Scalar elasticity

Modulus-frequency master curves have been constructed by applying appropriate time dependent renormalisation factors to the frequency and modulus individual data. From the scaling of these factors with reaction time, the static scaling exponents t and s have been calculated and observed to be independent of the chemical nature of the midblock, suggesting a unique gelation mechanism. For all the samples, 1.84[Pg.298]

This article presents the evaluation of the fracton dimensions via the calculations of the densities of states (DOS) for both percolating elastic and antiferromagnetic networks d = 2-4). The latter belongs to a different universality class that for scalar elasticity. We claim the fracton dimension Saf for antiferromagnetic fractons to be very close to unity independent of the Euclidean dimension d,... 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

The transmission of forces through a lattice as a function of the fraction p, of bonds in the lattice has been analyzed by Kantor and Webman [63], Feng and colleagues [64-66], Thorpe et al. [68-72] and others [73]. The normalized elastic modulus E/Eo, of the lattice as a function of p was found to obey relations similar to scalar percolation... 

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

External stress, locally applied, can have nonlocal static effects in ferroelastics (see Fig. 4 of Ref. [7]). Dynamical evolution of strains under local external stress can show striking time-dependent patterns such as elastic photocopying of the applied deformations, in an expanding texture (see Fig.5 of Ref. [8]). Since charges and spins can couple linearly to strain, they are like internal (unit-cell) local stresses, and one might expect extended strain response in all (compatibility-linked) strain-tensor components. Quadratic coupling is like a local transition temperature. The model we consider is a (scalar) free energy density term... 

Considering that f, g, x, and u are vectors, the differentiation leads to formation of matrices. The matrix A is well known in stability analysis as the jacobian matrix it quantifies the effects of all state variables on their rates of change. A matrix similar to B turns up in metabolic control analysis, as N3v/3p [48, 108], where it denotes the immediate effects of parameter perturbations on the rates of change of all variables. If the function y is scalar and denotes a rate, then C becomes a row vector c harboring unsealed elasticity coefficients and D becomes a row vector d containing so-called n-elasticities - sensitivities of the rates with respect to the parameters [109]. The linearized system is ... 

The main difference between a solid and a liquid is that the molecules in a solid are not mobile. Therefore, as Gibbs already noted, the work required to create new surface area depends on the way the new solid surface is formed [ 121. Plastic deformations are possible for solids too. An example is the cleavage of a crystal. Plastic deformations are described by the surface tension y also called superficial work, The surface tension may be defined as the reversible work at constant elastic strain, temperature, electric field, and chemical potential required to form a unit area of new surface. It is a scalar quantity. The surface tension is usually measured in adhesion and adsorption experiments. 

Think about what happens when, say, an elastomer is under tensile stress. The elastic constants, s and c, cannot be scalar quantities, otherwise Eqs. 10.5 and 10.6 would not completely describe the elastic response. When the elastomer is stretched, a contraction... 

Green s functions appear as the solutions of seismic field equations (acoustic wave equation or equations of dynamic elasticity theory) in cases where the right-hand side of those equations represents the point pulse source. These solutions are often referred to as fundamental solutions. For example, in the case of the scalar wave equation (13.54), the density of the distribution of point pulse forces is given as a product,... 

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G" (r, t), which we call the Green s tensor of the vector wave equation. 

The Born model [74], for example, satisfies the first condition however, it does not satisfy the second one because in it the longitudinal and transverse elastic constants of the linear chain of bonds (the lattice analog of a rod) decrease /V 1 however, the rods must behave more pliably in relation to transverse shifts (the elastic constant decreases L-3). Therefore, the Born scalar model leads to enhanced rigidity in the vicinity of pc. 

The conservation laws in Eqs. (1) and (2), related to the elastic and exciting collisions, represent four scalar equations connecting the two vectorial velocities V, V before the collision event with the corresponding ones v, V after the event. 

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