Solids, stress-strain relations

Stress, C7 Fig. 2.41 The standard linear solid Stress-Strain Relations As shown earlier the stress-strain relations are (2.44)... 

Equations (2.9) and (2.10) are representative of all isotropic, homogeneous solids, regardless of the stress-strain relations of a solid. What is strongly materials specific and uncertain is the appropriate value for shear stress, particularly if materials are in an inelastic condition or anisotropic, inhomogeneous properties are involved. The limiting shear stress controlled by strength is termed r. ... 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

Stress-Strain Relations for Ideal Solids and Ideal Liquids... 

The postulate used in formulating the equations above is not wholly arbitrary the format of these equations has a close analogy with the stress-strain relations for hydrostatic pressure in an elastic solid [5], the linearity of which follows directly from Hooke s law. 

The complicated morphology of crystalline polymer solids and the coexistence of crystalline and amorphous phases make the stress and strain fields extremely nonhomogeneous and anisotropic. The actual local strain in the amorphous component is usually greater and that in the crystalline component is smaller than the macroscopic strain. In the composite structure, the crystal lamellae and taut tie molecules act as force transmitters, and the amorphous layers are the main contributors to the strain. Hence in a very rough approximation, the Lennard-Jones or Morse type force field between adjacent macro-molecular chain sections (6, 7) describes fairly well the initial reversible stress-strain relation of a spherulitic polymer solid almost up to the yield point, i.e. up to a true strain of about 10%. 

The stress strain relation (generalized linear Hooke s law) incorporating the thermal expansion of the solid is... 

A general treatment of the stress-strain relations of rubberlike solids was developed by Rivlin (1948, 1956), assuming only that the material is isotropic in elastic behavior in the unstrained state and incompressible in bulk. It is quite surprising to note what far-reaching conclusions follow from these elementary propositions, which make no reference to molecular structure. 

It is informative to note that when the deformation resistance of a solid becomes significantly rate-dependent, but a tensile stress-strain relation F(eP,eP) at different strain rates is still available for different cases of monotonic straining at different strain rates eP, without interruptions, holdings, or reversals in the deformation, the associated-flow-rule relations of eqs. (3.28a-f) can still serve as a useful guide. However, dealing with more complex resistances and paths of... 

By solving the differential eqn 4.47 show that the stress-strain relation crCy) for constant-strain-rate deformation of a Zener solid has the following form... 

In Sect. 1.2 above, the stress-strain relation in uniaxial tension tests was given in Eq. (1.5), indicating a Hookean behavior. This section now considers linear elastic solids, as described by Hooke, according to which (Ty is linearly proportional to the strain, y. Each stress component is expected to depend linearly on each strain component. For example, the Cn may be expressed as follows ... 

First, when finite strains are imposed on solids (especially those soft enough to be deformed substantially without breaking), the stress-strain relations are much more complicated (non-Hookean deformation) similarly, in steady flow with finite strain rates, many fluids (especially polymeric solutions and undiluted uncross-linked polymers) exhibit marked deviations from Newton s law (non-Newtonian flow). The dividing line between infinitesimal and finite depends, of course, on the level of precision under consideration, and it varies greatly from one material to another. 

Differential Stress-Strain Relations and Solutions for a Kelvin Solid... 

Thus, the properties E and v for PVP and mannitol must be determined as well as E denotes the elastic modulus, which relates to the stress-strain curve obtained through uniaxial tensile testing [29]. As the mechanical behavior of dried mannitol is unknown, in this study, a linear stress-strain relation is assumed. If a dried particle is deformed, its deformation is assumed to be permanent. The ultimate tensile stress of the material is denoted as R. Here, is considered to be the stress at which the material (i.e., the solid layer) fails and a crack develops, independently of the type of failure. The negative ratio of the transverse to axial strain is given by Poisson s ratio, v, cf. Eq. (9.7). 

Of particular interest are situations where the strain tensor is essentially two-dimensional such situations are often encountered in cross-sections of solids which can be considered infinite in one dimension, by convention assigned the z axis or the x axis, and due to the symmetry of the problem only the strain on a plane perpendicular to this direction matters. This is referred to as plane strain . We discuss the form that the stress-strain relations take in the case of plane strain for an isotropic solid, using the notation X3 for the special axis and X, X2 for the plane perpendicular to it. 

From the general stress-strain relations of an isotropic solid, Eq. (E.31), and the fact that... 

Derive the equations of motion for an isotropic solid without any external forces, Eq. (E.41), starting with the general relations Eq. (E.18) and using the relevant stress-strain relations,... 

The theory relevant for the study of mechanical properties of such objects is taken from solid mechanics. This theory is based on, among other things, fundamental mechanics, and it contains theoretical analyses of dynamic properties of various kinds of objects bars, plates and other geometric configurations with plastic or elastic properties. Within this theory we have massive knowledge about, say, stress-strain relations of various idealized objects. This knowledge is presented in a mathematical form and is about model objects which are idealized to such a degree that a mathematical analysis is feasible. The research object is represented by such a model object. 

The finite element procedures for the analysis of elastic>plastic solids at large strain have been given by Lee [9] and implemented by Chiou [11] and Chiou et al. [10]. In this work, only a few comments on the finite element procedures will be made. Equation (16), which links the Truesdell stress rate tensor and the deformation rate tensor, may be regarded as the stress-strain relation in rate fom with a being the "slope" at a particular point in stress space. However, in nonlinear finite element analysis, one has to have a stress-strain relation in incremental form which enables the increments in displacements, strains, and stresses not to be infinitesimally small. Therefore, it is proposed to adopt the following incremental stress-strain relation... 

Figure 1 Diagram showing the viscoelastic behavior of a mucous gel. The first two panels illustrate stress-strain relations in idealized materials, namely an elastic solid, for which the displacement or strain is proportional to the applied force or stress, and a viscous liquid, for which the rate of strain (displacement/time) is proportional to the stress. Mucus is a viscoelastic semisolid. It responds instantaneously as a solid, with a very rapid displacement in response to an applied force. This is followed by a transition to a liquidlike response, in which the rate of strain is constant with time. Finally a zone of viscous response is reached, in which the rate of displacement is constant with time. After release of the applied force, the mucous gel recoils only partially to its initial position.

In solids of cubic symmetry or in isotropic, homogeneous polycrystalline solids, the lateral component of stress is related to the longitudinal component of stress through appropriate elastic constants. A representation of these uniaxial strain, hydrostatic (isotropic) and shear stress states is depicted in Fig. 2.4. Such relationships are thought to apply to many solids, but exceptions are certainly possible as in the case of vitreous silica [88C02]. 

Polymers are viscoelastic materials meaning they can act as liquids, the visco portion, and as solids, the elastic portion. Descriptions of the viscoelastic properties of materials generally falls within the area called rheology. Determination of the viscoelastic behavior of materials generally occurs through stress-strain and related measurements. Whether a material behaves as a viscous or elastic material depends on temperature, the particular polymer and its prior treatment, polymer structure, and the particular measurement or conditions applied to the material. The particular property demonstrated by a material under given conditions allows polymers to act as solid or viscous liquids, as plastics, elastomers, or fibers, etc. This chapter deals with the viscoelastic properties of polymers. 

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