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**Publisher's Note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Tanné E, Li T, Bourdin B, Marigo J-J, Maurini C (2018) Crack nucleation in variational phase-field models of brittle fracture. J Mech Phys Solids 110:80–99

Ulloa J, Wambacq J, Alessi R, Samaniego E, Degrande G, Francois S (2022) A micromechanics-based variational phase-field model for fracture in geomaterials with brittle-tensile and compressive-ductile behavior. J Mech Phys Solids 159:104684

van Dijk NP, Espadas-Escalante JJ, Isaksson P (2020) Strain energy density decompositions in phase-field fracture theories for orthotropy and anisotropy. Int J Solids Struct 196:140–153

Vu B-T, Le Quang H, He Q-C (2022) Modelling and simulation of fracture in anisotropic brittle materials by the phase-field method with novel strain decompositions. Mech Res Commun 124:103936

Wang T, Ye X, Liu Z, Chu D, Zhuang Z (2019) Modeling the dynamic and quasi-static compression-shear failure of brittle materials by explicit phase field method. Comput Mech 64:1537–1556

Wu T, Carpiuc A, Poncelet M, De Lorenzis L (2017) Phase-field simulation of interactive mixed-mode fracture tests on cement mortar with full-field displacement boundary conditions. Eng Fract Mech 182:658–688

You T, Zhu Q-Z, Li P-F, Shao J-F (2020) Incorporation of tension-compression asymmetry into plastic damage phase-field modeling of quasi brittle geomaterials. Int J Plast 124:71–95

Zhang X, Sloan SW, Vignes C, Sheng D (2017) A modification of the phase-field model for mixed mode crack propagation in rock-like materials. Comput Methods Appl Mech Eng 322:123–136

Zolesi C, Maurini C (2023) Crack nucleation in variational phase-field models of fracture with softening modulation, in progress

**Publisher's Note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.