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Continuum Mechanics and Thermodynamics

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ISSN (Print) 1432-0959 - ISSN (Online) 0935-1175
Published by Springer-Verlag

[2216 journals]

Thermodynamics from an observer’s viewpoint (on the example of the viscous fluid)
- Abstract: Abstract The development of the non-equilibrium thermodynamics without local equilibrium hypothesis is considered. The theory is based on the causal mechanics of the heat conducting continuum, which includes the 1st law of thermodynamics as a theorem. The conditions of applicability of the 2nd law of thermodynamics and the dissipation of the kinetic energy problem are discussed. The reasoning is carried out in the framework of the causal model of the viscous fluid. Main conclusions are illustrated using examples from the numerical analysis.
  PubDate: 2013-05-22
The splitting of intrinsic energy and the origin of mass density in continuum mechanics
- Abstract: Abstract We show that the total intrinsic energy of a body must split into the sum of two terms—an internal energy which depends upon ‘state’ and a kinetic energy which is quadratic in the square of the particle speed.We use the non-relativistic group invariance structure of a generalized form of the balance of energy in continuum thermomechanics, together with a fundamental axiomatic requirement. The fundamental concepts of motion, force, power, heating and intrinsic energy are introduced as primitive, and we derive the notion of mass and its balance. When James Serrin died on August 23, 2012, this work had just been completed. Jim was my close, personal and treasured friend for over 40 years. We collaborated on several works over those years, and we often talked together and socialized on various occasions. I had highest respect for him in all human and professional ways, and there was a definite mutual expression of affection and appreciation. A friendship could not contain more. This paper drew him back to a subject he had worked on years ago, and he was happy to be involved again with a fundamental issue in continuum mechanics. My efforts in this work are dedicated to the memory of James Serrin. He was such a scholar of great breadth and depth—He was wise and witty, and I benefited greatly from his presence. Roger Fosdick
  PubDate: 2013-05-18
The thermodynamics of gradient elastoplasticity
- Abstract: Abstract A thermomechanical framework for the modelling of gradient plasticity is developed within the range of linear strains. Full anisotropy is considered. Special focus is given to the restrictions imposed by the Clausius–Duhem inequality. A rather general example gives a complete anisotropic model and shows its thermodynamic consistency. This is finally particularized for the isotropic case by using isotropic tensor-function representations.
  PubDate: 2013-05-16
Description of liquid–gas phase transition in the frame of continuum mechanics
- Abstract: Abstract A new method of describing the liquid–gas phase transition is presented. It is assumed that the phase transition is characterized by a significant change of the particle density distribution as a result of energy supply at the boiling point that leads to structural changes but not to heating. Structural changes are described by an additional state characteristics of the system—the distribution density of the particles which is presented by an independent balance equation. The mathematical treatment is based on a special form of the internal energy and a source term in the particle balance equation. The presented method allows to model continua which have different specific heat capacities in liquid and in gas state.
  PubDate: 2013-04-23
The principle of the minimum of the dissipation potential for non-isothermal processes
- Abstract: Abstract In this paper, we contribute to the methodology of material modeling by presenting a potential-based approach for non-isothermal inelastic processes. It is based on the principle of the minimum of the dissipation potential which was used previously only in the isothermal context. In contrast to the principle of maximum dissipation, the presented procedure results in mathematically simplified equations. Due to its variational character, the inclusion of constraints is very simple. After derivation of our method, we use the examples of non-isothermal perfect plasticity and shape memory alloys for demonstration of the validity and performance of the concept.
  PubDate: 2013-04-23
Direct determination of multi-axial elastic potentials for incompressible elastomeric solids: an accurate, explicit approach based on rational interpolation
- Abstract: Abstract With a novel approach based on certain logarithmic invariants, we demonstrate that a multi-axial elastic potential for incompressible, isotropic rubber-like materials may be obtained directly from two one-dimensional elastic potentials for uniaxial case and simple shear case, in a sense of exactly matching finite strain data for four benchmark tests, including uniaxial extension, simple shear, bi-axial extension, and plane-strain extension. As such, determination of multi-axial elastic potentials may be reduced to that of two one-dimensional elastic potentials. We further demonstrate that the latter two may be obtained by means of rational interpolating procedures for uniaxial data and shear data displaying strain-stiffening effects. Numerical examples are presented in fitting Treloar’s data and other data.
  PubDate: 2013-03-27
Hydrodynamic simulation of a n ⁺ − n − n ⁺ silicon nanowire
- Abstract: Abstract Non-equilibrium electron transport in silicon nanowires has been tackled with a hydrodynamic model. This model has been formulated by taking the moments of the multisubband Boltzmann equation, coupled to the Schrödinger–Poisson system. Closure relations are obtained by means of the maximum entropy principle (MEP) of extended thermodynamics, including scattering of electrons with acoustic and nonpolar optical phonons. Simulation results for a quantum n + − n − n + silicon diode are shown.
  PubDate: 2013-03-20
Methods for incorporating particle rearrangement into compaction using thermodynamic approaches
- Abstract: Abstract This paper presents a novel, yet thermodynamically consistent, model of the isothermal compaction of loose granular material based on the principle of maximum dissipation rate. The method is first tested out on a simple version of the Bingham model and a hard particle model of rate-independent granular flow where it is seen that only the dissipation function and dilatancy rule are required in either case and the procedures are identical. This hard particle model is subsequently modified by the introduction of damage. Yield surface and flow rules are produced that are broadly in accordance with experimental findings. The key to the above modification is the concept of a dilatancy rule with two contributions. (1) A shear induced negative dilatancy, where any shear deformation has a tendency to produce densification. (2) Under many circumstances, this is countered by positive dilatancy such as at the critical state where the two mechanisms balance. This modification uses the idea that the first contribution is encouraged by microscopic damage local to the particle contacts that might permit compaction to occur under hydrostatic pressure alone. A mechanism is postulated whereby shear stresses operating at the microscopic level, while cancelling out at the macroscopic level, might occur with low levels of damage but produce no overall shear strains.
  PubDate: 2013-03-12
Rate constitutive theories for ordered thermoviscoelastic fluids: polymers
- Abstract: Abstract This paper presents development of rate constitutive theories for compressible as well as in incompressible ordered thermoviscoelastic fluids, i.e., polymeric fluids in Eulerian description. The polymeric fluids in this paper are considered as ordered thermoviscoelastic fluids in which the stress rate of a desired order, i.e., the convected time derivative of a desired order ‘m’ of the chosen deviatoric Cauchy stress tensor, and the heat vector are functions of density, temperature, temperature gradient, convected time derivatives of the chosen strain tensor up to any desired order ‘n’ and the convected time derivative of up to orders ‘m−1’ of the chosen deviatoric Cauchy stress tensor. The development of the constitutive theories is presented in contravariant and covariant bases, as well as using Jaumann rates. The polymeric fluids described by these constitutive theories will be referred to as ordered thermoviscoelastic fluids due to the fact that the constitutive theories are dependent on the orders ‘m’ and ‘n’ of the convected time derivatives of the deviatoric Cauchy stress and conjugate strain tensors. The highest orders of the convected time derivative of the deviatoric Cauchy stress and strain tensors define the orders of the polymeric fluid. The admissibility requirement necessitates that the constitutive theories for the stress tensor and heat vector satisfy conservation laws, hence, in addition to conservation of mass, balance of momenta, and conservation of energy, the second law of thermodynamics, i.e., Clausius–Duhem inequality must also be satisfied by the constitutive theories or be used in their derivations. If we decompose the total Cauchy stress tensor into equilibrium and deviatoric components, then Clausius–Duhem inequality and Helmholtz free-energy density can be used to determine the equilibrium stress in terms of thermodynamic pressure for compressible fluids and in terms of mechanical pressure for incompressible fluids, but the second law of thermodynamics provides no mechanism for deriving the constitutive theories for the deviatoric Cauchy stress tensor. In the development of the constitutive theories in Eulerian description, the covariant and contravariant convected coordinate systems and Jaumann measures are natural choices. Furthermore, the mathematical models for fluids require Eulerian description in which material point displacements are not measurable. This precludes the use of displacement gradients, i.e., strain measures, in the development of the constitutive theories. It is shown that compatible conjugate pairs of convected time derivatives of the deviatoric Cauchy stress and strain measures in co-, contravariant and Jaumann bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of constitutive theories for ordered thermofluids in Eulerian description. This framework has a foundation based on the basic principles and axioms of continuum mechanics, but the resulting constitutive theories for the deviatoric Cauchy stress tensor must satisfy the condition of positive work expanded, a requirement resulting from the entropy inequality. The paper presents a general theory of constitutive equations for ordered thermoviscoelastic fluids which is then specialized to obtain commonly used constitutive equations for Maxwell, Giesekus and Oldroyd-B constitutive models in contra- and covariant bases and using Jaumann rates.
  PubDate: 2013-03-08
Thermodynamics of continuous media with intrinsic rotation and magnetoelectric coupling
- Abstract: Abstract The thermodynamics of an electrically charged, multicomponent continuous medium with intrinsic rotation is analysed in the presence of electromagnetic fields with a weak linear magnetoelectric coupling in the non-relativistic limit. Taking into account the chemical composition of the current densities and stress tensors yields scalar dissipation terms accounting for chemical reactivities and vectorial dissipation terms accounting for transport. Three equations characterising the continuous medium are derived: a thermostatic equilibrium equation, a reversible and an irreversible thermodynamic evolution equation. Explicit expressions for the temperature and the chemical potentials are derived in terms of the electromagnetic fields and the magnetoelectric coupling. The transport equations contain electromagnetic terms normally not included in a standard thermodynamic phenomenology.
  PubDate: 2013-03-06
Inhomogeneous spherical configurations of inflated membranes
- Abstract: Abstract Based on physically meaningful choice of the strain measures, we study the equilibrium and stability of an inflated spherical membrane. First, we obtain general results deduced by global geometric properties and then we analyze the possibility of inhomogeneous configurations. The stability analysis shows that under special constitutive assumptions the global energy minimum can be attained by inhomogeneous spherical configurations that we analytically describe. We argue that these deformations can reproduce well-known experimental results.
  PubDate: 2013-03-01
On a stress-power-based characterization of second-gradient elastic fluids
- Abstract: Abstract An algebraic characterization of fluidity applicable to second-gradient materials is argued, individuating a collection of deformations from a reference placement that entail no pointwise stress-power expenditure. For simplicity, the characterization in question is developed in the context of elastic materials, within which general representations for the stress response, both Cauchy-like and Piola-like, of elastic second-gradient fluids are derived.
  PubDate: 2013-03-01
Shape recovery behaviour of NiTi strips in bending: experiments and modelling
- Abstract: Abstract We present a theoretical and experimental investigation of the bending recovery performances for a commercial NiTi shape memory alloy strip. We evaluate the mechanical properties and the shape setting parameters and estimate the evolution of the curvature during heating in an Ethylene Glycol-based water solution. To model the strip bending response, we use a one-dimensional phenomenological constitutive equation for the shape memory material, based on the introduction of (twinned and detwinned) martensite and austenite volume fractions as internal variables. Under the assumption of uniform bending, we calculate a quasi-closed-form solution for the stress and martensite fraction distributions in a shape memory beam during bending and subsequent shape recovery. Using our characterisation data as input parameters of the model, we find that the theoretical curvature evolution is in good agreement with experimental data.
  PubDate: 2013-03-01
Moving interfaces that separate loose and compact phases of elastic aggregates: a mechanism for drastic reduction or increase in macroscopic deformation
- Abstract: Abstract Granular materials such as sand may be viewed as continuous bodies composed of much smaller elastic bodies. The multiscale geometry of structured deformations captures the contribution at the macrolevel of the smooth deformation of each small body in the aggregate (deformation without disarrangements) as well as the contribution at the macrolevel of the non-smooth deformations such as slips and separations between the small bodies in the aggregate (deformation due to disarrangements). When the free energy response of the aggregate depends only upon the deformation without disarrangements, is isotropic, and possesses standard growth and semi-convexity properties, we establish (i) the existence of a compact phase in which every small elastic body deforms in the same way as the aggregate and, when the volume change of macroscopic deformation is sufficiently large, (ii) the existence of a loose phase in which every small elastic body expands and rotates to achieve a stress-free state with accompanying disarrangements in the aggregate. We show that a broad class of elastic aggregates can admit moving surfaces that transform material in the compact phase into the loose phase and vice versa and that such transformations entail drastic changes in the level of deformation of transforming material points.
  PubDate: 2013-03-01
A direct approach to fiber and membrane reinforced bodies. Part I. Stress concentrated on curves for modelling fiber reinforced materials
- Abstract: Abstract An approach is outlined to the equilibrium in fiber-reinforced materials in which the fibers are modeled as curves or lines with concentrated material properties. The system of forces representing the interaction of the fibers with the bulk matter is analyzed, and equilibrium of forces is derived from global laws. The displacements of the bulk matter are assumed to have continuous extension to the fibers. This forces the set of admissible deformations superquadratically integrable. This in turn forces the energy of the bulk of superquadratic growth. The material of the bulk matrix therefore cannot be linearly elastic. The energy of fibers can have a slower growth and can be quadratic. A formal set of assumptions is given under which an equilibrium state of minimum energy exists in the given external conditions. A weak form of equilibrium equations is derived for this equilibrium state. An explicitly calculable axisymmetric example is presented with an isotropic and quadratic energy of the matrix (linear elasticity) and linearly stretchable fiber. Since the superquadratic growth assumption is not satisfied, some peculiar features of the solution arise, such as the infinite limit of the radial displacement near the fiber. Nevertheless, from the obtained solution, we can compute the normal force in the fiber and the shear stress at the interface.
  PubDate: 2013-03-01
Gianpietro Del Piero: a continuator of the Italian tradition in continuum mechanics, as started by Gabrio Piola
- PubDate: 2013-03-01
Singular stress fields for masonry-like vaults
- Abstract: Abstract In the present paper, we apply the theorems of limit analysis to vaults modeled as masonry-like materials, that is, unilateral continuous bodies. On allowing for singular stresses, we consider statically admissible stress field concentrated on surfaces lying inside the masonry. Such structures are unilateral membranes, whose geometry is described a la Monge, and the equilibrium of them, under vertical loads, is formulated in the Pucher form. The problem is reduced to a singular partial differential equation of the second order where the shape f and the stress function F appear symmetrically. The unilateral restrictions require that the membrane surface lies in between the extrados and intrados surfaces of the vault and that the stress function be concave. Such a constraint is, in general, not satisfied on a given shape for given loads: in such a case, the shape has to be modified to fit the constraint. In a sense, the unilateral assumption renders the membrane an underdetermined structure that must adapt its shape in order to satisfy the unilateral restrictions. A number of simple examples are presented to illustrate how the method works.
  PubDate: 2013-03-01
Gianpietro Del Piero
- PubDate: 2013-03-01
A tribute to my “Maestro” Gianpietro Del Piero
- PubDate: 2013-03-01
Gianpietro Del Piero: a scientist on the edge between engineering sciences and functional analysis
- PubDate: 2013-03-01