Evaluation of transformation region around crack tip in shape memory alloys

Abstract

When an edge cracked shape memory alloy (SMA) plate is loaded, phase transformation around the crack tip results in a non-homogeneous region composed of austenite and martensite phases that affects its fracture behavior. In this work, the size of the phase transformation region surrounding the tip of an edge crack in a thin SMA plate is calculated analytically using a transformation function that governs forward phase transformation, together with crack tip asymptotic stress equations. Stress intensity factors required in the asymptotic equations are obtained from a least squares fit of full displacement field, calculated using finite elements, to asymptotic near-tip opening displacement equation. The present work predicts the size and shape of the transformation region in closed form. For comparison purposes, the region is also calculated using ABAQUS with user defined material subroutines (UMAT) for plane strain and plane stress. Transformation regions calculated analytically and computationally are plotted with experimental, analytical and numerical results available in the literature; the results show a good agreement with the experimental results.

Appendix

The Helmholtz free energy used in ZM model is formulated as follows:

$$\begin{aligned}&\varPsi \left( \varepsilon _{ij}^{A},\varepsilon _{ij}^{M},T,\zeta ,\varepsilon _{ij}^{ori}\right) =(1-\zeta )\left[ \frac{1}{2}\varepsilon _{ij}^{A} E_{ijkl}^{A}{\varepsilon }_{kl}^{A}\right] \nonumber \\&\qquad +\,\zeta \left[ \frac{1}{2}\left( {\varepsilon }_{ij}^{M} - {\varepsilon }_{ij}^{ori}\right) E_{ijkl}^{M}\left( {\varepsilon }_{kl}^{M} - {\varepsilon }_{kl}^{ori}\right) + C(T)\right] \\&\qquad +\,\varGamma \frac{\zeta ^{2}}{2} + \frac{\zeta }{2}\left[ \alpha \zeta + \beta (1 - \zeta )\right] \frac{2}{3} {\varepsilon }_{ij}^{ori} {\varepsilon }_{ij}^{ori}\text {.} \nonumber \end{aligned}$$

(21)

In the equation above ${\varepsilon }_{ij}^{M}$ and ${\varepsilon }_{ij}^{A}$ are local strain tensors of martensite and austenite phases. $\varGamma$ is defined as

$$\begin{aligned} \varGamma= & {} \frac{1}{2} \left[ \left( \frac{1}{E_M}-\frac{1}{E_A}\right) \frac{\sigma ^2_{MF}-\sigma ^2_{MS} + \sigma ^2_{AS}-\sigma ^2_{AF}}{2} \right. \nonumber \\&\left. +\, (\sigma _{MF}-\sigma _{MS}+\sigma _{AS}-\sigma _{AF} ) \varepsilon _0 - 2(\alpha -\beta ) \varepsilon ^2_0 \right] \nonumber \\ \end{aligned}$$

(22)

In the model, Reuss scheme is used to relate the total strain ${\varepsilon }_{ij}$ to the strains of austenite and martensite phases as given below:

$$\begin{aligned} {\varepsilon }_{ij}=(1 - \zeta ){\varepsilon }_{ij}^{A}+\zeta {\varepsilon }_{ij}^{M} \text {,} \end{aligned}$$

(23)

the internal state variable $\zeta$ should be bounded in the interval [0, 1], therefore:

$$\begin{aligned} \zeta \geqslant 0 \quad \text {and} \quad (1 - \zeta ) \geqslant 0 \text {,} \end{aligned}$$

(24)

and the equivalent orientation strain, $\varepsilon _{0}$ as shown in Fig. 1b, has the following maximum value:

$$\begin{aligned} {\varepsilon }_{0}-\sqrt{\frac{2}{3} {\varepsilon }_{ij}^{ori}{\varepsilon }_{ij}^{ori}} \geqslant 0 \text {.} \end{aligned}$$

(25)

Equations (23), (24) and (25) are used to build the following constraints potential ${\varPsi }_{c}$ (Moumni 1995):

$$\begin{aligned} \begin{aligned} {\varPsi }_{c}=&-\lambda _{ij} [{(1 - \zeta ) {\varepsilon }_{ij}^A+\zeta {\varepsilon }_{ij}^M-{\varepsilon }_{ij}}] \\&-\,\left( { {\varepsilon }_{0} -\sqrt{\frac{2}{3} {\varepsilon }_{ij}^{ori}{\varepsilon }_{ij}^{ori}}} \right) - \nu _{1} \zeta - \nu _{2} ({1 - z}) \text {,} \end{aligned} \end{aligned}$$

(26)

where the Lagrange multipliers $\lambda$, $\mu$, $\nu _{1}$, $\nu _{2}$ and $\mu$ obey the following conditions:

$$\begin{aligned} \begin{aligned}&\nu _{1} \geqslant 0 , \quad \nu _{1} \zeta = 0 ,\quad \nu _{2} \geqslant 0 , \quad \nu _{2} (1 - \zeta ) = 0\\&\text {and} \quad \mu \geqslant 0 , \mu \left( {\varepsilon }_{0} - \sqrt{\dfrac{2}{3} {\varepsilon }_{ori} {\varepsilon }_{ori} }\right) =0\text {.} \end{aligned} \end{aligned}$$

(27)

The sum of the Helmholtz energy density, Eq. (21), and the potential ${\varPsi }_{c}$, Eq. (26), gives the Lagrangian, which is then used to derive the state equations, and the following stress–strain relation is obtained:

$$\begin{aligned} \sigma _{ij} ={S_{ijkl}^{-1}}:\left( {{\varepsilon _{kl}}-\zeta {\varepsilon }_{kl}^{{ori}}}\right) \text {,} \end{aligned}$$

(28)

in which ${S_{ijkl}}$ is the compliance tensor defined as:

$$\begin{aligned} S_{ijkl} =(1 - \zeta ) S_{ijkl}^{A}+\zeta S_{ijkl}^{M}\text {,} \end{aligned}$$

(29)

where $S_{ijkl}^{A}$ and $S_{ijkl}^{M}$ are the compliance tensors of austenite and martensite phases respectively.

According to the theory of generalized standard materials with internal constraints represented by Halphen and Nguyen (1974), the thermodynamic forces that are related to the internal state variables $\zeta$ and ${\varepsilon }_{ij}^{{ori}}$ are sub-gradients of a pseudo-potential. The pseudo-potential of dissipation defined by Zaki and Moumni (2007) is given as follows:

$$\begin{aligned} \mathcal {D}(\dot{\zeta },\dot{\varepsilon }_{ij}^{{ori}}) =[{a (1 - \zeta )+ b \zeta }] |\dot{\zeta }| + \zeta ^{2} Y \sqrt{ \frac{2}{3} \dot{\varepsilon }_{ij}^{{ori}}\dot{\varepsilon }_{ij}^{{ori}} }\text {.} \end{aligned}$$

(30)

In the equation above Y is a positive material constant associated with $\sigma _{rs}$. The pseudo-potential of dissipation, $\mathcal {D}(\dot{\zeta },\dot{\varepsilon }_{ij}^{{ori}})$, allows the definition of transformation function, $F_{\zeta }$ given in Eq. (1).