Visualising elastic anisotropy: theoretical background and computational implementation

Nordmann, Joachim; Aßmus, Marcus; Altenbach, Holm

doi:10.1007/s00161-018-0635-9

Cited by 58 publications

(23 citation statements)

References 25 publications

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“…8 is still a line integral as integration is 160 performed along the normal to the fracture surface. Such aspects are beyond the scope of this work; an intuitive approach on the rotation of anisotropic tensors is provided in [52].…”

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confidence: 99%

Phase-Field Material Point Method for dynamic brittle fracture with isotropic and anisotropic surface energy

Kakouris

Triantafyllou

2019

Computer Methods in Applied Mechanics and Engineering

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A novel phase field material point method is introduced for robust simulation of dynamic fracture in elastic media considering the most general case of anisotropic surface energy. Anisotropy is explicitly introduced through a properly defined crack density functional. The particular case of impact driven fracture is treated by employing a discrete field approach within the material point method setting. In this, the equations of motion and phase field governing equations are solved independently for each discrete field using a predictor-corrector algorithm. Contact at the interface is resolved through frictional contact conditions. The proposed method is verified using analytical predictions. The influence of surface energy anisotropy and loading conditions on the resulting crack paths is assessed through a set of benchmark problems. Comparisons are made with the standard Phase Field Finite Element Method and experimental observations. 65 3 (PF-MPM) has been successfully introduced by the authors in [44] for quasistatic brittle fracture problems while a variant accounting for anisotropy in the quasi-static regime has been developed in [45].Moving beyond the state-of-the-art, we present a phase field MPM method for the solution of dynamic fracture considering materials with anisotropic frac-70 ture energy; isotropy emerges as a special case of the proposed formulation.Following, the method is extended to also account for frictional contact fracture problems. We use as point of departure the MPM contact algorithm introduced in Bardenhagen et al. [46] where multiple fields, termed discrete fields, are introduced in the non-deforming Eulerian mesh so that each contact body 75 corresponds to a different field. We define the variational structure of our phase field implementation of impact driven fracture at each discrete field from which the coupled weak form of the contact problem naturally emerges. Finally, we develop a predictor corrector solution algorithm for the solution of the governing equations over time. 80This paper is organized as follows. In section 2 phase-field modelling is briefly described in both isotropic and anisotropic brittle fracture. The discrete field formulation for phase field fracture due to impact is presented in section 3. The Material Point Method implementation for frictional contact fracture is presented in section 4. Finally, in section 5, a set of benchmark problems are 85 examined to demonstrate the accuracy and robustness of the proposed method. Preliminaries Phase field modellingIn the following, the case of an arbitrary deformable domain Ω is considered, with an external boundary ∂Ω and a crack path Γ as shown in Fig. (1a). 90The deformable domain Ω with domain volume V , is subjected to body forces b = b 1 b 2 b 3 T . Furthermore, a set of traction/pressure loadst is applied on the boundary ∂Ωt ⊆ ∂Ω. A prescribed displacement field, denoted asū, is imposed on the boundary ∂Ωū ⊆ ∂Ω.

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confidence: 99%

Phase-Field Material Point Method for dynamic brittle fracture with isotropic and anisotropic surface energy

Kakouris

Triantafyllou

2019

Computer Methods in Applied Mechanics and Engineering

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“…As mentioned earlier that the MAX phases compounds studied in this work have an elastic anisotropic nature, while the elastic anisotropy is not apparent. Figure 14 plots the 3D Young’s modulus surfaces obtained by an open-source software package (AnisoVis) 97 of Zr 2 GaC (a, b, c) and Hf 2 GaC (d, e, f) at 0, 10, and 50 GPa, respectively. It can be observed that the shape begins to deviate from the sphere with an increase in pressure, and color varies in different regions, which indicates the elastic anisotropy of M 2 GaC MAX phases.…”

Section: Resultsmentioning

confidence: 99%

Ab initio predictions of structure and physical properties of the Zr2GaC and Hf2GaC MAX phases under pressure

Qureshi

Tang

et al. 2021

Sci Rep

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The electronic structure, structural stability, mechanical, phonon, and optical properties of Zr2GaC and Hf2GaC MAX phases have been investigated under high pressure using first-principles calculations. Formation enthalpy of competing phases, elastic constants, and phonon calculations revealed that both compounds are thermodynamically, mechanically, and dynamically stable under pressure. The compressibility of Zr2GaC is higher than that of Hf2GaC along the c-axis, and pressure enhanced the resistance to deformation. The electronic structure calculations reveal that M2GaC is metallic in nature, and the metallicity of Zr2GaC increased more than that of Hf2GaC at higher pressure. The mechanical properties, including elastic constants, elastic moduli, Vickers hardness, Poisson’s ratio anisotropy index, and Debye temperature, are reported with fundamental insights. The elastic constants C11 and C33 increase rapidly compared with other elastic constants with an increase in pressure, and the elastic anisotropy of Hf2GaC is higher than that of the Zr2GaC. The optical properties revealed that Zr2GaC and Hf2GaC MAX phases are suitable for optoelectronic devices in the visible and UV regions and can also be used as a coating material for reducing solar heating at higher pressure up to 50 GPa.

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“…In more recent publications, e.g. in Nordmann et al [19], only 8 symmetry classes are distinguished. The difference to ten comes from two cyclic trigonal and tetragonal groups C3 and C4, which are simple threefold and fourfold rotational symmetries around an axis we call c. Both leave seven different, independent components in the stiffness tetrad.…”

Section: Discussionmentioning

confidence: 99%

A systematic approach to reduce the independent tensor components by symmetry transformations: a commented translation of “Tensors and Crystal Symmetry” by Carl Hermann

Glüge

Aßmus

2021

Continuum Mech. Thermodyn.

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Visualising elastic anisotropy: theoretical background and computational implementation

Cited by 58 publications

References 25 publications

Phase-Field Material Point Method for dynamic brittle fracture with isotropic and anisotropic surface energy

Phase-Field Material Point Method for dynamic brittle fracture with isotropic and anisotropic surface energy

Ab initio predictions of structure and physical properties of the Zr2GaC and Hf2GaC MAX phases under pressure

A systematic approach to reduce the independent tensor components by symmetry transformations: a commented translation of “Tensors and Crystal Symmetry” by Carl Hermann

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