Coleman Noll Procedure Research Articles

Extended Coleman–Noll procedure and thermodynamics of viscoelastic solids revisited

Extended Coleman–Noll procedure and thermodynamics of viscoelastic solids revisited

On the entropy inequality and its exploitation in continuum physics

AbstractThe paper deals with the statement and the application of the entropy principle, through the Clausius–Duhem inequality, in continuum physics. The conceptual role is taken from the Coleman–Noll paper of 1963 thus leading to the physical admissibility of constitutive equations. The statement is generalized by letting the rate of entropy production be a constitutive property per se. This generalization proves essential in connection with the modelling of hysteretic phenomena. As to the application, the view of the Coleman–Noll procedure is maintained but a representation formula is shown to generalize the consequences of the entropy principle; as an example the modelling of heat conduction is investigated. Furthermore, while applying the entropy principle to magnetic materials, it is shown an interesting connection between the balance of angular momentum and the thermodynamic restrictions. Also, the modelling through rate-type equations shows the need of Lagrangian fields to obey objectivity and that of the entropy production as a constitutive function to account for the difference between loading and unloading processes.

Modeling of heat conduction through rate equations

AbstractStarting from a classical thermodynamic approach, we derive rate-type equations to describe the behavior of heat flow in deformable media. Constitutive equations are defined in the material (Lagrangian) description where the standard time derivative satisfies the principle of objectivity. The statement of the Second Law is formulated in the classical form and the thermodynamic restrictions are then developed following a variant of the Coleman-Noll procedure where the entropy production too is given by a non-negative constitutive equation. Both the free energy and the entropy production are assumed to depend on a common set of independent variables involving, in addition to temperature, both temperature gradient and heat-flux vector together with their time derivatives. This approach results in rate-type constitutive function for the heat flux that are intrinsically consistent with the Second Law and easily amenable to analysis. In addition to providing already known models (e.g., Maxwell-Cattaneo-Vernotte’s and Jeffreys-like heat conductors), this scheme allows the formulation of new models of heat transport that are likely to apply also in nanosystems. This is consistent with the fact that higher-order time derivatives of the heat flux are in order when high-rate regimes occur.

Thermohyperelastic constitutive response for semilinear materials with application to deformation of a rotating thin disk

Constitutive modeling of soft elastomers is a well researched subject due to its numerous engineering applications. This study derives a thermodynamically consistent constitutive model for semilinear thermoelastic solids for which the strain energy density depends on the deformation gradient through the Biot stretch tensor. The Frechet derivatives, Coleman-Noll procedure, and product decomposition-based technique are employed to formulate the constitutive model. The combination of the derived model and thermomechanical field equations is specialized for application to problem of radial deformation of a rotating thin disk. The formulated model provides simple closed-form expressions for simulating thermomechanical responses of solids. Under suitable conditions, the derived three-configuration-based constitutive model reduces to the two-configuration-based constitutive equations. Finally, and among other things, the numerical solutions illustrate the effects of angular speed of rotation on thermal stress and displacement fields at points within the radially deforming rotating disk.

Thermodynamics of Composition Graded Thermoelastic Solids.

We propose a thermodynamic model describing the thermoelastic behavior of composition graded materials. The compatibility of the model with the second law of thermodynamics is explored by applying a generalized Coleman-Noll procedure. For the material at hand, the specific entropy and the stress tensor may depend on the gradient of the unknown fields, resulting in a very general theory. We calculate the speeds of coupled first- and second-sound pulses, propagating either trough nonequilibrium or equilibrium states. We characterize several different types of perturbations depending on the value of the material coefficients. Under the assumption that the deformation of the body can produce changes in its stoichiometry, altering locally the material composition, the possibility of propagation of pure stoichiometric waves is pointed out. Thermoelastic perturbations generated by the coupling of stoichiometric and thermal effects are analyzed as well.

A thermodynamically consistent non-isothermal phase-field model for selective laser sintering

Selective laser sintering (SLS) additive manufacturing possesses the characteristics of extremely non-uniform temperature distribution and high temperature gradient, which are unmanageable by the traditional isothermal models. In this work, we propose a thermodynamically consistent non-isothermal phase-field model (PFM) to investigate the microstructure evolution during SLS process with the full consideration of temperature gradient effect. The model is derived from a thermodynamic framework which invokes the microforce theory and Coleman–Noll procedure. The temperature-dependent parameters in the free energy formulation are determined by the experimentally measurable surface energy and grain boundary energy with a given interface width. Benchmark simulations of sintering two particles under temperature gradient indicate that the grain boundary has the trend to migrate to the higher temperature region, which is further verified by molecular dynamics simulations. This phenomenon cannot be captured by the traditional isothermal PFM. Grain boundary migrations can also be driven by curvature for two unequally-sized particles. Simulations on SLS of powder bed show the dependence of microstructure evolution and final porosity on the laser power and scanning speed. The developed non-isothermal PFM could be a practicable toolkit for the microstructure simulation of SLS process.

An acoustic Riemann solver for large strain computational contact dynamics

AbstractThis article presents a vertex‐centered finite volume algorithm for the explicit dynamic analysis of large strain contact problems. The methodology exploits the use of a system of first order conservation equations written in terms of the linear momentum and a triplet of geometric deformation measures (comprising the deformation gradient tensor, its co‐factor, and its Jacobian) together with their associated jump conditions. The latter can be used to derive several dynamic contact models ensuring the preservation of hyperbolic characteristic structure across solution discontinuities at the contact interface, a clear advantage over the standard quasi‐static contact models where the influence of inertial effects at the contact interface is completely neglected. Taking advantage of the conservative nature of the formalism, both kinetic (traction) and kinematic (velocity) contact interface conditions are explicitly enforced at the fluxes through the use of appropriate jump conditions. Specifically, the kinetic condition is enforced in the usual linear momentum equation, whereas the kinematic condition can now be easily enforced in the geometric conservation equations without requiring a computationally demanding iterative algorithm. Additionally, a total variation diminishing shock capturing technique can be suitably incorporated in order to improve dramatically the performance of the algorithm at the vicinity of shocks. Moreover, and to guarantee stability from the spatial discretization standpoint, global entropy production is demonstrated through the satisfaction of semi‐discrete version of the classical Coleman–Noll procedure expressed in terms of the time rate of the so‐called Hamiltonian energy of the system. Finally, a series of numerical examples is examined in order to assess the performance and applicability of the algorithm suitably implemented in OpenFOAM. The knowledge of the potential contact loci between contact interfaces is assumed to be known a priori.

Open system peridynamics

We propose, for the first time, a thermodynamically consistent formulation for open system (continuum-kinematics-inspired) peridynamics. In contrast to closed system mechanics, in open system mechanics mass can no longer be considered a conservative property. In this contribution, we enhance the balance of mass by a (nonlocal) mass source. To elaborate a thermodynamically consistent formulation, the balances of momentum, energy and entropy need to be reconsidered as they are influenced by the additional mass source. Due to the nonlocal continuum formulation, we distinguish between local and nonlocal balance equations. We obtain the dissipation inequality via a Legendre transformation and derive the structure and constraints of the constitutive expressions based on the Coleman–Noll procedure. For the sake of demonstration, we present an example for a nonlocal mass source that can model the complex process of bone remodelling in peridynamics. In addition, we provide a numerical example to highlight the influence of nonlocality on the material density evolution.

Mathematical modelling of axonal cortex contractility

The axonal cortex is composed of a regular structure of F-actin and spectrin able to contract thanks to myosin II motors. Such an active tension is of fundamental importance in controlling the physiological shape of axons. Recent experiments show that axons modulate the contraction of the cortex when subject to mechanical deformations, exhibiting a non-trivial coupling between the hoop and the axial active tension. However, the underlying mechanisms are still poorly understood. In this paper, we propose a continuum model of the axon based on the active strain theory. By using the Coleman–Noll procedure, we shed light on the coupling between the hoop and the axial active strain through the Mandel stress tensor. We propose a qualitative analysis of the system under the simplifying assumption of incompressibility, showing the existence of a stable equilibrium solution. In particular, our results show that the axon regulates the active contraction to maintain a homeostatic stress state. Finally, we propose numerical simulations of the model, using a more suitable compressible constitutive law. The results are compared with experimental data, showing an excellent quantitative agreement.Statement of Significance The mechanics of cortical contractility in axons is still poorly understood. Unravelling the mechanisms underlying axial and hoop stress generation in the cortex will give insight on the active regulation of axon diameter. The understanding of this phenomenon may shed new light on the physical causes of axonal morphological degeneration as a consequence of neurodegenerative diseases, viral infections, and traumatic brain injuries.

Феноменологический вывод термомеханической модели развития канала электрического пробоя типа «диффузной границы»

In this work we derive diffuse-interface type model for electric breakdown evolution in solid dielectrics which accounts for non-isothermal and mechanical effects. The proposed model consists of mass, momentum and energy conservation equation, Maxwell’s equations in quasi(electro)static approximation and Allen-Cahn type equation which describes phase-field evolution. The derivation of the model is based on the rational thermomechanics framework, M. Gurtin’s microforce and microstress theory and Coleman-Noll procedure.

Математическая модель двухфазной гиперупругой среды. «Скалярный» случай

The work is devoted to the phenomenological derivation of the multivelocity multifluid model to describe dynamics of the multiphase medium with hyperelastic phase behavior using the Coleman-Noll procedure. The proposed model can be considered as a generalization of the Baer and Nunziato type multiphase models with isotropic stress tensor. The paper presents the well-known particular cases to which the obtained multiphase model can be reduced. The resulting model is completely non-equilibrium, and the joint deformation of the phases is described by a single scalar parameter, which is the volume fraction of the phases.

Phase field model for electrically induced damage using microforce theory

In this paper, we present a consistent derivation of the phase field model for electrically induced damage. The derivation is based on Gurtin’s microstress and microforce theory and the Coleman–Noll procedure. The resulting model accounts for Ohmic currents, includes charge conservation law and allows for finite electric permittivity and conductivity distribution in the medium. Special attention is devoted to the case when the damaged region is a codimension-two object, i.e., a curve in three dimensions. It is shown that in this case the free energy of the model necessarily includes a high-order term, which ensures the well-posedness of the problem. A special problem setting is proposed to account for the prescribed charge distribution. Local features of the phase field distribution are illustrated with one-dimensional axisymmetric numerical experiments.

A thermo-viscoplasticity model for metals over wide temperature ranges- application to case hardening steel

In this contribution, a model for the thermomechanically coupled behaviour of case hardening steel is introduced with application to 16MnCr5 (1.7131). The model is based on a decomposition of the free energy into a thermo-elastic and a plastic part. Associated viscoplasticity, in terms of a temperature-depenent Perzyna-type power law, in combination with an isotropic von Mises yield function takes respect for strain-rate dependency of the yield stress. The model covers additional temperature-related effects, like temperature-dependent elastic moduli, coefficient of thermal expansion, heat capacity, heat conductivity, yield stress and cold work hardening. The formulation fulfils the second law of thermodynamics in the form of the Clausius–Duhem inequality by exploiting the Coleman–Noll procedure. The introduced model parameters are fitted against experimental data. An implementation into a fully coupled finite element model is provided and representative numerical examples are presented showing aspects of the localisation and regularisation behaviour of the proposed model.

Thermomechanical Modeling of Microstructure Evolution Caused by Strain-Induced Crystallization.

The present contribution deals with the thermomechanical modeling of the strain-induced crystallization in unfilled polymers. This phenomenon significantly influences mechanical and thermal properties of polymers and has to be taken into consideration when planning manufacturing processes as well as applications of the final product. In order to simultaneously capture both kinds of effects, the model proposed starts by introducing a triple decomposition of the deformation gradient and furthermore uses thermodynamic framework for material modeling based on the Coleman–Noll procedure and minimum principle of the dissipation potential, which requires suitable assumptions for the Helmholtz free energy and the dissipation potential. The chosen setup yields evolution equations which are able to simulate the formation and the degradation of crystalline regions accompanied by the temperature change during a cyclic tensile test. The boundary value problem corresponding to the described process includes the balance of linear momentum and balance of energy and serves as a basis for the numerical implementation within an FEM code. The paper closes with the numerical examples showing the microstructure evolution and temperature distribution for different material samples.

Nonlinear resultant theory of shells accounting for thermodiffusion

The complete nonlinear resultant 2D model of shell thermodiffusion is developed. All 2D balance laws and the entropy imbalance are formulated by direct through-the-thickness integration of respective 3D laws of continuum thermodiffusion. This leads to a more rich thermodynamic structure of our 2D model with several additional 2D fields not present in the 3D parent model. Constitutive equations of elastic thermodiffusive shells are discussed in more detail. They are formulated from restrictions imposed by the resultant 2D entropy imbalance according to Coleman–Noll procedure extended by a set of 2D constitutive equations based on heuristic assumptions.

Model of the dynamics of an interface between a smectic phase and an isotropic phase of different density

Soft modulated phases have been shown to undergo complex morphological transitions, in which layer remodeling induced by mean and Gaussian curvatures plays a major role. This is the case in smectic films under thermal treatment, where focal conics can be reshaped into conical pyramids and concentric ring structures. We build on earlier research on a smectic-isotropic, two phase configuration in which diffusive evolution of the interface was driven by curvature, while mass transport was neglected. Here, we explicitly consider evaporation-condensation processes in a smectic phase with mass transport through a coexisting isotropic fluid phase, as well as the hydrodynamic stresses at the interface and the resulting flows. By employing the Coleman-Noll procedure, we derive a phase-field model that accounts for a varying density field coupled to smectic layering of the order parameter. The resulting equations govern the evolution of an interface between a modulated phase and an isotropic fluid phase with distinct densities, and they capture compressibility effects in the interfacial region and topological transitions. We first verify a numerical implementation of the governing equations by examining the dispersion relation for interfacial transverse modes. The inverse decay rate is shown to scale as $Q^{2}$ ($Q$ is the wavenumber of the perturbation) due to hydrodynamic effects, instead of the $Q^{4}$ expected for diffusive decay. Then, by integrating the equations forward in time, we investigate fluid flow on distorted layers and focal conics, and show how interfacial stresses and density contrast significantly determine the structure of the flow and the evolution of the configuration.

On interface conditions on a material singular surface

The presence of grain boundaries significantly influences the overall mechanical behavior of materials with an underlying crystalline microstructure. They act as an obstacle against the movement of dislocations and, thus, significantly contribute to size effects as for example the Hall–Petch effect. Hence, the thermodynamically consistent modeling of the behavior of the plastic slip at a grain boundary is of utmost interest. To this end, balance equations at a grain boundary are derived from an extended energy balance by means of invariance considerations. The grain boundary is considered as a material singular surface with own internal and kinetic energy as well as energy supply. Consequently, the balances at the grain boundary depend on its mean curvature. The framework presented is applied to a small strain slip gradient crystal plasticity theory, regarding single slip. Accounting for the derived balance equations, thermodynamically consistent flow rules for the plastic slip at the grain boundary are obtained by exploitation of the Coleman–Noll procedure. In this context, a classification of flow rules for the plastic slip at the grain boundary is provided. Finally, the distribution of the plastic slip is presented regarding a three-phase laminate material with two elastoplastic phases and one elastic phase. The two elastoplastic phases represent the grains of a bicrystal. A grain boundary effect is discussed which is based on a variation of the internal length scale in one of the two elastoplastic phases.

A three-dimensional constitutive model for the martensitic transformation in polycrystalline shape memory alloys under large deformation

This work presents a three-dimensional constitutive model for the martensitic transformation in polycrystalline shape memory alloys (SMAs) under large deformation. By utilizing the logarithmic strain and rate, the model is able to account for large strains and rotations that SMA-based components may undertake, but also resolves the stress errors caused by the non-integrable objective rates that are widely used in current commercial finite element software. The model is developed through classical thermodynamic laws combined with the standard Coleman–Noll procedure. The scalar martensitic volume fraction and the second-order transformation strain tensor are chosen as the internal state variables to capture the material response exhibited by polycrystalline SMAs. A detailed implementation procedure of the proposed model is described through a user-defined material subroutine. Numerical experiments considering SMA components including a bar, a beam, a torque tube and a solid flexible structure under stress/thermally-induced phase transformations are investigated via the proposed model, and the results under cyclic loading are compared against the predictions provided by the Abaqus nonlinear solver. The development framework of the proposed model and its implementation procedure can be extended to incorporate other nonlinear phenomena exhibited by SMAs, such as transformation-induced plasticity, viscoplasticity, and damage under large deformation.

Multi-Component Multiphase Flow Through a Poroelastic Medium

An axiomatic development for a continuum description of a multi-component multiphase porous flow in an elastic medium is developed. The Coleman–Noll procedure is used to derive constitutive restrictions which guarantee that the resulting model satisfies an appropriate statement of the second law of thermodynamics and a corresponding dissipation inequality. Many of the models and formulations appearing in the engineering literature are shown to be special cases of the model developed here.

Differential consequences of balance laws in extended irreversible thermodynamics of rigid heat conductors.

We consider a system of balance laws arising in extended irreversible thermodynamics of rigid heat conductors, together with its differential conse- quences, namely the higher-order system obtained by taking into account the time and space derivatives of the original system. We point out some mathematical properties of the differential consequences, with particular attention to the problem of the propagation of thermal perturbations with finite speed. We prove that, under an opportune choice of the initial conditions, a solution of the Cauchy problem for the system of differential consequences is also a solution of the Cauchy problem for the original system. We investigate the thermodynamic compatibility of the system at hand by applying a generalized Coleman-Noll procedure. On the example of a generalized Guyer-Krumhansl heat-transport model, we show that it is possible to get a hyperbolic system of evolution equations even when the state space is non-local.