Helmholtz Free Energy Density Research Articles

A thermo-chemo-mechanically coupled peridynamics for investigating crack behavior in solids

In engineering applications, the phenomenon of cracking is often accompanied by a coupled multiphysics effect. Peridynamics (PD) is an effective approach for solving cracking problems, but currently, no general PD model accounts for the coupling of multiple physical fields. In this work, we develop a PD model of coupled deformation, heat conduction, species diffusion, and chemical reactions. First, we establish the equations for mass, linear momentum, and energy. Then we establish fully coupled constitutive laws that interpret the interactions between the various fields and formulate evolution equations that govern the flux of species and heat. These laws and equations are developed based on the inequality of energy dissipation and the principles of chemical kinetics. Species diffusion and chemical reactions are treated as separate processes to study their effects on the Helmholtz free energy density of solids and the subsequent formation and propagation of cracks. In addition, certain coupling coefficients are calibrated by equating the corresponding physical quantities in the PD model with those in continuum mechanics. Four specific cases are simulated to validate the model, including redistribution of vacancies in ceramics, hydrogen traps, and embrittlement in metals.

Mesoscopic irreversible thermodynamics of aging kinetics of alpha polypeptides [DNA] under various constraints: Special reference to the simple spring mechanics

The mesoscopic irreversible thermodynamic treatment of α-polypeptides and the helical polynucleotides (DNA) furnishes two sets of analytical expressions, which allow us not only to analyze the reversible force–extension experiments performed by atomic force microscopy (AFM) but also to predict the irreversible “aging” kinetics of the single-stranded and double-stranded polynucleotides (ssDNA and dsDNA) helical conformations exposed to aqueous solutions and applied static stress systems under the various constraints. The present physicochemical cage model emphasizes the fact that the global Helmholtz free energy of the helical conformation acts not only under the stored “intrinsic” unusual torsional and bending elastic energies inherited by the unfolded helical structure of the amino-acid (peptides) or the nucleic-acid (nucleotide) backbone but also reveals the importance of the interfacial Helmholtz free energy density associated with the interaction of the side-wall branches within the surrounding aqueous solutions. The analytical expression obtained for the unfolding force vs extension (FE) shows a strong non-linear elasticity behavior under the twist angle constraint when the interfacial Helmholtz energy term is incorporating into the scenario. This behavior is in excellent quantitative agreement with the AFM test results obtained by Idiris et al. (2000) on the poly-L-glutamic acid [Glu(n)-Cys] exposed to aqueous solutions, which show that acidity increases the degrees of helicity.

Elastic properties and constitutive behaviour of graphene at finite temperature and large deformation

In the earlier reported theoretical studies, the bending modulus of the graphene sheet predicted with the inclusion of the dihedral energy term is observed to be almost double as compared to those reported without considering the dihedral energy term. The bending modulus reported considering dihedral energy term is observed to be more accurate when compared with the quantum mechanical calculations. The findings of these studies for bending modulus (with/without considering dihedral energy terms) are reported without considering the effect of finite temperature. In the present work, a multiscale computational framework incorporating the dihedral energy term and finite temperature in the constitutive model is presented for the first time. A temperature-dependent bond length is expressed in terms of strain and curvature tensor through the temperature-related quadratic-type Cauchy-Born rule. The total Helmholtz free energy of the unit cell is expressed as a sum of the potential energy of the unit cell and thermal energy due to the thermal vibrations of atoms arising from the finite temperature. The atomic interactions are modelled through the Tersoff-Brenner potential considering different sets of empirical parameters, including the authors’ recently proposed set of empirical parameters. The stress/moment tensors and the tangent constitutive matrix are obtained from the derivatives of the Helmholtz free energy density with strains and curvatures. The thermal properties, elastic properties, and tangent stiffness coefficients for armchair and zigzag graphene with/without dihedral energy term at finite strain, curvature, and temperature are investigated and discussed in detail. The proposed multiscale model facilitates the accurate prediction of the constitutive behaviour of the graphene in a thermal environment with remarkable improvements for the prediction of bending modulus due to the inclusion of the dihedral energy term at finite temperature.

Localisation of stress-affected chemical reactions in solids described by coupled mechanics-diffusion-reaction models

Chemical reactions in solids can induce chemical expansion of the solid that causes the emergence of the mechanical stresses, which, in turn, can affect the rate of the reaction. A typical example of this is the reaction of Si lithiation, where the stresses can inhibit the reaction up to the reaction locking. The reactions in solids can take place within some volume (bulk reactions) or localise at a chemical reaction front (localised reactions). These cases are typically described by different thermo-chemo-mechanical theories that contain the source/sink terms either in the bulk or at the propagating infinitely-thin interface, respectively. However, there are reactions that can reveal both regimes; hence, there is a need to link the theories describing the bulk and the localised (sharp-interface) reactions. The present paper bridges this gap and shows that when a certain structure of the Helmholtz free energy density is adopted (based on the ideas from the phase-field methods), it is possible to obtain (in the limit) the same driving force for the chemical reaction (hence, the same reaction kinetics) as derived within the theory of the sharp-interface chemical reactions based on the chemical affinity tensor.

Isogeometric smooth crack-band model (isCBM) using spress–sprain relations adapted to microplane theory

Recent optimum fitting of the data from eleven different types of distinctive fracture tests of quasibrittle materials, including the gap tests and the size effect tests, revealed that the crack band model (CBM) with the M7 damage constitutive law outperforms all the other existing computational fracture models for concrete. This is attributed to physically correct boundary conditions, especially those at the crack faces,to the crack front of finite width which allows using a triaxial tensorial damage constitutive law, and to choosing for this law a realistic material behavior description, the M7. Nevertheless the CBM has limitations—the width of the crack band cannot be varied, the statistically smooth damage distribution across the band cannot be resolved, and the propagation direction is biased by the mesh orientation when a regular mesh is used. These limitations have been overcome by the smooth crack band model (sCBM), which uses unconventional homogenization of damage mechanics to yield an additional Helmholtz free energy density, called the “sprain energy”. This energy is a function of the curvature tensor (or a hessian) of the vectorial displacement field, called the “sprain curvature tensor” (or sprain tensor for brevity), which is a sum of the strain gradient tensor and, importantly, the rotation gradient tensor. But, in a previous study, difficulties arose in computational implementation—the use of constant-strain elements required the curvature to be resisted by sprain forces that represented the derivatives of sprain energy with respect to nodal displacements and had to be applied on the nodes of adjacent elements. The need for adjacent nodes is here avoided by a variational derivation of the Isogeometric Smooth Crack-Band Model (isCBM). The microplane damage constitutive model M7 for concrete is adopted, which is made possible by establishing the relations of the sprain stress (or spress, for brevity) and sprain on microplanes of arbitrary orientations. Using Isogeometric Analysis (IGA) based on non-uniform rational B-Splines (NURBS), the excessive displacement curvature and its resistance (spress) can be described by the shape functions with C1-continuity. Computational results show that the orientation bias of regular meshes is eliminated while keeping the constitutive law intact, which is important for compatibility with the existing material laws in general. In addition, comparisons with experimental data reveal that isCBM with M7 can reproduce the correspondence between the size of the fracture process zone and the fracture energy, as indicated by gap tests. Finally, the isCBM does not compromise the strain-path-dependence of material response.

A coupled model of transport-reaction-mechanics with trapping, Part II: Large strain analysis

A coupled finite strain chemo-transport-mechanical formulation with trapping is here proposed to extend a previous work set in the realm of small strain theory in continuum mechanics. The theory is rooted in non-equilibrium rational thermodynamics. The kinematics is based on a multiplicative decomposition of the deformation gradient to account for swelling and shrinking, thermal, elastic and inelastic contributions. Mass balance laws and balance of linear and angular momentum, as well as the laws of thermodynamics for a convecting body, are directly formulated in their material description, after specifications of some standard transformation rules between current and reference configuration. Thermodynamic restrictions are identified based on the functional dependence of the referential Helmholtz free energy density, which is chosen as the thermodynamic potential, and further subjected to a constitutive additive decomposition. Constitutive prescriptions for the chemical potentials, referential heat and mass fluxes, chemical kinetics and the generalized heat equation lead to the establishment of the governing equations. The theoretical framework is complemented by numerical simulations, highlighting the potential of the proposed formulation in multi-physics applications.

An energy-stable and conservative numerical method for multicomponent Maxwell–Stefan model with rock compressibility

Numerical simulation of gas flow in porous media is becoming increasingly attractive due to its importance in shale and natural gas production and carbon dioxide sequestration. In this paper, taking molar densities as the primary unknowns rather than the pressure and molar fractions, we propose an alternative formulation of multicomponent Maxwell–Stefan (MS) model with rock compressibility. Benefiting from the definitions of gas and solid free energies, this MS formulation has a distinct feature that it follows an energy dissipation law, and namely, it is consistent with the second law of thermodynamics. Additionally, the formulation obeys the famous Onsager's reciprocal principle. An efficient energy-stable numerical scheme is constructed using the stabilized energy factorization approach for the Helmholtz free energy density and certain carefully designed formulations involving explicit and implicit mixed treatments for the coupling between molar densities, pressure, and porosity. We rigorously prove that the scheme inherits the energy dissipation law at the discrete level. The fully discrete scheme has the ability to ensure the mass conservation law for each component as well as preserve the Onsager's reciprocal principle. Numerical tests are conducted to verify our theories, and in particular, to demonstrate the good performance of the proposed scheme in energy stability and mass conservation as expected from our theories.

Strain gradient viscoelasticity theory of polymer networks

A physically-based strain gradient viscoelasticity theory is proposed specially for polymer networks, accounting for the strain gradient effect and the history-dependent behavior, where the microstructure-dependence and history-dependence of stress and hyperstress are interpreted physically and quantitatively. The chain representation of the Helmholtz free energy density is transferred to the strain gradient continuum field representation through the geometric (spatial) connection between the chain stretch ratio and the continuum deformation measure. The necessity of using strain gradients to characterize asymmetric deformation of the microstructure of polymer networks serves as the origin of microstructure-dependent hyperstress, which is work-conjugated to the strain gradient field. The free energy per chain is assumed history-dependent to account for chain-environment interactions (e.g., friction between segments), which ultimately results in the history-dependent behavior of polymeric solids. A general constitutive relation is provided, which together with the assumed network structure, gives a concrete one. It is shown that independent of the assumed network structure, the stress, and the hyperstress share the same dimensionless relaxation function. A strain gradient viscoelastic constitutive relation based on the eight-chain network is applied to the analysis of polymer nanocomposites, where the complex moduli are rationally defined and calculated numerically. It is found that the strain gradient effect can be viewed as an amplifier that transforms the strain-induced stress into the effective stress.

Crack tip enhanced phase-field model for crack evolution in crystalline Ti6Al from concurrent crystal plasticity FE-molecular dynamics simulations

Dislocations nucleating from a crack-tip contribute to the plasticity evolution in the vicinity of crack, as well as to the crack propagation process. This paper systematically develops a method for enhancing the defect energy, accounting for crack-tip nucleated dislocations in the Helmholtz free energy density functional, associated with the phase-field formulation of crack evolution, in a coupled crystal plasticity finite element phase field (CPFE-PF) model. A concurrent crystal plasticity FE- hyper-dynamics accelerated molecular dynamics (CPFE-MD) model is created. The model that identifies, transfers, and propagates dislocations from the atomistic to continuum domain, is used to generate a dataset of crack tip dislocation density evolution along with different state variables. The paper focuses on crack tip plasticity mechanisms for the crystalline alloy Ti6Al. Using a Bayesian inference approach the critical state variables that affect the evolution of crack tip nucleated dislocation density are inferred. A functional form of the evolution of dislocation density in terms of the state variables is derived by employing a genetic programming based symbolic regression (GPSR) approach. The contribution of nucleated dislocation densities to effective plastic strain evolution at the crack tip is validated using the CPFE-MD model simulations. A comparison of the crack path and other state variables in the vicinity of the crack with and without contributions from the nucleated dislocations shows the importance of this augmentation on crack evolution.

FE{}^textrm{ANN}: an efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining

Herein, we present a new data-driven multiscale framework called FE{}^textrm{ANN} which is based on two main keystones: the usage of physics-constrained artificial neural networks (ANNs) as macroscopic surrogate models and an autonomous data mining process. Our approach allows the efficient simulation of materials with complex underlying microstructures which reveal an overall anisotropic and nonlinear behavior on the macroscale. Thereby, we restrict ourselves to finite strain hyperelasticity problems for now. By using a set of problem specific invariants as the input of the ANN and the Helmholtz free energy density as the output, several physical principles, e. g., objectivity, material symmetry, compatibility with the balance of angular momentum and thermodynamic consistency are fulfilled a priori. The necessary data for the training of the ANN-based surrogate model, i. e., macroscopic deformations and corresponding stresses, are collected via computational homogenization of representative volume elements (RVEs). Thereby, the core feature of the approach is given by a completely autonomous mining of the required data set within an overall loop. In each iteration of the loop, new data are generated by gathering the macroscopic deformation states from the macroscopic finite element simulation and a subsequently sorting by using the anisotropy class of the considered material. Finally, all unknown deformations are prescribed in the RVE simulation to get the corresponding stresses and thus to extend the data set. The proposed framework consequently allows to reduce the number of time-consuming microscale simulations to a minimum. It is exemplarily applied to several descriptive examples, where a fiber reinforced composite with a highly nonlinear Ogden-type behavior of the individual components is considered. Thereby, a rather high accuracy could be proved by a validation of the approach.

An improved branch and bound algorithm for phase stability testing of multicomponent mixtures

We investigate the phase stability of a multicomponent mixture at constant volume, temperature and moles (VTN stability). Our work is based on the TPD criterion derived in Mikyška and Firoozabadi (2012), Investigation of Mixture Stability at Given Volume, temperature and number of moles, Fluid Phase Equilibria and the branch and bound algorithm from Smejkal and Mikyška (2020), VTN-phase stability testing using the Branch and Bound strategy and the convex–concave splitting of Helmholtz free energy density, Fluid Phase Equilibria. In this contribution, we improve the algorithm with more effective bounding strategy. This improvement is achieved using the necessary condition of optimality. In the bounding step of the algorithm, before solving an underestimated convex optimization, we check whether the pressure (given by the Peng–Robinson equation of state) is feasible. If it is not the case, we can exclude the corresponding part of the feasible set from the search. The pressure function given by the Peng–Robinson equation of state is not convex and therefore leads to a nonconvex optimization problem which is computationally expensive. We propose to use a less precise estimate of the global maximum of the pressure. This estimate can be found by comparing the finite number of the values of the tangent plane to a concave overestimate of the Peng–Robinson equation of state. Another benefit of this additional step is to avoid the optimization of the underestimated objective function. The proposed method is tested on several specific examples.

An efficient bound-preserving and energy stable algorithm for compressible gas flow in porous media

Due to the great significance of the natural gas and shale gas, it is becoming increasingly important to simulate compressible gas flow in porous media. The model obeys an energy dissipation law as well as molar density must be positive and bounded in terms of the equation of state. For the purpose of eliminating nonphysical solutions as well as improving the stability in practical simulation, preservation of these properties is essential for a promising numerical method, but it is actually challenging due to the strong nonlinearity and complexity of the model. In this paper, we propose an efficient linearized numerical scheme that inherits the energy dissipation law as well as preserves the boundedness of molar density. Specifically, to treat the logarithmic type Helmholtz free energy density determined by the Peng-Robinson equation of state, we propose a novel adaptive stabilization approach involving the second derivatives of the convex energy terms. At each time step, the stabilization parameter is adaptively updated by a simple and explicit formula to ensure the energy dissipation law. The stabilized and linearized chemical potential allows to formulate the local mass conservation equation as an equivalent convection-diffusion form, and from this, an adaptive time step strategy is proposed to preserve the positivity and boundedness of molar density. The calculation of the time step size is fully explicit and easy to implement. Additionally, the fully discrete scheme is constructed using the conservative cell-centered finite difference method with the upwind strategy, and thus, it enjoys the local mass conservation. Numerical results are also presented to demonstrate the excellent performance of the proposed scheme.

Thermodynamically consistent modeling of immiscible gas–liquid flow in porous media

Modeling of immiscible gas–liquid two-phase flow with gas compressibility in porous media plays an important role in shale gas production, geological sequestration of carbon dioxide, and underground gas storage. The second law of thermodynamics is universally recognized as an essential principle any promising model should obey. The existing models have no proper concept of free energies for such a problem, thereby failing to obey this law. In this paper, we first introduce free energies to account for the liquid–gas capillarity effect and gas compressibility, and then using the second law of thermodynamics, we rigorously derive a thermodynamically consistent model for immiscible gas–liquid two-phase flow in porous media. The proposed free energy that describes the capillarity effect is verified by the laboratory data. For gas flow, we use molar density rather than pressure as the primary variable and take the Helmholtz free energy density determined by a realistic equation of state to characterize the gas compressibility. Numerical simulation results are also presented to demonstrate the thermodynamical consistency of the model and the applicability to simulate the liquid and gas displacement processes.

Third‐Order Padé Thermoelastic Constants of Solid Rocks

AbstractClassical third‐order thermoelastic constants are generally derived from the theory of small‐amplitude acoustic waves in isotropic materials during heat treatments. Investigating higher‐order thermoelastic constants for higher temperatures is challenging owing to the involvement of the number of unknown parameters. These Taylor‐type thermoelastic constants from the classical thermoelasticity theory are formulated based on the Taylor series of the Helmholtz free energy density for preheated crystals. However, these Taylor‐type thermoelastic models are limited even at low temperatures in characterizing the temperature‐dependent velocities of elastic waves in solid rocks as a polycrystal compound of different mineral lithologies. Thus, we propose using the Padé rational function to the total thermal strain energy function. The resulting Padé thermoelastic model gives a reasonable theoretical prediction for acoustic velocities of solid rocks at a higher temperature. We formulate the relationship between the third‐order Padé thermoelastic constants and the corresponding higher‐order Taylor thermoelastic constants with the same accuracy. Two additional Padé coefficients and can be calculated using the second‐, third‐, and fourth‐order Taylor thermoelastic constants associated with the Brugger's constants, which are consistent with those obtained by fitting the experimental data of polycrystalline material. The third‐order Padé thermoelastic model (with four constants) is validated by the fourth‐order Taylor thermoelastic prediction (with six constants) with ultrasonic measurements for polycrystals (olivine samples) and solid rocks (sandstone, granite, and shale). The results demonstrate that the third‐order Padé thermoelastic model can characterize thermally induced velocity changes more accurately than the conventional third‐order Taylor thermoelastic prediction (with four constants), especially for solid rocks at high temperatures. The Padé approximation could be considered a more accurate and universal model in describing thermally induced velocity changes for polycrystals and solid rocks.

Energy Stable and Mass Conservative Numerical Method for Gas Flow in Porous Media with Rock Compressibility

Taking rock compressibility into consideration brings up a quite challenging issue to modeling and numerical simulation of gas flow in porous media. To develop efficient numerical methods for such a problem, on the basis of the existing model, we, for the first time, propose an alternative formulation by introducing a free energy accounting for rock compressibility and by using chemical potential gradient instead of pressure gradient as the primary driving force. This formulation has a distinct feature that it is proved to satisfy an energy dissipation law. We further propose an efficient energy stable time discrete scheme using the improved energy factorization approach to tackle the Helmholtz free energy density determined by the Peng--Robinson equation of state. The discrete pressure is taken carefully using the discrete chemical potential and free energy such that the discrete energy dissipation law can be preserved. The fully discrete scheme is constructed based on the cell-centered finite difference method with the upwind strategy, which features full mass conservation and positivity preservation of molar density under proper conditions. Numerical results are also provided to show the performance of the proposed scheme and validate theoretical analysis.

Rotational Inertial Physics in Non-Classical Thermoviscous Fluent Continua Incorporating Internal Rotation Rates

In this paper, we derive non-classical continuum theory for physics of compressible and incompressible thermoviscous non-classical fluent continua using the conservation and balance laws (CBL) by incorporating additional physics of internal rotation rates arising from the velocity gradient tensor as well as their time varying rates and the rotational inertial effects. In this non-classical continuum theory time dependent deformation of fluent continua results in time varying rotation rates i.e., angular velocities and angular accelerations at material points. Resistance offered to these by deforming fluent continua results in additional moments, angular momenta and inertial effects due to rotation rates i.e., angular velocities and angular accelerations at the material points. Currently, this physics due to internal rotation rates and inertial effects is neither considered in classical continuum mechanics (CCM) nor in non-classical continuum mechanics (NCCM). In this paper, we present a derivation of conservation and balance laws in Eulerian description: conservation of mass (CM), balance of linear momenta (BLM), balance of angular momenta (BAM), balance of moment of moments (BMM), first and second laws of thermodynamics (FLT, SLT) that include: (i) Physics of internal rotation rates resulting from the velocity gradient tensor; (ii) New physics resulting due to angular velocities and angular accelerations due to spatially varying and time dependent rotation rates. The balance laws derived here are compared with those that only consider the rotational rates but neglect rotational inertial effects and angular accelerations to demonstrate the influence of the new physics. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and principle of equipresence when appropriate. Constitutive theories are derived using Helmholtz free energy density as well as representation theorem and integrity (complete basis). It is shown that the mathematical model consisting of the conservation and balance laws and constitutive theories presented in this paper has closure. Influence of new physics in the conservation and balance laws on compressible and incompressible thermoviscous fluent continua is demonstrated due to presence of angular velocities and angular accelerations arising from time varying rotation rates when the deforming fluent continua offer rotational inertial resistance. The fluent continua are considered homogeneous and isotropic. Model problem studies are considered in a follow-up paper.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

Chemical Potential-Based Modeling of Shale Gas Transport

Shale gas plays an increasingly important role in the current energy industry. Modeling of gas flow in shale media has become a crucial and useful tool to estimate shale gas production accurately. The second law of thermodynamics provides a theoretical criterion to justify any promising model, but it has been never fully considered in the existing models of shale gas. In this paper, a new mathematical model of gas flow in shale formations is proposed, which uses gas density instead of pressure as the primary variable. A distinctive feature of the model is to employ chemical potential gradient rather than pressure gradient as the primary driving force. This allows to prove that the proposed model obeys an energy dissipation law, and thus, the second law of thermodynamics is satisfied. Moreover, on the basis of energy factorization approach for the Helmholtz free energy density, an efficient, linear, energy stable semi-implicit numerical scheme is proposed for the proposed model. Numerical experiments are also performed to validate the model and numerical method.

Crystal Plasticity Phase-Field Model with Crack Tip Enhancement Through a Concurrent Atomistic-Continuum Model

This paper develops a method for physics-based augmentation of the Helmholtz free energy density functionals, used in coupled crystal plasticity phase-field finite element (CP-PF FE) models of fracture in crystalline metallic materials. Specifically, the defect and crack surface energy components are enhanced with terms that mechanistically account for the presence of atomic-scale, crack-tip nucleated dislocations. The additional terms in the free energy representation are motivated and calibrated by energy equivalence between a concurrent atomistic–continuum scale model and the CP-PF FE model. The atomistic domain of the concurrent model incorporates a time-accelerated, molecular dynamics (MD) LAMMPS code, while the continuum domain is modeled by a dislocation-density crystal plasticity FE model. The concurrent model transfers and transforms discrete dislocations in the atomistic domain to dislocation densities in the crystal plasticity domain. Dislocation densities are transported in the continuum domain by solving the advection equation using a particle-based reproducing kernel particle method with collocation. A new form of the defect energy density is proposed by considering the effect of crack-tip nucleated dislocations. Parameters in the augmented defect and surface energies are evaluated by comparing with the concurrent model results. A comparison of crack propagation with and without contributions from the nucleated dislocations demonstrates a significant effect of nucleated dislocations on the evolution of the crack.

A Thermodynamic-Consistent Model for the Thermo-Chemo-Mechanical Couplings in Amorphous Shape-Memory Polymers

Chemically-responsive amorphous shape-memory polymers (SMPs) can transit from the temporary shape to the permanent shape in responsive to solvents. This effect has been reported in various polymer-solvent systems. However, limited attention has been paid to the constitutive modeling of this behavior. In this work, we develop a fully thermo-chemo-mechanical coupled thermodynamic framework for the chemically-responsive amorphous SMPs. The framework shows that the entropy, the chemical potential and the stress can be directly obtained if the Helmholtz free energy density is defined. Based on the entropy inequality, the evolution equation for the viscous strain, the temperature and the number of solvent molecules are also derived. We also provide an explicit form of Helmholtz free energy density as an example. In addition, based on the free volume concept, the dependence of viscosity and diffusivity on the temperature and solvent concentration is defined. The theoretical framework can potentially advance the fundamental understanding of chemically-responsive shape-memory effect. Meanwhile, it can also be used to describe other important physical processes such as the diffusion of solvents in glassy polymers.