A Shear Compression Disk Specimen with Controlled Stress Triaxiality Under Dynamic Loading

Abstract

This paper presents an experimental and numerical study of the potential of the Shear Compression Disk specimen (SCD) to characterize the plastic flow and fracture of metals under various levels of stress triaxiality at strain rates of up to 10⁴ 1/s. The main loading mode in that specimen is shear with triaxiality ranging from 0 to -0.8. The specimen is relatively small and fits into a standard split Hopkinson pressure bar system. Aluminum 7075-T651 alloy was chosen for a test case study. Experimental and numerical investigations reveal the adequacy of the SCD specimen for the study of mechanical properties of materials under high strain-rates and low, though wide, range of stress triaxialities.

Appendix A: A Method for Constitutive Relation Extraction

As an additional confirmation of the obtained stress–strain curve for the high strain-rate loading, the experimental results were interpreted directly using a similar procedure to the one which is applied for SCS specimens impacted in a SHPB apparatus. The procedure requires a transformation of the axial force and displacements on the SCD faces to the averaged von Mises stress and averaged total strain of the working line of the gauge (Fig. 3(a)). The procedure and its results are detailed here.

For that transformation the following derivation and relations have been used. We first verify numerically that, during impact the averaged Mises stress and the averaged total equivalent strain along the midline of the gauge (Fig. 3(a)), represent the $\sigma - \varepsilon$ curve which was inputted into the numerical simulation as the material property. This is similar to what has been shown for the shear compression specimen (SCS) [8] at strain rate of 3,000 1/s. Hence a similar data reduction technique can be used here for the shear compression disk (SCD). The averaged total equivalent strain on the mid-line of the gauge is related to the applied displacements (d) on the upper face of the specimen during the impact in the SHPB according to the approximate relation

$$ \widehat{\varepsilon }(t) = {c_1}\frac{{d(t)}}{w} + {c_2}{\left( {\frac{{d(t)}}{w}} \right)^2} + {c_3}{\left( {\frac{{d(t)}}{w}} \right)^3} $$

(A1)

where w is the width of the slot. The averaged von Mises stress on the mid-line of the gauge is related to the applied force (P) by:

$$ \widehat{\sigma }(t) = \left[ {{c_4}\frac{{P(t)}}{{{\sigma_y}{A^s}}} + {c_5}{{\left( {\frac{{P(t)}}{{{\sigma_y}{A^s}}}} \right)}^2} + {c_6}{{\left( {\frac{{P(t)}}{{{\sigma_y}{A^s}}}} \right)}^3}} \right]\,{\sigma_y} $$

(A2)

The applied force P(t) is related to the measured transmitted strain $\varepsilon (t)$ by: $P(t) = {\varepsilon_t}(t)\,E\,A$ where E and A are elastic modulus and cross sectional area of the transmitting bar respectively. The sheared area of the gauge can be estimated by ${A^s} \cong Lh$ where L is the circumferential line:$L = \pi \left( {D + w} \right)$ where D is the inner diameter of the specimen’s slots (equal to the incident bar diameter) and h is the vertical height of the specimen gauge. An average shear stress in the gauge is $P(t)/{A^s}$. Equation (A2) simply approximates the normalized von Mises stress on the mid-line of the gauge to the normalized averaged applied shear $\frac{{P(t)}}{{{\sigma_y}{A^s}}}$ where the static yield stress of the specimen is ${\sigma_y}$. The coefficients ${c_i}\,,\,\,i = 1...6$ are calculated from the numerical analysis. Using for example the known static properties of Aluminum 7075-T651 and simulating the impact of a specimen with β = 15° (specimen 17 - Table 1) the coefficients are: c₁ = 0.14442, c₂ = 0.81074, c₃ = -0.40775, c₄ = 3.6816, c₅ = -4.4228 and c₆ = 2.4896. Using (Equations (A1) and (A2)) with these coefficients and the experimentally measured (d(t), P(t)) yield the first approximation for the dynamic $\sigma - \varepsilon$ curve. The resulting stress–strain curve is shown here in Fig. 13 along with the quasi-static curve and the dynamic curve obtained by the iterative process employing full numerical simulation (Fig. 7). An agreement in the results obtained by the two analyses is noted.

Fig. 13

Comparison of the quasi-static (blue) and dynamic (green, red) $\sigma - \varepsilon$ curves for Aluminum 7075-T651 obtained by specimen 17. The resulting dynamic experimental $\sigma - \varepsilon$ curve (green) is obtained by application of (equations (A1) and (A2)) with the experimental results (d(t),P(t)). The resultant dynamic $\sigma - \varepsilon$ curve (red) is numerical modification of the green line