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Journal ArticleDOI

A phase-field description of dynamic brittle fracture

TL;DR: It is shown that the combination of the phase-field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions.
About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2012-04-01. It has received 1260 citations till now. The article focuses on the topics: Temporal discretization & Fracture mechanics.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors provide an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities, and propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem.
Abstract: In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities. Within the formulations stemming from Griffith's theory, we focus on quasi-static models featuring a tension-compression split, which prevent cracking in compression and interpenetration of the crack faces upon closure, and on the staggered algorithmic implementation due to its proved robustness. In this paper, we establish an appropriate stopping criterion for the staggered scheme. Moreover, we propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem. This enables a significant reduction of computational cost--about one order of magnitude--with respect to the available (non-linear) models. The conceptual and structural similarities of the hybrid formulation to gradient-enhanced continuum damage mechanics are outlined as well. Several benchmark problems are solved, including one with own experimental verification.

880 citations


Cites background or methods from "A phase-field description of dynami..."

  • ...The early quasi-static formulations have been extended to the dynamic case in [20–25]....

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  • ...[22], 2012, Hofacker and Miehe [23,24], 2012, 2013, Schlüter et al....

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  • ...[10,15,16] and [22], enabling to test the phasefield method in general, as well as the performance of the various particular formulations....

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  • ...in [22], the difference between decompositions Eqs....

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  • ...The pre-existing crack is modeled by defining an initial strainhistory field as in [22]....

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Journal ArticleDOI
TL;DR: This contribution focuses in mechanical problems and analyze the energetic format of the PDE, where the energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem.

721 citations


Cites background from "A phase-field description of dynami..."

  • ...The initial strain history function (H(x, 0)) could be defined as a function of the closest distance from x to the line (l), which represents the discrete crack [35]....

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Journal ArticleDOI
TL;DR: In this article, a phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime is proposed, which captures the entire range of behavior of a ductile material exhibiting $$J_2$$J2-PLasticity, encompassing plasticization, crack initiation, propagation and failure.
Abstract: Phase-field modeling of brittle fracture in elastic solids is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths including branching and merging. We propose a novel phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime. The formulation is shown to capture the entire range of behavior of a ductile material exhibiting $$J_2$$J2-plasticity, encompassing plasticization, crack initiation, propagation and failure. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of ductile fracture as reported in the experimental literature.

522 citations


Cites background from "A phase-field description of dynami..."

  • ...Phase-field modeling of brittle fracture [1-4] is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths including branching and merging....

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Journal ArticleDOI
TL;DR: An introduction to IGA applied to simple analysis problems and the related computer implementation aspects is presented, and implementation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is presented.

522 citations


Cites methods from "A phase-field description of dynami..."

  • ..., [18, 83], and the thick level set method [85]....

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  • ...A phase field model for dynamic fracture was presented in [18] using adaptive T-spline refinement to provide an effective method for simulating fracture in three dimensions....

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Journal ArticleDOI
TL;DR: It is shown that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features of NURBS, which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions.

435 citations

References
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Journal ArticleDOI
TL;DR: In this article, a displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method.
Abstract: SUMMARY An improvement of a new technique for modelling cracks in the nite element framework is presented. A standard displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique. Copyright ? 1999 John Wiley & Sons, Ltd.

5,815 citations


"A phase-field description of dynami..." refers background in this paper

  • ...• They obey the convex hull property, see Piegl and Tiller (1997). • Local refinement is possible. For additional details on basic T-spline analysis technology see Scott et al. (2010). The spline-based analysis strategy of isogeometric analysis has shown some advantages when compared to standard C finite elements. First, isogeometric analysis allows for the efficient and exact geometric representation of many objects of engineering interest. Also, when using T-spline-based isogeometric analysis, efficient and automatic local mesh refinement strategies exist that preserve the exact geometry. Secondly, the isogeometric basis is in general smooth. Although the phase-field model permits the use of traditional C finite elements, the use of a smooth base is anticipated to have favorable effects. One is that stresses are represented more accurately than with traditional C finite elements, which has been observed to yield efficient spatial discretizations for discrete (see Verhoosel et al. (2010)) and smeared (see Verhoosel et al. (2011)) fracture models....

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  • ...• They obey the convex hull property, see Piegl and Tiller (1997). • Local refinement is possible. For additional details on basic T-spline analysis technology see Scott et al. (2010). The spline-based analysis strategy of isogeometric analysis has shown some advantages when compared to standard C finite elements....

    [...]

  • ...• They obey the convex hull property, see Piegl and Tiller (1997). • Local refinement is possible. For additional details on basic T-spline analysis technology see Scott et al. (2010). The spline-based analysis strategy of isogeometric analysis has shown some advantages when compared to standard C finite elements. First, isogeometric analysis allows for the efficient and exact geometric representation of many objects of engineering interest. Also, when using T-spline-based isogeometric analysis, efficient and automatic local mesh refinement strategies exist that preserve the exact geometry. Secondly, the isogeometric basis is in general smooth. Although the phase-field model permits the use of traditional C finite elements, the use of a smooth base is anticipated to have favorable effects. One is that stresses are represented more accurately than with traditional C finite elements, which has been observed to yield efficient spatial discretizations for discrete (see Verhoosel et al. (2010)) and smeared (see Verhoosel et al....

    [...]

  • ...• They obey the convex hull property, see Piegl and Tiller (1997). • Local refinement is possible....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision, and study their application in computer vision.
Abstract: : This reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision. In computer vision, a fundamental problem is to appropriately decompose the domain R of a function g (x,y) of two variables. This problem starts by describing the physical situation which produces images: assume that a three-dimensional world is observed by an eye or camera from some point P and that g1(rho) represents the intensity of the light in this world approaching the point sub 1 from a direction rho. If one has a lens at P focusing this light on a retina or a film-in both cases a plane domain R in which we may introduce coordinates x, y then let g(x,y) be the strength of the light signal striking R at a point with coordinates (x,y); g(x,y) is essentially the same as sub 1 (rho) -possibly after a simple transformation given by the geometry of the imaging syste. The function g(x,y) defined on the plane domain R will be called an image. What sort of function is g? The light reflected off the surfaces Si of various solid objects O sub i visible from P will strike the domain R in various open subsets R sub i. When one object O1 is partially in front of another object O2 as seen from P, but some of object O2 appears as the background to the sides of O1, then the open sets R1 and R2 will have a common boundary (the 'edge' of object O1 in the image defined on R) and one usually expects the image g(x,y) to be discontinuous along this boundary. (JHD)

5,516 citations


"A phase-field description of dynami..." refers methods in this paper

  • ...The variational formulation for quasi-static brittle fracture leads to an energy functional that closely resembles the potential presented by Mumford and Shah (1989), which is encountered in image segmentation....

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Journal ArticleDOI
TL;DR: In this article, the concept of isogeometric analysis is proposed and the basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model.

5,137 citations


"A phase-field description of dynami..." refers methods in this paper

  • ...In contrast to earlier work on phase-field models, we have chosen to use isogeometric spatial discretizations as introduced by Hughes et al. (2005), which are based on NURBS and T-splines....

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Book
25 Aug 1995
TL;DR: This chapter discusses the construction of B-spline Curves and Surfaces using Bezier Curves, as well as five Fundamental Geometric Algorithms, and their application to Curve Interpolation.
Abstract: One Curve and Surface Basics.- 1.1 Implicit and Parametric Forms.- 1.2 Power Basis Form of a Curve.- 1.3 Bezier Curves.- 1.4 Rational Bezier Curves.- 1.5 Tensor Product Surfaces.- Exercises.- Two B-Spline Basis Functions.- 2.1 Introduction.- 2.2 Definition and Properties of B-spline Basis Functions.- 2.3 Derivatives of B-spline Basis Functions.- 2.4 Further Properties of the Basis Functions.- 2.5 Computational Algorithms.- Exercises.- Three B-spline Curves and Surfaces.- 3.1 Introduction.- 3.2 The Definition and Properties of B-spline Curves.- 3.3 The Derivatives of a B-spline Curve.- 3.4 Definition and Properties of B-spline Surfaces.- 3.5 Derivatives of a B-spline Surface.- Exercises.- Four Rational B-spline Curves and Surfaces.- 4.1 Introduction.- 4.2 Definition and Properties of NURBS Curves.- 4.3 Derivatives of a NURBS Curve.- 4.4 Definition and Properties of NURBS Surfaces.- 4.5 Derivatives of a NURBS Surface.- Exercises.- Five Fundamental Geometric Algorithms.- 5.1 Introduction.- 5.2 Knot Insertion.- 5.3 Knot Refinement.- 5.4 Knot Removal.- 5.5 Degree Elevation.- 5.6 Degree Reduction.- Exercises.- Six Advanced Geometric Algorithms.- 6.1 Point Inversion and Projection for Curves and Surfaces.- 6.2 Surface Tangent Vector Inversion.- 6.3 Transformations and Projections of Curves and Surfaces.- 6.4 Reparameterization of NURBS Curves and Surfaces.- 6.5 Curve and Surface Reversal.- 6.6 Conversion Between B-spline and Piecewise Power Basis Forms.- Exercises.- Seven Conics and Circles.- 7.1 Introduction.- 7.2 Various Forms for Representing Conics.- 7.3 The Quadratic Rational Bezier Arc.- 7.4 Infinite Control Points.- 7.5 Construction of Circles.- 7.6 Construction of Conies.- 7.7 Conic Type Classification and Form Conversion.- 7.8 Higher Order Circles.- Exercises.- Eight Construction of Common Surfaces.- 8.1 Introduction.- 8.2 Bilinear Surfaces.- 8.3 The General Cylinder.- 8.4 The Ruled Surface.- 8.5 The Surface of Revolution.- 8.6 Nonuniform Scaling of Surfaces.- 8.7 A Three-sided Spherical Surface.- Nine Curve and Surface Fitting.- 9.1 Introduction.- 9.2 Global Interpolation.- 9.2.1 Global Curve Interpolation to Point Data.- 9.2.2 Global Curve Interpolation with End Derivatives Specified.- 9.2.3 Cubic Spline Curve Interpolation.- 9.2.4 Global Curve Interpolation with First Derivatives Specified.- 9.2.5 Global Surface Interpolation.- 9.3 Local Interpolation.- 9.3.1 Local Curve Interpolation Preliminaries.- 9.3.2 Local Parabolic Curve Interpolation.- 9.3.3 Local Rational Quadratic Curve Interpolation.- 9.3.4 Local Cubic Curve Interpolation.- 9.3.5 Local Bicubic Surface Interpolation.- 9.4 Global Approximation.- 9.4.1 Least Squares Curve Approximation.- 9.4.2 Weighted and Constrained Least Squares Curve Fitting.- 9.4.3 Least Squares Surface Approximation.- 9.4.4 Approximation to Within a Specified Accuracy.- 9.5 Local Approximation.- 9.5.1 Local Rational Quadratic Curve Approximation.- 9.5.2 Local Nonrational Cubic Curve Approximation.- Exercises.- Ten Advanced Surface Construction Techniques.- 10.1 Introduction.- 10.2 Swung Surfaces.- 10.3 Skinned Surfaces.- 10.4 Swept Surfaces.- 10.5 Interpolation of a Bidirectional Curve Network.- 10.6 Coons Surfaces.- Eleven Shape Modification Tools.- 11.1 Introduction.- 11.2 Control Point Repositioning.- 11.3 Weight Modification.- 11.3.1 Modification of One Curve Weight.- 11.3.2 Modification of Two Neighboring Curve Weights.- 11.3.3 Modification of One Surface Weight.- 11.4 Shape Operators.- 11.4.1 Warping.- 11.4.2 Flattening.- 11.4.3 Bending.- 11.5 Constraint-based Curve and Surface Shaping.- 11.5.1 Constraint-based Curve Modification.- 11.5.2 Constraint-based Surface Modification.- Twelve Standards and Data Exchange.- 12.1 Introduction.- 12.2 Knot Vectors.- 12.3 Nurbs Within the Standards.- 12.3.1 IGES.- 12.3.2 STEP.- 12.3.3 PHIGS.- 12.4 Data Exchange to and from a NURBS System.- Thirteen B-spline Programming Concepts.- 13.1 Introduction.- 13.2 Data Types and Portability.- 13.3 Data Structures.- 13.4 Memory Allocation.- 13.5 Error Control.- 13.6 Utility Routines.- 13.7 Arithmetic Routines.- 13.8 Example Programs.- 13.9 Additional Structures.- 13.10 System Structure.- References.

4,552 citations