Showing papers in "Continuum Mechanics and Thermodynamics in 2008"

Asymptotic expansions by Γ-convergence

Abstract: Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.

Crystallographically based model for transformation-induced plasticity in multiphase carbon steels

Abstract: The microstructure of multiphase steels assisted by transformation-induced plasticity consists of grains of retained austenite embedded in a ferrite-based matrix. Upon mechanical loading, retained austenite may transform into martensite, as a result of which plastic deformations are induced in the surrounding phases, i.e., the ferrite-based matrix and the untransformed austenite. In the present work, a crystallographically based model is developed to describe the elastoplastic transformation process in the austenitic region. The model is formulated within a large-deformation framework where the transformation kinematics is connected to the crystallographic theory of martensitic transformations. The effective elastic stiffness accounts for anisotropy arising from crystallographic orientations as well as for dilation effects due to the transformation. The transformation model is coupled to a single-crystal plasticity model for a face-centered cubic lattice to quantify the plastic deformations in the untransformed austenite. The driving forces for transformation and plasticity are derived from thermodynamical principles and include lower-length-scale contributions from surface and defect energies associated to, respectively, habit planes and dislocations. In order to demonstrate the essential features of the model, simulations are carried out for austenitic single crystals subjected to basic loading modes. To describe the elastoplastic response of the ferritic matrix in a multiphase steel, a crystal plasticity model for a body-centered cubic lattice is adopted. This model includes the effect of nonglide stresses in order to reproduce the asymmetry of slips in the twinning and antitwinning directions that characterizes the behavior of this type of lattices. The models for austenite and ferrite are combined to simulate the microstructural behavior of a multiphase steel. The results of the simulations show the relevance of including plastic deformations in the austenite in order to predict a more realistic evolution of the transformation process.

A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy

Abstract: We present a micromechanical model for polycrystalline shape-memory alloys which is capable of reproducing important aspects of the material behavior such as pseudoelasticity, pseudoplasticity, tension–compression asymmetry and the influence of texture inhomogeneities which may occur from the production process of components or specimens. Our model is based on the optimization of the material’s free energy density and uses a dissipation ansatz which is homogeneous of first order. Considering the full anisotropic material properties of both the austenite and the martensite phase, we compute the evolution of the orientation distributions of austenite and martensite as internal variables of our model.

Linear Γ-limits of multiwell energies in nonlinear elasticity theory

Abstract: We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to Γ-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.

Derivation of a new free energy for biological membranes

Abstract: A new free energy for thin biomembranes depending on chemical composition, degree of order and membranal-bending deformations is derived in this paper. This is a result of constitutive and geometric assumptions at the three-dimensional level. The enforcement of a new symmetry group introduced in (Deseri et al., in preparation) and a 3D--2D dimension reduction procedure are among the ingredients of our methodology. Finally, the identification of the lower order term of the energy (i.e. the membranal contribution) on the basis of a bottom-up approach is performed; this relies upon standard statistical mechanics calculations. The main result is an expression of the biomembrane free energy density, whose local and non-local counterparts are weighted by different powers of the bilayer thickness. The resulting energy exhibits three striking aspects: (i) the local (purely membranal) energy counterpart turns out to be completely determined through the bottom-up approach mentioned above, which is based on experimentally available information on the nature of the constituents; (ii) the non-local energy terms, that spontaneously arise from the 3D--2D dimension reduction procedure, account for both bending and non-local membranal effects; (iii) the non-local energy contributions turn out to be uniquely determined by the knowledge of the membranal energy term, which in essence represents the only needed constitutive information of the model. It is worth noting that the coupling among the fields appearing as independent variables of the model is not heuristically forced, but it is rather consistently delivered through the adopted procedure.

A new approach for the ellipsoidal statistical model

Abstract: In this paper we aim to introduce a systematic way to derive relax-ation terms for the Boltzmann equation based under minimization problem of the entropy under moments constraints [7], [11]. In partic-ular the moment constraints and corresponding coefficients are linked with the eigenfunctions and eigenvalues of the linearized collision operator through the Chapman-Enskog expansion. Then we deduce from this expansion a single relaxation term of BGK-type. Here we stop the moments constraints at the order 2 in the velocity and recover the Ellipsoidal Statistical model [8]

Landau theory for biaxial nematic liquid crystals with two order parameter tensors

Abstract: We present a phenomenological theory for the homogeneous phases of nematic liquid crystals constituted by biaxial molecules. We propose a general polynomial potential in two macroscopic order parameter tensors that reproduces the mean-field phase diagram confirmed by Monte Carlo simulations [De Matteis et al. in Phys Rev E 72:041706 (2005)] and recently recognized to be universal [Bisi et al. in Phys Rev E 73:051709 (2006)] for dispersion force molecular pair-potentials enjoying the D 2h symmetry. The requirement that the phenomenological theory comply uniquely with this phase diagram reduces considerably the admissible phenomenological coefficients, both in their number and in the ranges where they can vary.

Mixed analytical–numerical relaxation in finite single-slip crystal plasticity

Abstract: The modeling of the finite elastoplastic behaviour of single crystals with one active slip system leads to a nonconvex variational problem, whose minimization produces fine structures. The computation of the quasiconvex envelope of the energy density involves the solution of a nonconvex optimization problem and faces severe numerical difficulties from the presence of many local minima. In this paper, we consider a standard model problem in two dimensions and, by exploiting analytical relaxation results for limiting cases and the special structure of the problem at hand, we obtain a fast and efficient numerical relaxation algorithm. The effectiveness of our algorithm is demonstrated with numerical examples. The precision of the results is assessed by lower bounds from polyconvexity.

Dynamic fracture: an example of convergence towards a discontinuous quasistatic solution

Abstract: Considering a one-dimensional problem of debonding of a thin film in the context of Griffith’s theory, we show that the dynamical solution converges, when the speed of loading goes down to 0, to a quasistatic solution including an unstable phase of propagation. In particular, the jump of the debonding induced by this instability is governed by a principle of conservation of the total quasistatic energy, the kinetic energy being negligible.

Towards multi-scale continuum elasticity theory

Abstract: We propose a new method of constructing a series of nested quasicontinuum models, which describe linear elastic behavior of crystal lattices at successively smaller scales. The relevant scales are dictated by the interatomic interactions and are not arbitrary. The novelty of the model is in the use of a decomposition of the displacement field into the coarse part and the micro-level corrections. The coarse contribution is the conventional homogenized displacement field used in classical continuum elasticity. The micro-level corrections are sub-continuum fields representing the fine structure of the boundary layers exhibited by the discrete equilibrium configuration. The model is based on a multi-point Pade approximation in the Fourier space of the discrete Green’s function. We systematically compare the new model with the conventional strain gradient model.

Cauchy’s stress theorem for stresses represented by measures

Abstract: A version of Cauchy’s stress theorem is given in which the stress describing the system of forces in a continuous body is represented by a tensor valued measure with weak divergence a vector valued measure. The system of forces is formalized in the notion of an unbounded Cauchy flux generalizing the bounded Cauchy flux by Gurtin and Martins (Arch Ration Mech Anal 60:305–324, 1976). The main result of the paper says that unbounded Cauchy fluxes are in one-to-one correspondence with tensor valued measures with weak divergence a vector valued measure. Unavoidably, the force transmitted by a surface generally cannot be defined for all surfaces but only for almost every translation of the surface. Also conditions are given guaranteeing that the transmitted force is represented by a measure. These results are proved by using a new homotopy formula for tensor valued measure with weak divergence a vector valued measure.

Dynamics of martensitic phase boundaries: discreteness, dissipation and inertia

Abstract: We study a fully inertial model of a martensitic phase transition in a one-dimensional crystal lattice with long-range interactions. The model allows one to represent a broad range of dynamic regimes, from underdamped to overdamped. We systematically compare the discrete model with its various continuum counterparts including elastic, viscoelastic and viscosity-capillarity models. Each of these models generates a particular kinetic relation which links the driving force with the phase boundary velocity. We find that the viscoelastic model provides an upper bound for the critical driving force predicted by the discrete model, while the viscosity-capillarity model delivers a lower bound. We show that at near-sonic velocities, where inertia dominates dispersion, both discrete and continuum models behave qualitatively similarly. At small velocities, and in particular near the depinning threshold, the discreteness prevails and predictions of the continuum models cannot be trusted.

A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material. I. On thermodynamically consistent evolution

Abstract: In the present study an evolution equation for the Cauchy stress tensor is proposed for an isotropic elasto-visco-plastic continuum. The proposed stress model takes effects of elasticity, viscosity and plasticity of the material simultaneously into account. It is ascribed with some scalar coefficient functions and, in particular, with an unspecified tensor-valued function N, which is handled as an independent constitutive quantity. It is demonstrated that by varying the values and the specific functional forms of these coefficients and N, different known models in non-Newtonian rheology can be reproduced. A thermodynamic analysis, based on the MA¼ller-Liu entropy principle, is performed. The results show that these coefficients and N are not allowed to vary arbitrarily, but should satisfy certain restrictions. Simple postulates are made to further simplify the deduced general results of the thermodynamic analysis. They yield justification and thermodynamic consistency of the existing models for a class of materials embracing thermoelasticity, hypoelasticity and in particular hypoplasticity, of which the thermodynamic foundation is established successively for the first time in literature. The study points at the wide applicability and practical usefulness of the present model in different fields from non-Newtonian fluid to solid mechanics. In this paper the thermodynamic analysis of the proposed evolution-type stress model is discussed, its applications are reported later.

Director reorientation and order reconstruction: competing mechanisms in a nematic cell

Abstract: We propose a model to explore the competition between two mechanisms possibly at work in a nematic liquid crystal confined within a flat cell with strong uniaxial planar conditions on the bounding plates and subject to an external field. To obtain an electric field perpendicular to the plates, a voltage is imposed across the cell; no further assumption is made on the electric potential within the cell, which is therefore calculated together with the nematic texture. The Landau-de Gennes theory of liquid crystals is used to derive the equilibrium nematic order tensor Q. When the voltage applied is low enough, the equilibrium texture is nearly homogeneous. Above a critical voltage, there exist two different possibilities for adjusting the order tensor to the applied field within the cell: plain director reorientation, i.e., the classical Freedericksz transition, and order reconstruction. The former mechanism entails the rotation of the eigenvectors of Q and can be described essentially by the orientation of the ordinary uniaxial nematic director, whilst the latter mechanism implies a significant variation of the eigenvalues of Q within the cell, virtually without any rotation of its eigenvectors, but with the intervention of a wealth of biaxial states. Either mechanism can actually occur, which yields different nematic textures, depending on material parameters, temperature, cell thickness and the applied potential. The equilibrium phase diagram illustrating the prevailing mechanism is constructed for a significant set of parameters.

Perfect plasticity and hyperelastic models for isotropic materials

Abstract: We study the introduction of multi-dimensional plasticity in hyperelastic models. We construct a potential (the internal energy) which models the isotropic elastic and perfectly plastic behavior of materials. We apply this method to the analysis of the flyer-plate experiment. The mathematical analysis exhibits a strong similarity with phase transition issues. Numerical experiments confirm the theoretical analysis.

A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material

Abstract: This paper continues Part I, in which a unified evolution equation for the Cauchy stress tensor, which takes elastic, viscous, and plastic features of the material simultaneously into account, was proposed. Hypoplasticity in particular was incorporated to account for the plastic characteristics. In the present paper, the stress model is applied to study normal stress differences in the context of viscometric flow, and the unsteady flow characteristics of an elasto-visco-plastic fluid between two infinite parallel plates driven by a sudden motion of the plate, to estimate the performance and limitations of the proposed method. Numerical calculations show that, in the context of viscometric flow, different degrees of plasticity and the associated first and second normal stress differences can be addressed appropriately by the stress model. For the unsteady flow situation the results show that the complex behavior of the fluid, in particular after the start of the driving motion, can be described to some extent by the model. In addition, different relaxation and retardation spectra with plastic characteristics can be simulated by varying the model parameters. These findings suggest the applicability of the proposed stress model, for example, in the fields of granular/debris and polymeric flows.

Volume-weighted mixture theory for granular materials

Abstract: In the present work we treat granular materials as mixtures composed of a solid and a surrounding void continuum, proposing then a continuum thermodynamic theory for it. In contrast to the common mass-weighted balance equations of mass, momentum, energy and entropy for mixtures, the volume-weighted balance equations and the associated jump conditions of the corresponding physical quantities are derived in terms of volume-weighted field quantities here. The evolution equations of volume fractions, volume-weighted velocity, energy, and entropy are presented and explained in detail. By virtue of the second law of thermodynamics, three dissipative mechanisms are considered which are specialized for a simple set of linear constitutive equations. The derived theory is applied to the analysis of reversible and irreversible compaction of cohesionless granular particles when a vertical oscillation is exerted on the system. In this analysis, a hypothesis for the existence of a characteristic depth within the granular material in its closely compacted state is proposed to model the reversible compaction.

Hamiltonian and Lagrangian theory of viscoelasticity

Abstract: The viscoelastic relaxation modulus is a positive-definite function of time. This property alone allows the definition of a conserved energy which is a positive-definite quadratic functional of the stress and strain fields. Using the conserved energy concept a Hamiltonian and a Lagrangian functional are constructed for dynamic viscoelasticity. The Hamiltonian represents an elastic medium interacting with a continuum of oscillators. By allowing for multiphase displacement and introducing memory effects in the kinetic terms of the equations of motion a Hamiltonian is constructed for the visco-poroelasticity.

A physico-mechanical approach to modeling of metal forming processes—part I: theoretical framework

Abstract: A combined physico-mechanical approach to research and modeling of forming processes for metals with predictable properties is developed. The constitutive equations describing large plastic deformations under complex loading are based on both plastic flow theory and continuum damage mechanics. The model which is developed in order to study strongly plastically deformed materials represents their mechanical behavior by taking micro-structural damage induced by strain micro-defects into account. The symmetric second-rank order tensor of damage is applied for the estimation of the material damage connected with volume, shape, and orientation of micro-defects. The definition offered for this tensor is physically motivated since its hydrostatic and deviatoric parts describe the evolution of damage connected with a change in volume and shape of micro-defects, respectively. Such a representation of damage kinetics allows us to use two integral measures for the calculation of damage in deformed materials. The first measure determines plastic dilatation related to an increase in void volume. A critical amount of plastic dilatation enables a quantitative assessment of the risk of fracture of the deformed metal. By means of an experimental analysis we can determine the function of plastic dilatation which depends on the strain accumulated by material particles under various stress and temperature-rate conditions of forming. The second measure accounts for the deviatoric strain of voids which is connected with a change in their shape. The critical deformation of ellipsoidal voids corresponds to their intense coalescence and to formation of large cavernous defects. These two damage measures are important for the prediction of the meso-structure quality of metalware produced by metal forming techniques. Experimental results of various previous investigations are used during modeling of the damage process.

A new approach for the limit to tree height using a liquid nanolayer model

Abstract: Liquids in contact with solids are submitted to intermolecular forces inferring density gradients at the walls. The van der Waals forces make liquid heterogeneous, the stress tensor is not any more spherical as in homogeneous bulks and it is possible to obtain stable thin liquid films wetting vertical walls up to altitudes that incompressible fluid models are not forecasting. Application to micro tubes of xylem enables to understand why the ascent of sap is possible for very high trees like sequoias or giant eucalyptus.

Fluid penetration effects in porous media contact

Abstract: The contact boundary conditions at the interface between two fluid-saturated porous bodies are derived. The general derivation is performed within the well-founded framework of the Theory of Porous Media (TPM) based on the constituent balance relations of mass, momentum, and energy accounting for finite discontinuities at the contact surface. Particular attention is drawn to the effects associated with the interstitial fluid flux across the interface. The derived contact conditions include two kinematic continuity conditions for the solid velocity and the fluid seepage velocity as well as two jump conditions for the effective solid stress and the pore-fluid pressure. As an application, the common case of biphasic porous media contact proceeding from materially incompressible constituents and inviscid fluid properties is discussed in detail.

Microcontinuum derivation of Goodman–Cowin theory for granular materials

Abstract: This study presents a derivation of the Goodman–Cowin (GC) equation using the microcontinuum field theory. Through the decomposition of various microcontinuum field quantities into the straining, dilatant, and rotational parts, a microcontinuum can be classified into seven subclasses. One of the subclasses, called a microdilatation continuum, is introduced when only the dilatant motion in a macroelement is taken into account. The balance equation of equilibrated force in the GC theory can be derived while introducing the equilibrated intrinsic body force in the energy balance equation of the microdilatation continuum. The internal length of granular materials, appearing in the modified GC equation, is interpreted as the gyration radius of a macroelement. This study also obtains the evolution equation of the internal length from the microcontinuum point of view.

What does an ideal wall look like

Abstract: This paper deals with the interface between a solid and an ideal gas. The surface of the solid is considered to be an ideal wall, if the flux of entropy is continuous, i.e., if the interaction between wall and gas is non-dissipative. The concept of an ideal wall is discussed within the framework of kinetic theory. In particular it is shown that a non-dissipative wall must be adiabatic and does not exerts shear stresses to the gas, if the interaction of a gas atom with the wall is not influenced by the presence of other gas atoms. It follows that temperature jumps and slip will be observed at virtually all walls, although they will be negligibly small in the hydrodynamic regime (i.e., for small Knudsen numbers).

Spatially resolved piezo-spectroscopic characterizations for the validation of theoretical models of notch-root stress fields in ceramic materials

Abstract: The main aim of this work is a precise experimental assessment of the local stress fields developed at the notch-root in a ruby crystal, selected as a paradigm brittle material, by means of photo- and electron-stimulated luminescence techniques. Our approach takes advantage of the piezo-spectroscopic (PS) effect, which consists of a spectral shift of the luminescence emitted by the material due to lattice strain. Highly spatially resolved stress maps were extensively collected at the notch-root and spectral shifts monitored for the chromophoric (R-lines) fluorescence observed in a single-crystalline ruby sample. Experimental data were analyzed and compared to the theoretical solutions of notch-root stress fields given by Filippi and by Creager-Paris. Due to its inherent simplifications, the Creager–Paris solution was found leading to underestimation of the maximum stress value piled up in the material, while the Filippi’s solution represented a more suitable approximation for the stress field developed at the notch-root.

On variational features of vortex flows

Abstract: Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones, the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997; Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property. This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion is discussed.

Maximum-entropy principle for nonlinear hydrodynamic transport in semiconductors

Abstract: We present, in a strong nonlinear context, a full-band hydrodynamic approach by using the first 13 moments of the distribution function in the framework of extended thermodynamics. Following this approach we show that: (1) the full-band effects of the band structure are described accurately up to high electric fields both in homogeneous and nonhomogeneous conditions; (2) the effectiveness of the dissipation processes can be properly investigated, in homogeneous conditions, only in a strong nonlinear context; and (3) the hyperbolicity region of the system is very large, also in the nonlinear conditions. In this way, by using a strong nonlinear closure, it is possible to describe accurately the transport phenomena in submicron devices, when very high electric fields and field gradients occur (E ≈ 220 kV/cm, E/(dE/dx) ≈ 100 A).

On the definitions of failure and critical states in hypoplasticity

Abstract: A granular body is said to be at failure or in a critical state if the stress state does not change while the body is continuously deformed. Within the framework of hypoplasticity, such states, generally called stationary states,are conventionally defined by the condition that an objective (the Jaumann) stress rate vanishes. However, not all stationary states attained under monotonic deformation lie within the scope of this definition. Simple shear is an example. In fact, stationary states are characterized by zero material time derivative of the stress tensor rather than zero Jaumann rate. In the present paper, we give a generalized definition of stationarity by the condition of zero material time derivative of the stress tensor. The new definition extends the set of possible stationary states and includes those which are not covered by the previous definition. Stationary states are analysed quantitatively using calibrated hypoplastic equations. It is shown numerically that, if the norm of the spin tensor is of the same order as, or smaller than, the norm of the stretching tensor, the old definition approximates all possible sationary states with sufficient accuracy.