Stress-strain relationship, idealized

Figure 10.4. Schematic view of reversible and irreversible stress-strain relationships in a compression-deeompression cycle. Left, ideal linear elasticity (small deformation) center, nonlinear elastieity and right, a relationship showing a hysteresis loop.

The complex relationship between the configurational distortion produced by a perturbation field in polymers and the Brownian motion that relaxes that distortion make it difficult to establish stress-strain relationships. In fact, the stress at a point in the system depends not only on the actual deformation at that point but also on the previous history of deformation of the material. As a consequence the relaxation between the stress and strain or rate of strain cannot be expressed by material constants such as G or /, as occurs in ideal elastic materials, but rather by time-dependent material functions, G t) and J t). It has been argued that the dynamics of incompressible liquids may be characterized by a function of the evolution of the strain tensor from the beginning up to the present time. According to this criterion, the stress tensor would be given by (3,4)... 

Fig. 8.1 Ideal yield behaviour. The full lines represent the stress-strain relationship on loading and the dotted line represents the relationship when unloading takes place starting at the point A.

Reduction of moduli with increasing strain amplitude is a major characteristic displayed by the nonlinear nature of the stress-strain relationship of soils. An idealized shear modulus reduction curve is given in Figure 9.21, whereby extrapolating the curve to zero strain, the maximum shear modulus, can be estimated at the intercept. Hardin and Dmevich (1972) and Hardin (1978) suggested the use of the following form of empirical equation for calculation of laboratory for many imdisturbed cohesive soils as well as sands ... 

Figure 11.9 Stress-strain relationship for an ideal rigid-plastic material...

Usually, sealants and adhesive materials for construction applications are evaluated by looking at the engineering side, butnotthe chemistry of the material. As a result, only tests that measure the mechanical properties are used. Most of the studies on the viscoelastic properties use traditional tests such as tensile testing to obtain data, which can be used in complicated mathematical equations to obtain information on the viscoelastic properties of a material. For example, Tock and co-workers studied the viscoelastic properties of stmctural silicone rubber sealants. According to the author, the behavior of silicone mbber materials subjected to uniaxial stress fields carmotbe predicted by classical mechanical theory which is based on linear stress-strain relationship. Nor do theories based on ideal elastomers concepts work well when extensions exceed... 

In a fluid under stress, the ratio of the shear stress, r. to the rate of strain, y, is called the shear viscosity, rj, and is analogous to the modulus of a solid. In an ideal (Newtonian) fluid the viscosity is a material constant. However, for plastics the viscosity varies depending on the stress, strain rate, temperature etc. A typical relationship between shear stress and shear rate for a plastic is shown in Fig. 5.1. 

If a material exhibits linear-elastic stress-strain behavior prior to rupture (an ideal behavior approximated by many thermosets), then a simple relationship exists between the material s fracture toughness and its fracture surface energy, J (or G),... 

FIGURE 10.1 Rheological properties of an ideal solid (a) depicts the rheological representation of an ideal solid, whereas (b) displays the relationship between stress, strain, and time for an ideal solid. 

A nonlinear path-dependent constitutive model for the soil mainly depends on the shear stress-shear strain relationship, which is extended to three-dimensional generic conditions and assumed to follow Masing s rule for the soil hysteresis. The soil is idealized as an assembly of a finite number of elasto-perfectly plastic elements connected in parallel as shown in Fig. 25.3 (Okhovat et al. 2009, Mohammed and Maekawa 2012 Mohammed et al. 2012a). The nonlinear behavior of the soil system in liquefaction is assumed as in undrained state, since its drainage takes much longer than the duration of an earthquake (Towhata 2008). The soil undrained behavior is shown in Fig. 25.4. 

The relationship between the tensile stress and tensile strain that a fiber displays is known as that fiber s stress-strain behavior. If a fiber obeys the Hooke s law, its tensile stress is directly proportional to the tensile strain, up to the point of failure, where the fiber breaks without yielding or plastic deformation (Figure 15.5). In this case, the fiber exhibits ideal stress-strain behavior. 

If the solid does not shows time-dependent behavior, that is, it deforms instantaneously, one has an ideal elastic body or a Hookean solid. The symbol E for the modulus is used when the applied strain is extension or compression, while the symbol G is used when the modulus is determined using shear strain. The conduct of experiment such that a linear relationship is obtained between stress and strain should be noted. In addition, for an ideal Hookean solid, the deformation is instantaneous. In contrast, all real materials are either viscoplastic or viscoelastic in nature and, in particular, the latter exhibit time-dependent deformations. The rheological behavior of many foods may be described as viscoplastic and the applicable equations are discussed in Chapter 2. 

Generalized Strain-Stress Relationships for Ideal Elastic Systems 170... 

Both E, in ideal solids, and rj, in ideal liquids, are material functions independent of the size and shape of the material they describe. This holds for isotropic and homogeneous materials, that is, materials for which a property is the same at all directions at any point. Isotropic materials are so characterized because their degree of symmetry is infinite. In contrast, anisotropic materials present a limited number of elements of symmetry, and the lower the number of these elements, the higher the number of material functions necessary to describe the response of the material to a given perturbation. Even isotropic materials need two material functions to describe in a generalized way the relationship between the perturbation and the response. In order to formulate the mechanical behavior of ideal solids and ideal liquids in terms of constitutive equations, it is necessary to establish the concepts of strain and stress. 

GENERALIZED STRAIN-STRESS RELATIONSHIPS FOR IDEAL ELASTIC SYSTEMS... 

In the ideal case of a Hookean body, the relationship between stress and strain is fully linear, and the body returns to its original shape and size, after the stress applied has been relieved. The proportionality between stress and strain is quantified by the modulus of elasticity (unit Pa). The proportionality factor under conditions of normal stress is called modulus of elasticity in tension or Young s modulus E), whereas that in pure shear is called modulus of elasticity in shear or modulus of rigidity (G). The relationships between E, G, shear stress, and strain are defined by ... 

Direct information on elastic recovery, relative hardness, work of indentation, and strain rate-stress relationship (Fig. 2.15) can provide a comprehensive fingerprint of a particular sample resulting, for example, from a change in either a production process or a wear test procedure. It is ideally suited to the comparison of one sample with a control or reference. The wider assumptions that are needed to derive indirect... 

The question of whether microhardness is a property related to the elastic modulus E or the yield stress T is a problem which has been commented on by Bowman Bevis (1977). These authors found an experimental relationship between microhardness and modulus and/or yield stress for injection-moulded semicrystalline plastics. According to the classical theory of plasticity the expected microindentation hardness value for a Vickers indenter is approximately equal to three times the yield stress (Tabor s relation). This assumption is only valid for an ideally plastic solid showing sufficiently large deformation with no elastic strains. PE, as we have seen, can be considered to be a two-phase material. Therefore, one might anticipate a certain variation of the H/ T 3 ratio depending on the proportion of the compliant to the stiff phase. 

A fluid in which the shear stress is proportional to the shear velocity, corresponding to this law, is called an ideal viscous or Newtonian fluid. Many gases and liquids follow this law so exactly that they can be called Newtonian fluids. They correspond to ideal Hookeian bodies in elastomechanics, in which the shear strain is proportional to the shear. A series of materials cannot be described accurately by either Newtonian or Hookeian behaviour. The relationship between shear stress and strain can no longer be described by the simple linear rule given above. The study of these types of material is a subject of rheology. 

FIGURE 10.6 Relationships between stress and time and between strain and time for an ideal sobd in oscillatory analysis. 

In the cube shown in Fig. 3.5. the tensor components for the strain-stress relationship of a 3D-body can be seen. Neglecting the z-coordinate, the tensor reduces from a 3x4 to a 2x2 matrix. The use of the 3D-rheology for related surface problems is only valid if a 3D-analogue for the relaxation is introduced. This is the only way to learn about the surface state in the absence of ideally elastic behaviour of the adsorption layer. 

An ideal elastic body (also called Hooke s body) is defined as a material that deforms reversibly and for which the strain is proportional to the stress, with recovery to the original volume and shape occurring immediately upon release of the stress. In a Hooke body, stress is directly proportional to strain, as illustrated in Fig. 3. The relationship is known as Hooke s law, and the behavior is referred to as Hookean behavior. 

In an ideal elastic solid, a one-to-one relationship between stress and strain is expected. In practice, however, there are often small deviations. These are termed anelastic effects and result from internal friction in the material. Part of the strain develops over a period of time. One source of anelasticity is thermoelasticity, in which the volume of a body can be changed by both temperature and applied stress. The interaction will depend on whether a material has time to equilibrate with the surroundings. For example, if a body is rapidly dilated, the sudden... 

Hooke s law Hooke s law describes the relationship between applied stress and strain in an ideal elastic solid body... 

It is important to appreciate that plasticity is different in kind from elasticity, where there is a unique relationship between stress and strain defined by a modulus or stiffness constant. Once we achieve the combination of stresses required to produce yield in an idealized rigid plastic material, deformation can proceed without altering stresses and is determined by the movements of the external constraints, e.g. the displacement of the jaws of the tensometer in a tensile test. This means that there is no unique relationship between the stresses and the total plastic deformation. Instead, the relationships that do exist relate the stresses and the incremental plastic deformation, as was first recognized by St Venant, who proposed that for an isotropic material the principal axes of the strain increment are parallel to the principal axes of stress. 

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