Strain rate relations

An analogous expression in the stress description is derived by inserting the strain rate relation (5.16) into the expression for the loading function (5.8) and using (5.61) and (5.62) to obtain... 

The remaining equations of the referential formulation may be translated into spatial terms by proceeding in the same way. Using (A.36) and (A.45) in the strain rate relation (5.131)... 

Equation (4) can be used to compare the change in creep behavior at zero time and for an infinitely long time. Consider that a composite is initially loaded at an infinitely rapid rate to a constant creep stress [Pg.170]

For very high velocities, the reduced creep resistance and the forming strain rate-related hardening properties necessitate the inclusion of material behavior. 

An interesting three-parameter model (the Burger model has four parameters) was proposed by Hsueh [6] and is shown in Fig. 3b. He demonstrated that for a Hookean elastic element (Ei) in series with a Kelvin solid (E2,ry), the stress-strain rate relations for constant strain rate and constant stress creep tests are,... 

Norton s power law and an Arrhenius-type equation are the most common expressions describing the stress and temperature dependence of the steady-state creep rate. Of the many relations suggested to describe creep-rupmre life, the steady-state strain-rate relation was extended for the calculation of rupture strength and is used to predict service lifetimes. The Norton Bailey concept is used in regard to many solids, including ceramics (e.g., see Headrick et al. [50]). 

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... 

Leroueil, S., Kabbaj, M., Tavenas, F. and Bouchard, R. Stress tiain strain rate relation for the compressibility of sensitive natural clays. Geotechnique, 35(2) 159 180,1985. 

To obtain the constant strain rate relation, the above constitutive equation is solved using the condition, de/df = R = constant, to result in ... 

In 1954, Bagnold performed one of the earliest quantitative studies of granular materials [170]. He measured the stress versus strain-rate relation for a neutrally buoyant suspension of spheres in a Couette shear device and found two regimes... 

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... 

Extensional flows occur when fluid deformation is the result of a stretching motion. Extensional viscosity is related to the stress required for the stretching. This stress is necessary to increase the normalized distance between two material entities in the same plane when the separation is s and the relative velocity is ds/dt. The deformation rate is the extensional strain rate, which is given by equation 13 (108) ... 

A sliding plate rheometer (simple shear) can be used to study the response of polymeric Hquids to extension-like deformations involving larger strains and strain rates than can be employed in most uniaxial extensional measurements (56,200—204). The technique requires knowledge of both shear stress and the first normal stress difference, N- (7), but has considerable potential for characteri2ing extensional behavior under conditions closely related to those in industrial processes. 

From the second of these, using (5.62) and (5.I82), the inelastic contribution to the stress rate and the inelastic contribution to the strain rate are related by... 

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. 

Since the yield function is independent of p, the yield surface reduces to a cylinder in principal stress space with axis normal to the 11 plane. If the work assumption is made, then the normality condition (5.80) implies that the plastic strain rate is normal to the yield surface and parallel to the II plane, and must therefore be a deviator k = k , k = 0. It follows that the plastic strain is incompressible and the volume change is entirely elastic. Assuming that the plastic strain is initially zero, the spherical part of the stress relation (5.85) becomes... 

An explicit relation for the plastic strain rate may be obtained by using (5.80) through (5.82). The partial derivatives of/are, from (5.92)... 

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... 

The objectivity of the spatial stress rate relation (5.154) may be investigated by applying the coordinate transformation (A.50) representing a rotation and translation of the coordinate frame. The spatial strain and its convected rate are indifferent by (A.58) and (A.62). So are the stress and its Truesdell rate. It is readily verified from (5.151), (5.152), and the fact that K has been assumed to be invariant, that k and its Truesdell rate are also indifferent. Using these facts together with (A.53) in (5.154) with c and b given by (5.155)... 

To tackle any of these we need constitutive equations which relate the strain-rate e or time-to-failure tf for the material to the stress ct and temperature T to which it is exposed. These come next. 

However, for the high strain rates appropriate for the analysis of typical extrusion and injection moulding situations it is often found that the simple Power Law is perfectly adequate. Thus equations (5.22), (5.23) and (5.27) are important for most design situations relating to polymer melt flow. 

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

A common feature of the three PTEB samples is that the yield stress decreases as the drawing temperature increases (Table 2), whereas it does not change significantly with the strain rate. The Young modulus does not change with the strain rate but it decreases and the break strain increases as the drawing temperature increases. The main conclusion is that the behavior of PTEB-RT is intermediate between the other two samples, with the advantage of a considerable increase in the modulus in relation to sample PTEB-Q and without much decrease in the break strain (Table 2). 

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