A Direct Method for Predicting the High-Cycle Fatigue Regime of Shape-Memory Alloys Structures

Abstract

Shape Memory Alloys (SMAs) belong to the class of so-called smart materials that offer promising perspectives in various fields such as aeronautics, robotics, biomedicals or civil engineering. For elastic-plastic materials, there is an established correlation between fatigue and energy dissipation. In particular, high-cycle fatigue occurs when the energy dissipation remains bounded in time. Although the physical mechanisms in SMAs differ from plasticity, the hysteresis that is commonly observed in the stress-strain response of those materials shows that some energy dissipation occurs. It can be reasonably assumed that situations where the energy dissipation remains bounded are the most favorable for fatigue durability. In this contribution, we present a direct method for determining if the energy dissipation in a SMA structure (submitted to a prescribed loading history) is bounded or not. That method is direct in the sense that nonlinear incremental analysis is completely bypassed. The proposed method rests on a suitable extension of the well-known Melan theorem. An application related to biomedical stents is presented to illustrate the method.

Appendix

For the sake of completeness, we give in this Appendix a proof of Theorem 1. Consider a solution $({\varvec{\varepsilon }},{\varvec{\alpha }},{\varvec{\sigma }},{\varvec{A}}^r,{\varvec{A}}^d)$ to the evolution problem (1)–(13). By (11) we have

$$ {\varvec{A}}= {\varvec{A}}^d + {\varvec{A}}^r$$

with

$$\begin{aligned} \dot{{\varvec{\alpha }}}\in \partial I_{{\varvec{\mathscr {C}}}}({\varvec{A}}^d );\ {\varvec{A}}^r \in \partial I_{{\varvec{\mathscr {T}}}}({\varvec{\alpha }}). \end{aligned}$$

(23)

The positive quantity

$$D(t)=\int _{\varOmega } {\varvec{A}}^d.\dot{{\varvec{\alpha }}}d{\varvec{x}}$$

can be interpreted as the rate of dissipated energy. Note that D(t) is positive because of the principle of maximum dissipation (6) and the fact that the elasticity domain ${\varvec{\mathscr {C}}}$ contains the origin. Under the condition (16), we show in the following that the total dissipated energy $\int _0^T D(t)dt$ remains bounded as $T\rightarrow \infty$. To that purpose, consider the positive functional W(t) defined as

$$ W(t) = \displaystyle \int _{\varOmega } w({\varvec{\varepsilon }}(t)-{\varvec{\varepsilon }}^E(t),{\varvec{\alpha }}(t)) \, d{\varvec{x}}.$$

By time-differentiation we have

$$ \dot{W}(t) = \int _{\varOmega } [({\varvec{\sigma }}-{\varvec{\sigma }}^E):(\dot{{\varvec{\varepsilon }}}-\dot{{\varvec{\varepsilon }}}^E-{\varvec{K}}:\dot{{\varvec{\alpha }}})+ f'({\varvec{\alpha }}):\dot{{\varvec{\alpha }}} ]\, d{\varvec{x}}.$$

Since ${\text {div}}({\varvec{\sigma }}-{\varvec{\sigma }}^E)=0$ in $\varOmega$, $({\varvec{\sigma }}-{\varvec{\sigma }}^E). {\varvec{n}}=0$ on $\varGamma _T$ and ${\varvec{u}}-{\varvec{u}}^E=0$ on $\varGamma _u$, the principle of virtual power gives $\int _{\varOmega }({\varvec{\sigma }}-{\varvec{\sigma }}^{E}):(\dot{{\varvec{\varepsilon }}}-\dot{{\varvec{\varepsilon }}}^E) \, d{\varvec{x}}= 0$. Therefore

$$ \dot{W}(t) = \int _{\varOmega } [-{\varvec{K}}^T:({\varvec{\sigma }}-{\varvec{\sigma }}^E)+f'({\varvec{\alpha }})]:\dot{{\varvec{\alpha }}} \, d{\varvec{x}}$$

which using (3) and (23) can be rewritten as

$$\begin{aligned} \dot{W}(t) = -D(t) + \int _{\varOmega } [-{\varvec{A}}^r+{\varvec{K}}^T:{\varvec{\sigma }}^E]:\dot{{\varvec{\alpha }}} \, d{\varvec{x}}. \end{aligned}$$

(24)

Let $({\varvec{A}}_*^r,m)$ satisfying (16). Setting ${\varvec{A}}_*^d=m{\varvec{K}}^T:{\varvec{\sigma }}^E(t)-{\varvec{A}}_{*}^r$, we find

$$\begin{aligned} \dot{W}(t) = -D(t) + \int _{\varOmega } [-{\varvec{A}}^r+\frac{1}{m}({{\varvec{A}}^d_{*}}+{{\varvec{A}}^r_{*}})]:\dot{{\varvec{\alpha }}} \, d{\varvec{x}}. \end{aligned}$$

(25)

The property (16) shows that ${\varvec{A}}_*^d \in {\varvec{\mathscr {C}}}$ for $t>{\tau }$. Since $\dot{{\varvec{\alpha }}}\in \partial I_{{\varvec{\mathscr {C}}}}({\varvec{A}}^d)$, the principle of maximum dissipation (6) gives

$$\begin{aligned} ({\varvec{A}}^d-{\varvec{A}}_{*}^d):\dot{{\varvec{\alpha }}}\ge 0 . \end{aligned}$$

(26)

Moreover, since ${\varvec{A}}^r\in \partial I_{{\varvec{\mathscr {T}}}}({\varvec{\alpha }})$ and ${\varvec{\alpha }}\in {\varvec{\mathscr {T}}}$, Eq. (12) gives ${\varvec{A}}^r(t):({\varvec{\alpha }}(t)-{\varvec{\alpha }}(t'))\ge 0$ for any $t'$. In the limit $t' \longrightarrow t$ with $t'<t$, we obtain

$$\begin{aligned} {\varvec{A}}^r:\dot{{\varvec{\alpha }}} \ge 0 . \end{aligned}$$

(27)

Combining (26)–(27) with (25) gives

$$\begin{aligned} \dot{W}(t) \le \frac{1-m}{m}D(t) + \frac{1}{m}\int _{\varOmega } {{\varvec{A}}^r_{*}}:\dot{{\varvec{\alpha }}} \, d{\varvec{x}}. \end{aligned}$$

(28)

Since ${\varvec{A}}^r_{*}$ is time-independent, the time-integration of (28) on a time interval $[{\tau },T]$ yields

$$\begin{aligned} \begin{array}{ll} (m-1)\displaystyle \int _{\tau }^T D(t)\, dt \le &{} mW({\tau }) + \displaystyle \int _\varOmega {{\varvec{A}}^r_{*}}:({{\varvec{\alpha }}}(T)-{\varvec{\alpha }}({\tau }))\, d{\varvec{x}}\\ \end{array} \end{aligned}$$

(29)

where the property $W(T)\ge 0$ has been used. The set ${\varvec{\mathscr {T}}}$ being bounded, there exists a positive constant K such that $\Vert {\varvec{\alpha }}\Vert \le K$ for any ${\varvec{\alpha }}\in {\varvec{\mathscr {T}}}$. Therefore

$$\int _\varOmega {\varvec{A}}_{*}^r:({\varvec{\alpha }}(t)-{\varvec{\alpha }}({\tau }))\, d{\varvec{x}}\le \displaystyle 2K \int _\varOmega \Vert {\varvec{A}}_{*}^r\Vert \, d{\varvec{x}}.$$

Combining that inequality with (29) gives

$$ (m-1)\displaystyle \int _{\tau }^T D(t)\, dt \le mW({\tau }) + \displaystyle 2K \int _\varOmega \Vert {\varvec{A}}_{*}^r\Vert \, d{\varvec{x}}. $$

The right-hand side of that inequality is independent of T. This proves that the dissipated energy $\int _{\tau }^T D(t)$ remains bounded as $T\longrightarrow +\infty$.

From there we can show (under some technical assumptions) that ${\varvec{\alpha }}(t)$ tends to a limit as $t\longrightarrow +\infty$. Assume that the elasticity domain ${\varvec{\mathscr {C}}}$ contains a ball of radius $r>0$ centered at the origin. In such a condition, we have $r \dot{{\varvec{\alpha }}}(t) / \Vert \dot{{\varvec{\alpha }}}(t)\Vert \in {\varvec{\mathscr {C}}}$ for any t. Using the principle of maximum dissipation (6), we find

$$ 0 \le \dot{{\varvec{\alpha }}}: ({\varvec{A}}^d - r \frac{ \dot{{\varvec{\alpha }}}}{ \Vert \dot{{\varvec{\alpha }}}\Vert }).$$

Hence

$$ \Vert \dot{{\varvec{\alpha }}} \Vert \le \frac{1}{r} {\varvec{A}}^d:\dot{{\varvec{\alpha }}} $$

which after space integration gives

$$\begin{aligned} \int _{\varOmega } \Vert \dot{{\varvec{\alpha }}} \Vert dx \le \frac{1}{r} D(t) . \end{aligned}$$

(30)

Let $\mathbb A$ be the vectorial space in which ${\varvec{\alpha }}({\varvec{x}})$ takes values and let $L_1(\varOmega ,\mathbb A)$ be the space of integrable functions with values in $\mathbb A$. The inequality (30) can be rewritten as

$$\Vert \dot{{\varvec{\alpha }}}\Vert _{L_1(\varOmega ,\mathbb A)} \le \frac{1}{r} D(t)$$

where $\Vert \cdot \Vert _{L_1(\varOmega ,\mathbb A)}$ is the norm in $L_1(\varOmega ,\mathbb A)$. Since $\int _{0}^T D(t)$ is bounded as $T\longrightarrow +\infty$, the integral $\int _0 ^T \Vert \dot{{\varvec{\alpha }}}(t)\Vert _{L_1(\varOmega ,\mathbb A)} dt$ converges as $T\longrightarrow +\infty$. From Riesz-Fischer theorem, the space $L_1(\varOmega ,\mathbb {A})$ is a Banach space. It follows (see [29] or Theorem 97 in [32]) that the integral $\int _0 ^T \dot{{\varvec{\alpha }}}(t) dt$ converges as $T\longrightarrow \infty$. Hence ${\varvec{\alpha }}(t)$ converges towards a limit as $T\longrightarrow \infty$.    $\square$