Stress volume relation

What are the characteristic mechanical responses of solids to shock loading This question is most clearly addressed through the relation between stress-volume relations and wave structures. 

Figure 1.1 shows a typical stress-volume relation for a solid which remains in a single structural phase, along with a depiction of idealized wave profiles for the solid loaded with different peak pressures. The first-order picture is one in which the characteristic response of solids depends qualitatively upon the material properties relative to the level of loading. Inertial properties determine the sample response unlike static high pressure, the experimenter does not have independent control of stresses within the sample. 

Fig. 2.2. The characteristic stress pulses produced by shock loading differ considerably depending upon the stress range of the loading. The first-order features of the stress pulses can be anticipated from critical features of the stress-volume relation. In the figure, P is the applied pressure and HEL is the Hugoniot elastic limit. Characteristic regimes of materials response can be categorized as elastic, elastic-plastic, or strong shock.

The rise times of the elastic wave may be quite narrow in elastic single crystals, but in polycrystalline solids the times can be significant due to heterogeneities in physical and chemical composition and residual stresses. In materials such as fused quartz, negative curvature of the stress-volume relation can lead to dispersive waves with slowly rising profiles. 

Within the elastic regime, the conservation relations for shock profiles can be directly applied to the loading pulse, and for most solids, positive curvature to the stress volume will lead to the increase in shock speed required to propagate a shock. The resulting stress-volume relations determined for elastic solids can be used to determine higher-order elastic constants. The division between the elastic and elastic-plastic regimes is ideally marked by the Hugoniot elastic limit of the solid. 

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

Fig. 2.9. The measured stress-volume relation of shock-loaded sapphire reveals a substantial reduction in strength, but a small finite strength is retained. The reduction in strength is indicated by the small high pressure offset between the static and shock data, and from extrapolation of high pressure shock data to atmospheric pressure conditions (Graham and Brooks [71G01]).

Fig. 5.9. The piezoelectric polarization of Fig. 5.7 is found to increase with volume compression. Hence, the large decrease in change of charge with stress is a manifestation of the highly nonlinear stress-volume relation of PVDF, not nonlinear piezoelectricity.

Curran [61C01] has pointed out that under certain unusual conditions the second-order phase transition might cause a cusp in the stress-volume relation resulting in a multiple wave structure, as is the case for a first-order transition. His shock-wave compression measurements on Invar (36-wt% Ni-Fe) showed large compressibilities in the low stress region but no distinct transition. 

Fig. 5.13. The stress-volume relations of fee and bee alloys show the strong compressibility anomaly in the fee phase below 25 kbar (2.5 GPa) associated with the magnetic interactions. Above 25 kbar, the fee alloy has a normal value for compressibility (after Graham et al. [67G01]).

The resulting stress-volume relations for the 28.5-at. % Ni alloys are shown in Figure 5.13. The cusp in the fee curve at 430 MPa (4.3 kbar) is the mean value observed for the Hugoniot elastic limit, whereas the dashed line shown for the fee alloy indicates the stress region for which some strain hardening is indicated from the stress profiles. It is readily apparent that below 2.5 GPa (25 kbar) the fee alloy shows a much larger compressibility than the bcc alloy. 

Conservation relations are used to derive mechanical stress-volume states from observed wave profiles. Once these states have been characterized through experiment or theory they may, in turn, predict wave profiles for the material in question. For the case of a well-defined shock front propagating at constant speed L/ to a constant pressure P and particle velocity level u, into a medium at rest at atmospheric pressure, with initial density, p, the conservation of momentum, mass, and energy leads to the following relations ... 

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

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. 

For more information on Pulay stress and related complications associated with finite sets of plane waves and k points when calculating forces in supercells with varying volumes, see G. P. Francis and M. C. Payne, /. Phys. Condens. Matter 2 (1990), 4395. 

Bremner, J.D., Randall, P., Vermetten, E., Staib, L., Bronen, R.A., Mazure, C., et al. (1997) Magnetic resonance imaging—based measurement of hippocampal volume in posttraumatic stress disorder related to childhood physical and sexual abuse—a preliminary report. Biol Psychiatry 41 23-32. 

The fundamental equations treated in structural analyses are the mechanical equilibrium, strain-displacement relation, and stress-strain relation. The equilibrium equations in an elementary volume can be expressed ... 

Meier (9) has modeled the spherical domain morphology by a simple cubic lattice in which domains are arranged on the lattice sites. The tie molecules run between nearest-neighbor domains and are assumed to be confined by pairs of infinite, parallel walls. The extension ratio for the interdomain region is set equal to the macroscopic extension ratio divided by the volume fraction of the interdomain material. The ratio of the initial interdomain dimension to the domain dimension is set equal to the ratio of the volume fractions of the interdomain and domain material. Using this three-chain model, Meier calculates the stress-strain relation by differentiating his entropy expression with respect to the interdomain extension ratio. The Meier calculation has some difficulties the interdomain deformation fails to vanish in the absence of an applied macroscopic deformation the relation between the ratio of the domain dimension to the initial interdomain dimension and the ratio of volume fractions is incorrect and the differentiation should be carried out with respect to the macroscopic extension ratio. 

Gaylord and Lohse (10) have calculated the stress-strain relation for cilia and tie molecules in a spherical domain morphology using the same type of three-chain model as Meier. It is assumed that the overall sample deformation is affine while the domains are undeformable. It is predicted that the stress increases rapidly with increasing strain for both types of chains. The rate of stress rise is greatly accelerated as the ratio of the domain thickness to the initial interdomain separation increases. The results indicate that it is not correct to use the stress-strain equation obtained by Gaussian elasticity theory, even if it is multiplied by a filler effect correction term. No connection is made between the initial dimensions and the volume fractions of the domain and interdomain material in this theory. 

The strain to- fracture is expectai to be related to the craze flow stress, with craze fibril breakdown occurring more rapidly at higher stresses. Since the flow stresses are related to the volume fraction of PB through the craze growth velocities, the strains to fracture are also expected to relate to the rubber content. 

The forces acting on a surface are due to pressure and viscous effects. In two-dimensional flow, the viscous stress at any point on an imaginary surface within the fluid can be resolved into two perpendicular components one normal to the surface called normal stress (which should not be confused with pressure) and another along the surface called shear stress. The normal stress is related to the velocity gradients dir/d.v and Ou/dy, that are much smaller than du/dy. to which shear stress is related. Neglecting the nonnal stresses for simplicity, the surface forces acting on the control volume in the. v-direciion arc as shown in Fig. 6 -25. Then the net surface force acting jii the.t-direction becomes... 

As before, the convention used here is that T denotes the total stress in the j direction exerted on a plane perpendicular to the i axis. Recall that the shear stresses are related to the time rate of change of the shearing deformation of the fluid element, whereas the normal stresses are related to the time rate of change of volume of the fluid element. As a result, both shear and normal stresses depend on velocity gradients in the flow. In most viscous flows, normal stresses are much smaller than shear stresses and usually neglected. The normal stresses may however become important when the normal velocity gradients are very large. 

Figure 4 illustrates the typical volume dilatation-strain behavior along with its first and second derivatives. Clearly these measures are realistic in that the derivatives do take on the character of cumulative and instantaneous frequency distributions. Similar models can be constructed to relate the loss in stiffness to the number of vacuoles that have formed resulting in very simple but accurate stress-strain relations (1). 

The treatment of mechanical deformation in elastomers is simplified when it is realized that the Poisson ratio is almost 0.5. This means that the volume of an elastomer remains constant when deformed, and if one also assumes that it is essentially incompressible (XjXjXj = 1), the stress-strain relations can be derived for simple extension and compression using the stored energy fimction w. 

It also leads to certain quantitative predictions on the stress-strain relations of filaments taken at various degrees of swelling which appeared to be particularly well fulfilled It is further capable of explaining in first approximation the course of the volume changes observed on stretching swollen filaments at various degrees of swelling. 

The elastic stress is related to the change in the free energy si (per volume) for a virtual deformation 6e p as... 

Fig. 9.2 Calculated compressive stress-strain relations as grain size decreases from 20 pm to 10 nm at low porosity levels a Cpore % 6 % and b Cpore % 11 % [13]. With kind permission of Elsevier and Dr. Weng (Note that Cpp e indicates volume concentration of pores)...

Based on a convention consolidation experiment conducted in a geotechnical Lab, the following relation between the void ratio e and the effective volume stress cr, can be found as < (,-< = log in which e and C(, are current and initial void ratios, respectively cr and are the current and initial effective volume stresses (> 0) denotes the slope of the c-log(o ) curve. This relation, after divided by the term 1 + < (,. can be alternatively expressed as a volume stress-strain relation (i.e., cr, vs. ,) in a form of an exponential function ... 

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