We present a new phase field framework for modelling fracture and fatigue in Shape Memory Alloys (SMAs). The constitutive model captures the superelastic behaviour of SMAs and damage is driven by the elastic and transformation strain energy densities. We consider both the assumption of a constant fracture energy and the case of a fracture energy dependent on the martensitic volume fraction. The framework is implemented in an implicit time integration scheme, with both monolithic and staggered solution strategies. The potential of this formulation is showcased by modelling a number of paradigmatic problems. First, a boundary layer model is used to examine crack tip fields and compute crack growth resistance curves (R-curves). We show that the model is able to capture the main fracture features associated with SMAs, including the toughening effect associated with stress-induced phase transformation. Insight is gained into the role of temperature, material strength, crack density function and fracture energy homogenisation. Secondly, several 2D and 3D boundary value problems are addressed, demonstrating the capabilities of the model in capturing complex cracking phenomena in SMAs, such as unstable crack growth, mixed-mode fracture or the interaction between several cracks. Finally, the model is extended to fatigue and used to capture crack nucleation and propagation in biomedical stents, a paradigmatic application of nitinol SMAs.

中文翻译：

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形状记忆合金中断裂和疲劳的相场建模

我们提出了一种新的相场框架，用于对形状记忆合金 (SMA) 中的断裂和疲劳进行建模。本构模型捕捉 SMA 的超弹性行为，损伤由弹性和转换应变能量密度驱动。我们考虑了恒定断裂能的假设和断裂能取决于马氏体体积分数的情况。该框架以隐式时间积分方案实现，具有整体式和交错式解决方案策略。通过对许多典型问题进行建模，展示了该公式的潜力。首先，边界层模型用于检查裂纹尖端场并计算裂纹扩展阻力曲线（R 曲线）。我们表明该模型能够捕获与 SMA 相关的主要断裂特征，包括与应力诱导相变相关的韧化效应。深入了解温度、材料强度、裂纹密度函数和断裂能量均匀化的作用。其次，解决了几个 2D 和 3D 边界值问题，展示了该模型在捕捉 SMA 中复杂裂纹现象方面的能力，例如不稳定裂纹扩展、混合模式断裂或多个裂纹之间的相互作用。最后，该模型扩展到疲劳并用于捕获生物医学支架中的裂纹成核和扩展，这是镍钛诺 SMA 的典型应用。解决了几个 2D 和 3D 边界值问题，展示了该模型在捕捉 SMA 中复杂裂纹现象方面的能力，例如不稳定裂纹扩展、混合模式断裂或多个裂纹之间的相互作用。最后，该模型扩展到疲劳并用于捕获生物医学支架中的裂纹成核和扩展，这是镍钛诺 SMA 的典型应用。解决了几个 2D 和 3D 边界值问题，展示了该模型在捕捉 SMA 中复杂裂纹现象方面的能力，例如不稳定裂纹扩展、混合模式断裂或多个裂纹之间的相互作用。最后，该模型扩展到疲劳并用于捕获生物医学支架中的裂纹成核和扩展，这是镍钛诺 SMA 的典型应用。