Dilation strain

As discussed in Appendix A, symmetric tensors have properties that are important to the subsequent derivation of conservation laws. As illustrated in Fig. 2.9, there is always some orientation for the differential element in which all the shear strain rates vanish, leaving only dilatational strain rates. This behavior follows from the transformation laws... 

Since the range of the cohesive forces is much larger than the range of the repulsive forces, the surface layer will suffer, relative to bulk layers, a dilatation strain. In our model of quasi-independent (001) layers, we are interested only in intralayer quantities, which are thus amenable to the calculation of effects of the dilatation strain, to which the surface layer is assumed to respond linearly, with the elastic constatns of the bulk. Dilatation along the c axis will provide variations on our three parameters (the c distance, the monoclinic angle p, and the orientation of the surface molecules in their unit cells) as well as on the frequencies of the associated librations. 

Soft surface We assume in this model that the surface-layer energy shifts are proportional to the dilatation strain (3.37) caused by the missing interactions.1 With attractive forces of the type (3.39) in r- 5, the ratio p may be calculated for various values of the parameters a and d of the model (the spacing and size of the molecules , Fig. 3.23). It is concluded that p is practically independent of these parameters (from p = 25.2 for a = d = 0 it becomes 19.8 for a = d = 1). The typical values a = 6 and d = 10 lead to the ratio p = 20. On this point, we make the remark that only forces in r-5 are capable of yielding values of p compatible with the experimental value... 

Crazes usually form under tensile stress when a critical strain is surpassed they do not occur under compressive stress applying hydrostatic pressure during tensile deformation can even inhibit their development. Crazes always nucleate preferentially at points of triaxial stress concentration. It is the dilatational strain which initiates crazes and cracks. 

The total strain developed in all of these materials is essentially entirely from crazing and therefore at all points prior to fracture the imposed strain rate of the experiment must be matched by the specimen through its dilatational strain rate 6 which is a product of three terms... 

Here u is the position of a layer plane and z is the position coordinate locally parallel to the director n, where n is parallel to the average molecular axis, which is assumed to remain normal to the layer plane, du/dz = e is the compressional (or dilational) strain. Thus, layer bending and layer compression are characterized by a splay (or layer-bend) modulus K and a compression modulus B. Other kinds of distortion present in nematics, such as bend or twisting of the director n, are not compatible with layers that remain nearly parallel, and hence are forbidden. Equation (10-36) is not invariant to rotations of frame, and its validity is limited to weak distortions a rotationally invariant expression has been given by -Grinstein and Pelcovits (1981).---------------------------------------------------------... 

To avoid these mathematical details and focus on the key concepts of tablet stress analysis this discussion will examine the simplest of viscoelastic models using the method outlined by Fluggie (97). To begin the analysis, the boundary conditions which apply to tablet compaction, will be used to set up the stress and strain tensors Equations (26) and (27). Then the dilation and distortion uations (28-31) will be used lo obtain dilation and distortion tensors. After obtaining the dilational and distortional stress and strain tensors, a Kelvin viscoelastic model will be used to relate the distortional stress to distortional strain and the dilational stress to dilational strain. 

Using the bulk and shear modulus to relate dilational stress to dilational strain and distortional stress to distortional strain yields ... 

Figure 4 illustrates the typical volume dilatation-strain behavior along with its first and second derivatives. Clearly these measures are realistic in that the derivatives do take on the character of cumulative and instantaneous frequency distributions. Similar models can be constructed to relate the loss in stiffness to the number of vacuoles that have formed resulting in very simple but accurate stress-strain relations (1). 

Figure 4. Schematic of the dilatation-strain relationship and its first and second...

Below this yield stress, no continued shear strain can occur, only a small elastic (dilational) strain that corresponds to the distortion of the... 

The effect of the dilation strains is to reduce the stress intensity at the crack tip A tip by a shielding factor K, such that... 

There is experimental evidence, for many rubber-toughened polymers, that the rubber particles cavitate early in the deformation. The degree of cross-linking is kept relatively low in the polybutadiene phase of ABS to aid cavitation, and sometimes silicone oil is added for the same reason. Figure 4.12 shows both the conventional stress-strain curve and the volumetric strain versus tensile strain for rubber-modified polystyrene. When the polystyrene yields, the volume strain increases at a higher rate. Majority of the dilatational strain is due to cavitation in the rubber phase. 

One can also form linear combinations dj. of these components that correspond to the symmetry strains e. For isotropic bands with E k) = ( it ) the deformation potential dp of a purely dilational strain Sp = By xx + has no... 

A spherical rubber particle of radius R is subjected to a small dilatational strain, and cavitates to form a void of radius r. Show that a rubber obeying eqn 3.29 gives the following ejqiiession for the function F(A ) in eqn 5.41 ... 

Fig. 11.3 Influence of the dilation strain parameter, upon a critical size below which a fracture will not occur (from Ref. 73). Different numbered curves stand for different fresh solid phases with various fracture toughness parameters. The dilation strain parameter accounts for the volume mismatch between the pristine and lithiated phases...

To avoid shear failure, horizontal FRP wrapping should limit the column dilation in the loading direction up to a dilation strain <0.004. 

Diffusion of moisture or solvent into the adhesive can significantly change the state of the stress field and hence the reliability of the bonded joint. Various researchers modeled the effects of viscoelasticity and diffusion in the analysis of bonded joints.( 9-42) in all these works, either the effects of moisture or the coupling between the viscoelastic stress field and the moisture/solvent concentration were not considered. In general, the diffusion coefficient is a function of the dilatational strain, temperature field, and solvent concentration. Recently,... 

Hence, the two sources of nonlinearity in Eq. (54) are moisture concentration c and dilatational strain Consequently, Eqs. (48) and (54) are coupled. The diffusion boundary-value problem must be solved in conjunction with the nonlinear viscoelasticity boundary-value problem by using an iterative procedure. Finite-element formulation of Eq. (54) is standard (see Reddy(47)) and is given by... 

A review of the theoretical basis, finite-element model, and sample applications of the program NOVA are presented. The updated incremental Lagrangian formulation is used to account for geometric nonlinearity (i.e., small strains and moderately large rotations), the nonlinear viscoelastic model of Schapery is used to account for the nonlinear constitutive behavior of the adhesive, and the nonlinear Fickean diffusion model in which the diffusion coefficient is assumed to depend on the temperature, penetrant concentration, and dilational strain is used. Several geometrically nonlinear, linear and nonlinear viscoelastic and moisture... 

Vector containing the difference of kinematic and dilatational strains... 

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