Crack kinking in a variational phase-field model of brittle fracture with strongly anisotropic surface energy

2020

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Summary The surface energy a phase‐field approach to brittle fracture in anisotropic materials is also anisotropic and gives rise to second‐order gradients in the phase field entering the energy functional. This necessitates C1 continuity of the basis functions which are used to interpolate the phase field. The basis functions which are employed in isogeometric analysis (IGA), such as nonuniform rational B‐splines and T‐splines naturally possess a higher order continuity and are therefore ideally suited for phase‐field models which are equipped with an anisotropic surface energy. Moreover, the high accuracy of spline discretizations, also relative to their computational demand, significantly reduces the fineness of the required discretization. This holds a fortiori if adaptivity is included. Herein, we present two adaptive refinement schemes in IGA, namely, adaptive local refinement and adaptive hierarchical refinement, for phase‐field simulations of anisotropic brittle fracture. The refinement is carried out using a subdivision operator and exploits the Bézier extraction operator. Illustrative examples are included, which show that the method can simulate highly complex crack patterns such as zigzag crack propagation. An excellent agreement is obtained between the solutions from global refinement and adaptive refinement, with a reasonable reduction of the computational effort when using adaptivity.

ℰ_{ℓ} false(u, c false) = \int_{Ω} a (c) 𝒲 (u) d Ω - \int_{Γ_{t}} u \cdot \hat{t} d Γ + \frac{𝒢_{0}}{β ℓ_{c}} \int_{Ω} false(w (c) + ℓ_{c}^{4} \nabla^{2} c : C : \nabla^{2} c false) d Ω,

where a ( c ) = (1 − c ) 2 is a degradation function, w ( c ) = 9 c is a monotonically increasing function which represents the energy dissipation per unit volume, and

β = 4 \int_{0}^{1} \sqrt{w (c)} d c = 96 / 5

is a normalization parameter. ∇ 2 c is a Hessian, that is,

{(\nabla^{2} c)}_{i j} = \frac{\partial^{2} c}{\partial x_{i} \partial x_{j}}

and C is a positive‐definite fourth‐order tensor with the same symmetries as the linear elastic stiffness tensor 29 .…”

Section: Phase‐field Approximations Of Anisotropic Fracturementioning

confidence: 99%

“…Assuming a cubic symmetry, three material constants, C 1111 , C 1122 , and C 1212 , suffice to define C . The damage evolution then follows from (in a strong format): 20,23

\frac{2 ℓ^{4} 𝒢_{0}}{β} (2 false(C_{1122} + 2 C_{1212} false) \frac{\partial^{4} c}{\partial x^{2} \partial y^{2}} + C_{1111} (\frac{\partial^{4} c}{\partial x^{4}} + \frac{\partial^{4} c}{\partial y^{4}})) + 𝒲 (u) a^{'} (c) ℓ + \frac{𝒢_{0}}{β} w^{'} (c) = 0,

complemented by the irreversibility condition

ċ \geq 0

. The resulting anisotropic surface energy

𝒢_{c} (θ)

then takes the form: 20,23

𝒢_{c} (θ) = 𝒢_{0} \sqrt[4]{C (θ)},

with

C (θ) = \frac{1}{4} false(3 C_{1111} + C_{1122} + 2 C_{1212} false) (1 + C_{1111})

…”

Section: Phase‐field Approximations Of Anisotropic Fracturementioning

confidence: 99%

“…A significant advantage can thus be gained by using adaptive mesh refinement. Using Equation (6), the effective internal length

ℓ (θ)

then takes the form: 20,23

ℓ (θ) = ℓ_{c} \sqrt[4]{C (θ)} .

…”

Section: Phase‐field Approximations Of Anisotropic Fracturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Adaptive isogeometric analysis for phase‐field modeling of anisotropic brittle fracture

2020

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“…10 In addition, stress fields can be improved when enriching stress fields with higher-order terms, 11 while crack tracking algorithms can also help to better simulate complex dynamic crack patterns, such as crack branching. 12,13 Recently, phase-field models have been introduced to describe brittle fracture, [14][15][16] allowing for a straightforward treatment of crack branching and merging. 17 Isogeometric analysis has also been introduced in the context of crack propagation analysis.…”

Section: Introductionmentioning

confidence: 99%

Energy conservation during remeshing in the analysis of dynamic fracture

2019

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Summary The analysis of (dynamic) fracture often requires multiple changes to the discretisation during crack propagation. The state vector from the previous time step must then be transferred to provide the initial values of the next time step. A novel methodology based on a least‐squares fit is proposed for this mapping. The energy balance is taken as a constraint in the mapping, which results in a complete energy preservation. Apart from capturing the physics better, this also has advantages for numerical stability. To further improve the accuracy, Powell‐Sabin B‐splines, which are based on triangles, have been used for the discretisation. Since scriptC1 continuity of the displacement field holds at crack tips for Powell‐Sabin B‐splines, the stresses at and around crack tips are captured much more accurately than when using elements with a standard Lagrangian interpolation, or with NURBS and T‐splines. The versatility and accuracy of the approach to simulate dynamic crack propagation are assessed in two case studies, featuring mode‐I and mixed‐mode crack propagation.

Weak imposition of Dirichlet boundary conditions for analyses using Powell–Sabin B‐splines

2021

Powell–Sabin B‐splines are enjoying an increased use in the analysis of solids and fluids, including fracture propagation. However, the Powell–Sabin B‐spline interpolation does not hold the Kronecker delta property and, therefore, the imposition of Dirichlet boundary conditions is not as straightforward as for the standard finite elements. Herein, we discuss the applicability of various approaches developed to date for the weak imposition of Dirichlet boundary conditions in analyses which employ Powell–Sabin B‐splines. We take elasticity and fracture propagation using phase‐field modeling as a benchmark problem. We first succinctly recapitulate the phase‐field model for propagation of brittle fracture, which encapsulates linear elasticity, and its discretization using Powell–Sabin B‐splines. As baseline solution we impose Dirichlet boundary conditions in a strong sense, and use this to benchmark the Lagrange multiplier, penalty, and Nitsche's methods, as well as methods based on the Hellinger‐Reissner principle, and the linked Lagrange multiplier method and its modified version.