Stress-strain theory

For concentrated emulsions and foams, Prin-cen [1983, 1985] proposed a stress-strain theory based on a two-dimensional cell model. Consider a steady state shearing of such a system. Initially, at small values of strain, the stress increases linearly as in elastic body. As the strain value increases, the stress reaches its yield value, then at stiU higher deformation, it catastrophically drops to the negative values. The reason for the latter behavior is the creation of unstable cell structure that provides the recoil mechanism. The predicted dependencies for modulus and the yield stress were expressed as ... 

Stress-strain cycling Stress-strain theory Stress-strain... 

Lade, P.V. Elasto-Plastic Stress-Strain Theory for Cohesionless Soil with Curved Yield Surface, Int. J. Solids and Structure 13(1977), 1019/1035... 

As a pipeline is heated, strains of such a magnitude are iaduced iato it as to accommodate the thermal expansion of the pipe caused by temperature. In the elastic range, these strains are proportional to the stresses. Above the yield stress, the internal strains stiU absorb the thermal expansions, but the stress, g computed from strain 2 by elastic theory, is a fictitious stress. The actual stress is and it depends on the shape of the stress-strain curve. Failure, however, does not occur until is reached which corresponds to a fictitious stress of many times the yield stress. 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

It is consistent with the approximations of small strain theory made in Section A.7 to neglect the higher-order terms, and to consider the elastic moduli to be constant. Stated in another way, it would be inconsistent with the use of the small deformation strain tensor to consider the stress relation to be nonlinear. The previous theory has included such nonlinearity because the theory will later be generalized to large deformations, where variable moduli are the rule. 

The theory relating stress, strain, time and temperature of viscoelastic materials is complex. For many practical purposes it is often better to use an ad hoc system known as the pseudo-elastic design approach. This approach uses classical elastic analysis but employs time- and temperature-dependent data obtained from creep curves and their derivatives. In outline the procedure consists of the following steps ... 

Perhaps the most significant complication in the interpretation of nanoscale adhesion and mechanical properties measurements is the fact that the contact sizes are below the optical limit ( 1 t,im). Macroscopic adhesion studies and mechanical property measurements often rely on optical observations of the contact, and many of the contact mechanics models are formulated around direct measurement of the contact area or radius as a function of experimentally controlled parameters, such as load or displacement. In studies of colloids, scanning electron microscopy (SEM) has been used to view particle/surface contact sizes from the side to measure contact radius [3]. However, such a configuration is not easily employed in AFM and nanoindentation studies, and undesirable surface interactions from charging or contamination may arise. For adhesion studies (e.g. Johnson-Kendall-Roberts (JKR) [4] and probe-tack tests [5,6]), the probe/sample contact area is monitored as a function of load or displacement. This allows evaluation of load/area or even stress/strain response [7] as well as comparison to and development of contact mechanics theories. Area measurements are also important in traditional indentation experiments, where hardness is determined by measuring the residual contact area of the deformation optically [8J. For micro- and nanoscale studies, the dimensions of both the contact and residual deformation (if any) are below the optical limit. 

The analysis of such a laminate by use of classical lamination theory revolves about the stress-strain relations of an individual orthotropic lamina under a state of plane stress in principal material directions... 

Vectors are commonly used for description of many physical quantities such as force, displacement, velocity, etc. However, vectors alone are not sufficient to represent all physical quantities of interest. For example, stress, strain, and the stress-strain iaws cannot be represented by vectors, but can be represented with tensors. Tensors are an especially useful generalization of vectors. The key feature of tensors is that they transform, on rotation of coordinates, in special manners. Tsai [A-1] gives a complete treatment of the tensor theory useful in composite materials analysis. What follows are the essential fundamentals. 

However, these transverse shearing stresses were neglected implicitly when we adopted the Kirchhoff hypothesis of lines that were normal to the undeformed middle surface remaining normal after deformation in Section 4.2.2 on classical lamination theory. That hypothesis is interpreted to mean that transverse shearing strains are zero, and, hence, by the stress-strain relations, the transverse shearing stresses are zero. The Kirchhoff hypothesis was also adopted as part of classical plate theory in Section 5.2.1. 

More or less implicit in the theory of materials of this type is the assumption that all the fibers are straight and unstressed or that the initial stresses in the individual fibers are essentially equal. In practice this is quite unlikely to be true. It is expected, therefore, that as the load is increased some fibers will reach their breaking points first. As they fail, their loads will be transferred to other as yet unbroken fibers, so that the successive breaking of fibers rather than the simultaneous breaking of all of them will cause failure. As reviewed in Chapter 2 (SHORT TERM LOAD BEHAVIOR, Tensile Stress-Strain, Modulus of elasticity) the result is usually the development of two or three moduli. 

It is also possible to estimate the cross-link density from the stress-strain data, using the statistical theory of rubber-like elasticity [47,58]. For a swollen rubber the relationship is... 

Figure 18.17 shows that the characteristics of the stress-strain curve depend mainly on the value of n the smaller the n value, the more rapid the upturn. Anyway, this non-Gaussian treatment indicates that if the rubber has the idealized molecular network strucmre in the system, the stress-strain relation will show the inverse S shape. However, the real mbber vulcanizate (SBR) that does not crystallize under extension at room temperature and other mbbers (NR, IR, and BR at high temperature) do not show the stress upturn at all, and as a result, their tensile strength and strain at break are all 2-3 MPa and 400%-500%. It means that the stress-strain relation of the real (noncrystallizing) rubber vulcanizate obeys the Gaussian rather than the non-Gaussian theory. 

Stress upturn in the stress-strain relation of carbon black-fiUed rubbers can be reasonably revealed in terms of the non-Gaussian treatment, by regarding the distance between adjacent carbon particles as the distance between cross-links in the theory. 

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. 

Work done by L. Mullins on the prestressing of filler-loaded vulcanisates showed that such prestressing gives a stress-strain curve approaching that of an unfilled rubber. This work has thrown much light on so called permanent set and the theory of filler reinforcement. See Stress Softening. 

I want to emphasize, above all, that these theories of matter as stresses, strains, singularities, or vortices of ether, were mechanical (and even hydro-dynamic) theories. When scientists such as Crookes and Lodge, and Theoso-phists such as Besant and Leadbeater, melded physics with spiritual and psychic forces via theories of the ether (and the additional particles that Theosophy added to the equation), they were lending scientific credibility to spiritual ideas. Paradoxically, in their critique of scientific materialism, they asserted a mechanical theory of spirituality. Theosophy thus required a form of vitalism to counterbalance the mechanistic tendencies of its physics. 

In the [ 45]j tensile test (ASTM D 3518,1991) shown in Fig 3.22, a uniaxial tension is applied to a ( 45°) laminate symmetric about the mid-plane to measure the strains in the longitudinal and transverse directions, and Ey. This can be accomplished by instrumenting the specimen with longitudinal and transverse element strain gauges. Therefore, the shear stress-strain relationships can be calculated from the tabulated values of and Ey, corresponding to particular values of longitudinal load, (or stress relations derived from laminated plate theory (Petit, 1969 Rosen, 1972) ... 

Figure 10. Dependence of the reduced equilibrium modulus of POP triol - MDI networks prepared in the presence of diluent. POP triol Mu= 708 stress-strain measurements in the presence of diluent (o) and after evaporation of the diluent ( ). Flory theory for the values of the front factor A indicated, theoretical dependence including trapped interchain constraints Numbers at curves Indicate the value of ry.

Furnace black-reinforced EPDM and NBR blends were compounded with different concentrations of azodicarbonamide foaming agent to produce EPDM and NBR foamed composites. All the mechanical parameters measured were found to decrease as the foaming agent concentration and/or temperature increased. The stress-strain results were discussed with reference to the continuum mechanics theory for compressible materials. 16 refs. 

The mechanical response of polypropylene foam was studied over a wide range of strain rates and the linear and non-linear viscoelastic behaviour was analysed. The material was tested in creep and dynamic mechanical experiments and a correlation between strain rate effects and viscoelastic properties of the foam was obtained using viscoelasticity theory and separating strain and time effects. A scheme for the prediction of the stress-strain curve at any strain rate was developed in which a strain rate-dependent scaling factor was introduced. An energy absorption diagram was constructed. 14 refs. 

The value of Ci is obtained from the plot of o/2(X - A ) vs. 1A and extrapolating to 1A = 0. By comparison with the theory of elasticity, it has been proposed that Cl = 1/2 NRT, where N is cross-link density, R the gas constant, and T the absolute temperature (of the measurement). To assure near-equilibrium response, stress-strain measurements are carried out at low strain rate, elevated temperature, and sometimes in the swollen state. °... 

---