Hugoniot elastic limit

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

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.

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. 

Table 2.1. Third-order elastic constants determined from Hugoniot elastic limits (after Davison and Graham [79D01]).

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. 

The thorough and persistent work on precursor decay (the dependence of Hugoniot elastic limit on propagation distance) of Duvall s Washington State University group was successful in demonstrating that precursor attenuation was due to both stress relaxation and hydrodynamic attenuation. Typical data on crystalline LiF is shown in Fig. 2.7. Observed plastic strain... 

Table 2.3. Selected values of Hugoniot elastic limits. (See Jones and Graham [71J02], Gust [80G03].)...

Perhaps the most dramatic exception to the perfectly elastic, perfectly plastic materials response is encountered in several brittle, refractory materials that show behaviors indicative of an isotropic compression state above their Hugoniot elastic limits. Upon yielding, these materials exhibit a loss of shear strength. Such behavior was first observed from piezoelectric response measurements of quartz by Neilson and Benedick [62N01]. The electrical response observations were later confirmed in mechanical response measurements of Waekerle [62W01] and Fowles [61F01]. 

Fig. 2.10. Certain high strength solids with low thermal conductivity show a loss or reduction of shear strength when loaded above the Hugoniot elastic limit. The idealized behavior of such solids upon loading is shown here. The complex, heterogeneous nature of such yield phenomena probably results in processes that are far from thermodynamic equilibrium.

Fig. 2.12. If solids undergo a shock-induced polymorphic transformation, the volume change at the transformation causes significant changes in the wave profile produced by shock loading. In the figure, is the applied pressure, Pj is the pressure of the phase transition, and HEL is the Hugoniot elastic limit.

The sample-polarity anomaly in current pulses from x-quartz shocked above the Hugoniot elastic limit gave the first indication of unusual conduc-... 

The piezoelectric response investigation also provides direct evidence that significant inelastic deformation and defect generation can occur well within the elastic range as determined by the Hugoniot elastic limit. In quartz, the Hugoniot elastic limit is 6 GPa, but there is clear evidence for strong nonideal mechanical and electrical effects between 2.5 and 6 GPa. The unusual dielectric breakdown phenomenon that occurs at 800 MPa under certain... 

Fig. 5.1. The electrostatic configurations of the Neilson-Benedick three-zone model describe a piezoelectric solid subject to elastic-inelastic shock deformation which divides the crystal into three distinct zones. Zone 1, ahead of the elastic wave, is unstressed. Zone 2 is elastically stressed at the Hugoniot elastic limit. Zone 3 is isotropically pressurized to the input pressure value (after Graham [74G01]).

A limited number of minus-x orientation samples were impact loaded in the vicinity of the Hugoniot elastic limit at stresses from 5.9 to 6.7 GPa. The principal observation of these experiments was that positive currents were observed from negative polarity disks when a stress of 5.9 GPa was exceeded. Such an observation confirms that quartz responds as predicted by the model, and that the elastic limit is in the vicinity of 6 GPa. 

Observations of current pulses from shock-loaded, x-cut quartz in the vicinity of and above the Hugoniot elastic limit provided rather remarkable confirmation of the nature of the phenomena resulting from mechanical yielding and shock-induced conduction. Lithium niobate provides another opportunity to test the generality of the models. 

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. 

Kato et al. [26] directly observed the shock-pressure history by using an in-material piezoresistive carbon gauge to study the ice Hugoniot in detail below 1 GPa. They found that the HEL (Hugoniot Elastic Limit) of ice was between... 

The choice of a particular simulation timescale enables determination of the velocity of the first shock wave and the thermodynamic state where the transition between the first and second waves occurs. For example, simulations performed for 60ps show plastic deformation for a 2.815 km/sec shock speed but no plastic deformation for 2.8 km/sec. Therefore the first (elastic) wave speed for a double wave simulation is 2.815 km/sec and the thermodynamic state at the transition between the two waves is the state where the slowest volume change occurs in the elastically compressed portion of the 2.815 km/sec simulation. (These choices were utilized to produce Figure 9, to be discussed later.) The dependence of the Hugoniot elastic limit on the simulation time is discussed in more detail in the next section. 

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