Generalized stress-strain relations

Many materials, particularly polymers, exhibit both the capacity to store energy (typical of an elastic material) and the capacity to dissipate energy (typical of a viscous material). When a sudden stress is applied, the response of these materials is an instantaneous elastic deformation followed by a delayed deformation. The delayed deformation is due to various molecular relaxation processes (particularly structural relaxation), which take a finite time to come to equilibrium. Very general stress-strain relations for viscoelastic response were proposed by Boltzmann, who assumed that at low strain amplitudes the effects of prior strains can be superposed linearly. Therefore, the stress at time t at a given point in the material depends both on the strain at time t, and on the previous strain history at that point. The stress-strain relations proposed by Boltzmann are [4,39] ... 

If the stresses and strains along three orthogonal axes are considered, then the general stress-strain relation can be written as,... 

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

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

This phenomenon gives us the very important information when we consider the orientation and extensibility of cross-linked molecules under extension. Generally, for the modeling or calculation for the stress-strain relation of cross-hnked molecules, the perfect regular network is adopted a... 

Moreover, we must pay attention to the points that in the cross-linked rubber, the cross-link stops the sliding of molecules and has its own excluded volume. Generally, in the calculation of the stress-strain relation, the four-chain model is used for the cross-link junction and recently the eight-chain model is considered to be more realistic and available. Thus, it is quite reasonable to consider that the bulky excluded volume that a cross-link junction possesses must be a real obstacle for the orientation of molecules, just like the case observed in Figure 18.16B. 

Ultimate properties of toughness (energy to rupture), tensile strength, and maximum extensibility are all affected by strain-induced crystallization. In general, the higher the temperature the lower the extent of crystallization and consequently the lower these stress/strain related properties. There is also a parallel result brought about by the presence of increased amounts of diluent since this also discourages stress-related crystallization. 

Figures 29 A and B show the original data from general biaxial extension measurements on the NR sample. Here the measured stresses at and a2 at a series of fixed Xx are plotted against X2. All these values are isochronal (10 min). The graphs have been displayed to illustrate the accuracy of our measurements. In Fig. 30, the observed at are compared with the predictions from other stress-strain relations.

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

For all materials (other than fabrics, for which the concept is not relevant) the basic parameter is a measure of stiffness or modulus derived from the stress-strain curve. As with tensile tests, because the stress -strain relation is generally not linear, care must be taken to compare only measures of stiffness defined in the same way. With rigid foams and plastics there are additionally measures of yield or strength. 

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. 

In Eq. (46), the x(t) are the spatial (deformed) coordinates and the partial differentiation is performed with respect to the material (undeformed) coordinates. Expression (45) was introduced by Blatz et al. and Ogden, independently, who adopted the idea of a generalized strain measure to predict stress-strain relations of crosslinked samples of elastomers under various types of deformation. Representation in the principal axes system yields for the components of the Lagrangian stress tensor... 

For a unidirectional laminate the elastic stress-strain relations define an orthotropic material for which the generalized form of Hooke s Law, relating the stress o to the strain e,... 

Argyris JH (1954) Energy theorems and structural analysis a generalized discourse with applications on energy principles of structural analysis including the effects of temperature and non-linear stress-strain relations part I. Aircr Eng Aerosp Technol 26(10) 347-356... 

When a composite is treated as an anisotropic elastic material, the stress-strain relations are based on a general approach, which is valid in a region of small deformations. These theoretical relations were developed for application... 

The general factorable single integral constitutive equation is an equation that describes well the viscoelastic properties of a large class of crosslinked rubbers > and the elasticoviscous properties of many polymer melt8 under various types of deformation An appropriate way to compare the stress-strain relation for cured elastomers and polymer melts therefore is to calculate the strain-dependent function contained in this constitutive equation from experimental results and to compare the strain measures so obtained. 

For a general state of stress and deformation at a material point, how are individual components of plastic strain rate related to stress components in this framework An answer is provided through the work of Rice (1970) on the general structure of stress-strain relations for time-dependent plastic deformation. In the present setting, it is most conveniently expressed in terms of deviatoric stress components Sij defined in terms of stress in... 

The latter equation is the uniaxial stress-strain relation for a polymer analogous to Hooke s law for a material that is time independent but is valid only for the case of a constant input of strain. The relaxation test provides the defining equation for the material property identified as the relaxation modulus. More general differential and integral stress-strain relations for an arbitrary loading will be developed in later Chapters. 

In general 3D condition, the stress-strain relation is given by the scalar damage elasticity equation ... 

We can also use the stress-strain relations, Eq. (E.32), and the general expression Eq. (E.26), to arrive at the following expression for the strain energy density of the... 

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 general formulation of the evolution of linear isotropic viscoelastic material in time is governed by the stress-strain relation, in which the stress can be expressed as a convolution of the strain rate with a relaxation function as in... 

---