On crack tip stress fields in pseudoelastic shape memory alloys

Abstract

In a domain of reasonable accuracy around the crack tip, asymptotic equations can provide closed form expressions that can be used in formulation of crack problem. In some studies on shape memory alloys (SMAs), although the pseudoelastic behavior results in a nonlinear stress–strain relation, stress distribution in the vicinity of the crack tip is evaluated using asymptotic equations of linear elastic fracture mechanics (LEFM). In pseudoelastic (SMAs), upon loading, stress increases around the crack tip and martensitic phase transformation occurs in early stages. In this paper, using the similarity in the loading paths of a pseudoelastic SMA and a strain hardening material, the stress–strain relation is represented by nonlinear Ramberg–Osgood equation which is originally proposed for strain hardening materials, and the stress distribution around the crack tip of a pseudoelastic SMA plate is reworked using the Hutchinson, Rice and Rosengren (HRR) solution, first studied by Hutchinson. The size of the transformation region around the crack tip is calculated in closed form using a thermodynamic force that governs the martensitic transformation together with the asymptotic equations of HRR. A UMAT is written to separately describe the thermo-mechanical behavior of pseudoelastic SMAs. The results of the present study are compared to the results of LEFM, computational results from ABAQUS, and experimental results for the case of an edge cracked NiTi plate. Both set of asymptotic equations are shown to have different dominant zones around the crack tip with HRR equations representing the martensitic transformation zone more accurately.

Appendix: Material systems used in finite elements

The thermodynamic force $F_z$ in ZM Model is derived from the following free energy $\psi$, and pseudo-potential of dissipation D (Zaki and Moumni 2007):

$$\begin{aligned} \psi= & {} (1-z)\left( \frac{1}{2}\varvec{\epsilon _A}:\varvec{K_A}:\varvec{\epsilon _A}\right) \nonumber \\&+\,z\left[ \frac{1}{2}\left( \varvec{\epsilon _M}-\varvec{\epsilon _{ori}} \right) :\varvec{K_M}:\left( \varvec{\epsilon _M}-\varvec{\epsilon _{ori}} \right) +C(T)\right] \nonumber \\&+\,G\frac{z^2}{2}+\frac{z}{2}[\alpha {z}+\beta (1-z)] \left( \frac{2}{3}\varvec{\epsilon _{ori}}:\varvec{\epsilon _{ori}}\right) \end{aligned}$$

(16)

$$\begin{aligned} D= & {} [a(1-z)+bz]{\dot{z}}sgn({\dot{z}})+{z^2}Y\sqrt{\frac{2}{3}\varvec{{\dot{\epsilon }}_{ori}}:\varvec{{\dot{\epsilon }}_{ori}}}\nonumber \\ \end{aligned}$$

(17)

where ${\varvec{\epsilon _A}}$ and ${\varvec{\epsilon _M}}$ are the local deformation tensors and $\varvec{K_A}$ and ${\varvec{K_M}}$ are elastic module tensors of austenite and martensite, respectively, ${\varvec{\epsilon _{ori}}}$ is the orientation strain tensor of martensite, z is the volume fraction of martensite, and C(T) is the latent heat density calculated as follows:

$$\begin{aligned} C(T)=\zeta \left( T-A_f^{0}\right) +\kappa \end{aligned}$$

(18)

where $A_f^{0}$ is the phase change temperature from austenite to martensite.

The material constants in the following tables are calculated as explained in Zaki and Moumni (2007), and used in UMAT.