On material interfaces in thermomechanical solids

This contribution derives thermodynamically consistent balance equations for material interfaces in thermomechanical solids. Thereby, both displacement and temperature jumps are admissible. To this end, the interface is equipped with its own thermodynamic life, i.e. we assume separate interface free energy, entropy and the like. The thermodynamical admissibility then follows from proper definitions of the interface displacement and the interface temperature, whereby in particular, the latter follows in an unexpected format. The formulation is exemplified for the example of thermoelasticity.     

Large scale numerical simulations for multi-phase fluid dynamics with moving interfaces

This short communication presents our recent studies to implement numerical simulations for multi-phase flows on top-ranked supercomputer systems with distributed memory architecture. The numerical model is designed so as to make full use of the capacity of the hardware. Satisfactory scalability in terms of both the parallel speed-up rate and the size of the problem has been obtained on two high rank systems with massively parallel processors, the Earth Simulator (Earth simulator research center, Yokohama Kanagawa, Japan) and the TSUBAME (Tokyo Institute of Technology, Tokyo, Japan) supercomputers.     

Thermomechanics of solids with general imperfect coherent interfaces

The objective of this contribution is to develop a thermodynamically consistent theory for general imperfect coherent interfaces in view of their thermomechanical behavior and to establish a unified computational framework to model all classes of such interfaces using the finite element method. Conventionally, imperfect interfaces with respect to their thermal behavior are often restricted to being either highly conducting (HC) or lowly conducting (LC) also known as Kapitza. The interface model here is general imperfect in the sense that it allows for a jump of the temperature as well as for a jump of the normal heat flux across the interface. Clearly, in extreme cases, the current model simplifies to HC and LC interfaces. A new characteristic of the general imperfect interface is that the interface temperature is an independent degree of freedom and, in general, is not a function of only temperatures across the interface. The interface temperature, however, must be computed using a new interface material parameter, i.e., the sensitivity. It is shown that according to the second law, the interface temperature may not necessarily be the average of (or even between) the temperatures across the interface. In particular, even if the temperature jump at the interface vanishes, the interface temperature may be different from the temperatures across the interface. This finding allows for a better, and somewhat novel, understanding of HC interfaces. That is, a HC interface implies, but is not implied by, the vanishing temperature jump across the interface. The problem is formulated such that all types of interfaces are derived from a general imperfect interface model, and therefore, we establish a unified finite element framework to model all classes of interfaces for general transient problems. Full details of the novel numerical scheme are provided. Key features of the problem are then elucidated via a series of three-dimensional numerical examples. Finally, we recall since the influence of interfaces on the overall response of a body increases as the scale of the problem decreases, this contribution has certain applications to nano-composites and also thermal interface materials.     

Analysis of plastic deformation in nanoscale metallic multilayers with coherent and incoherent interfaces

Nanoscale metallic multilayered (NMM) composites possess ultra high strength (order of GPa) and high ductility, and exhibit high fatigue resistance. Their mechanical behavior is governed mainly by interface properties (coherent and/or incoherent interfaces), dislocation mechanisms in small volume, and dislocation-interface interaction. In this work, we investigate these effects within a dislocation dynamics (DD) framework and analyze the mechanical behavior of two systems: (1) a bi-material system (CuNi) with coherent interface and (2) a newly developed tri-material system (CuNiNb) composed of both coherent and incoherent interfaces. For the bi-material case we analyze the influence of networks of interfacial dislocations whose nature and distribution are commensurate with the level of relaxation and loading of the structure. Misfit and pre-deposited interfacial dislocation arrays, as well as combinations of both, are studied and the dependence of strength on layer thickness is reported, along with observed dislocation mechanisms. It is shown that interfacial defect configurations significantly alter the strength and mechanical behavior of the material. Furthermore, it is shown that the implementation of penetrable interfaces in DD captures the strength dependence at layer thicknesses on the order of 3-7 nm. For the tri-material case we analyze the effects of coherent and incoherent interfaces in large-scale simulations. The results show that these materials have strong strength-size dependence but are limited by the strength of the incoherent (CuNb) interface which is weak in shear. The weak interface acts as a dislocation sink. This in turn induces an internal shear stress field that activates cross-slip in the adjacent CuNi interlace and thus causing softening. Moreover, it is shown that the yield stress of the CuNiNb system is controlled by the volume fraction of the Nb. Because Nb is the most compliant of the three materials, an increase in volume fraction of Nb decreases the overall yield strength of the material.     

Propagation of weak waves in elastic-plastic and elastic-viscoplastic solids with interfaces

A numerical technique based on the method of singular surfaces has been developed for the computation of wave propagation in solids exhibiting rate-independent elastic-plastic or rate-dependent elastic-viscoplastic behavior. The von Mises yield condition and associated flow rule is taken to represent the rate-independent behavior, while the Perzyna dynamic overstress model is taken to represent the rate-dependent behavior. For 1100-0 Al, a good empirical fit with published experimental data was found to be: $J_2{}^{1/2}?\kappa \left(W^p\right)=\left({\tau }_0/{\gamma }_0\right)\left(\stackrel{0}{W^p}/J_2{}^{1/2}\right)$ where:J₂ is the second invariant of the stress deviator;_k(W^p) is the static hardening curve;W^p is the plastic work and the parameter (τ₀/γ₀) = 0 (rate-independent model) or (80)^?1 to (70)^?1 MPa · s. In the numerical technique, the “connection equations” which provide relations between discontinuities in space and time derivatives lend themselves naturally to finite difference representations. A five-point space-time grid (center point coincident with the instantaneous location of the singular surface) is sufficient for the differenced form of the connection equations and suggests a natural marching scheme for the calculation of all necessary variables at each time step. Supplementing these equations which hold in the interior of the specimen are interface equations which assure continuity in stress and velocity across boundaries which separate materials with dissimilar properties. Application of the technique is made to wave propagation in pure shear for the purpose of comparing numerical predictions with relevant experimental data. The measurements of Duffyet al.[10] which are obtained from the torsional Kolsky apparatus (one dimensional torsional shear wave propagation in a thin-walled tube) were compared with predictions obtained numerically. By using the experimental input pulse history and the constitutive equation reported above, excellent agreement between the predicted and observed histories of reflected and transmitted pulses was obtained when the viscoplastic model was used. Poorer agreement was observed when the rate-independent model (τ₀/γ₀=0) was used. It is concluded that the Perzyna model gives good results for the behavior of 1100-0 Al at high rates of strain.     

Plasticity of damaged solids and shear band localization

Summary The main objective of the paper is the investigation of shear band localization conditions for finite elastic-plastic rate independent deformations of damaged solids. The first part of the paper is devoted to the formulation of the constitutive relations for elastic-plastic solids when isotropic and kinematic hardening effects and the micro-damage process are taken into consideration. The isotropic work-hardening effect is incorporated in the theory directly by defining the work-hardening-softening material function while the kinematic hardening effect and the softening effect generated by the micro-damage process are described by means of the internal state variable method. The second part of the paper aims at the investigation of the localization of plastic deformations. Different effects on the localization phenomenon are investigated. Particular attention is focused on kinematic hardening and micro-damage effects. It has been found that the influence of these two cooperative phenomena on the onset of localization within shear bands has synergetic nature. The results obtained are in good agreement with recent experimental observations.

---

Plastizität von geschädigten Feststoffen und Lokalisierung in Scherzonen

Übersicht Hauptgegenstand der Arbeit ist die Untersuchung der Bedingungen, die bei großen elastischplastischen Formänderungen von geschädigten, formänderungsgeschwindigkeitunabhängigen Feststoffen zur Lokalisierung in Scherzonen führen. Der erste Teil dient der Formulierung des Stoffgesetzes für elastischplastische Werkstoffe mit isotroper und kinematischer Verfestigung sowie Mikro-Schädigung. Die isotrope Verfestigung wird unmittelbar durch eine Verfestigungs-Entfestigungsfunktion berücksichtigt, während die kinematische Verfestigung und die Entfestigung infolge Mikro-Schädigung durch innere Zustandsgrößen beschrieben werden. Der zweite Teil befaßt sich mit der Lokalisierung der plastischen Formänderung, wobei verschiedene Einflüsse untersucht werden. Besondere Aufmerksamkeit wird auf die kinematische Verfestigung und Mikro-Schädigung gerichtet. Es stellt sich heraus, daß beide Erscheinungen bei der Lokalisierung in Scherzonen zusammenwirken. Die Ergebnisse stehen in guter Übereinstimmung mit neueren experimentellen Beobachtungen.

     

On thermoelastic fields of a multi-phase inhomogeneity system with perfectly/imperfectly bonded interfaces

The stress fields of cylindrical and spherical multi-phase inhomogeneity systems with perfect or imperfect interfaces under uniform thermal and far-field mechanical loading conditions are investigated by use of the Boussinesq displacement potentials. The radius of the core inhomogeneity and the thickness of its surrounding coatings are arbitrary. The discontinuities in the tangential and normal components of the displacement at the imperfect interfaces are assumed to be proportional to the associated tractions. In this work, for the problems where the phases of the inhomogeneity system are homogeneous, the exact closed-form thermo-elastic solutions are presented. These solutions along with a systematic numerical methodology are utilized to solve various problems of physical importance, where the constituent phases of the inhomogeneity system may be made of a number of different functionally graded (FG) and homogeneous materials, and each interface may have a perfect or imperfect boundary condition, as desired. Also, the effect of the interfacial sliding and debonding on the stress field and elastic energy of an FG-coated inhomogeneity is examined.     

From coherent to incoherent mismatched interfaces: A generalized continuum formulation of surface stresses

The equilibrium of coherent and incoherent mismatched interfaces is reformulated in the context of continuum mechanics based on the Gibbs dividing surface concept. Two surface stresses are introduced: a coherent surface stress and an incoherent surface stress, as well as a transverse excess strain. The coherent surface stress and the transverse excess strain represent the thermodynamic driving forces of stretching the interface while the incoherent surface stress represents the driving force of stretching one crystal while holding the other fixed and thereby altering the structure of the interface. These three quantities fully characterize the elastic behavior of coherent and incoherent interfaces as a function of the in-plane strain, the transverse stress and the mismatch strain. The isotropic case is developed in detail and particular attention is paid to the case of interfacial thermo-elasticity. This exercise provides an insight on the physical significance of the interfacial elastic constants introduced in the formulation and illustrates the obvious coupling between the interface structure and its associated thermodynamics quantities. Finally, an example based on atomistic simulations of Cu/Cu₂O interfaces is given to demonstrate the relevance of the generalized interfacial formulation and to emphasize the dependence of the interfacial thermodynamic quantities on the incoherency strain with an actual material system.     

Interfacial excess energy, excess stress and excess strain in elastic solids: Planar interfaces

In this paper, interfacial excess energy and interfacial excess stress for coherent interfaces in an elastic solid are reformulated within the framework of continuum mechanics. It is shown that the well-known Shuttleworth relationship between the interfacial excess energy and interfacial excess stress is valid only when the interface is free of transverse stresses. To account for the transverse stress, a new relationship is derived between the interfacial excess energy and interfacial excess stress. Dually, the concept of transverse interfacial excess strain is also introduced, and the complementary Shuttleworth equation is derived that relates the interfacial excess energy to the newly introduced transverse interfacial excess strain. This new formulation of interfacial excess stress and excess strain naturally leads to the definition of an in-plane interfacial stiffness tensor, a transverse interfacial compliance tensor and a coupling tensor that accounts for the Poisson's effect of the interface. These tensors fully describe the elastic behavior of a coherent interface upon deformation.     

Damping associated with porosity and crack in solids

When a material is subjected to alternating stresses, there are temperature fluctuations indicative of damping. Temperature effects give rise to entropy production. An analysis is made to obtain the entropy produced for a vibration cycle. This corresponds to the reciprocity of temperature rise and strain yielded that alter the material damping factor (MDF) as a function of shape and magnitude of material porosity or existing cracks. A homogeneous, isotropic, elastic bar is considered. It consists of uniformly distributed cavities or a single-edge crack subjected to alternating axial stresses. Dynamic characteristics of the porous or cracked medium are determined to evaluate the damping factor of the bar and/or of the material. The experimental data correlate well with the analytical results. The calculated damping factor in this work can be used as an indicator of structural integrity.     

Wave processes in solids with defects

The propagation of plane harmonic waves in viscoelastic and elastoviscoplastic materials are studied using the equations of the field theory of defects, the kinematic identities for an elastic continuum with defects, and the dynamic equations of gauge theory. Wave propagation velocities and refraction and absorption coefficients are determined. The structure of the waves and the correlation between the displacement waves and the defect-field waves determining plastic deformation are analyzed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 190–197, November–December, 2008.     

Elastic solids with different moduli in tension and compression

A new approach is given to the theory of non-linear elastic materials which have different behaviour in tension and compression. Two applications are made to incompressible non-linear materials using general forms for the strain energy functions. The linear form of the theory is shown to be equivalent to that used by previous writers.     

Temperature in heat generating solids with memory

The present paper is concerned with time dependent heat transport by wave propagation in an homogeneous isotropic elastic solid with memory. The energy generated in the material for electrical heating or chemical or nuclear reactions is propagated with a finite speed.The effect of this thermal wave speed is noticeable in many practical applications involving short time and high heat flux situations. The one dimensional, time dependent temperature distribution in a heat generating solid is analytically determined resorting to the Maxwell-Cattaneo-Vernotte equation, following the theory of complex functions of complex variables. Some results are reported and shortly discussed; a comparison with the classical Fourier theory is made.

---

Temperaturverteilung in einem festen Körper mit Wärmeerzeugung und thermischer Nachwirkung

Zusammenfassung Die Arbeit behandelt den zeitabhängigen Wärmetransport durch eine fortschreitende Welle in einem homogenen isotropen elastischen Festkörper mit Gedächtnis. Die im Material durch elektrische Heizung oder durch chemische oder nukleare Reaktionen erzeugte Wärme wird mit endlicher Geschwindigkeit fortgeleitet. Die Wirkung dieser thermischen Wellengeschwindigkeit ist für manche praktische Anwendungen von Bedeutung, soweit kurze Zeiten und hohe Wärmeflüsse eine Rolle spielen. Die eindimensionale zeitabhängige Temperaturverteilung in einem wärmeerzeugenden Körper ist analytisch bestimmt unter Benutzung der Maxwell-Cattaneo-Vernotte-Gleichung nach der Theorie komplexer Funktionen komplexer Variablen. Einige Ergebnisse werden mitgeteilt, kurz diskutiert und mit der klassischen Fourier-Theorie verglichen.

---

Nomenclature a heat flux relaxation function - b _n constant defined in (25) - b _n constant defined in (30) - c specific heat at constant pressure - E elasticity modulus - Fo Fourier number defined in (8) - i imaginary unity - K thermal conductivity - L slab half thickness - N integer number defined in (28) - p integration variable for the inverse Laplace transform - q heat flux vector - Q power produced in the unit volume - Q temperature defined in (8) - R _m residue associated to them-th pole - s time - S sound speed - t time - T temperature defined in (8) - T⁺ dimensionless temperature defined in (33) - u function defined in (11) - inverse Laplace transform of the functionu - U unit Heaviside step function - x dimensionless coordinate defined in (8) - thermal diffusivity - vector differential operator - ² Laplace operator - constant defined in (26) - cartesian coordinate - mass density - gq temperature - dimensionless material thermal relaxation time defined in (8) - _R material thermal relaxation time     

Monodimensional solids with constrained solutions

Summary By applying the extension of the abstract theory of variational inequalities on convex sets (see Lions and Stampacchia[5]) we formulate and resolve numerically some classical non linear problems in the theory of beams, in which the solutions must satisfy certain inequalities which give preassigned conditions of monolateral constraint or plasticity.

---

Sommario In applicazione dell'estensione della teoria astratta delle disuguaglianze variazionali su insiemi convessi (cfr. Lions e Stampacchia[5]) sono formulati e numericamente risolti alcuni classici problemi non lineari della teoria delle travi, in cui le soluzioni debbono soddisfare a certe disuguaglianze, che traducono preassegnate condizioni di vincolo monolaterale o di plasticità.

     

Viscoelastic solids with unbounded relaxation function

Linear viscoelastic solids are considered where the relaxation function is unbounded in that the initial value of the function and the derivative may be infinite but the function, freed from the equilibrium modulus, is integrable. The second law is given a general form for approximate cycles and then for cycles. Thermodynamic restrictions are derived in connection with cycles and shown to be equivalent to those obtained for bounded relaxation functions. Then the thermodynamic restrictions are shown to be also sufficient for the validity of the second law in the general case of approximate cycles. Finally, a functional is considered which proves to be endowed with the characteristic properties of the free energy.     

On thermomechanical solids with boundary structures

This paper, in line with the previous works (Javili and Steinmann, 2009, Javili and Steinmann, 2010), is concerned with the thermomechanically consistent theory and formulation of boundary potential energies and the study of their impact on the deformations of solids. Thereby, the main thrust in this contribution is the extension to thermomechanical effects. Although boundary effects can play a dominant role in the material behavior, the common modelling in continuum mechanics takes exclusively the bulk into account, nevertheless, neglecting possible contributions from the boundary. In this approach the boundary is equipped with its own thermodynamic life, i.e. we assume separate boundary energy, entropy and the like. Afterwards, the derivations of generalized balance equations, including boundary potentials, completely based on a tensorial representation is carried out. The formulation is exemplified for the example of thermohyperelasticity.     

Dislocation Patterns and Work-Hardening in Crystalline Plasticity

We propose a continuum model for the evolution of the total dislocation densities in fcc crystals, in the framework of rate-independent plasticity. The basic physical features which are taken into account are: (i) the role of dislocations in hardening; (ii) the relations between the slip velocity and dislocation mobility; (iii) the energetics of self and mutual interactions between dislocations; (iv) nonlocal effects in the interaction between dislocations. A set of reaction–diffusion equations is obtained, with mobilities depending on the slip velocities, which is able to describe the formation of dislocation walls and cells. To this effect, the results of numerical simulations in two special cases are presented. Mathematics Subject Classifications (2000) 74C15, 74C20.