Displacement field

A simple derivation of tliis equation based on tire lowest-order derivative (curvature) of tire layer displacement field u(r) has been provided [87]. A similar expression can be obtained for a uniaxial columnar phase [20] (witli tire columns lying in tire z direction) ... 

Depending on the structure of the optical probe, components of vector quantities (velocity field, displacement field) and their signs can be distinguished in measurements, ensuring directional sensitivity. 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

Upon substitution of the displacement field. Equation (4.163), in the stress-displacement relations and subsequently in the stress-equilibrium differential equations. Equation (4.164), the displacement-equilibrium equations are, for each layer,... 

Alternatively, one may attempt to estimate the integral over the derivative of the displacement field that entered in the expression for the coupling constant g= pc Jy cPr du/2. Since da is the divergence of a vector, the integral is reduced to that over a surface within the droplet s boundary ... 

If we expand the value of the displacement field c[) in terms of spherical harmonics according to = Jt/cj) t/(cos 0) /(r =0), it is then possible to write down equations of motion for the (/, m) components of both ripplon and phonon displacements ... 

The forward problem is the calculation of displacement fields from input viscoelasticity parameters. The latter describe correctly the investigated object if the calculated and measured displacement images converge. 

Fig. 8. Reconstruction of Young s modulus map in a simulated object. A 3D breast phantom was first designed in silico from MR anatomical images. Then a given 3D Young s modulus distribution was supposed with a 1 cm diameter stiff inclusion of 200 kPa (A). The forward problem was the computing of the 3D-displacement field using the partial differential equation [Eq. (5)]. The efficiency of the 3D reconstruction (inverse problem) of the mechanical properties from the 3D strain data corrupted with 15% added noise can be assessed in (B). The stiff inclusion is detected by the reconstruction algorithm, but its calculated Young s modulus is about 130 kPa instead of 200 kPa. From Ref. 44, reprinted by permission of Wiley-Liss, Inc., a subsidiary of John Wiley Sons, Inc.

To be specific regarding the formalism, let uqia(t) denote the a component of the displacement field associated with wave vector q and eigenmode i at time t. In the absence of external forces, which can simply be added to the equation, the equation of motion for the coordinates that are not thermo-statted explicitly uqia would read as follows ... 

The temperature, fiber tension, stresses, and strains vary only in the radial directions. An elasticity solution is employed to calculate the six components of the stresses and strains. The solution procedure follows the established techniques of elasticity solutions. A displacement field is assumed that satisfies the equilibrium equations and the compatibility conditions. The latter requires that at each interface the displacements and the normal stresses in adjacent... 

The behavior of an edge dislocation is more complicated since its displacement field produces both shear and normal stresses. The solution consists of the superposition of two terms, each of which behave relativistically with limiting velocities corresponding to the speed of transverse shear waves and longitudinal waves, respectively [2, 4, 5]. The relative magnitudes of these terms depend upon v. 

Up to now we have been discussing in this Chapter many-particle effects in bimolecular reactions between non-interacting particles. However, it is well known that point defects in solids interact with each other even if they are not charged with respect to the crystalline lattice, as it was discussed in Section 3.1. It should be reminded here that this elastic interaction arises due to overlap of displacement fields of the two close defects and falls off with a distance r between them as U(r) = — Ar 6 for two symmetric (isotropic) defects in an isotropic crystal or as U(r) = -Afaqjr-3, if the crystal is weakly anisotropic [50, 51] ([0 4] is an angular dependent cubic harmonic with l = 4). In the latter case, due to the presence of the cubic harmonic 0 4 an interaction is attractive in some directions but turns out to be repulsive in other directions. Finally, if one or both defects are anisotropic, the angular dependence of U(f) cannot be presented in an analytic form [52]. The role of the elastic interaction within pairs of the complementary radiation the Frenkel defects in metals (vacancy-interstitial atom) was studied in [53-55] it was shown to have considerable impact on the kinetics of their recombination, A + B -> 0. 

Various considerations when producing data have been discussed by Kim et al20. An experimental scheme for efficient characterization has been proposed21 and an intermediate approach between using simple uniaxial tension and two independent strains to obtain input data given22. A novel technique based on use of a speckle extensometer to give the whole displacement field in two dimensions has also been described23. 

When the electronic charge in the optical material is displaced by the electric field (E) of the light and polarization takes place, the total electric field (the "displaced" field, D) within the material becomes ... 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

Equations (2), (4), and (5) can be combined with the deviatoric elasticity equation and the equilibrium equations to form a set of field equations consisting of a Navier-type equation and two coupled diffusion equations. For the class of problems characterized by an irrotational displacement field with chemical and hydraulic loadings only, the two coupled diffusion equations simplify to... 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

Abstract These notes present the physics of gravitational lensing in various cosmological contexts. The equations and approximations that are commonly used to describe the displacement field or the amplification effects are presented. Several observational applications are discussed. They range from micro-lensing effects to cosmic shear detection that is a weak lensing effect induced by the large-scale structure of the Universe. The scientific perspectives of this latter application are presented in some details. 

As on Fig. 13.3 when 2 lenses contribute to the deflexion the total displacement field, , is obtained form the sum of the 2 contribution and... 

---