Making Shear Simple – Validation of the Shear Compression Specimen 0 (SCS0) for Shear Testing

Abstract

Objective

Validate and assess the limitations of the Shear Compression 0 Specimen (SCS0) as a simple shear specimen for quasi-static and dynamic large strain loading conditions. Propose a simple data reduction procedure, using a simple, back of the envelope method, as a first approximation for the strain, as opposed to cumbersome numerical simulations and avoid the use of ad-hoc data reduction factors.

Methods

Static and dynamic finite elements simulations were performed in which the large deformation options was turned on and off. Assessment of the Lode parameter in each case and evaluation of the accuracy of the specimen’s strains and stresses as determined through simple data reduction and full numerical simulations.

Results

The SCS0 was shown to undergo simple shear, both statically and dynamically, as evidenced from the very low values of the Lode parameter. The calculated stress is in excellent agreement with the measured one, determined using simple strength of materials definitions. When assuming the corresponding kinematics, it is observed that the calculated and the measured strain diverge to an extent of about 25%. This discrepancy is shown to result from the assumption of large geometrical deformations in the numerical model as opposed to the simple analytical kinematics.

Conclusion

The conclusion is that the SCS0 is now fully validated, and the experimentalist will decide which strain approximation is suitable, between analytical and numerical.

Appendix

Appendix A Specimen Drawing

Appendix Figure A - Specimen drawing

Appendix Table A - Modified gauge length

Material	CP-Ti	Ti-6-Al-4-V	CP-Zr
Gauge length $\left[mm\right]$	7	10	7.6

Appendix B Analytical Considerations

The calculations that were carried for the strains in equation (6):

$$\begin{aligned}&{\varepsilon }_{eq}=\sqrt{\frac{2}{3}{\varepsilon }_{ij}^{dev}{\varepsilon }_{ij}^{dev}}=\frac{2}{3}\sqrt{\frac{3\left({e}_{11}^{2}+{e}_{22}^{2}+{e}_{33}^{2}\right)}{2}+\frac{3\left({\gamma }_{12}^{2}+{\gamma }_{23}^{2}+{\gamma }_{31}^{2}\right)}{4}}& \\& {\text{where} \hspace{0.17em}}\left[\begin{array}{c}{e}_{11}=\frac{2}{3}{\varepsilon }_{11}-\frac{1}{3}{\varepsilon }_{22}-\frac{1}{3}{\varepsilon }_{33}\\ {e}_{22}=-\frac{1}{3}{\varepsilon }_{11}+\frac{2}{3}{\varepsilon }_{22}-\frac{1}{3}{\varepsilon }_{33}\\ {e}_{33}=-\frac{1}{3}{\varepsilon }_{11}-\frac{1}{3}{\varepsilon }_{22}+\frac{2}{3}{\varepsilon }_{33}\end{array}\right]\text{, }\left[\begin{array}{c}{\gamma }_{12}=2{\varepsilon }_{12}\\ \\ {\gamma }_{23}=2{\varepsilon }_{23}\\ \\ {\gamma }_{31}=2{\varepsilon }_{31}\end{array}\right]\\&\text{ and }{\varepsilon }_{ij}^{dev}={\varepsilon }_{ij}-\frac{1}{3}tr\left(\varepsilon \right)\cdot {\delta }_{ij}{\hspace{0.17em}\hspace{0.17em}}& \\& \Rightarrow {\varepsilon }_{eq}=\frac{2}{3}\sqrt{{\varepsilon }_{11}^{2}+{\varepsilon }_{22}^{2}+{\varepsilon }_{33}^{2}-{\varepsilon }_{11}{\varepsilon }_{22}-{\varepsilon }_{11}{\varepsilon }_{33}-{\varepsilon }_{22}{\varepsilon }_{33}+3\left({\varepsilon }_{12}^{2}+{\varepsilon }_{23}^{2}+{\varepsilon }_{31}^{2}\right)}& \\& 2-\text{D: }{\varepsilon }_{eq}=\frac{2}{3}\sqrt{\left[{\varepsilon }_{22}^{2}-{\varepsilon }_{22}{\varepsilon }_{33}+{\varepsilon }_{33}^{2}+3{\varepsilon }_{23}^{2}\right]}& \\& \text{Pure shear: }\;{\varepsilon }_{eq}=\frac{2}{\sqrt{3}}{\varepsilon }_{23}=\frac{{\gamma }_{23}}{\sqrt{3}}& \end{aligned}$$

(B.1)

The invariant identities for ${I}_{1},{I}_{2},{I}_{3}$ that were used are:

$$\begin{aligned}&{I}_{1}={\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}={\sigma }_{kk}=tr\left(\sigma \right)\\&{I}_{2}=\left|\begin{array}{ccc}{\sigma }_{22}& {\sigma }_{23}\\& {\sigma }_{32}& {\sigma }_{33}\end{array}\right|+\left|\begin{array}{ccc}{\sigma }_{11}& {\sigma }_{13}\\& {\sigma }_{31}& {\sigma }_{33}\end{array}\right|+\left|\begin{array}{ccc}{\sigma }_{11}& {\sigma }_{12}\\& {\sigma }_{21} {\sigma }_{22}\end{array}\right|\\&\quad\,={\sigma }_{11}{\sigma }_{22}+{\sigma }_{22}{\sigma }_{33}+{\sigma }_{11}{\sigma }_{33}-{\sigma }_{12}^{2}-{\sigma }_{23}^{2}-{\sigma }_{31}^{2}\\&\, \quad=\frac{1}{2}\left({\sigma }_{ii}{\sigma }_{jj}-{\sigma }_{ij}{\sigma }_{ji}\right)=\frac{1}{2}\left[tr\left({\sigma }^{2}\right)-tr{\left(\sigma \right)}^{2}\right]\\& {I}_{3}={\sigma }_{11}{\sigma }_{22}{\sigma }_{33}+2{\sigma }_{12}{\sigma }_{23}{\sigma }_{31}\\&\quad\quad-\,{\sigma }_{12}^{2}{\sigma }_{33}-{\sigma }_{23}^{2}{\sigma }_{11}-{\sigma }_{31}^{2}{\sigma }_{22}=det\left({\sigma }_{ij}\right)\end{aligned}$$

(B.2)

The invariant identities for ${J}_{1},{J}_{2},{J}_{3}$ that were used are:

$$\begin{aligned}&{s}_{ij}={\sigma }_{ij}-{\sigma }_{mean}{\delta }_{ij}={\sigma }_{ij}-\frac{1}{3}tr\left(\sigma \right){\delta }_{ij};\; {\hspace{0.17em}}{\sigma }_{mean}=\frac{{I}_{1}}{3}{\hspace{0.17em}}\\& {J}_{1}={s}_{kk}=0\\& {J}_{2}=\frac{1}{2}{s}_{ij}{s}_{ji}=\frac{1}{2}tr\left({s}^{2}\right)\\&\quad=\frac{1}{6}\left[{\left({\sigma }_{11}-{\sigma }_{22}\right)}^{2}+{\left({\sigma }_{22}-{\sigma }_{33}\right)}^{2}+{\left({\sigma }_{33}-{\sigma }_{11}\right)}^{2}\right]\\&\quad\quad\;+{\sigma }_{12}^{2}+{\sigma }_{23}^{2}+{\sigma }_{31}^{2}\\&\quad \;=\frac{1}{3}{I}_{1}^{2}-{I}_{2}=\frac{1}{2}\left[tr\left({\sigma }^{2}\right)-\frac{1}{3}tr{\left(\sigma \right)}^{2}\right]\\& {J}_{3}=det\left({s}_{ij}\right)=\frac{1}{3}{s}_{ij}{s}_{jk}{s}_{ki}=\frac{1}{3}tr\left({s}^{3}\right)\\&\quad\;\;\, =\frac{2}{27}{I}_{1}^{3}-\frac{1}{3}{I}_{1}{I}_{2}+{I}_{3}\\&\quad\;\;\,=\frac{1}{3}\left[tr\left({\sigma }^{3}\right)-tr\left({\sigma }^{2}\right)tr\left(\sigma \right)+\frac{2}{9}tr{\left(\sigma \right)}^{3}\right]\end{aligned}$$

(B.3)

Appendix C Material Properties

Appendix Table C - Elastic properties

	CP Ti	Ti6Al4V	CP Zr	Maraging Steel C300
Density $[\frac{kg}{{m}^{3}}]$	4.51 $\cdot \;{10}^{3}$	4.43 $\cdot \;{10}^{3}$	6.53 $\cdot\; {10}^{3}$	8 $\cdot\; {10}^{3}$
Quasi-static yield stress $[MPa]$	230	890	420	-
Dynamic yield stress $[MPa]$	280	1170	490	-
Young’s Modulus $[MPa]$	105,000	114,000	94,500	190,000
Poisson’s Ratio	0.37	0.33	0.34	0.3

For the plastic properties of our three materials, an overall bi-linear law was used, such that the plastic part is a linear extrapolation of our experimental data (both quasi-static and dynamic loading cases).

Appendix Figure B - Mechanical properties of all three materials in both quasi-static and dynamic loading. Top figure is for the quasi-static case and the bottom figure is for the dynamic case.

Appendix D Recorded Results of Dynamically Experimented SCS0

Appendix Figure D - Dynamically Fractured SCS0. (a) Ti6Al4V specimen; (b) CP Zr specimen