Stress relation with strain

To obtain a stress-strain relation with this formalism, it is necessary to keep track of the effect of the deformation as it is performed on each chain y, y = 1,..., Nc, in the system of Nc chains with controlled chain vectors. In the initially generated systems, the chain vectors R(y) have an isotropic distribution and the initial stress jf = Yly=i <7y() ) where ft iy) is the stress contribution of... 

Figure 5 demonstrates the different behaviour resulting from Eqs. (49) and (54) in the case of uniaxial compression. We also tested the elastic potential (Eq. (44)) in the two cases v = 1/2 and v = —1/4 by comparing the corresponding stress-strain relations with biaxial extension experiments which cover relatively small as well as large deformation regions for an isoprene rubber vulcanizate. In the rectan-... 

Since the laminate comprises a number of laminae oriented in different directions with respect to each other, having the same stress-strain relations, the stress-strain equation of the /cth layer of the laminate is as given by Hull [4] as ... 

Calculations of the full stress tensor is a method ideally suited to the derivation of elastic constants, since it contains up to six independent pieces of information that otherwise would require extensive calculations of total energy. The c- and c. 2 elastic constants can be found from the stress-strain relation with the application of an ei-strain. (The Voigt notation is used, see e.g. (Nye, 1957), i.e. 11- -1, 22- 2, 33- 3, 23 4, 13 5, 12- -6 thus... 

The different forms of the modulus are still widely disputed and different opinions are expressed in the literature. Hsu and Mark " " prepared networks of polybutadiene by endlinking. According to dynamic measurements polybutadiene has a high plateau modulus in viscoelasticity and one can expect a strong contribution to the modulus from entanglements. In this study experiments have been fitted to the stress-strain relations with the Flory constraint-fluctuation model as in ref. 245. The authors concluded that there is no contribution from entanglements to the modulus. The same conclusion was drawn in refs. 245-248. 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

In a given motion, a particular material particle will experience a strain history The stress rate relation (5.4) and flow rule (5.11), together with suitable initial conditions, may be integrated to obtain the eorresponding stress history for the particle. Conversely, using (5.16) instead of (5.4), may be obtained from by an analogous ealeulation. As before, may be represented by a continuous curve, parametrized by time, in six-dimensional symmetric stress spaee. 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

With the foregoing reduction from 36 to 21 independent constants, the stress-strain relations are... 

If there are two orthogonal planes of material property symmetry for a material, symmetry will exist relative to a third mutually orthogonal plane. The stress-strain relations in coordinates aligned with principal material directions are... 

However, as mentioned previously, orthotropic laminae are often constructed in such a manner that the principal material coordinates do not coincide with the natural coordinates of the body. This statement is not to be interpreted as meaning that the material itself is no longer orthotropic instead, we are just looking at an orthotropic material in an unnatural manner, i.e., in a coordinate system that is oriented at some angle to the principal material coordinate system. Then, the basic question is given the stress-strain relations In the principal material coordinates, what are the stress-strain relations in x-y coordinates ... 

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. 

The material cannot be described with linear elastic stress-strain relations... 

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... 

Once the deflections are known, the stresses are straightforwardly obtained by substitution in the stress-strain relations. Equation (4.16), after the strains are found from Equation (4.12). Note that the solution in Equation (5.31) is expressed in terms of only the laminate stiffnesses D., Di2. D22. and Dgg. This solution will not be plotted here, but will be used as a baseline solution in the following subsections and plotted there in comparison with more complicated results. 

The macromechanical behavior of a lamina was quantitatively described in Chapter 2. The basic three-dimensional stress-strain relations for elastic anisotropic and orthotropic materials were examined. Subsequently, those relations were specialized for the plane-stress state normally found in a lamina. The plane-stress relations were then transformed in the plane of the lamina to enable treatment of composite laminates with different laminae at various angles. The various fundamental strengths of a lamina were identified, discussed, and subsequently used in biaxial strength criteria to predict the off-axis strength of a lamina. 

The basic approaches as summarized by Ashton and Whitney [6-31] will now be discussed. First, a symmetric laminate with orthotropic laminae having principal material directions aligned with the plate axes will be treated. The transverse normal strain can be found from the orthotropic stress-strain relations, Equation (2.15), as... 

Designers unfamiliar with plastic products can use the suggested preliminary safety factor guidelines in Table 2-11. They provide for extreme safety. Any product designed with these guidelines in mind should conduct tests on the products themselves to relate the guidelines to actual performance (Chapter 4, RP PIPES, Stress-Strain Curves). With more experience, more-appropriate values will be developed targeting to use 1.5 to 2.5. After field service of... 

Figure 18.1 is the typical stress-strain curves of the filled rubber (SBR filled with fine carbon black, HAF),

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. 

Actually, some fluids and solids have both elastic (solid) properties and viscous (fluid) properties. These are said to be viscoelastic and are most notably materials composed of high polymers. The complete description of the rheological properties of these materials may involve a function relating the stress and strain as well as derivatives or integrals of these with respect to time. Because the elastic properties of these materials (both fluids and solids) impart memory to the material (as described previously), which results in a tendency to recover to a preferred state upon the removal of the force (stress), they are often termed memory materials and exhibit time-dependent properties. 

There are many types of deformation and forces that can be applied to material. One of the foundations of viscoelastic theory is the Boltzmann Superposition Principle. This principle is based on the assumption that the effects of a series of applied stresses acting on a sample results in a strain which is related to the sum of the stresses. The same argument applies to the application of a strain. For example we could apply an instantaneous stress to a body and maintain that stress constant. For a viscoelastic material the strain will increase with time. The ratio of the strain to the stress defines the compliance of the body ... 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

In Eq. (1), a is the equilibrium stress (Nm 2) supported by the swollen specimen a is the stretched specimen length divided by the unstretched length (extension ratio) v2 is the volume fraction of dry protein and p is the density of dry protein. In the common case of tetrafunctional crosslinks, the concentration of network chains n (mol network chains/g polymer) is exactly one-half the concentration of crosslinks, so that n = 2c. The hypothesis that a specimen behaves as if it were an ideal rubber can be confirmed by observing a linear relation with zero intercept between a and the strain function (a — 1/a2) and by establishing a direct proportionality between a and the absolute temperature at constant value of the extension ratio, as stipulated by Eq. (1). 

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