Continuum displacement field

The Burgers circuit concept introduced above from the discrete perspective has a continuum analog. Mathematically, the origin of this analog is the fact that there is a jump in the continuum displacement fields used to characterize the geometric state of the body. Recall that above we described the Volterra procedure in which the body is cut and rejoined after a relative translation operation. The resulting displacement jump reveals itself upon consideration of the integral... 

Fully Nonlocal QC The nonlocal formulation of the QC method was developed for modeling inhomogeneous structural features. A first formulation was presented in the original QC studies this method was later expanded (see, e.g.. Ref. 89) and, finally, the fully nonlocal QC (FNL-QC) method was developed by Knap and Ortiz.The key point of the nonlocal formulation is that each atom within the representative crystallite is displaced according to the actual continuum displacement field at its position. Thus, the displacement field considered when computing the energy (or force) of a repatom can be nonuniform. In the original formulation, the repatoms are... 

Broadly speaking, our description of continuum mechanics will be divided along a few traditional lines. First, we will consider the kinematic description of deformation without searching for the attributes of the forces that lead to a particular state of deformation. Here it will be shown that the displacement fields themselves do not cast a fine enough net to sufficiently distinguish between rigid body motions, which are often of little interest, and the more important relative motions that result in internal stresses. These observations call for the introduction of other kinematic measures of deformation such as the various strain tensors. Once we have settled these kinematic preliminaries, we turn to the analysis of the forces in continua that lead to such deformation, and culminate in the Cauchy stress principle. 

In light of this solution, we may now compute the continuum estimate for the elastic energy stored in the displacement fields as a result of the presence of the... 

Continuum Models of Point Defects with Tetragonal Symmetry One of the approaches we used to examine the structure and energetics of point defects from the continuum perspective was to represent the elastic consequences of the point defect by a collection of force dipoles. The treatment given in the chapter assumed that the point defect was isotropic. In this problem, derive an equation like that given in eqn (7.38). Assume that the forces at (a/2)ei and (a/2)e2 have strength /o while those at (a/2)e3 have strength /i. Evaluate the displacement fields explicitly and note how they differ from the isotropic case done earlier. 

A dislocated crystal has a distinct geometric character from one that is dislocation-free. From both the atomistic and continuum perspectives, the boundary between slipped and unslipped parts of the crystal has a unique signature. Whether we choose to view the material from the detailed atomic-level perspective of the crystal lattice or the macroscopic perspective offered by smeared out displacement fields, this geometric signature is evidenced by the presence of the so-called Burgers vector. After the passage of a lattice dislocation, atoms across the slip plane assume new partnerships. Atoms which were formerly across from... 

Our initial foray into the elastic theory of dislocations has revealed much about both the structure and energetics of dislocations. From the continuum standpoint, the determination of the displacement fields (in this case uf) is equivalent to solving for the structure of the dislocation. We have determined a generic feature of such fields, namely, the presence of long-range strains that decay as r. Another... 

Recall from our review of continuum mechanics that the divergence of the displacement field is a measure of the volume change associated with a given deformation. In this case, it is the volume change associated with the dislocation fields at the obstacle that gives rise to the interaction energy. 

With some appropriate definition of nearest neighbors (for example, the Voronoi construction, described in Section III.D, provides a unique definition of nearest neighbors), a bond angle field d can be defined, where 6 is the orientation of a bond between nearest neighbors with respect to some reference direction. In the continuum limit, the bond angle field is related to the displacement field by... 

The displacement field of an edge-dislocation in the SmA phase has been calculated by De Gennes in the framework of the elastic continuum theory... 

In the continuum limit the local deformation of a crystal can be described by the gradient of the displacement field (/, t) = Ri(t) - Ri where Ji, and JJ, are the temporary and equilibrium lattice positions, respectively. The gradient tensor is given by = dUi (i)ldR with k, l = x, y, z. The general deformation can be described as a pure strain deformation tj followed by a rotation D... 

While an explicit treatment of the microscopic, atomistic degrees of freedom is necessary when a locally realistic approach is required, on a macroscopic level the continuum framework is perfectly adequate when the relevant characteristics are scalar, vector, or tensor fields (e.g., displacement fields for mechanical properties). A combination of the two levels of description would be useful [29,30]. Here we focus on the deformation of solids The elastic mechanics of homogeneous materials is well understood on the atomistic scale and the continuum theory correctly describes it. Inelastic deformation and inhomogeneous materials, however, require techniques that bridge length and time scales. 

We will deal with electromagnetic phenomena in the electrostatic regime, that is, we disregard any magnetic and radiative effects. In accordance with the continuum hypothesis, the governing equations for continuous media are Maxwell equation. Here, the eleetric field E, the electric displacement field D, the magnetic field B, the polarization field P, the electrical current density and the electrical potential (p are averaged locally over their microscopic counterparts. The fundamental equations are... 

If the trajectories of all cracks are known beforehand (e.g., from symmetry considerations), the crack sizes take the part of generalized coordinates. When the number of cracks is bounded, the number of degrees of freedom will be bounded, too. Hence, we approach the problem of finite-degrees-of-freedom mechanical systems with unilateral constraints. Equations of continuum mechanics are to be used on the preparatory stage, when stress, strain, and displacement fields are evaluated for states Y.o and X ... 

Rybicki R (1986) Dislocations and their geophysical application. In Teisseyre R (ed) Continuum theories in solid earth physics. Elsevier, Amsterdam, pp 18-186 Satake K, Kanamori H (1991) Use of tsunami waveforms for earthquake source study. Nat Hazards 4 193-208 Savage JC (1998) Displacement field for an edge dislocation in a layered half-space. J Geophys Res 103 2439-2446... 

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

Soft biological structures exhibit finite strains and nonlinear anisotropic material response. The hydrated tissue can be viewed as a fluid-saturated porous medium or a continuum mixture of incompressible solid (s), mobile incompressible fluid (f), and three (or an arbitrary number) mobile charged species a, (3 = p,m, b). A mixed Electro-Mechano-Chemical-Porous-Media-Transport or EMCPMT theory (previously denoted as the LMPHETS theory) is presented with (a) primary fields (continuous at material interfaces) displacements, Ui and generalized potentials, ifi ( , r/ = /, e, to, b) and (b) secondary fields (discontinuous) pore fluid pressure, pf electrical potential, /7e and species concentration (molarity), ca = dna/dVf or apparent concentration, ca = nca and c = Jnca = dna/dVo. The porosity, n = 1 — J-1(l — no) and no = no(Xi) = dVj/dVo for a fluid-saturated solid. Fixed charge density (FCD) in the solid is defined as cF = dnF/dV , cF = ncF, and cF = cF (Xf = JncF = dnF/d o. 

According to the macroscopic Maxwell approach, matter is treated as a continuum, and the field in the matter in this case is the direct result of the electric displacement (electric induction) vector D, which is the electric field corrected for polarization [7] ... 

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