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Isogeometric cohesive elements for two and three dimensional compositedelamination analysis Vinh Phu Nguyen a,1 , Pierre Kerfriden a,2 , St´ephane P.A. Bordas a,3, ∗ a School of Engineering, Institute of Mechanics and Advanced Materials, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA Abstract Isogeometric cohesive elements are presented for modeling two and three dimensional delaminated composite structures.We exploit the knot insertion algorithm offered by NURBS (Non Uniform Rational B-splines) to generate cohesive ele-ments along delamination planes in an automatic fashion. A complete computational framework is presented includingpre-processing, processing and post-processing. They are explained in details and implemented in MIGFEM–an opensource Matlab Isogemetric Analysis code developed by the authors. The composite laminates are modeled using bothNURBS solid and rotation-free shell elements. Several two and three dimensional examples ranging from standarddelamination tests (the mixed mode bending test) to the L-shaped specimen with a fillet, three dimensional (3D)double cantilever beam and a 3D singly curved thick-walled laminate are provided. To the authors’ knowledge, it isthe first time that NURBS-based isogeometric analysis for two/three dimensional delamination modeling is presented.IGA provides a bi-directional system in which one can go forward from CAD to analysis and backwards from analysisto CAD. This is believed to facilitate the design of composite structures. Keywords: isogeometric analysis (IGA), B-spline, NURBS, finite elements (FEM), CAD, delamination, composite,cohesive elements, interface elements 1. Introduction Laminated composite materials are often used in fields like automotive, aerospace, and sport equipments due totheir high strength and stiffness in combination with low density. The caveat with laminated composite structures istheir low out-of-plane strength and delamination or interfacial cracking between composite layers is unarguably one of the predominant modes of failure in laminated composite which can change the structural stiffness significantly and isdifficult to detect during inspection. This failure mode has therefore been widely investigated both experimentally andnumerically. Delamination analyses have been traditionally performed using standard low order Lagrange elements ina FEM framework, see e.g., [1,2,3,4] and references therein. The two most popular computational methods for the analysis of delamination are the Virtual Crack Closure Technique (VCCT)[5,6] and interface elements with a cohesive law (also known as decohesion elements)[1,2,3,7]. The latter is adopted in this contribution for it can deal with initiation and propagation of delamination in a unified theory. It should be emphasized that inserting interface elements ∗ Corresponding author 1 n guyenpv@cardiff.ac.uk, ORCID: 0000-0003-1212-8311 2 p ierre@cardiff.ac.uk 3 s
[email protected], ORCID: 0000-0001-7622-2193 Preprint submitted to Elsevier May 17, 2013 into a Lagrange finite element (FE) mesh is a time-consuming task even with commercial FE packages. Due to thatfact, an open source mesh generator for cohesive modeling was developed by the first author and presented in [8]. BesideFEM, other numerical discretization methods have been utilized for delamination analyses as well. The Element FreeGalerkin, which is a meshfree method, with the smooth moving least square basis was also adopted for delaminationanalysis [9]. In order to alleviate the computational expense of cohesive elements, formulations with enrichment of theFE basis was proposed in[10,11]. The extended finite element method (XFEM)[12] has been adopted for delamination studies e.g.,[13,14,15,16] which makes the pre-processing simple for the delaminations can be arbitrarily located with respect to the FE mesh. The interaction between the delamination plane and the mesh is resolved during thesolving step by using enrichment functions. However, implementation of XFEM is more involved than other numericalmethods. For all aforementioned methods, curved geometries of the solids are not exactly represented.Isogeometric analysis (IGA) was proposed by Hughes and his co-workers[17] in 2005 to reduce the gap betweenComputer Aided Design (CAD) and Finite Element Analysis (FEA). The idea is to use CAD technology such B-splines,NURBS (Non Uniform Rational B-splines), T-splines etc. as basis functions in a finite element (FE) framework. Sincethis seminal paper, a monograph was published entirely on the subject [18] and applications were found in several fieldsincluding structural mechanics, solid mechanics, fluid mechanics and contact mechanics. It should be emphasized thatthe idea of using CAD technologies in finite elements dates back at least to[19] where B-splines were used as shapefunctions in FEM and subdivision surfaces were adopted to model shells[20].Due to the ultra smoothness provided by NURBS basis, IGA was successfully applied to many engineering problemsranging from contact mechanics, see e.g.,[21,22,23], optimisation problems[24,25], structural mechanics[26,27,28], structural vibration [29,30], to fluids mechanics[31,32], fluid-structure interaction problems [33]. In addition, due to the ease of constructing high order continuous basis functions, IGA has been used with great success in solvingPDEs that incorporate fourth order (or higher) derivatives of the field variable such as the Hill-Cahnard equation[34], explicit gradient damage models[35] and gradient elasticity [36]. IGA has been implemented in commercial FE packages– Abaqus [37] and LS-Dyna [38]. In the context of fracture mechanics, IGA has been applied to fracture using the partition of unity method (PUM)[39] to capture two dimensional strong discontinuities and crack tip singularitiesefficiently [40,41]. A phase field model for dynamic fracture has been presented in [42] where adaptive refinement with T-splines provides an effective method for simulating fracture in three dimensions. Cohesive fracture modeling in anIGA framework was presented in [43]. The method hinges on the ability to specify the continuity of NURBS/T-splinesthrough a process known as knot insertion. Highly accurate stress fields in cracked specimens were obtained with coarsemeshes. We refer to[44] for an overview of IGA and its implementation aspects.More recently, in[45] high order B-splines cohesive FEs with C 0 continuity across element boundary were utilized toefficiently model delamination of two dimensional (2D) composite specimens. In the referred paper, it was shown thatby using high order B-spline (order of up to 4) basis functions along the delamination path, relatively coarse meshes canbe used and 2D delamination benchmark tests such as the mixed mode bending test were solved within 10 seconds ona laptop. In this manuscript, prompted by our previous encouraging results reported in [45] plus the work in [43] and a practical motivation of a predictive tool not only for analyzing but also for designing composite laminates, we presentan isogeometric framework for two and three dimensional (2D/3D) delamination analysis of laminated composites.Both the geometry and the displacement field are approximated using NURBS, therefore curved geometries are exactlyrepresented. We use knot insertion algorithm of NURBS to duplicate control points along the delamination pathswhere delamination will take place. Meshes of zero-thickness interface elements can be straightforwardly generated.The proposed ideas are implemented in our open source Matlab IGA code, MIGFEM 4 , described in[44]. Severalexamples are provided including the mixed mode bending test, a L-shaped curved composite specimen test[46,47], 3D double cantilever beam and a 3D singly curved thick-walled laminate. Moreover, isogeometric shell elements are 4 available for download at https://sourceforge.net/projects/cmcodes/ 2 used for the first time, at least to the authors’ knowledge, to model delamination. Our findings are (i) the proposedIGA-based framework reduces significantly the time being spent on the pre-processing step to prepare FE models fordelamination analyses and (ii) from the analysis perspective, the smooth high order NURBS basis functions are able toproduce highly accurate stress fields which is very important in fracture modeling. The consequence is that relativelycoarse meshes (compared to meshes of lower order elements) can be adopted and thus the computational expense isreduced [43,45]. Moreover, IGA provides a bi-directional system in which one can go forward from CAD to analysis and backwards from analysis to CAD. This is believed to facilitate the design of composite structures.The remainder of the paper is organized as follows. Section2gives the strong and weak formulations of the studiedproblem. It also points out the key difficulties of standard Lagrange finite elements used for delamination analyses.Section3briefly presents NURBS curves, surfaces and solids. Section4is devoted to a discussion on knot insertion and automatic generation of cohesive interface elements followed by finite element formulations for solids with cohesivecracks given in Section5.Numerical examples are given in Section6.Finally, Section7ends the paper with some concluding remarks. 2. Problem description 2.1. Strong form Considering a solid Ω, as shown in Fig. (1), that is bounded by Γ and contains a cohesive crack Γ d . Prescribeddisplacements¯ u are imposed on the Dirichlet boundary Γ u and prescribed tractions¯ t are applied on the Neumannboundary Γ t , Γ u ∪ Γ t = Γ, Γ u ∩ Γ t = ∅ . Under the assumption of small displacements and gradients (note that finitedeformation theories can be used in the proposed framework without any difficulties), the deformation of the materialis characterized by the infinitesimal strain tensor ǫ ij = (1 / 2)(( ∂u i /∂x j )+( ∂u j /∂x i )) for the bulk and the displacement jump [[ u i ]] for the cohesive crack. The governing equations for quasi-static problems include the balance of linearmomentum, the natural, essential boundary conditions and the traction continuity on the crack surface ∇· σ + b = 0 x ∈ Ω (1a) n · σ =¯ t x ∈ Γ t (1b) u =¯ u x ∈ Γ u (1c) n + d · σ = t +c ; n − d · σ = t − c ; t +c = − t c = − t − c x ∈ Γ d (1d)where σ is the Cauchy stress tensor, b is the body force vector, n denotes the normal to the boundary Γ and t c is thecohesive traction across the crack Γ d with unit normal vector n d .Constitutive equations for the bulk and the cohesive crack can be written as σ = σ ( ǫ ( u ) , α ) , t c = t c ([[ u ]] , β ) (2)where α and β are history variables. Concrete constitutive models used in this paper will be presented later, seeSection5.3. 2.2. Weak form The weak formulation reads: finding the displacement field u such that Ω δ u · b dΩ + Γ t δ u · ¯ t dΓ t = Ω δ ǫ : σ ( u )dΩ + Γ d δ u · t c ([[ u ]])dΓ d (3)be satisfied for any admissible displacement field δ u subject to the Dirichlet boundary conditions on Γ u .3 Figure 1: A two dimension solid containing a cohesive crack. 2.3. Difficulties with standard interface elements For problems in which the crack path is known a priori such as delamination in composite laminates and debondingof the matrix/inclusion interface, interface elements, which are elements inserted at the common boundary of continuumelements (Fig. (2)) where potential fracture will occur, are usually the method of choice. The reason behind thepopularity of cohesive interface elements is probably due to the straightforward computer implementation.Figure 2: Discretization of the solid into continuum elements and zero-thickness interface elements.Interface elements, when applied to delamination analyses, suffer from two shortcomings namely (i) a long and cum-bersome pre-processing step (doubling nodes along each delamination path, modifying the connectivity of continuumelements above and below the delamination path) and (ii) a refined mesh has to be employed otherwise unphysicaloscillations in the global load-displacement behaviour of the structure are observed (in the worse case, this can cause theiterative solver to diverge). Furthermore, imagine that during the design process of composite laminates, the analystdecides to change a geometry parameter, he/she then has to go back to the CAD system to change the geometry modeland repeat the time-consuming mesh generation procedure again.In what follows, we present an isogeometric framework for delamination analyses that resolve all the aforementioned4 shortcomings of Lagrange finite elements. We will demonstrate how straightforward it is to insert interface elements ina NURBS mesh for both 2D and 3D. And since there is a two-way link between CAD and FEA using NURBS, changesto the geometry can be dealt with without difficulties. As far as the mesh density requirement is concerned, the highlysmooth NURBS basis functions can alleviate this to some extent as pointed out previously in[45]. 3. NURBS curves, surfaces and solids In this section, NURBS are briefly reviewed. We refer to the standard textbook [48] for details. A knot vector isa sequence in ascending order of parameter values, written Ξ = { ξ 1 ,ξ 2 ,...,ξ n + p +1 } where ξ i is the i th knot, n is thenumber of basis functions and p is the order of the B-spline basis. Open knots in which the first and last knots appear p +1 times are standard in the CAD literature and thus used in this manuscript i.e., Ξ = { ξ 1 ,...,ξ 1 p + 1 times ,ξ 2 ,...,ξ m ,...ξ m p + 1 times } .Given a knot vector Ξ, the B-spline basis functions are defined recursively starting with the zeroth order basisfunction ( p = 0) given by N i, 0 ( ξ ) = 1 if ξ i ≤ ξ < ξ i +1 0 otherwise(4)and for a polynomial order p ≥ 1 N i,p ( ξ ) = ξ − ξ i ξ i + p − ξ i N i,p − 1 ( ξ ) + ξ i + p +1 − ξ ξ i + p +1 − ξ i +1 N i +1 ,p − 1 ( ξ ) (5)This is referred to as the Cox-de Boor recursion formula. Note that when evaluating these functions, ratios of the form0 / 0 are defined as zero.Some salient properties of B-spline basis functions are (1) they constitute a partition of unity, (2) each basis functionis nonnegative over the entire domain, (3) they are linearly independent, (4) the support of a B-spline function of order p is p + 1 knot spans i.e., N i,p is non-zero over [ ξ i ,ξ i + p +1 ], (5) basis functions of order p have p − m i continuousderivatives across knot ξ i where m i is the multiplicity of knot ξ i and (6) B-spline basis are generally only approximants(except at the ends of the parametric space interval, [ ξ 1 ,ξ n + p +1 ]) and not interpolants.Figure3illustrates some quadratic B-splines functions defined on an open non-uniform knot vector. Note that thebasis functions are interpolatory at the ends of the interval thanks to the use of open knot vectors and also at ξ = 4,the location of a repeated knot where only C 0 -continuity is attained. Elsewhere, the functions are C 1 -continuous. Theability to control continuity by means of knot insertion is particularly useful for modeling discontinuities such as cracksor material interfaces as will be presented in this paper. In general, in order to have a C − 1 continuity at a knot, itsmultiplicity must be p + 1.NURBS basis functions are defined as R i,p ( ξ ) = N i,p ( ξ ) w i W ( ξ )= N i,p ( ξ ) w i nj =1 N j,p ( ξ ) w j (6)where N i,p ( ξ ) denotes the i th B-spline basis function of order p and w i are a set of n positive weights. Selectingappropriate values for the w i permits the description of many different types of curves including polynomials andcircular arcs. For the special case in which w i = c,i = 1 , 2 ,...,n the NURBS basis reduces to the B-spline basis. Notethat for simple geometries, the weights can be defined analytically see e.g., [48]. For complex geometries, they areobtained from CAD packages such as Rhino [49].5
