Initial Plastic Strain Rate Research Articles

Numerical studies of self-organization of shear bands in one and two dimensions

In the present manuscript we perform a numerical analysis of the self-organization processes of adiabatic shear bands formation in depleted uranium, aluminum alloy, and high-strength steel in one- and two-dimensional cases. Since the processes of shear bands formation are strongly nonlinear, three materials with significantly different parameters considered allow us to generalize obtained dependencies. For all materials, we investigate the evolution of stress, temperature and velocity fields from initial to final stage of localization. It is well known that localization processes occur during the high shear rate loads in the presence of initial microstructural defects in metallic materials. Starting with a random distribution of initial stress, which models microstructural defects, we obtained new dependencies for such significant parameters of the problem considered such as localization time and spacing between shear bands. The influence of initial plastic strain rate on these parameters was also studied. We obtain the linear dependence of localization time on initial strain rate and show that spatial dimension significantly influences on its value. Also, we introduce the method of shear bands selection during computations and present new statistical distributions for band spacing. We show that the interaction between shear bands significant influences on the spacing between them.

Numerical Simulation of Adiabatic Shear-Band Formation in Composites

The process of plastic flow localization in a composite material consisting of welded steel and copper plates under shear strain is considered. The mathematical model of this physical process is formulated. A new numerical algorithm based on the Courant–Isaacson–Rees scheme is proposed. The algorithm is verified using three test problems. The algorithm efficiency and performance are proved by the test simulations. The proposed algorithm is used for numerical simulation of plastic strain localization on composite materials. The influence of boundary conditions, the initial plastic strain rate, and the width of the materials forming the composite bar on the localization process is studied. It is demonstrated that at the initial stage, the shear velocity for the material layers varies. Theoretical estimates of the oscillation frequency and period are proposed; the calculations using these estimates agree completely with the numerical experiments. It is established that the deformation is localized in the copper part of the composite. One or two localization regions situated at a typical distance from the boundaries are formed, depending on the width of the steel and copper parts, as well as the initial plastic strain rate and the chosen boundary conditions. The dependence of this distance on the initial plastic strain rate is demonstrated and corresponding estimates for boundary conditions of two types are obtained. It is established that in the case of two localization regions, the temperature and deformation in one of them increase much faster than in the other, while at the initial stage these quantities are nearly equal in both regions.

Invariability of rate dependences of normalized flow stress in niobium and molybdenum under conditions of shock compression

The evolution of elastic-plastic shock waves has been studied in pure molybdenum and niobium at normal and elevated temperatures over propagation distances ranging from 0.03 to 5 mm. The experiments revealed that annealing of the metals substantially increases their Hugoniot elastic limits and, to a lesser degree, their spall strengths. Variations in the resistance of both the metals to fracture in tension with the test temperature can be described as modest. Measuring the decay of the elastic precursor waves with a propagation distance in the two metals has allowed a determining of the relationships between a flow stress τ and an initial plastic strain rate γ̇p. It was found that, at the plastic strain rates greater than 3÷4×104s−1, the temperature sensitivity of the transient values of τ is much lower than that at the strain rates below this range. The τ(γ̇p) data normalized on shear moduli of the metals have been approximated by simple functions that, despite substantial differences between the moduli and yield stresses, were found to be virtually identical for the two metals.

Numerical Simulation of Adiabatic Shear Bands Formation in Composites

The process of plastic flow localization under shear deformations of a composite material consisting from welded steel and copper is studied. A mathematical model describing this physical process is proposed. A new numerical approach based on Courant–Isaacson–Rees scheme is suggested. This algorithm was verified using three benchmark problems. Operability and effectiveness of this algorithm is confirmed. A numerical simulation of plastic flow localization in composite materials is performed. The influence on localization process of boundary conditions, of initial strain rate and materials width is studied. It is shown that at the initial stage the shear velocity of material layers oscillates. Theoretical estimates of frequency and oscillation period is given. Computational results coincide with these estimates. It is found that plastic flow localizes in the copper part of the composite. One or two areas of plastic flow localization appears depending on the width of steel and copper parts, as well as on the initial plastic strain rate and the selected type of a boundary conditions. The areas locate on characteristic distance from borders. The dependence of this distance and initial strain rate is shown and the corresponding estimates are obtained for two types of boundary conditions. When two areas of localization are formed, in one of them the temperature and the deformation increas faster than in another one.

Rate and temperature dependences of the yield stress of commercial titanium under conditions of shock-wave loading

The evolution of elastic-plastic shock waves with the propagation distance has been studied in annealed titanium of commercial purity at temperatures 20 °C and 600 °C. The free surface velocity histories of the shock-loaded samples 0.25–4.0 mm in thickness have been recorded using the Velocity Interferometer System for Any Reflector. The measured decays of the elastic precursor waves have been converted into relationships between the shear stress and the initial plastic strain rate at the Hugoniot elastic limit. It has been found that the temperature practically does not influence on the resistance to high-rate plastic deformation: the plastic strain rate varies with the shear stress as γ̇=2.5×106(τ/τ0)4.8 s−1 at 20 °C and γ̇=2.9×106(τ/τ0)4.9 s−1 at 600 °C. An analysis of the rise times of the plastic shock waves has shown that for the same level of shear stress, the plastic strain rate after small compressive strain is more than by order of magnitude higher than the initial plastic strain rate at the wave's foot. Such acceleration of the plastic deformation seems to be a result of an intense multiplication of the mobile dislocations or twins.

Deformation resistance and fracture of iron over a wide strain rate range

The results of measurements of the decay of an elastic precursor in iron at the distances from 0.13 to 10 mm and the spall strength of the samples with such thicknesses have been compared with similar data for the nanometer-scale samples. The decay has been described by a unique dependence whose differentiation gives the relationship between the initial plastic strain rate in the range from 103 to 109 s−1 and the compression stress in the elastic shock wave from 1.5 to 27.5 GPa. The dynamic breaking strength (spall strength) varies in this range of shock-wave load time from 1.5 to 20 GPa.

Plastic behavior of aluminum in high strain rate regime

The aim of this experiment was to study the plastic response of Al to dynamic loading at high strain rates. Dynamic loading was applied by direct laser ablation of the sample with pulse width of 3 ns long. The free surface velocity histories of shock loaded samples, 60-310 μm thick at room temperature, and 150 μm thick with initial temperature from 293 to 893 K, have been recorded using a line velocity interferometer for any reflections (VISAR) system. The line VISAR could measure free surface velocity profile with high temporal resolution (∼100 ps). The measured amplitudes of the elastic precursor waves have been approximated by power functions of the propagation distance with the power index of 0.581, and these data have been converted into relationships between the shear stress at Hugoniot elastic limit and the initial plastic strain rate. The peak longitudinal elastic stress and the strain rate meet a power law dependency with the power index of 0.44. Samples were recovered for post-shot metallographic analysis. The metallographic analysis leads to the conclusion that the spall strength of preheated aluminum is determined more by the rate of void nucleation rather than its growth.

Response of copper to shock-wave loading at temperatures up to the melting point

The evolution of elastic-plastic shock waves as a function of the propagation distance has been studied in 99.999% purity polycrystalline copper over the 300 to 1353 K temperature range. The free surface velocity histories of shock-loaded samples 0.1 to 2.0 mm in thickness have been recorded using the velocity interferometer. The measured decay of the elastic precursor waves has been converted into relationships between the shear stress at Hugoniot elastic limit and the initial plastic strain rate. Independently of the temperature, the initial densities of mobile dislocations in a range of 2.5×106 cm−2 to 5×108 cm−2 are required to provide observed initial strain rates varied from 2.3×103 s−1 to 2×106 s−1. Above 1100 K, the shape of the elastic precursor wave changes with the appearance of a sharp spike at its front part. This change is treated in terms of nucleation and multiplication of mobile dislocations. An analysis of the rise times of the plastic shock waves has shown that for the same level of shear stress, the plastic strain rates after a 2% compressive strain increase with respect to those just behind the elastic precursor front by a factor of 300 at 300 K and by a factor of 30 at 1353 K.

Effect of temperature, strain, and strain rate on the flow stress of aluminum under shock-wave compression

The evolution of elastic-plastic shock waves with the propagation distance has been studied in 99.99% purity aluminum and in annealed 6061 aluminum alloy. The free surface velocity histories of shock-loaded samples, 0.1–2.0 mm thick and with initial temperature from 296 to 932 K, have been recorded using velocity interferometer system for any reflector (VISAR). The measured amplitudes of the elastic precursor waves have been approximated by power functions of the propagation distance, and these data have been converted into relationships between the shear stress at the top of elastic precursor wave and the initial plastic strain rate. The latter was found to decrease from 106 to 104 s−1 over 0.1 to 2-mm precursor traverse, while the density of mobile dislocations corresponding to these strain rates varied from 2 × 108 to 5 × 106 cm−2. At fixed strain rates, the flow stress of aluminum grows linearly with temperature. An analysis of the rise times of the plastic shock waves has shown that, for the same level of shear stress, the plastic strain rate at the shock front is by an order of magnitude higher and the density of mobile dislocations is 2-3 times higher than their initial values behind the elastic precursor front.

Plastic flow in shock-loaded silver at strain rates from 104 s−1 to 107 s−1 and temperatures from 296 K to 1233 K

The evolution of elastic-plastic shock waves in 99.9% purity silver samples of 0.127 to 2.0 mm thickness has been studied in a series of VISAR-instrumented planar impact experiments with initial sample temperature varied from 296 to 1233 K. The decay of elastic precursor wave at 933, 1173, and 1233 K temperatures is approximately inversely proportional to the square root of the propagation distance. The latter corresponds to the cubic dependence of initial plastic strain rate, ranged from 104 s−1 to 106 s−1, on the shear stress. At fixed strain rates, the flow stress grows linearly with the temperature but the dependence becomes stronger near the silver melting point, 1234 K. An analysis of the rise times of the plastic shock waves shows that for the same level of shear stress the plastic strain rate at the shock front is significantly higher than that at the top of the elastic precursor wave.