Framework Of Thermomechanics Research Articles

New Dimensionless Number for the Transition from Viscous to Turbulent Flow

Within the framework of Classical Continuum Thermomechanics, we consider an unsteady isothermal flow of a simple isotropic linear viscous fluid in the liquid state to investigate the transient flow conditions. Despite the attention paid to this problem by several research works, it seems that the understanding of turbulence in these flow conditions is controversial. We propose a dimensionless procedure that highlights some aspects related to the transition from viscous to turbulent flow which occurs when a finite amplitude pressure wave travels through the fluid. This kind of transition is demonstrated to be described by a (first) dimensionless number, which involves the bulk viscosity. Furthermore, in the turbulent flow regime, we show the role played by a (second) dimensionless number, which involves the turbulent bulk viscosity, in entropy production. Within the frame of the 1D model, we test the performance of the dimensionless procedure using experimental data on the pressure waves propagation in a long pipe (water hammer phenomenon). The obtained numerical results show good agreement with the experimental data. The results’ inspection confirms the predominant role of the turbulent bulk viscosity on energy dissipation processes.

A constitutive level-set model for shape memory alloys

The present work proposes a phenomenological level-set model for shape memory alloys within the framework of internal variable theory and continuum thermomechanics. While the majority of constitutive models for shape memory alloys assume the martensitic volume fraction as an internal variable, in this study a level-set function is chosen to be the internal variable. Such a choice allows to account, on average, for the microstructural changes within the material by representing in an implicit manner the interfaces between different phases. Utilizing consistent thermodynamical arguments within the framework of canonical thermomechanics, the evolution equation for the level-set function is derived. Then, the model is implemented in a one dimensional case, where simple evolution equations are proposed reproducing certain characteristics of shape memory alloys, such as the pseudoelastic effect, the strain-temperature hysteresis, the one-way shape memory effect and hardening effects. Also, application of the model on functional fatigue phenomenon is thoroughly inspected. This new constitutive level-set model provides extreme flexibility and seems to be very promising for future research.

Micromechanical modeling of particle breakage of granular materials in the framework of thermomechanics

The particle breakage of granular materials under compression is a phenomenon of great importance. In this paper, a micromechanically based model for the compression of crushable granular materials is developed in the framework of thermomechanics. Both the internal and dissipative energies in the model are derived using the micro–macro volume averaging approach to ensure that all parameters involved have concrete physical meanings. The particle breakage is quantified by the change of the maximum particle size, the size polydispersity and the fractal dimension of the gradation. Compared to other breakage models, there is a major difference that highlights the novelty of the proposed model: neither the ultimate particle size distribution, nor the evolution path of the gradation is predefined in the model. The initiation, evolution and the attenuation of the breakage can be determined by the maximum dissipation principle using thermomechanics and micromechanics. Finally, it is demonstrated that the proposed model can predict the stress dependence of the elastic bulk modulus, the size dependence of the yielding stress and the elastic–plastic-pseudoelastic phase transition of granular materials.

An implicit representation of phase interface motion with internal variables

Internal variables in continuous media with a microstructure of multiple phases spatially distributed within its volume are discussed in this paper. However, in our analysis phase transition boundaries are represented implicitly under the use of the level-set formulation. Each level-set function representing an interface corresponds to an internal variable of state. Employing the general thermodynamic theory of internal variables within the framework of canonical thermomechanics, we derive the evolution equations for the level-set functions describing the motion of the interfaces inside a material. Finally, the concept of two internal variables and both dissipative and non-dissipative cases is discussed.

An advanced kinetic theory for morphing continuum with inner structures

Advanced kinetic theory with the Boltzmann-Curtiss equation provides a promising tool for polyatomic gas flows, especially for fluid flows containing inner structures, such as turbulence, polyatomic gas flows and others. Although a Hamiltonian-based distribution function was proposed for diatomic gas flow, a general distribution function for the generalized Boltzmann-Curtiss equations and polyatomic gas flow is still out of reach. With assistance from Boltzmann's entropy principle, a generalized Boltzmann-Curtiss distribution for polyatomic gas flow is introduced. The corresponding governing equations at equilibrium state are derived and compared with Eringen's morphing (micropolar) continuum theory derived under the framework of rational continuum thermomechanics. Although rational continuum thermomechanics has the advantages of mathematical rigor and simplicity, the presented statistical kinetic theory approach provides a clear physical picture for what the governing equations represent.

Continuum mechanics versus violations of the second law of thermodynamics

ABSTRACTSpontaneous, random violations of the second law become relevant as the length and/or timescales become very small. Modern statistical physics tells us that the second law then needs to be replaced by the fluctuation theorem and, mathematically, the irreversible entropy evolves as a submartingale. This is illustrated on the Couette flow of a molecular fluid. On the continuum level, such phenomena lead to a framework of thermomechanics relying on stochastic (rather than deterministic) functionals of energy and entropy, which are applied in heat diffusion and thermoelasticity settings. Counterintuitive thermomechanical behaviors are also discussed in (1) the evolution of acceleration wavefronts of nanoscale thickness and (2) the fluid mechanics which has to enter a permeability model of a poroelastic medium with nanoscale pores.

A note on the thermo-mechanics of elastic quasi-crystals

In recent years, many papers have been published concerning the elasticity of quasi-crystals. The present study has for main purpose to replace the proposed formulations into the framework of the modern thermo-mechanics of continua. Two types of modelling are envisaged in small deformations, those inspired by the physical descriptions proposed by Bak on the one hand and by Lubensky and co-workers on the other. While the first one fits well in a traditional variational formulation, the second one seems to be best accommodated in the frame of the thermo-mechanics of internal variables of state, the newly introduced “phason” field being then interpreted as such a vector variable. The inclusion of these two models in the theory of configurational forces—useful for the study of the expansion of defects frequent in such crystals—and the possibilities of including the effects of nonlinear elasticity and plasticity are briefly discussed. Important symmetry conditions are however left aside.

Fourier-based incremental homogenisation of coupled unilateral damage–plasticity model for masonry structures

In structural analysis of large masonry structures, nondemanding computation effort, numerical stability and simplified model assembly and meshing often have a higher priority over precise details of local stress or strain responses. This paper presents the development of a Fourier-based incremental homogenisation technique, where the macro–micro transformations of mechanical variables are derived by incremental variational problems to minimise the potential energy in representative volume elements (RVEs) with respect to local fluctuating displacement fields expanded in Fourier series. In addition to the proposed homogenisation technique, a unilateral damage–plasticity constitutive model for mortar joints in the RVE is developed within the framework of thermomechanics, which accounts for the stiffness and strength degradation (or recovery) due to the transverse crack opening/closing in the mortar joints. The numerical solution for the homogenisation problem and the performances of the proposed coupled-damage plastic mode and Fourier-based homogenisation scheme verified by detailed case studies are presented. It has been shown that the computational effort of the analysis with the proposed modelling technique can be considerably reduced by more than 20% as compared with that of the discrete modelling technique.

On Thermodynamic Restrictions in Peridynamics

This note examines restrictions imposed by the second law of thermodynamics on peridynamics in both the bond-based and the state-based formulations. Our study was carried out in the framework of thermomechanics with internal variables. In bond-based peridynamics, there are two possible thermomechanical interpretations of the dissipation function. One interpretation only admits a thermodynamic orthogonality of Ziegler, while the other admits powerless forces within a representation theory of Edelen. The latter interpretation is admissible in state-based peridynamics. Since the dissipation function of peridynamics is set up in velocity space, a link to statistical mechanics of irreversible phenomena does not appear possible.

Combining displacement, strain, temperature and heat source fields to investigate the thermomechanical response of an elastomeric specimen subjected to large deformations

Combining thermal and kinematic full-field measurements at the same time is gaining widespread acceptance in the community involved in the thermomechanical characterization of materials. However, obtaining such attractive information in the case of elastomers for which large deformations occur raises specific experimental issues which are addressed in this paper. The problem arises in mapping both the thermal and the kinematic quantities in the same reference coordinate system in order to write properly the heat diffusion equation. This equation provides heat sources which constitute a relevant quantity in the framework of thermomechanics. The procedure is presented and the underlying practical difficulties are detailed. It is then applied in the case of a specific test performed on a so-called three-branch specimen. Finally, the obtained strain and heat source distributions are briefly discussed.

Numerical simulation of martensitic phase transitions in shape memory alloys using an improved integration algorithm

AbstractThe numerical simulation of structures made of shape memory materials is of increasing interest in different fields. Among others, the computation of pipe connectors or medical devices like endoscopic instruments and stents is a challenge. In such practical applications the pseudoelastic effect as well as the one‐way and two‐way shape memory effects are utilized. These material properties are caused by martensitic phase transitions between austenite and martensite. In the present contribution, a recently proposed constitutive theory is numerically treated in the context of the finite element method. This constitutive theory is formulated in the framework of continuum thermomechanics for geometrically linear problems and is able to represent the occurring martensitic phase transitions in shape memory alloys. For the numerical integration of the evolution equations, the backward Euler method is applied. In spite of the complexity of the constitutive theory, it is shown that an improved integration procedure can be formulated, which merely involves the solution of three non‐linear equations for three scalar‐valued unknown variables. Numerical examples show the capability of the proposed model and the improved integration algorithm. Copyright © 2006 John Wiley & Sons, Ltd.

Stress computation in finite thermoviscoplasticity

In this article, a constitutive theory in the framework of continuum thermomechanics is introduced to represent the viscoplastic behavior of metals at finite deformations. In particular, the experimentally observed thermomechanical coupling phenomena are described by the theory. The model is based on the assumption that the occurring viscoplastic deformations are isochoric. For the numerical integration of the constitutive theory, a backward Euler method is applied. As a matter of fact, the application of the original backward Euler scheme does not preserve the property of the viscoplastic deformations to be isochoric. A major topic of the present article is the development of an improved numerical integrator on the basis of the original backward Euler method, which preserves exactly the incompressibility of the occurring viscoplastic deformations.

Numerical Modeling of Coupled Hygromechanical Degradation of Cementitious Materials

A coupled hygromechanical model for finite-element analyses of structures made of cementitious materials such as concrete or plaster is formulated within the framework of thermomechanics of partially saturated porous media. A multisurface elastoplastic-damage model, formulated in the space of plastic effective stresses, is employed to describe the nonlinear pre- and postfailure material behavior of concrete, taking the degradation of stiffness as well as the growth of inelastic strains as a consequence of the opening of microcracks into account. From relating stress and strain quantities defined on the mesolevel to respective homogenized quantities on the macrolevel, the hygromechanical coupling coefficients are identified. The effect of cracking on the isothermal liquid permeability is also accounted for. As a representative example, a two-dimensional simulation of a base restrained concrete wall subjected to both uniform drying and to rewetting at the foundation is described in the paper.

On the configurational force balance in thermomechanics

The aim of this paper is a new formulation of the configurational force balance in the general framework of thermomechanics. To this end, invariance requirements of the configurational working are used, following an approach proposed by Gurtin. To define the notion of working in a more general framework, the ideas of Green and Naghdi concerning the basic postulates of thermomechanics are invoked and used consistently. The main results are that the configurational forces balance in the general setting of thermomechanics without the invocation of constitutive relations and the pseudo-momentum equation for classical and dissipationless thermoelasticity. Finally, we give an application to fracture theory. We show that in classical thermoelasticity the derived pseudo-momentum balance provides a crack-driving force consistent with the well-known expression of the energy-release rate.

Models of hysteresis in the framework of thermomechanics with internal variables

The Ziegler–Green–Naghdi approach to the thermomechanics with internal variables is used to model the hysteretic behavior of a one-dimensional system with a single internal variable. In this framework, models of hysteresis arise from proper assumptions on the dissipation function. Two examples from elastoplasticity and pseudoelastic behavior of shape memory alloys are presented.

Application of thermodynamics to the global modelling of shallow foundations on frictional material

AbstractSoil–shallow foundation interaction has been theoretically analysed within the framework of thermomechanics. The design of a global interaction model has been achieved with an original treatment of the Clausius–Duhem inequality. The role of the gravity volume forces is emphasized. The paper is focused on a strip footing based on dense sand and subjected to time‐independent plastic processes. The theoretical approach has confirmed that an associated global flow rule cannot be expected to hold true. The analysis of the sources of dissipation has led to the development of a soil–footing interface model and a complete interaction model accounting for the interface constraints and the intrinsic frictional properties of the soil. Finally, the abilities of the complete model are checked by comparisons with experimental results found in the literature. Copyright © 2001 John Wiley & Sons, Ltd.

Microstructural Randomness Versus Representative Volume Element in Thermomechanics

Continuum thermomechanics hinges on the concept of a representative volume element (RVE), which is well defined in two situations only: (i) unit cell in a periodic microstructure, and (ii) statistically representative volume containing a very large (mathematically infinite) set of microscale elements (e.g., grains). Response of finite domains of material, however, displays statistical scatter and is dependent on the scale and boundary conditions. In order to accomplish stochastic homogenization of material response, scale-dependent hierarchies of bounds are extended to dissipative/irreversible phenomena within the framework of thermomechanics with internal variables. In particular, the free-energy function and the dissipation function become stochastic functionals whose scatter tends to decrease to zero as the material volume is increased. These functionals are linked to their duals via Legendre transforms either in the spaces of ensemble average velocities or ensemble-average dissipative forces. In the limit of infinite volumes (RVE limit (ii) above) all the functionals become deterministic, and classical Legendre transforms of deterministic thermomechanics hold. As an application, stochastic continuum damage mechanics of elastic-brittle solids is developed.

Coherent Solid-State Phase Transitions with Atomic Diffusion: A Thermomechanical Treatment

Using the framework of modern continuum thermomechanics, we develop sharp- and diffuse-interface theories for coherent solid-state phase transitions. These theories account for atomic diffusion and for deformation. Of essential importance in our formulation of the sharp-interface theory are a system of “configurational forces” and an associated “configurational force balance.” These forces, which are distinct from standard Newtonian forces, describe the intrinsic material structure of a body. The configurational balance, when restricted to the interface, leads to a generalization of the classical Gibbs–Thomson relation, a generalization that accounts for the orientation dependence of the interfacial energy density and also for a broad spectrum of dissipative transition kinetics. Our diffuse-interface theory involves nonstandard “microforces” and an associated “microforce balance.” These forces arise naturally from an interpretation of the atomic densities as macroscopic parameters that describe atomistic kinematics distinct from the motion of material particles. When supplemented by thermodynamically consistent constitutive relations, the microforce balance yields a generalization of the Cahn–Hilliard relation giving the chemical potentials as variational derivatives of the total free energy with respect to the atomic densities. A formal asymptotic analysis (thickness of the transition layer approaching zero) demonstrates the correspondence between versions of our theories specialized to the case of a single mobile species for situations in which the time scale for interface propagation is small compared to that for bulk diffusion. While the configurational force balance is redundant in the diffuse-interface theory, when integrated over the transition layer, the limit of this balance is the interfacial configurational force balance (i.e., generalized Gibbs–Thomson relation) of the sharp-interface theory.

On the large deformation behaviour of reinforced rubber at different temperatures

This essay investigates the temperature dependence of the mechanical properties of a filler-loaded tread compound experimentally and proposes a physically based method to represent this behaviour in the framework of non-linear continuum thermomechanics. To this end, we realise a series of monotonic and cyclic strain controlled tests on cylindrical specimens in tension at different temperature levels. The experimental data show the isothermal mechanical behaviour to be mainly influenced by non-linear elasticity in combination with non-linear rate dependence and weak equilibrium hysteresis. We observe that the rate sensitivity of the material depends strongly on the temperature : at low temperature levels, the rate sensitivity is essentially higher than at high temperatures. The elastic properties of the material depend comparatively less on the temperature. Nevertheless, higher temperature levels lead to higher equilibrium stresses. In order to represent the material behaviour, we start with a multiplicative split of the deformation gradient into a mechanical and a thermal part as proposed by Lu and Pister (1975). Physically, this idea corresponds to a stress-free thermal expansion followed by an isothermal stress-producing deformation. We suppose the thermal part of the deformation gradient to be isotropic. As a consequence of this, the velocity gradient decomposes additively into a pure thermal and a pure mechanical part. By using these elements, we exploit the Clausius Duhem inequality and assume the so-called ‘mechanical second Piola Kirchhoff stress tensor’ to be a functional of the ‘mechanical Green's strain tensor’. In a further step, we define this functional by a system of constitutive equations which are based on a rheological model. The evolution equations for the internal variables are formulated by using the concept of dual variables proposed by Haupt and Tsakmakis (1989, 1996). The rate sensitivity is modelled by a stress and temperature dependent viscosity function. The elastic part of the equilibrium stress is described by entropy elasticity in combination with a modified Mooney Rivlin strain energy function. The equilibrium hysteresis effects are represented by rate independent plasticity in arclength representation as proposed by Valanis (1971). The constitutive model is compatible with the dissipation principle of thermodynamics and describes the general trend of the experimental data fairly well.

Analysis of Bias-Type Actuator Using Shape Memory Alloy Based on Its Thermomechanical Constitutive Description

Dynamic behaviour of a bias type actuator using Ti-Ni shape memory alloy is examined theoretically as well as experimentally. The constitutive and the heat equations for the shape memory alloy, derived in the framework of continuum thermomechanics with phase transformation taken into account, are employed to describe the response of the considered actuator to applied electric input varying in time. Material parameters in the constitutive model adopted are determined from uniaxial loading unloading tests at different temperatures. An experimental setup of the actuator, consisting of a shape memory alloy wire and a bias spring connected serially, is also manufactured, and the response of the actuator to cyclic electric input is measured. The experimental results obtained are compared with the predictions of the analytical model, which proves promising to describe the behaviour of the actuator very well in terms of its dependence on various factors such as the rigidity of the bias spring, as well as the magnitude and the period of cyclic input.