On Beltrami-Michell-Like Equations for Nonlinear Elastic Dielectrics

1986 ◽  
Vol 10 (3) ◽  
pp. 167-173
Author(s):  
S. Dost ◽  
P.G. Glockner
Keyword(s):  
Stress Field ◽  
Deformation Gradient ◽  
Polarization Vector ◽  
Balance Equations ◽  
Independent Field ◽  
Polarization Gradient ◽  
Nonlinear Elastic

Beltrami-Michell-like equations for nonlinear elastic dielectrics are obtained by choosing the deformation gradient, the polarization gradient and the polarization vector as independent field variables, so as to yield linear compatibility equations. The corresponding stress field also yields linear balance equations. Two simple examples for the case of semilinear isotropic elastic dielectrics are solved to illustrate the theory.

scholarly journals Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers

2019 ◽  
Vol 25 ◽  
pp. 19 ◽  
Author(s):  
Carlos Mora-Corral ◽  
Marcos Oliva
Keyword(s):  
Sobolev Space ◽  
Elastic Energy ◽  
Lower Semicontinuous ◽  
Deformation Gradient ◽  
Variational Model ◽  
Nematic Elastomer ◽  
Nematic Elastomers ◽  
Lower Semicontinuous Envelope ◽  
Nonlinear Elastic

We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an Oseen–Frank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuous envelope, which turns out to be the quasiconvexification of the mechanical term plus the tangential quasiconvexification of the nematic term. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation is in the Sobolev space W1,p (with p > n − 1 and n the dimension of the space) and does not present cavitation.

Effect of membrane bending stiffness on the deformation of capsules in simple shear flow

2001 ◽  
Vol 440 ◽  
pp. 269-291 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Curvature Tensor ◽  
Deformation Gradient ◽  
Bending Stiffness ◽  
Boundary Element Methods ◽  
Simple Shear Flow ◽  
Normal Vector ◽  
Membrane Thickness ◽  
Bending Moments ◽  
Balance Equations ◽  
Capsule Deformation

The effect of interfacial bending stiffness on the deformation of liquid capsules enclosed by elastic membranes is discussed and investigated by numerical simulation. Flow-induced deformation causes the development of in-plane elastic tensions and bending moments accompanied by transverse shear tensions due to the non-infinitesimal membrane thickness or to a preferred configuration of an interfacial molecular network. To facilitate the implementation of the interfacial force and torque balance equations involving the hydrodynamic traction exerted on either side of the interface and the interfacial tensions and bending moments developing in the plane of the interface, a formulation in global Cartesian coordinates is developed. The balance equations involve the Cartesian curvature tensor defined in terms of the gradient of the normal vector extended off the plane of the interface in an appropriate fashion. The elastic tensions are related to the surface deformation gradient by constitutive equations derived by previous authors, and the bending moments for membranes whose unstressed shape has uniform curvature, including the sphere and a planar sheet, arise from a constitutive equation that involves the instantaneous Cartesian curvature tensor and the curvature of the resting configuration. A numerical procedure is developed for computing the capsule deformation in Stokes flow based on standard boundary-element methods. Results for spherical and biconcave resting shapes resembling red blood cells illustrate the effect of the bending modulus on the transient and asymptotic capsule deformation and on the membrane tank-treading motion.

scholarly journals Homogenized Balance Equations for Nonlinear Poroelastic Composites

2021 ◽  
Vol 11 (14) ◽  
pp. 6611
Author(s):  
Laura Miller ◽  
Raimondo Penta
Keyword(s):  
Deformation Gradient ◽  
Strain Energy Function ◽  
Biological Tissues ◽  
Type Model ◽  
Nonlinear Behaviour ◽  
Pore Scale ◽  
Balance Equations ◽  
The Matrix ◽  
Elastic Composites ◽  
Porous Composites

Within this work, we upscale the equations that describe the pore-scale behaviour of nonlinear porous elastic composites, using the asymptotic homogenization technique in order to derive the macroscale effective governing equations. A porous hyperelastic composite can be thought of as being comprised of a matrix interacting with a number of subphases and percolated by a fluid flowing in the pores (which is chosen to be Newtonian and incompressible here). A general nonlinear macroscale model is derived and is then specified for a particular choice of strain energy function, namely the de Saint-Venant function. This leads to a macroscale system of PDEs, which is of poroelastic type with additional terms and transformations to account for the nonlinear behaviour of the material. Our new porohyperelastic-type model describes the effective behaviour of nonlinear porous composites by prescribing the stress balance equations, the conservation of mass and Darcy’s law. The coefficients of these macroscale equations encode the detailed microstructure of the material and are to be found by solving pore-scale differential problems. The model reduces to the following limit cases of (a) linear poroelastic composites when the deformation gradient approaches the identity, (b) nonlinear composites when there are no pores and (c) nonlinear poroelasticity when only the matrix–fluid interaction is considered. This model is applicable when the interactions between various hyperelastic solid phases occur at the pore-scale, as in biological tissues such as artery walls, the myocardium, lungs and liver.

scholarly journals Nonlinear morphoelastic plates I: Genesis of residual stress

2011 ◽  
Vol 16 (8) ◽  
pp. 812-832 ◽  
Author(s):  
Joseph McMahon ◽  
Alain Goriely ◽  
Michael Tabor
Keyword(s):  
Residual Stress ◽  
Elastic Body ◽  
Response Function ◽  
Deformation Gradient ◽  
Global Constraint ◽  
Balance Equations ◽  
Multiplicative Decomposition ◽  
Rigorous Analysis ◽  
Volumetric Growth ◽  
Residual Strains

Volumetric growth of an elastic body may give rise to residual stress. Here a rigorous analysis is given of the residual strains and stresses generated by growth in the axisymmetric Kirchhoff plate. Balance equations are derived via the Global Constraint Principle, growth is incorporated via a multiplicative decomposition of the deformation gradient, and the system is closed by a response function. The particular case of a compressible neo-Hookean material is analyzed, and the existence of residually stressed states is established.

scholarly journals COMPARATIVE ANALYSIS OF THE PROPERTIES OF SODIUM NIOBATE AND SODIUM - LITHIUM NIOBATE CERAMICS

2021 ◽  
pp. 278-285
Author(s):  
Ольга Витальевна Малышкина ◽  
Кирилл Валерьевич Пацуев ◽  
Александра Ивановна Иванова ◽  
Майс Али
Keyword(s):  
Comparative Analysis ◽  
Lithium Niobate ◽  
Polarization Vector ◽  
Sodium Niobate ◽  
Ceramic Samples ◽  
State Of Polarization ◽  
Continuous Mass ◽  
Polarization Gradient ◽  
Individual Grains ◽  
Niobate Lithium

Авторами исследовано влияние температуры синтеза ниобата натрия, на состояние поляризации в образцах керамики чистого ниобата натрия и модифицированного литием. Проведено сравнительное исследование структуры и пироэлектрических свойств полученных образцов. Показано, что введение в качестве модификатора лития приводит к существенному изменению структуры в глубине образцов керамики на основе ниобата натрия. Если в глубине образцов чистого ни ниобата натрия, как и на поверхности, различаются отдельные зерна, то центральная часть керамики ниобата натрия-лития представляет собой сплошной массив, в котором отдельные зерна не наблюдаются. Во всех образцах, кроме чистого ниобата натрия, синтезированного двойным синтезом (первый при 650 °C, второй при 700 °C), установлено существование градиента поляризации по толщине образцов, направленного от стороны, соответствующей положительному концу вектора поляризации к стороне, соответствующей отрицательному концу вектора поляризации. The authors studied the effect of the temperature of sodium niobate synthesis on the state of polarization in ceramic samples of pure sodium niobate and modified with lithium. A comparative study of the structure and pyroelectric properties of the obtained samples has been carried out. It is shown that the introduction of lithium as a modifier leads to a significant change in the structure in the depth of ceramic samples based on sodium niobate. If in the depth of the pure sodium niobate samples, as well as on the surface, there are individual grains, then the central part of the sodium niobate-lithium niobate ceramics is a continuous mass in which individual grains are not observed. In all samples, except for pure sodium niobate, which was synthesized by double synthesis (the first at 650 °C, the second at 700 °C), the existence of a polarization gradient along the thickness of the samples was established. The gradient is directed from the side corresponding to the positive end of the polarization vector to the side corresponding to the negative end of the polarization vector.

Effects of Ambipolar Diffusion on Prominence Thread Models

1994 ◽  
Vol 144 ◽  
pp. 315-321 ◽  
Author(s):  
M. G. Rovira ◽  
J. M. Fontenla ◽  
J.-C. Vial ◽  
P. Gouttebroze
Keyword(s):  
Energy Balance ◽  
Transition Region ◽  
Ambipolar Diffusion ◽  
Line Profiles ◽  
Balance Equations ◽  
Model Calculations ◽  
Partial Frequency ◽  
Frequency Redistribution ◽  
Partial Frequency Redistribution ◽  
Theoretical Structure

AbstractWe have improved previous model calculations of the prominence-corona transition region including the effect of the ambipolar diffusion in the statistical equilibrium and energy balance equations. We show its influence on the different parameters that characterize the resulting prominence theoretical structure. We take into account the effect of the partial frequency redistribution (PRD) in the line profiles and total intensities calculations.

Observation of secondary slip systems in tungsten bicrystals

1984 ◽  
Vol 42 ◽  
pp. 478-479
Author(s):  
J. R. Fekete ◽  
R. Gibala
Keyword(s):  
Stress Field ◽  
Grain Boundaries ◽  
Deformation Behavior ◽  
Stress Axis ◽  
Polycrystalline Materials ◽  
Slip Systems ◽  
Metallic Materials ◽  
Twist Boundary ◽  
Secondary Slip ◽  
Plane Normal

The deformation behavior of metallic materials is modified by the presence of grain boundaries. When polycrystalline materials are deformed, additional stresses over and above those externally imposed on the material are induced. These stresses result from the constraint of the grain boundaries on the deformation of incompatible grains. This incompatibility can be elastic or plastic in nature. One of the mechanisms by which these stresses can be relieved is the activation of secondary slip systems. Secondary slip systems have been shown to relieve elastic and plastic compatibility stresses. The deformation of tungsten bicrystals is interesting, due to the elastic isotropy of the material, which implies that the entire compatibility stress field will exist due to plastic incompatibility. The work described here shows TEM observations of the activation of secondary slip in tungsten bicrystals with a [110] twist boundary oriented with the plane normal parallel to the stress axis.

The need of electron microscopy in the study of domains and domain walls in ferroelectrics

1994 ◽  
Vol 52 ◽  
pp. 550-551
Author(s):  
Wenwu Cao
Keyword(s):  
Domain Wall ◽  
Domain Walls ◽  
High Mobility ◽  
Continuum Theory ◽  
Polarization Vector ◽  
Ferroelectric Materials ◽  
Dielectric Constants ◽  
Nonlocal Coupling ◽  
Cubic Symmetry ◽  
Gradient Energy

Domain structures play a key role in determining the physical properties of ferroelectric materials. The formation of these ferroelectric domains and domain walls are determined by the intrinsic nonlinearity and the nonlocal coupling of the polarization. Analogous to soliton excitations, domain walls can have high mobility when the domain wall energy is high. The domain wall can be describes by a continuum theory owning to the long range nature of the dipole-dipole interactions in ferroelectrics. The simplest form for the Landau energy is the so called ϕ model which can be used to describe a second order phase transition from a cubic prototype,where Pi (i =1, 2, 3) are the components of polarization vector, α's are the linear and nonlinear dielectric constants. In order to take into account the nonlocal coupling, a gradient energy should be included, for cubic symmetry the gradient energy is given by,

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