In this manuscript, the computational homogenisation of phase-field fractures is addressed. To this end, a variationally consistent two-scale phase-field fracture framework is developed, which formulates the coupled momentum balance and phase-field evolution equations at the macro-scale as well as at the Representative Volume Element (RVE¹) scale. The phase-field variable represent fractures at the RVE scale, however, at the macro-scale, it is treated as an auxiliary variable. The latter interpretation follows from the homogenisation of the phase-field through volume or a surface-average. For either homogenisation choices, the set of macro-scale and sub-scale equations, and the pertinent macro-homogeneity satisfying boundary conditions are established. As a special case, the concept of selective homogenisation is introduced, where the phase-field is chosen to live only in the RVE domain, thereby eliminating the macro-scale phase-field evolution equation. Numerical experiments demonstrate the local macro-scale material behaviour of the selective homogenisation based two-scale phase-field fracture model, while its non-selective counterpart yields a non-local macro-scale material behaviour.

在此手稿中，解决了相场裂缝的计算均一化问题。为此，开发了一个变分一致的两尺度相场裂缝框架，该框架在宏观尺度和代表性体积元（RVE ^{1）上建立}了耦合的动量平衡和相场演化方程。） 规模。相场变量代表RVE尺度的裂缝，但是，在宏观尺度上，它被视为辅助变量。后一种解释来自于通过体积或表面平均的相场均化。对于任一均质化选择，都建立了宏观尺度和次尺度方程组以及满足边界条件的相关宏观均匀性。作为一种特殊情况，引入了选择性均质化的概念，其中选择了相场以仅存在于RVE域中，从而消除了宏观尺度的相场演化方程。数值实验证明了基于选择性均质化的两尺度相场断裂模型的局部宏观尺度行为，