such as that of Phan-Thien and Tanner (PTT), connecting the free energy to the second moment of the end-to-end distance of a multisegment chain. Through a series of 3D nonequilibrium Monte Carlo simulations of both the amorphous and the semicrystalline microscopic morphologies, it is shown that the interaction of the flow-induced deformation with the semicrystalline microstructure is nonlinear: the amorphous interlamellar structure changes significantly from its corresponding homogeneous bulk amorphous state, even far away from the crystalline interface. Our approach allows for a quantitative estimation of this effect on both thermodynamic quantities, like the extended microscopic free energy, as well as various statistics of the chain conformations.     

The Debye Theory of Rotary Diffusion: History, Derivation, and Generalizations

Motivated by the Debye theory of rotary diffusion in a dipolar fluid, we systematically develop a continuum mechanical theory of rotary diffusion. This theory generalizes classical kinematics to include continuous rotary degrees of freedom and introduces an additional balance law associated with the rotary degrees of freedom. Various constitutive relations are proposed in accordance with standard procedures of nonlinear continuum mechanics. The resulting set of equations provides a properly invariant and thermodynamically consistent theory that allows for constitutive nonlinearities. In particular, the classical Debye theory along with the Nernst-Einstein relations are shown to follow from a special case of linear constitutive relations and an assumption of ideality in which the free energy consists only of a classical entropic contribution. Within our theory, the notion of osmotic pressure arises naturally as a consequence of accounting for forces that act conjugate to the rotary degrees of freedom and serves as the driving force for rotary diffusion. Accepted November 18, 2000?Published online April 23, 2001     

A finite-deformation-based phenomenological theory for shape-memory alloys

In this work we develop a finite-deformation-based, thermo-mechanically-coupled and non-local phenomenological theory for polycrystalline shape-memory alloys (SMAs) capable of undergoing austenite ↔$\leftrightarrow$ martensite phase transformations. The constitutive model is developed in the isotropic plasticity setting using standard balance laws, thermodynamic laws and the theory of micro-force balance (Fried and Gurtin, 1994). The constitutive model is then implemented in the ABAQUS/Explicit (2009) finite-element program by writing a user-material subroutine. Material parameters in the constitutive model were fitted to a set of superelastic experiments conducted by Thamburaja and Anand (2001) on a polycrystalline rod Ti–Ni. With the material parameters calibrated, we show that the experimental stress-biased strain–temperature-cycling and shape-memory effect responses are qualitatively well-reproduced by the constitutive model and the numerical simulations. We also show the capability of our constitutive mode in studying the response of SMAs undergoing coupled thermo-mechanical loading and also multi-axial loading conditions by studying the deformation behavior of a stent unit cell.     

Incorporation of polymer diffusivity and migration into constitutive equations

Given a general one-particle constitutive equation for the stress tensor, we discuss how to incorporate the additional effects of polymer diffusivity and migration into that constitutive equation within the framework of continuum mechanics. For the example of an upper-convected Maxwell model representing the polymer contribution to the stress tensor of a dilute polymer solution, we describe i) how to modify the constitutive equation for the stress tensor to include diffusion and migration effects, ii) how to formulate a balance equation for the polymer mass density in order to describe the nonhomogeneous composition of the polymer solution resulting from migration, and iii) how to close the extended set of coupled equations by means of further constitutive equations for the migration velocity and the diffusion tensor. In order to guarantee the material objectivity for all equations, we formulate them in the body tensor formulation of continuum mechanics (and then translate them into Cartesian space). The proposed equations are compared to results of a recent kinetic-theory approach.Dedicated to Professor Arthur S. Lodge on the occasion of his 70th birthday and his retirement from the University of Wisconsin.     

Scission induced by circular shear and heat conduction in an elastomeric cylinder

Constitutive equations for the thermo-mechanics of elastomeric materials generally assume that they do not undergo microstructural change. A constitutive theory is discussed here which accounts for such changes arising from continuous scission of macromolecular junctions of elastomeric networks due to deformation and high temperatures and the subsequent cross-linking of molecules into new networks with new reference states. The total stress is the superposition of the stresses in the remainder of the original network and in each subsequently formed network. Each network acts as a temperature-dependent non-linear elastic material. The interaction of this material response with inhomogeneous deformation and temperature fields is studied for finite circular shear of a cylinder. Numerical results illustrate how the mechanical response of the cylinder depends on the temperature dependence of both the scission–cross-linking process and the properties of the elastic networks.     

Analysis of the rate-dependent coupled thermo-mechanical response of shape memory alloy bars and wires in tension

In this paper, the coupled thermo-mechanical response of shape memory alloy (SMA) bars and wires in tension is studied. By using the Gibbs free energy as the thermodynamic potential and choosing appropriate internal state variables, a three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is derived. Taking into account the effect of generated (absorbed) latent heat during the forward (inverse) martensitic phase transformation, the local form of the first law of thermodynamics is used to obtain the energy balance relation. The three-dimensional coupled relations for the energy balance in the presence of the internal heat flux and the constitutive equations are reduced to a one-dimensional problem. An explicit finite difference scheme is used to discretize the governing initial-boundary-value problem of bars and wires with circular cross-sections in tension. Considering several case studies for SMA wires and bars with different diameters, the effect of loading–unloading rate and different boundary conditions imposed by free and forced convections at the surface are studied. It is shown that the accuracy of assuming adiabatic or isothermal conditions in the tensile response of SMA bars strongly depends on the size and the ambient condition in addition to the rate dependency that has been known in the literature. The data of three experimental tests are used for validating the numerical results of the present formulation in predicting the stress–strain and temperature distribution for SMA bars and wires subjected to axial loading–unloading.     

A non isothermal Ginzburg-Landau model for phase transitions in shape memory alloys

A thermodynamical model for martensitic phase transitions in shape memory alloys is formulated in this paper in the framework of the Ginzburg-Landau approach to phase transitions. A single order parameter is chosen to represent the austenite parent phase and two mirror related martensite variants. A free energy previously proposed in the literature (Levitas et al. in Phys. Rev. B 66:134206, 2002; Phys. Rev. B 66:134207, 2002; Phys. Rev. B 68:134201, 2003) is employed, in its simplest form, as the main constitutive content of the model. In this paper we treat time-dependent Ginzburg-Landau equation as a balance law on the structure order and we couple it to a energy balance equation, thus allowing to account of heat transfer processes. We obtain a coupled thermo-mechanical problem whose consistency with the Second Law is verified.     

Coupled Solvent and Heat Transport of a Mixture of Swelling Porous Particles and Fluids: Single Time-Scale Problem

A three-spatial scale, single time-scale model for both moisture and heat transport is developed for an unsaturated swelling porous media from first principles within a mixture theoretic framework. On the smallest (micro) scale, the system consists of macromolecules (clay particles, polymers, etc.) and a solvating liquid (vicinal fluid), each of which are viewed as individual phases or nonoverlapping continua occupying distinct regions of space and satisfying the classical field equations. These equations are homogenized forming overlaying continua on the intermediate (meso) scale via hybrid mixture theory (HMT). On the mesoscale the homogenized swelling particles consisting of the homogenized vicinal fluid and colloid are then mixed with two bulk phase fluids: the bulk solvent and its vapor. At this scale, there exists three nonoverlapping continua occupying distinct regions of space. On the largest (macro) scale the saturated homogenized particles, bulk liquid and vapor solvent, are again homogenized forming four overlaying continua: doubly homogenized vicinal fluid, doubly homogenized macromolecules, and singly homogenized bulk liquid and vapor phases. Two constitutive theories are developed, one at the mesoscale and the other at the macroscale. Both are developed via the Coleman and Noll method of exploiting the entropy inequality coupled with linearization about equilibrium. The macroscale constitutive theory does not rely upon the mesoscale theory as is common in other upscaling methods. The energy equation on either the mesoscale or macroscale generalizes de Vries classical theory of heat and moisture transport. The momentum balance allows for flow of fluid via volume fraction gradients, pressure gradients, external force fields, and temperature gradients.     

液晶高分子各向异性粘弹性流体本构方程理论

将液晶高分子－各向异性流体的本构方程,建立在Oldroyd随体导数观点基础上。推广上随机Oldroyd B流体模型,提出共转OldroydB流体模型,同时将微观结构的影响通过宏观参数表示出来,使在宏观理论中包含微观结构的贡献,即引入取向物质函数,非线性各向异性黏度函数和各向异性松弛时间及推迟时间等,表征取向运动对黏度和松弛及推迟现象的影响,在此基础上开展了一类新的液晶高分子－Oldroyd型本构方程理论,由该类型本构方程得出的物质函数,液晶高分子流体的第一、第二法向应力差与实验结果一致,解释了液晶高分子溶液的第一、第二法向应力差的特殊流变学行为。     

A micropolar mixture theory of multi-component porous media

A mixture theory is developed for multi-component micropolar porous media with a combination of the hybrid mixture theory and the micropolar continuum theory. The system is modeled as multi-component micropolar elastic solids saturated with multi- component micropolar viscous fluids. Balance equations are given through the mixture theory. Constitutive equations are developed based on the second law of thermodynamics and constitutive assumptions. Taking account of compressibility of solid phases, the volume fraction of fluid as an independent state variable is introduced in the free energy function, and the dynamic compatibility condition is obtained to restrict the change of pressure difference on the solid-fluid interface. The constructed constitutive equations are used to close the field equations. The linear field equations are obtained using a linearization procedure, and the micropolar thermo-hydro-mechanical component transport model is established. This model can be applied to practical problems, such as contaminant, drug, and pesticide transport. When the proposed model is supposed to be porous media, and both fluid and solid are single-component, it will almost agree with Eringen＇s model.