Stress, single-valued

Whereas a linear relation between flow stress and lattice-parameter change is obeyed for any single solute element in nickel, the change in yield stress for various solutes in nickel is not a single-valued function of the lattice parameter, but depends directly on the position of the solute in the Periodic Table... 

For a single-value toughness material, dT/dc = 0. Accordingly, if the applied stress intensity factor is always increasing with crack length, equation 4 is always satisfied. Thus, the condition for fracture is equation 5, where is given by the applied loading conditions. 

In TPE, the hard domains can act both as filler and intermolecular tie points thus, the toughness results from the inhibition of catastrophic failure from slow crack growth. Hard domains are effective fillers above a volume fraction of 0.2 and a size <100 nm [200]. The fracture energy of TPE is characteristic of the materials and independent of the test methods as observed for rubbers. It is, however, not a single-valued property and depends on the rate of tearing and test temperature [201]. The stress-strain properties of most TPEs have been described by the empirical Mooney-Rivlin equation... 

Because of the previously mentioned inadequacy of the function a —l/a, a different value for the parameter %i is required for the set of points (Fig. 135) at each elongation a. These values are —0.90, — 0.73, and —0.56 for a = 1.4, 2.0, and 3.0, respectively. If the function a — l/a were replaced by an empirical representation of the shape of the stress-strain curve, a single value of xi would suffice to represent all of the data within experimental error. This limitation of Eq. (41) relates to an unexplained feature of the stress-strain curve and is... 

Poison s ratio is used by engineer s in place of the more fundamental quality desired, the bulk modulus. The latter is in fact determined by r for linearly elastic systems—h ncc the widespread use of v engineering equation for large deformations, however, where the Strain is not proportional to the stress, a single value of the hulk modulus may still suffice even when the value of y is not- constant,... 

Concret does not have well defined elastic and plastic regions due to its brittle nature. A maximum compressive stress value is reached at relatively low strains and is maintained for small deformations until crushing occurs. The stress-strain relationship for concrete is a nonlinear curve. Thus, the elastic modulus varies continuously with strain. The secant modulus at service load is normally used to define a single value for the modulus of elasticity. This procedure is given in most concrete texts. Masonry lias a stress-strain diagram similar to concrete but is typically of lower compressive strength and modulus of elasticity. 

Let us consider a homogeneously, but not hydrostatically, stressed solid which is deformed in the elastic regime and whose structure elements are altogether immobile. If we now isothermally and reversibly add lattice molecules to its different surfaces (with no shear stresses) from the same reservoir, the energy changes are different. This means that the chemical potential of the solid is not single valued, or, in other words, a non-hydrostatically stressed solid with only immobile components does not have a unique measurable chemical potential [J. W. Gibbs (1878)]. 

Thermodynamics of mobile species uptake. Operationally, one seeks a linear, or at least single valued, relationship between the film mass change and analyte composition. In this section we stress the importance of characterising the mass change / composition relationship, and illustrate circumstances under which the desired behaviour will not prevail. 

Materials that exhibit a direct proportionality between shearing stress and rate of shear are called Newtonian materials. These include water and aqueous solutions, simple organic liquids, and dilute suspensions and emulsions. Most foods are non-Newtonian in character, and their shearing stress-rate-of-shear curves are either not straight or do not go through the origin, or both. This introduces a considerable difficulty, because their flow behavior cannot be expressed by a single value, as is the case for Newtonian liquids. 

Limitations to the effectiveness of mechanical models occur because actual polymers are characterized by many relaxation times instead of single values and because use of the models mentioned assumes linear viscoelastic behavior which is observed only at small levels of stress and strain. The linear elements are nevertheless useful in constructing appropriate mathematical expressions for viscoelastic behavior and for understanding such phenomena. 

As can be seen from these results and from Fig. 10 (for HD), satisfactory agreement caumot be achieved by the use of a single value of the slip parameter for both tangential and normal stresses. 

The stress-strain diagram gives considerably more information about the product tested than the single-value result, which is all that can be obtained from the present unit ... 

There are obvious parallels between eqn. (10.5) and eqn. (10.2) in fact eqn. (10.5) suggests that continuum mechanics is just one more set of applications of the idea that materials tend to move down gradients of chemical potential. But there is also a conspicuous difference ideas in group A embody the idea that, at any point in space, a material component just has a chemical potential—a single value by contrast, group B embodies the idea that, under nonhydrostatic stress, it is a plane or a direction i that has a chemical potential associated with it, and that the potential associated with one plane can be different from the potential associated with another plane at the same point in space. The objective of this book is to treat situations where deformation and interdiffusion are occurring simultaneously that is to say, we want to combine eqns. (10.2) and (10.5), and hence the question, Is chemical potential single-valued or multi-valued must be faced. 

The purpose of the ensuing Part II is to work on the question just set out. It will be proposed that, in general, a component s chemical potential at a point is multivalued (being single-valued only in the special case of hydrostatic stress). In other words, eqn. (10.5) correctly shows a material s linear strain being related to the second derivative of its potential with respect to a variable 0 that specifies orientation it is eqn. (10.2), relating the quantity of material in a cell to the second derivative of its potential with respect to position in space, that needs to be examined and refined. When refined, its resemblance to eqn. (10.5) will be even closer. 

As regards history, let us look at the use of multi-valued potentials (I know too little to survey the extensive literature in which a component s potential is taken as single-valued). Key papers are two by Hans Ramberg The Gibbs free energy of crystals under anisotropic stress, a possible cause for... 

Time-independent fluids in which the shear stress is a nonlinear and single-valued function of the strain rate. 

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