ρ z; t ð Þ¼ρ0 exp εz z; t ½ ð ð ÞÞ 2υ z;t ð Þ 1 ð34:4Þ This equation is proposed assuming the principle of mass conservation, and allows for the calculation of the local mass density, ρ(z, t), as a function of initial density, ρ0, local axial compressive strain, εz(z, t), and local Poisson’s ratio, υ(z, t). Local Poisson’s ratio can also be evaluated using the local radial and axial strain values as: υ z; t ð Þ¼ dεr z; t ð Þ dεz z; t ð Þ ð 34:5Þ Further details on the compressibility model used in the present work can be found in Ref. [7]. 34.5 Full-Filed Deformation Response Considering the longitudinal elastic wave speed in the material (cl ¼1180 m/s), specimen characteristic time, i.e. time for a single stress wave traverse along the axis, is calculated as 16μs for the short specimen and 22μs for the long specimen. It is well-established that at least three consecutive stress wave reverberations are required before the quasi-static equilibrium is achieved in a dynamically deformed material [10]. Accordingly, minimum durations of 48μs and66μs are required for the stress to equilibrate in the short and long specimens, respectively. Failure in the long specimen initiates well before stress equilibrium. Such a condition makes the conventional way of determining the constitutive response based on boundary measurements quite inaccurate, thus necessitating the inclusion of inertia loading effects during the transient stress state. Although less stress variability along the axis of the short specimen is expected due to the short time required to reach equilibration, it will still be necessary to take into account the development of inertia stresses to acquire more meaningful and more accurate constitutive information for this specimen. Axial strain distribution over the gauge area is determined using the displacement fields obtained from DIC. Figure 34.4 shows typical axial strain maps obtained from DIC. Considerable spatial variability in the axial strain is observed in both specimens, while the degree of this spatial variability is lower in the short specimen. In both specimens, the highest magnitude of strain is developed at the middle sections of each specimen. To explain this observation, note that the applied impact produces two distinct waves: (1) an elastic wave traveling at the speed of sound in the specimen, and (2) a plastic wave moving at a significantly slower velocity. The elastic wave created at the instant of impact does not plastically deform the material. However, the plastic wave traverses over the length of the specimen reducing its velocity to zero while plastically deforming the material. The stress developed behind the plastic wave deforms the material axially, as well as laterally and forms the mushroom configuration at the impacted side of the specimen, as clearly shown in Fig. 34.4. Furthermore, the highest strain magnitudes are also developed at the location of the plastic wave front. Fig. 34.4 Full-field axial strain maps obtained for (a) short and (b) long specimens at different times after the impact 256 B. Koohbor et al.
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