Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of cracking-induced failure is, therefore, a major concern in structural designs. Computational modeling of fracture constitutes an indispensable tool not only to predict the failure of cracking structures but also to shed insights into understanding the fracture processes of many materials such as concrete, rock, ceramic, metals, and biological soft tissues. This chapter provides an extensive overview of the literature on the so-called phase-field fracture/damage models (PFMs), particularly, for quasi-static and dynamic fracture of brittle and quasi-brittle materials, from the points of view of a computational mechanician. PFMs are the regularized versions of the variational approach to fracture which generalizes Griffith's theory for brittle fracture. They can handle topologically complex fractures such as initiation, intersecting, and branching cracks in both two and three dimensions with a quite straightforward implementation. One of our aims is to justify the gaining popularity of PFMs. To this end, both theoretical and computational aspects are discussed and extensive benchmark problems (for quasi-static and dynamic brittle/cohesive fracture) that are successfully and unsuccessfully solved with PFMs are presented. Unresolved issues for further investigations are also documented.

Phase-field modeling of fracture

SUMMARY

A novel approach based on a combination of isogeometric analysis (IGA) and extended FEM is presented for fracture analysis of structures. The extended isogeometric analysis is capable of an efficient analysis of general crack problems using nonuniform rational B-splines as basis functions for both the solution field approximation and the geometric description, and it can reproduce crack tip singular fields and discontinuity across a crack. IGA has attracted a lot of interest for solving different types of engineering problems and is now further extended for the analysis of crack stability and propagation in two-dimensional isotropic media. Concepts of the extended FEM are used in IGA to avoid the necessity of remeshing in crack propagation problems and to increase the solution accuracy around the crack tip. Crack discontinuity is represented by the Heaviside function and isotropic analytical displacement fields near a crack tip are reproduced by means of the crack tip enrichment functions. Also, the Lagrange multiplier method is used to impose essential boundary conditions. Moreover, the subtriangles technique is utilized for improving the accuracy of integration by the Gauss quadrature rule. Several two-dimensional static and quasi-static crack propagation problems are solved to demonstrate the efficiency of the proposed method and the results of mixed-mode stress intensity factors are compared with analytical and extended FEM results. Copyright © 2011 John Wiley & Sons, Ltd.

https://chpc.ut.ac.ir/download/paper1/Extended%20isogeometric%20analysis%20for%20simulation%20of%20stationary%20and.pdf

Extended isogeometric analysis for simulation of stationary and propagating cracks

Mechanical modelling of 3D woven composites considering realistic unit cell geometry

We present an efficient model based on finite elements for viscoelastic, highly flexible surfaces. It is particularly designed for numerically stiff materials such as textiles because it yields linear equations in each time step and allows fast time stepping in an implicit integration method. This is achieved by reducing the nonlinear elasticity problem to a planar, linear one in each step. We apply this model to a garment simulation in which we assemble garments from CAD cloth patterns, seam these together, and animate the cloth in dynamic scenes with any chosen material properties. This results in a physically accurate but also fast simulation.

A fast finite element solution for cloth modelling

The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill–Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss’ and Voigt’s bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE calculations is presented. The performances of the proposed schemes are illustrated via a series of twoand three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks. [DOI: 10.1115/1.4034024]

/pdf/aspects-of-computational-homogenization-at-finite-4jj9ulnomt.pdf

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

A nonlinear fractional viscoelastic material model for polymers

Inelastic material behavior of polymers – Experimental characterization, formulation and implementation of a material model

This paper addresses the application of the extended finite element method (XFEM) to the modeling of two-dimensional coupled magneto-mechanical field problems Continuum formulations of the stationary magnetic and the coupled magneto-mechanical boundary problem are outlined, and the corresponding weak forms are derived The XFEM is applied to generate numerical models of a representative volume element, characterizing a magnetoactive composite material Weak discontinuities occurring at material interfaces are modeled numerically by an enriched approximation of the primary field variables In order to reduce the complexity of the representation of curved interfaces, an element local approach is proposed which allows an automated computation of the level set values The composite’s effective behavior and its coupled magneto-mechanical response are computed numerically by a homogenization procedure The scale transition process is based on the energy equivalence condition, which is satisfied by using periodic boundary conditions

XFEM modeling and homogenization of magnetoactive composites

The FE-simulation of inhomogeneous structures, such as composite materials, biological tissues or foams, requires the generation of respective finite element meshes. With increasing complexity of the inner architecture of such structures, this becomes a time-consuming and laborious task. Additionally, the risk of forming bad-shaped elements that may lead to ill-conditioned numerical problems grows significantly. A solution to this problem provides the extended finite element method (XFEM). Thereby, the interface between different materials is represented by a local enrichment of the displacement approximation. As a consequence of this, the element boundary need not be aligned to the interface.



In order to improve the accuracy of the interface approximation, the development of a plane element based on the XFEM and quadratic shape functions will be presented. This element allows for the description of curved material interfaces. The computation of the element stiffness matrix requires a numerical integration process that accounts for discontinuous fields. Regarding a linear element formulation, this can be achieved by an adapted triangulation of the element domain. However, in the case of a curved interface this solution is not applicable. Hence, non-uniform rational B-Spline (NURBS) surfaces are used to evaluate the integrals numerically.



Finally, the results of different examples will show the general properties such as the accuracy of the numerical integration procedure and the convergence behavior of this element formulation. Copyright © 2011 John Wiley & Sons, Ltd.

Development of a quadratic finite element formulation based on the XFEM and NURBS

This paper addresses the multiscale simulation of fibre-reinforced polymers. The considered composite materials exhibit a hierarchical material structure with three distinct length scales—micro, meso and macro. This feature of the morphology allows for the application of homogenization techniques based on a representative volume element (RVE) that is entirely typical for the local, periodic material structure. The effective macroscopic material behaviour of the composite can be predicted from the properties of the individual constituents and the geometric arrangement of the reinforcing fibres based on the simulation of the material behaviour in the RVE.

The heterogeneous material structure in an RVE is modelled by the eXtended finite element method (XFEM). To this, two special element types, called X-element and 2X-element, are derived. They can represent one or two material interfaces within the element domain. For an efficient modelling process, an automated model generation procedure that determines the required element types, locates the material interfaces and performs the subdivision of the X-elements into tetrahedral integration domains has been developed. Problems related to a consistent interface approximation and a continuous displacement field are discussed.

In the generated RVE models, a viscoplastic material model accounts for the inelastic material behaviour of the polymeric matrix, whereas the glass-fibres are assumed to have a linear elastic stress–strain behaviour. Using periodic displacement boundary conditions, effective stress–strain curves are computed for glass-fibre-reinforced polypropylene with unidirectional and woven arrangements of the reinforcing fibres. Copyright © 2010 John Wiley & Sons, Ltd.

Volker Ulbricht

Papers

A nonlinear fractional viscoelastic material model for polymers

Inelastic material behavior of polymers – Experimental characterization, formulation and implementation of a material model

XFEM modeling and homogenization of magnetoactive composites

Development of a quadratic finite element formulation based on the XFEM and NURBS

Multiscale XFEM‐modelling and simulation of the inelastic material behaviour of textile‐reinforced polymers