Displacement gradient

Clarity requires that a distinction be made between elastic strain and plastic deformation. They both have units of length/length, but they are physically different entities. Elastic strain is recoverable (conservative) plastic deformation is not (non-conservative). At a dislocation core, where atoms exchange places via shear, the plastic displacement gradient is a maximum as it passes from zero some distance ahead of the core, up to the maximum, and then back to zero some distance back of the core. In crystals with distinct bonds, the gradient becomes indefinite (infinite) at the core center. 

If a dislocation line lies parallel to the x-axis of an xy-plane, and is kinked, the kink lies parallel to the y-axis. Therefore, if the line is of edge character, the kink is of screw character. If the line is of screw character, the kink is of edge character. In either case, the displacement gradient is indefinite at the center of the kink. This means that whatever symmetry exists in the undislocated crystal, structure is destroyed at a kink. 

It is assumed here that the axial displacements are independent of the radial position, and the stress components in the radial and circumferential directions are neglected for Eqs. (4.8) and (4.9). Also, the radial displacement gradient with respect to the axial direction is neglected compared to the axial displacement gradient with respect to the radial direction in Eq. (4.10). Combination of Eqs. (4.10) and (4.16) for the boundary condition of the axial displacement continuity at the bonded interface (i.e. i/ (a,z) = u z)) and integration gives ... 

Axux - Ayuy)6 is a sixth-power, displacement gradient term. Monte... 

Another combination of the displacement gradient tensors which are often used are the Cauchy strain tensor and the Finger strain tensor defined by B —1 = Afc A and B = EEt, respectively. 

In this section we consider the simplest approach to the thermodynamics of a deformed network, for which a tensor of displacement gradients is given by... 

In the ideal case, when one considers the network of chains of equal lengths, the stresses under the given deformation can be obtained in a very simple way. In virtue of the speculations of the previous section, free energy of the whole network can be represented as the sum of free energy of all the chains, while each of the equal chains of the network can be characterised by the same equilibrium distribution function W(s), where s is the separation between adjacent junctions. In the state without deformation, the function has the form (1.5), while in a deformed state, it depends on the displacement gradients (1.39). The free energy of the whole network can be written down simply as... 

Since the velocity gradient is related to the displacement gradient by the expression v 2 = —iojA12, it follows that, instead of the dynamic viscosity, the use may be made of another characteristic - the dynamic modulus... 

Instead of velocity gradients, displacement gradients can be used in relation (8.38). In this form, relations of the kind (8.38) are established on the basis of the phenomenological theory of so-called simple materials (Coleman and Nolle 1961). To put the theory into practice, function (8.38) should be, for example, represented by an expansion into a series of repeated integrals, so that, in the simplest case, one has the first-order constitutive relation (8.37). Let us note that the first person who used functional relations of form (8.38) for the description of the behaviour of viscoelastic materials was Boltzmann (see Ferry 1980). 

As an example, we shall consider simple shear when z/12 0, and find components of the tensor of the recoverable displacement gradients A12, An, A22, A33 the components of the tensor are calculated from the relaxation equations (9.49) or (9.58). In this case the matrix of the deformation tensor is determined as follows... 

Because simple translation of the entire solid is not of interest, this class of motion is eliminated to give a parameter related only to local deformations of the solid this parameter is the displacement gradient, V . The gradient of a vector field Vu is a second-rank tensor, specified by a 3 by 3 matrix. The elements of this displacement gradient matrix are given by (Vu),y = dujdxj, also denoted Uij in which i denotes the i" displacement element and j denotes a derivative with respect to the y spatial coordinate, i.e. [1],... 

The displacement gradient represents changes in interparticle distance as well as local rotations caused by the displacement. 

Solution From Equation 2.2, ui,i == kh, while all the other elements of the displacement gradient are zero. As a result, the only non-zero strain element is Sn = fc, This represents a fractional change in length, in particular an elongation, in the x direction of magnitude kt,. 

Figure 3.28 Deformation generated by a SAW (a) in an acoustically thin (R << 1) film, in which in-plane displacement gradients (due to sinusoidal wave variation) dominate, and (b) in an acoustically thick (R 1) film, where cross-film gradients (due to inertial fllm lag) also arise. (Reprinted with permission. See Ref. [50]. O 1994 American CJiemical Society.)...

In the case of a simple shear deformation, schematically indicated in Figure 4.6b, the only nonzero components of the displacement gradient and strain tensors are given by... 

E displacement gradient tensor N. numberofprimitivepathstepsoc-ctqpied by the matrix chains in a... 

Fig. 10 Reprinted from [157], with permission from AAAS. (a) Shear-induced displacements of particles in a colloidal packing between z = 0 (the cover slide) and z = 23 pm after 50 min of shear (3% average accumulated strain), (b) Strain distribution and STZs in a 7pm thick section in the shear-displacement gradient plane. Particle colour indicates the value of the local shear strain...

Here we have used the fact that Fij = 5 + Uij. In addition, we have invoked the summation convention in which all repeated indices (in this case the index k) are summed over. For the case in which all the displacement gradient components satisfy Uij 1, the final term in the expression above may be neglected, resulting in the identification of the small strain (or infinitesimal strain) tensor,... 

To give a flavor of the types of results that can be obtained using this method, fig. 12.12 shows results on nanoindentation. The nanoindentation calculations were carried out using a pseudo-two-dimensional geometry which allows for out-of-plane displacements but not out-of-plane displacement gradients. As the... 

Alternatively, the centrifuge tube may be punctured at its base using a fine hollow needle. As the drops of gradient pass from the tube through the needle they may be collected using a fraction collector and further analyzed. Analysis of the contents of the displaced gradient can be achieved by ultraviolet spectrophotometry, refractive index measurements, scintillation counting or chemical analysis. 

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