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Massimiliano Zingales - One of the best experts on this subject based on the ideXlab platform.

  • the mechanically based approach to 3d non local Linear Elasticity Theory long range central interactions
    International Journal of Solids and Structures, 2010
    Co-Authors: M Di Paola, Giuseppe Failla, Massimiliano Zingales
    Abstract:

    Abstract This paper presents the generalization to a three-dimensional (3D) case of a mechanically-based approach to non-local Elasticity Theory, recently proposed by the authors in a one-dimensional (1D) case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range forces exerted by non-adjacent elements. Specifically, the long-range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements, measured along the line connecting the centroids. Further, the long-range forces are assumed to be proportional to a proper, material-dependent, distance-decaying function and to the products of the interacting volumes. Consistently with the modelling of the long-range forces as central body forces, the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces. The proposed 3D formulation is developed both in a mechanical and in a variational context. For this the elastic energy functionals of the solid with long-range interactions are introduced, based on the principle of virtual work to set the proper correspondence between the mechanical and the kinematic variables of the model. Numerical applications are reported for 2D solids under plane stress conditions.

  • Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory
    Journal of Elasticity, 2009
    Co-Authors: M Di Paola, Giuseppe Failla, Massimiliano Zingales
    Abstract:

    In this paper the physically-based approach to non-local Elasticity Theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.

  • physically based approach to the mechanics of strong non local Linear Elasticity Theory
    Journal of Elasticity, 2009
    Co-Authors: M Di Paola, Giuseppe Failla, Massimiliano Zingales
    Abstract:

    In this paper the physically-based approach to non-local Elasticity Theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.

Roger Fosdick - One of the best experts on this subject based on the ideXlab platform.

  • A Stokes theorem for second-order tensor fields and its implications in continuum mechanics
    International Journal of Non-linear Mechanics, 2004
    Co-Authors: Roger Fosdick, Gianni Royer-carfagni
    Abstract:

    Abstract We give a constructive proof of a particular Stokes theorem (1.4) for tensor fields in R 3 ⊗ R 3 . Its specialization to symmetric tensor fields, given in (1.5), bears a close relation to compatibility in Linear Elasticity Theory and to the generalized Beltrami representation of symmetric tensor fields in continuum mechanics. These issues are discussed.

  • About Clapeyron's Theorem in Linear Elasticity
    Journal of Elasticity, 2003
    Co-Authors: Roger Fosdick, Lev Truskinovsky
    Abstract:

    We examine some elementary interpretations of the classical theorem of Clapeyron in Linear Elasticity Theory. As we show, a straightforward application of this theorem in the purely mechanical setting leads to an apparent paradox which can be resolved by referring either to dynamics or to thermodynamics. These richer theories play an essential part in understanding the physical significance of this theorem.

  • About Clapeyron’s Theorem in Linear Elasticity
    Journal of Elasticity, 2003
    Co-Authors: Roger Fosdick, Lev Truskinovsky
    Abstract:

    We examine some elementary interpretations of the classical theorem of Clapeyron in Linear Elasticity Theory. As we show, a straightforward application of this theorem in the purely mechanical setting leads to an apparent paradox which can be resolved by referring either to dynamics or to thermodynamics. These richer theories play an essential part in understanding the physical significance of this theorem.

  • The constraint of local injectivity in Linear Elasticity Theory
    Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2001
    Co-Authors: Roger Fosdick, Gianni Royer-carfagni
    Abstract:

    There are problems in classical Linear Elasticity Theory whose known solutions must be rejected because they predict unacceptable deformation behaviour, such as the interpenetration of material regions. What has been missing is a proper account of the constraint that allowable deformations must be injective. This type of constraint is highly nonLinear and nonconvex, even within the classical Linear Theory, and it is expected to give rise to the existence of an appropriate constraint reaction field. We propose to determine the displacement field u() : B Rn (n 2, 3) of an elastic body B Rn such that the potential energy is minimized subject to the constraint that the deformation y f(x) x u(x), x B, is locally invertible, i.e. det(1 u) > 0 in B. In Linear Elasticity Theory, the strain energy (assumed positive definite) is a quadratic function of u and, in the context of plane problems where the dimension n 2, the constraint is properly closed, which allows us to prove, at least in this case, an existence the...

Alain Claverie - One of the best experts on this subject based on the ideXlab platform.

  • Direct Mapping of Strain in a Strained Silicon Transistor by High-Resolution Electron Microscopy
    Physical Review Letters, 2008
    Co-Authors: Florian Hüe, Martin Hÿtch, Hugo Bender, Florent Houdellier, Alain Claverie
    Abstract:

    Aberration-corrected high-resolution transmission electron microscopy (HRTEM) is used to measure strain in a strained-silicon metal-oxide-semiconductor field-effect transistor. Strain components parallel and perpendicular to the gate are determined directly from the HRTEM image by geometric phase analysis. Si 80 Ge 20 source and drain stressors lead to uniaxial compressive strain in the Si channel, reaching a maximum value of ÿ1:3% just below the gate oxide, equivalent to 2.2 GPa. Strain maps obtained by Linear Elasticity Theory, modeled with the finite-element method, agree with the experimental results to within 0.1%.

Sergey A Akimov - One of the best experts on this subject based on the ideXlab platform.

  • monolayerwise application of Linear Elasticity Theory well describes strongly deformed lipid membranes and the effect of solvent
    Soft Matter, 2020
    Co-Authors: Timur R Galimzyanov, Pavel V Bashkirov, Paul S Blank, Joshua Zimmerberg, Oleg V Batishchev, Sergey A Akimov
    Abstract:

    The Theory of Elasticity of lipid membranes is used widely to describe processes of cell membrane remodeling. Classically, the functional of a membrane's elastic energy is derived under assumption of small deformations; the membrane is considered as an infinitely thin film. This functional is quadratic on membrane surface curvature, with half of the splay modulus as its proportionality coefficient; it is generally applicable for small deformations only. Any validity of this functional for the regime of strong deformations should be verified experimentally. Recently, research using molecular dynamics simulations challenged the validity of this classic, Linear model, i.e. the constancy of the splay modulus for strongly bent membranes. Here we demonstrate that the quadratic energy functional still can be applied for calculation of the elastic energy of strongly deformed membranes without introducing higher order terms with additional elastic moduli, but only if applied separately for each lipid monolayer. For cylindrical membranes, both classic and monolayerwise models yield equally accurate results. For cylindrical deformations we experimentally show that the elastic energy of lipid monolayers is additive: a low molecular weight solvent leads to an approximately twofold decrease in the membrane bending stiffness. Accumulation of solvent molecules in the inner monolayer of a membrane cylinder can explain these results, as the solvent partially prevents lipid molecules from splaying there. Thus, the Linear Theory of Elasticity can be expanded through the range from weak to strong deformations—its simplicity and physical transparency describe various membrane phenomena.

M Di Paola - One of the best experts on this subject based on the ideXlab platform.

  • the mechanically based approach to 3d non local Linear Elasticity Theory long range central interactions
    International Journal of Solids and Structures, 2010
    Co-Authors: M Di Paola, Giuseppe Failla, Massimiliano Zingales
    Abstract:

    Abstract This paper presents the generalization to a three-dimensional (3D) case of a mechanically-based approach to non-local Elasticity Theory, recently proposed by the authors in a one-dimensional (1D) case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range forces exerted by non-adjacent elements. Specifically, the long-range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements, measured along the line connecting the centroids. Further, the long-range forces are assumed to be proportional to a proper, material-dependent, distance-decaying function and to the products of the interacting volumes. Consistently with the modelling of the long-range forces as central body forces, the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces. The proposed 3D formulation is developed both in a mechanical and in a variational context. For this the elastic energy functionals of the solid with long-range interactions are introduced, based on the principle of virtual work to set the proper correspondence between the mechanical and the kinematic variables of the model. Numerical applications are reported for 2D solids under plane stress conditions.

  • Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory
    Journal of Elasticity, 2009
    Co-Authors: M Di Paola, Giuseppe Failla, Massimiliano Zingales
    Abstract:

    In this paper the physically-based approach to non-local Elasticity Theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.

  • physically based approach to the mechanics of strong non local Linear Elasticity Theory
    Journal of Elasticity, 2009
    Co-Authors: M Di Paola, Giuseppe Failla, Massimiliano Zingales
    Abstract:

    In this paper the physically-based approach to non-local Elasticity Theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.