Displacement fields defined

The concept of potential energy in mechanics is one example of a scalar field, defined by a simple number that represents a single function of space and time. Other examples include the displacement of a string or a membrane from equilibrium the density, pressure and temperature of a fluid electromagnetic, electrochemical, gravitational and chemical potentials. All of these fields have the property of invariance under a transformation of space coordinates. The numerical value of the field at a point is the same, no matter how or in what form the coordinates of the point are expressed. 

Polarization density (of a dielectric medium) — The polarization density P is the difference between the electric displacement in a - dielectric medium Dc and the electric displacement in a vacuum Do- (Note The electric displacement is defined as the product of the electric field strength E and the - permittivity e.) The polarization density of a dielectric medium may also be... 

Fig. 3. Craze-crack resistance curves for isothermal conditions with = 30 MPav /s and temperature dependent stress-displacement fields for ki = 300 MPa- /m/s and Ki = 3000 MPa- /m/s. The triangle indicates the initiation of crazing while the square corresponds to the onset of unstable crack propagation, defining...

Here the index I is used to denote a particular charge distribution (i.e. a particular electronic state of the system). The displacement field P/(r) represents a charge distribution p/(r) according to the Poisson equation V T>i = npi. In (16.79) D(r ) and the associated p(r) represent a fluctuation in the nuclear polarization, defined by the equilibrium relationship between the nuclear polarization and the displacement vector (cf. Eqs (16.14) and (16.15))... 

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... 

These differences are shown in the following examples where measurements of the dynamic moduli, G and G" are used to monitor the structure of gel networks. Measurements are performed by imposing an oscillatory shear field on the material and measuring the oscillatory stress response. The stress is decomposed into a component in phase with the displacement (which defines the storage modulus G ) and a component 90 out of phase (which defines the loss modulus G"). The value of G indicates the elastic and network structure in the system (15, 17, 18) and can be interpreted by using polymer kinetic theories. 

At the walls of the nanochannels, the surface electric charge is utilized to define the electrical boundary condition (Eq. 3) here, D is the electric displacement field that is defined by Eq. 4 ... 

The element common to the function of all piezoelectric transducers is particle displacement within a solid. Relative particle displacements cause the generation of restoring stress forces and for piezoelectric materials cause the generation of electrical fields as well. Displacement is defined as the vector indicating the difference between equilibrium and perturbed positions of a solid particle. However, this vector is not invariant to translational motion so that strain is used to represent relative displacement. Strain and displacement are related by the equation ... 

The displacement field within each element is defined in Fig. 3.1 as... 

The effect of an external electric field on a dielectric matcrbl (vacuum field, E, in V/m) b described by the displacement field D f tE, where t b the vacuum permittivity (8.854 X 10 AS/Vm) and c b the relative permittivity (dimensionless). The polarization field P b defined as... 

In order to obtain the stress field of a screw dislocation in an isotropic solid, we can define the displacement field as... 

We begin with the definitions of the strain and stress tensors in a solid. The reference configuration is usually taken to be the equilibrium structure of the solid, on which there are no external forces. We define strain as the amount by which a small element of the solid is distorted with respect to the reference configuration. The arbitrary point in the solid at (x, y, z) cartesian coordinates moves to x + u, y + v, z + w) when the solid is strained, as illustrated in Fig. E.l. Each of the displacement fields u,v,w is a function of the position in the solid u x, y, z), v(x, y, z), w(x, y, z). 

There is a different notation which makes the symmetric or antisymmetric nature of the strain and rotation tensors more evident we define the three cartesian axes as Xi, X2, Xs (instead of x, y, z) and the three displacement fields as u, U2, us (instead of u, v, w). Then the strain tensor takes the form... 

Thus, defining the displacement field as D = cqE + P leads to equation (6.1a). We may note that Pe should not comprise the polarization charge density / p ,. For liquids and isotropic solids, the polarization is proportional to the electrical field, and the following expressions introducing the susceptibility x tho relative dielectric constant e can be used ... 

In fact, the inverse problem is of great interest. This means the attempt to retrieve, also in a quantitative way, the source parameters starting from the knowledge of the fiiSAR surface displacement field. In particular, some useful information to define the fault geometry (dip and strike angle width and length), the extension of the rupture, and the slip distribution on the fault plane can be obtained. 

FETI methods are iterative methods where the subdomains communicate with each other through a set of Lagrange multipliers defined at subdomain boundaries. In a static problem, the equilibrium of each subdomain is satisfied at each iteration whereas the continuity of the displacement field is achieved at convergence. 

Since the wavelengths of sound waves are very large compared with the lattice constant, the variation of the displacements u( ) are very slight from cell to cell. We therefore define a spatially slowly-changing displacement field u(x) which is equal to u( ) at the lattice sites r( ) (Fig.3.15)... 

Let us consider a homogeneous, isotropic solid body. Under the action of applied forces, the solid body exhibit deformation. A point of initial position vector r (with components (x, y, z)) has, after the deformation, a new position r = r + u where u(x, y, z) is the displacement field. The strain tensor Uik is defined as (Landau and Lifshitz, 1986)... 

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