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High order Bsplines based finite elements for the delamination analysis of
laminated composites
Vinh Phu Nguyena,1,∗, Hung Nguyen-Xuana,b,2
aDivision of Computational Mechanics, Ton Duc Thang University, HCM, VietnambDepartment of Mechanics, Faculty of Mathematics and Computer Science, University of Science-VNU, 227 Nguyen Van Cu Street,
HCM, Vietnam
Abstract
Finite element analysis of the delamination process in delaminated composites is addressed using high order Bezierelements that rely on the Bernstein polynomials which are commonly used in Computer Aided Design. Laminae of acomposite laminate are modelled with two dimensional Bezier elements. Zero-thickness cohesive elements, which areinserted along the interface in between the laminae to model the interface cracking, employ the high order univariateBernstein basis functions. Quadratic, cubic and quartic Bernstein basis functions are examined. An open sourcepre-processing code is presented to generate Bezier meshes and interface elements. Implementation of a simple yethighly efficient arc-length solver which is able to trace equilibrium paths having multiple snapbacks is given. Numericalexamples are presented for the analyses of benchmark composite delamination problems which include the doublecantilever beam, the end notch flexure and the mixed mode bending tests. Additionally multiple delamination analysisof a composite specimen is also presented. For all given examples, an efficient performance of the proposed high orderfinite element formulation was observed.
Keywords: delamination, composite, Bsplines, interface elements, Bezier elements
1. Introduction
Delamination or interfacial cracking between composite layers is unarguably one of the predominant modes offailure in laminated composite. This failure mode has therefore been widely investigated both experimentally andnumerically. Finite element procedures for delamination analyses are often divided in two categories. The first is basedon the theory of Linear Elastic Fracture Mechanics (LEFM), while the second formulates the problem in the frameworkof cohesive zone model. The most adopted LEFM-based technique for delamination analyses is the Virtual CrackClosure Technique VCCT [1, 2]. Although the VCCT has been implemented into commercial FE codes Abaqus R© andMarcTM, among others, it has limitations– the VCCT cannot be used to predict the initiation of the delamination orin other words, it is restricted to problems in which the initial position of the crack is known. Problems also arise whenmore than one crack propagate simultaneously. There are some difficulties when using the VCCT in the simulation ofdelamination growth. The calculation of fracture parameters requires nodal variables and topological information ofnodes ahead and behind the crack front. Such calculations may require remeshing as crack advances.
∗Corresponding [email protected]@hcmus.edu.vn
Preprint submitted to Elsevier November 4, 2013
Interface elements equipped with a cohesive law (also known as a traction-separation law), which are also namedcohesive elements, are more recent than VCCT. Cohesive elements can overcome the aforementioned shortcomingsof the VCCT–they can predict both the initiation and propagation of delamination. Multiple delamination can bemodelled with ease. We refer to e.g., [3, 4, 5] and references therein.
Difficulties also arise when interface elements are used for progressive delamination analysis. They include (1) aproper definition for the elastic stiffness of the interface elements, (2) extremely refined meshes are needed to resolve thecohesive zone ahead of the crack tip and (3) convergence issues associated with the descending branch of the cohesivelaw. Engineering solutions to the first two shortcomings have been reported in [6, 7]. A self-adaptive enrichmentscheme was presented in [8] . A Partition-of-Unity based enrichment finite element formulation has been given in[9]. The Element Free Galerkin, which is a meshfree method, with the smooth MLS (moving least square) basis wasadopted for delamination analysis [10]. In [11], a viscous regularization technique and in [12] a local arc-length methodcombined with line search techniques were presented to resolve the convergence issues. A discrete cohesive zone modelwas proposed in [13] which claims to be an effective numerical model for delamination analysis.
B-splines have been utilized in finite element methods for many reasons see e.g., [14, 15, 16]. One reason is thehigher continuity of derivatives and smoothness of B-splines. Another reason is the possibility of reducing the requirednumber of degrees of freedom compared to a conventional finite element analysis. Furthermore, if B-splines are utilizedto represent the geometry of a finite element model, interfacing a finite element analysis program with existing computeraided design programs (which make extensive use of B-splines) is possible. Recently Hughes and his co-workers hasgeneralized B-splines finite element method to the so-called Isogeometric Analysis [17] in which B-splines, NURBS(non-uniform rational B-splines) are employed.
As the isogeometric analysis are becoming more widely popular, there is a desire to apply them quantitativelyand to a broader class of problems. In this paper, for the first time, we apply this novel method to the delaminationanalysis of composites. Specifically we use two dimensional solid Bezier elements [18] to model the composite pliesand one dimensional Bezier elements for the cohesive interface. Bezier elements are elements defined by the Bernsteinpolynomials [18]. We exploit the ultra-smooth Bernstein basis functions in solving delamination problems. Moreprecisely quadratic, cubic and quartic basis are adopted to discretize the cohesive elements. In order to resolve unstableresponses with snap-backs, the energy-based arc-length method [19], which is a global arc-length method is adopted.The computer implementation of this energy-based control in an arc-length method is also given. An open sourcepre-processing code that is capable of inserting interface elements into existing finite element meshes (consisting ofstandard two/three dimensional Lagrange finite elements or two dimensional Bezier elements) is presented. Numericalexamples show the excellent performance of this energy-based arc-length method in solving delamination problemsexhibiting multiple elastic snap-back instabilities. Our findings are (1) basis functions commonly used in CAD such asthe Bernstein polynomials can be well applied to delamination analysis, (2) for a given problem, the utilization of highorder Bezier elements results in significantly reduced computational resources (number of nodes, number of integrationpoints) compared to analyses using linear elements. This is expected to have positive impact on coupled multiscaleanalyses of thin heterogeneous layers [20, 21].
The remainder of the paper is structured as follows: Section 2 presents the finite element formulation employingthe Bernstein polynomials. This includes the presentation of the Bernstein basis functions, the solid and interfaceBezier elements. Section 3 discusses some implementation issues–a pre-processing tool is described and the computerimplementation of the energy-based arc-length method [19] is given. Numerical examples are given in Section 4. Section5 ends the paper with potential directions for future works.
2. Finite element formulation
This section presents the proposed finite element formulation for delamination analysis. As mentioned in theintroductory part, the novelty of our work lies in the adoption of a new element kinematics that adopts the Bernstein
2
polynomials. Other parts such as formulations for the tangent matrix, internal force vector, derivatives of the shapefunctions with respect to the global coordinates, cohesive laws etc. are standard and thus not discussed herein or onlybriefly touched upon.
2.1. Bernstein polynomials
The Bernstein polynomials form a basis for the Bezier elements [18]. In CAD, Bernstein polynomials are used toconstruct the so-called Bezier curves/surfaces [22]. The univariate Bernstein basis functions of order p are defined overthe biunit interval [−1, 1] as
Bi,p(ξ) =1
2p
(
pi− 1
)
(1 − ξ)p−(i−1)(1 + ξ)i−1 (1)
where the binomial coefficient
(
pi− 1
)
= p!(i−1)!(p+1−i)! , 1 ≤ i ≤ p + 1. We emphasize that, in CAD Bernstein
polynomials are defined in the interval [0, 1]. However, in a finite element setting, the biunit interval [−1, 1], where theGauss quadrature is defined, is preferable. The Bernstein basis functions for p = 1, 2, 3 and 4 are plotted in Figure1. Note that for p = 1, the Bernstein basis resemble the linear Lagrange basis. The Bernstein polynomials have thefollowing properties
−1 −0.6 −0.2 0.2 0.6 10
0.2
0.4
0.6
0.8
1
−1 −0.6 −0.2 0.2 0.6 10
0.2
0.4
0.6
0.8
1
−1 −0.6 −0.2 0.2 0.6 10
0.2
0.4
0.6
0.8
1
1 21
23
1
4
5
2
−1 −0.6 −0.2 0.2 0.6 10
0.2
0.4
0.6
0.8
1
1
2 3
4
3
ξ
ξξ
ξ
p = 1 p = 2
p = 3 p = 4
Figure 1: Bernstein basis functions for polynomial degree p = 1, 2, 3, 4.
3
1. Partition of unity∑n
i=1 Bi,p(ξ) = 1
2. Pointwise nonnegativity Bi,p(ξ) ≥ 0, ∀ξ
3. Linear independence i.e.,∑n
i=1 αiBi,p(ξ) = 0⇔ αk = 0, k = 1, 2, . . . , n
4. Endpoint interpolation B1,p(−1) = Bp+1,p(1) = 1
5. Symmetry Bi,p(ξ) = Bp+2−i,p(−ξ)
The fourth property and the fact that Bi,p(−1) = 0 (i 6= 1), Ni,p(1) = 0 (i 6= p + 1) ensures the satisfaction of theKronecker delta property at the external boundaries where the Dirichlet boundary conditions are usually applied. Thus,imposing the Dirichlet boundary conditions does not require any special treatment as it is the case for NURBS basedfinite elements. Although it would be possible to generate the Bernstein basis on the fly, however, for performancereasons we have chosen to implement the basis and its derivatives separately for each value of p.
2.2. Solid Bezier elements
The bivariate Bernstein basis functions are formed as the tensor-product of univariate basis functions Bi,p(ξ1)
(corresponds to the ξ1 direction) and Bj,q(ξ2) (corresponds to the ξ2 direction)
Ba,p,q(ξ) = Bi,p(ξ1)Bj,q(ξ
2) (2)
with 1 ≤ i ≤ p+ 1 and 1 ≤ j ≤ q + 1 and
a(i, j) = (p+ 1)(j − 1) + i (3)
Note that ξ = (ξ1, ξ2) denotes the coordinates in the parent element [−1, 1] × [−1, 1]. Figure 2 plots some bivariateBernstein basis functions. From Equation 3, it is clear that the basis functions are numbered from left to right inone dimension. In two dimensions each row is numbered from left to right, starting with the bottom row and movingupward. We refer to Figure 3. Due to the tensor-product nature, multi-variate Bernstein basis can be constructedstraightforwardly and hence extension to 3D Bezier elements should not pose any difficulty. It should be emphasizedthat the Bezier elements can be seamlessly incorporated into existing FE codes by simply writing a new shape functionroutine. Note also that the Bezier elements are only C0 continuous across the element boundary.
The physical domain is denoted by Ω and the parent domain by Ω. The mapping from the parent domain to thephysical domain is then given by
x =
n∑
I
BI(ξ)xI (4)
where the shape functions (a common terminology in the FEM community) BI(ξ) refers to the bivariate Bernsteinbasis functions, see Equation 2, xI is the coordinates of node I and n = (p + 1)(q + 1) denotes the number of nodesper element. Derivatives of the shape functions (Equation 2) with respect to ξ are computed using the chain rule. Werefer to [23] for details in a context of the more general IGA framework.
In an isoparametric formulation, the displacement field is approximated by the same shape functions. That is
u(x) =
n∑
I
BI(ξ)uI (5)
where uI denotes the nodal displacements vector. The stiffness matrix is numerically integrated using the standardGauss quadrature i.e., a (p+ 1)× (q + 1) Gauss rule.
4
−1−0.5
00.5
1
−1−0.5
00.5
10
0.005
0.01
0.015
0.02
0.025
−1−0.5
00.5
1
−1−0.5
00.5
10
0.020.040.060.08
0.10.120.140.16
Figure 2: Bivariate Bernstein basis functions: bicubic basis B2,3(ξ1)B2,3(ξ
2) (left) and cubic-linear basisB2,3(ξ
1)B1,1(ξ2) (right).
12 31
4 5 6
8 97
2 3 4
5 6 7 8
1 2 3 4
5 6 7
9 10 11 12
8ξ1
ξ2
Figure 3: Bezier elements: a bi-quadratic B3×3 (or B9) element (p = q = 2) (left), a cubic-linear B4×2 element(p = 3, q = 1) (middle) and a cubic-quadratic B4×3 element (p = 3, q = 2) (right).
5
2.3. Zero-thickness interface Bezier elements
For a comprehensive treatment of the interface element formulation, we refer to, among others, [24, 5] and referencestherein. Here the basic of the method is illustrated by considering the six-node quadratic interface element shown inFigure 4, the displacement of its upper and lower surfaces is given by
u+ = B1,2(ξ)a1 +B2,2(ξ)a2 +B3,2(ξ)a3u− = B1,2(ξ)a4 +B2,2(ξ)a5 +B3,2(ξ)a6
(6)
where Bi,2(ξ) are the univariate Bernstein basis functions of order 2 and ai (i = 1, 6) denote the nodal displacements.
solid elements
solid elements
1 2 3
64 5ξ
Figure 4: A quadratic six node interface elements inserted in between two solid elements.
Having defined the displacement of the upper and lower faces of the interface, the displacement jump can becomputed according to
Ju(x)K = u+ − u− = Nint(a+ − a−) (7)
where use was made of Equation (6). In the above, Nint is given by
Nint =
[
B1,2 0 B2,2 0 B3,2 00 B1,2 0 B2,2 0 B3,2
]
(8)
and a+ = [a1x, a1y, a2x, a2y, a3x, a3y]T. A similar definition for a− can be defined. From the global displacement jump
Ju(x)K, a local displacement jump defined in a coordinate system aligned with the cohesive element can be determinedwhich is then substituted into a cohesive law to compute the corresponding traction.
2.4. Cohesive laws
In this work, the bilinear cohesive law, which is one of the most widely used cohesive laws, is adopted, see Figure 5.The fracture energy is denoted by Gc (precisely, GIc for mode I and GIIc for mode II), k is the interface stiffness, τ0
represents the ultimate strength (τ01 is the mode II interface strength and τ03 denotes the mode I interface strength),∆0 is the delamination onset displacement and ∆f is the delamination growth displacement. For more details, we referto [25].
6
Gc
∆0 ∆f ∆
τ0
τ
k
Figure 5: A bilinear cohesive law.
3. Implementation issues
Two practical issues relevant to delamination analyses using interface elements are (1) a pre-processing tool thatis able to insert interface elements into an existing FE mesh and (2) a powerful path following method to trace theresponse that usually contains severe snap-backs. For analysts who possess a commercial FE package like ABAQUSthe aforementioned issues are solved. In this section, we present options to solve these issues for analysts with in-houseFE codes.
3.1. Mesh generation
A majority of researches on delamination analysis has used commercial FE packages, for example Abaqus R© andMarcTM that provide powerful pre-processing tools for mesh generation–solid elements for the composite layers andthen cohesive elements along the delamination path. In this section, a simple mesh generation program is presented.The meshing procedure is performed in two steps. In the first step, a Bezier mesh for the composite sample is generated.In the second step, this mesh is adjusted to take into account the presence of the interface elements.
Bezier meshes for the composite samples are generated using a Matlab code3. Firstly, a mesh of bilinear quadrilateralelements is built (see Figure 6a). From this mesh, using order evaluation [22], meshes of higher order elements (seeFigure 6b and c) can be obtained straightforwardly.
In what follows, a simple open source pre-processing program4 is presented that reads a FE mesh and modifiesthat mesh so that one and two dimensional interface elements can be inserted along either material interfaces or grainboundaries of polycrystals or along a surface where the crack is assumed to grow. The program starts by reading a FEmesh including nodal coordinates, element connectivities and element groups. It then builds the support for all nodesand the neighbors of all elements. After that, it adapts the mesh by duplicating nodes and changing connectivities.Finally it generates a set of interface elements. Figure 7 shows some application examples of this pre-processing code.Detailed description of the algorithms is presented elsewhere [26]. This program currently supports Gmsh [27] format.For Lagrange finite elements, it works with two and three dimensional (2D/3D) linear and quadratic elements whilefor Bezier elements, for the moment, it only supports 2D elements of order ranging from one to four. Extension to 3DBezier elements is, however, straightforward.
3which is freely available at http://cmechanicsos.users.sourceforge.net.4This program can be downloaded at https://sites.google.com/site/phuvinhnguyensite/home/programs.
7
0 2 4 6 8 100123456789
10
0 2 4 6 8 100123456789
10
0 2 4 6 8 100123456789
10
(a) (b) (c)
Figure 6: An example of generation of finite element meshes of Bezier elements: (a) mesh with linear elements fromwhich meshes of high order elements (b,c) are generated using order evaluation (analog to p-refinement) [22].
(b)
(c)
(a)
(d)
Figure 7: Capabilities of the presented pre-processing program: generate (a) one dimensional interface elements alongbimaterial interfaces, (b,d) one and two dimensional interface elements along a pre-defined surface and (c) one dimen-sional interface elements along the grain boundaries in a polycrystalline solid.
8
3.2. Energy-based arc-length method
The basic idea of path following methods, often called arc-length methods, is to consider the load parameter λ asan additional unknown governed by a constraint function [28, 29]. Hence, the system of equations to be solved reads
[
f int(u)− λgφ(u, λ)
]
= 0 (9)
where u denotes the nodal displacement vector in solid mechanics, φ is the constraint function and g is the referenceconstant load vector. Different choices for φ result in different arc-length methods.
In [19, 30], an energy-based arc-length method was proposed. The basic idea is that, by the second law of ther-modynamics, the amount of energy dissipated in a system is positive and monotonically increasing. The discretizedenergy release rate reads [19]
G =1
2λuTg −
1
2λuTg (10)
Hence the above equation is an ideal candidate to be a constraint equation φ. The forward Euler integration schemefor the above rate equation gives the discrete arc-length function
φ =1
2
[
λ(n)(uT(n+1) − uT
(n))− (λ(n+1) − λ(n))uT(n)
]
g −∆τ = 0 (11)
where ∆τ [Nm] is the incremental path following parameter that represents the amount of energy to be released whengoing from load step n to load step n + 1. In the above, T denotes the transpose operation. The key advantage ofthis method is that it is no need to track where cracking is taking place. Yet, the method is simple and very fast forthe evaluation of the constraint function φ and its derivatives involves only certain nodes upon which there are appliedforces i.e., the non-zero components of the reference load vector g are usually one or two.
It should be emphasized that the energy release control is only applicable when there is an amount of dissipatedenergy i.e., during the delamination propagation phase. Therefore it is not suitable for the initial phase of the analysisor for possible elastic loading stages which may occur after a snapback. For these cases, a load control or a displacementcontrol must be used. Therefore, a hybrid solution strategy is adopted in which load or displacement control is usedwhen there is no dissipated energy and energy-based arc-length for the other case. For convenience of computerimplementation of this powerful arc-length, a detailed flowchart is given in Box 1. For the sake of simplicity, thealgorithm given in Box 1 corresponds to the response that consists of an elastic ascending branch and one softeningdescending branch with one snap-back (in Figure 8, this means the portion up to point A of the entire equilibriumpath).
A
Figure 8: A complicated equilibrium path successfully traced by the energy-based arc-length control [30].
9
Box 1 Flowchart for solution procedure with energy-based control
1. Initialization: λ = 0, set nd, ∆λ, ∆τmin,∆τmax
2. Solve an elastic load step from λ0 to λ (using load control)
i. Solving the equilibrium f int = λg using Newton-Raphson methodii. Check number of iterations: n > nd. If no, do λ0 = λ, λ = λ+∆λ and goto next load step (2i). Elseiii. Compute the release energy G = 1
2
[
λ0(uT − uT
0 )− (λ− λ0)uT0
]
g
iv. ∆τ = 0.5γG, γ = 0.25(n− nd)∗∗
v. Set ∆τmin ≤ ∆τ ≤ ∆τmax∗∗∗
3. Switch to arc-length control
i. Update K , f int from element contributions using a FE formulationii. Update v, ω, φ
v = 0.5λ0g
ω = −0.5uT0 g
φ = 12
[
λ0(uT − uT
0 )− (λ − λ0)uT0
]
g−∆τ
iii. Start the iteration procedure
r ← λg − f int
uI ← K−1r
uII ← K−1g
α ←vTuI + φ
vTuII + wu ← u+ uI − αuII
λ ← λ− α
iv. Check convergence, if not return to step 3i. Elsev. Do ∆τ = ∆τ ∗ 0.5γ , γ = 0.25(n− nd) and λ0 = λ
∗ In the above the superscript 0 denotes converged values of previous load step.∗∗ Automatic load incrementation [19, 30].∗∗∗To avoid unnecessarily small steps or too large steps.
10
3.3. Numerical integration for interface elements
It has been observed by many authors [31, 32] that integrating the cohesive elements with the familiar Gaussquadrature leads to responses with spurious oscillations when large stress gradients are present on the cohesive elements.In [31], the Newton-Cotes integration scheme was adopted for integrating the tangent stiffness matrix and the internalforce vector of the cohesive elements and spurious oscillations disappeared. In this manuscript, only full integrationschemes are used as recommended in [33] i.e., for solid elements of order p × q, a (p + 1)× (q + 1) Gauss quadraturerule is adopted and for cohesive elements of order p, a (p+ 1) Gauss or Newton-Cotes rule is utilized.
4. Numerical examples
The finite element model described in the previous sections has been implemented using the commercial numericaltoolkit jem/jive [34]. Geometrical nonlinearities are not taken into account. Benchmark problems that consist ofthe double cantilever beam (DCB), the end notch flexure (ENF) and mixed mode bending (MMB) fracture testconfigurations are studied. Additionally, a multiple mixed-mode delamination analysis of a composite specimen is alsoperformed to demonstrate the effectiveness of the arc-length control. Mode I fracture occurs in the DCB, mode IIoccurs in the ENF and mixed modes (mode I and mode II) occur in the MMB. Figure 9 show the studied fracturespecimens. The geometry data are L = 100 mm, h = 3 mm; the beam thickness B is equal to 20 mm for the DCB and10 mm for the ENF and MMB. The initial crack length a0 is 30 mm for the DCB and ENF and a0 = 20 mm for theMMB.
4.1. Double cantilever beam
As a pure mode I problem a double cantilever beam is studied. The composite plies are modelled as an orthotropicelastic material. The bilinear cohesive law in Section 2.4 is used for the interface elements. Material data are presentedin Table 1 which are taken from [5]. A plane strain condition is assumed. The interface stiffness is k = 57 × 10−7
N/mm3. As spurious oscillation can occur for coarse meshes, the energy-based arc-length method given in Section 3.2is utilized to trace the equilibrium curve of the beam.
E11 E22 G12 G23
135.3 GPa 9.0 GPa 5.2 GPa 5.2 GPa
ν12 ν23 GIc τ03
0.24 0.46 0.28 N/mm 57 MPa
Table 1: Material data for the DCB problem.
4.1.1. Mesh refinement
For linear elements, two meshes are considered. Firstly a mesh of 2×800 four-noded quadrilateral elements (Q4elements) is used. This mesh corresponds to 800 elements along the beam length and two layers of elements along thewidth. There are 560 four-noded linear interface elements which results in a total of 3204 nodes and thus 6408 degreesof freedom (dofs). This fine mesh provides the reference solution. The second mesh consists of 2×400 Q4 elements. Thebeam response is given in Figure 10 together with the theoretical curve obtained from a classical beam theory (CBT)[35]. As can be seen, while the 2×800 Q4 mesh can capture well the response, the coarse mesh 2×400 exhibits spuriousoscillation. It should be noted that the CBT leads to an overestimation of the stiffness of the undamaged laminate
11
L
a0 h
P, u
P, u
(a) Double cantilever beam (DCB)
L
a0 h
P, u
(b) End Notch Flexure (ENF)
L
a0 h
P1, u1P2, u2
(c) Mixed Mode Bending (MMB)
Figure 9: Fracture test configurations analysed: (a) DCB, (b) ENF and (c) MMB. The thick line a0 denotes the initialcracks while the dash lines represent the potential delamination path.
12
because the beam is not perfectly built-in, that is, rotation may occur at the delamination front. This explains themismatch of the theoretical and numerical elastic branches.
Next, quadratic Bezier elements are considered. To this end, two meshes–2×800 and 2×400 bicubic Bezier elements(B9 elements)– are studied. The beam response shown in Figure 10 confirms again that spurious oscillation appearsfor coarse meshes. It is clear that results from the quadratic elements deteriorate less than linear elements for too largemesh sizes.
In order to analyze the performance of cubic-linear elements two meshes–2×800, 2×400 B4×2 elements (cubicBernstein basis in the horizontal direction and linear basis in the vertical direction) are considered. From Figure 10 itcan be concluded that by using a cubic basis along the delamination direction, a coarse mesh can be adopted. Notethat the use of bicubic Bezier elements (B16 elements) is not recommended due to a high number of nodes per element(16) compared to 8 nodes per element for 4×2B elements.
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16
reac
tion
at P
[N]
displacement u [mm]
2x800 Q42x400 Q4
beam theory
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16
reac
tion
at P
[N]
displacement u [mm]
2x800 B92x400 B9
beam theory
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16
reac
tion
at P
[N]
displacement u [mm]
2x800 B4x22x400 B4x2beam theory
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16
reac
tion
at P
[N]
displacement u [mm]
2x400 B16beam theory
Figure 10: Double cantilever beam: load-displacement curves obtained with linear, quadratic and cubic-linear Bezierelements.
13
4.1.2. Effect of integration schemes
In this subsection, we study the influence of two commonly used numerical integration schemes namely the Gaussscheme and the Newton-Cotes scheme on the performance of the interface elements. It is well established that for linearinterface elements, the use of the Gauss integration scheme results in a spurious traction profile [31] and adoption of theNewton-Cotes scheme can avoid this. For quadratic and cubic/linear Bezier elements, a smooth response was obtainedwith both integration schemes. However they have different impacts on the performance of the Newton-Raphson solver,see Figure 11. The Gauss scheme requires more iterations per load steps and exhibits more diverged steps than theNewton-Cotes scheme (see the bottom left figure in which out of 135 steps, only 90 steps are converged while in thebottom right figure, all of 135 steps are converged.).
0
1
2
3
4
5
6
7
8
9
10
11
12
0 27 54 81 108 135
Num
ber
of it
erat
ions
per
ste
p
Load step number
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0 26 52 78 104 130N
umbe
r of
iter
atio
ns p
er s
tep
Load step number
0
1
2
3
4
5
6
7
8
9
10
11
0 18 36 54 72 90
Num
ber
of it
erat
ions
per
ste
p
Load step number
0
1
2
3
4
5
6
7
8
9
0 27 54 81 108 135
Num
ber
of it
erat
ions
per
ste
p
Load step number
Gauss Newton−Cotes
Figure 11: Double cantilever beam: performance of the solver for the Gauss and Newton-Cotes integration schemes.Top row is for 2 × 800 B9 elements and bottom row corresponds to 2 × 400 B16 elements. The left column is for theGauss scheme and the right column is for the Newton-Cotes scheme.
4.1.3. Effect of interface strength
In [6, 7] a solution that allows the use of coarse meshes in the simulation of delamination was proposed. The keyidea is to increase the cohesive zone length by artificially lowering the interfacial strength while keeping the fracturetoughness constant. The interface strength is varied from 57 MPa to 7 MPa. Two fine meshes–2 × 800 Q4 elementsand 2× 400 B4x2 elements are considered and the results are shown in Figure 12. Next, analyses are performed withthe reduced interface strength of 17 MPa and the result is given in Figure 13. With this reduced strength, a mesh of2×100 B42 elements (1204 nodes and 70 cubic interface elements) can be used without a significant loss of accuracy.
14
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8
reac
tion
at P
[N]
displacement u [mm]
2x800 Q4, 57MPa2x800 Q4, 37MPa2x800 Q4, 17MPa
2x800 Q4, 7MPabeam theory
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7 8
reac
tion
at P
[N]
displacement u [mm]
2x400 B4x2, 57MPa2x400 B4x2, 37MPa2x400 B4x2, 17MPa
2x400 B4x2, 7MPabeam theory
Figure 12: Double cantilever beam: load-displacement curves obtained with linear, quadratic and cubic Bezier elements.
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Figure 13: Double cantilever beam: responses of different Q4 and B42 elements with an artificially reduced interfacestrength of 17 MPa.
15
4.2. The ENF test
The ENF specimen, see Figure 9b, is modeled with isotropic material to make a fair comparison with analyticalsolutions [35] which are valid for isotropic materials only. The properties for the isotropic material are E = 150 GPaand ν = 0.25. The properties for the cohesive elements are GIIc = 1.45 N/mm and τ01 = 80 MPa. The interfacestiffness is k = 106 N/mm3. In order to prevent interpenetration of the two arms, in addition to cohesive elements,contact elements are placed along the initial crack. Contact elements are simply interface elements equipped with africtionless interpenetration law.
In Figure 14a the load displacement responses obtained with different meshes of Q4 elements are given. At leasta mesh that consists of 4×400 elements (2406 nodes, 400 linear cohesive elements, 6400 Gauss points (GPs) for solidelements and 800 Newton-Cotes integration points for cohesive elements) is needed. Results obtained with quadraticand cubic-quadratic Bezier elements, denoted by B9 and B4×3 respectively, are presented in Figures 14b-14c. Withquadratic elements, a mesh of 2×100 elements (1206 nodes, 100 quadratic cohesive elements, 1800 GPs and 300Newton-Cotes integration points) is required to obtain a good solution and with cubic-quadratic elements, a mesh of2×50 elements (906 nodes, 50 cubic cohesive elements, 1200 GPs and 200 Newton-Cotes integration points) is sufficient.It is worthy noting that the less integration points are needed the more efficient multi-level FE schemes (FE2) [36].The proposed formulation is therefore expected to yield a more efficient multiscale model for thin layers [20, 21] wherethe number of microscopic FE problems are equal to the number of cohesive integration points. Figure 14d shows theresult obtained with the quartic-linear Bezier elements. Different delamination stages are given in Figure 15.
In what follows, the performance of the energy based arc-length control is studied. Figure 16 gives the load-displacement response and the number of iterations per load step. The analysis starts with a displacement controlwith a constant increment of 0.2 mm. This is the case for the first fifteen steps. The sixteenth step was diverged byattempting to use a displacement control. This step was solved again with the energy based arc-length control whichremains since then. In order to avoid this diverged step, we switch a displacement control to an arc-length control whenthe elastic energy 0.5λ0gu
T0 (the subscript 0 denotes quantities at the beginning of a given load step) is approximately
closed to GIIca0. Only about 60 steps (without no divergence) are needed as shown in Figure 16b and the elapsed timewas therefore only 20 seconds on a personal computer with one processor used and a RAM of 4 GB under a LINUXenvironment.
In Figure 17, the load-displacement response of the ENF test obtained with various interface strengths which varyfrom 80 MPa to 20 MPa is presented. It is clear that using a too low interface strength (20 MPa in this case) results inan incorrect detection of the delamination propagation. The above findings indicate that reducing interfacial strengthas a means of reducing the required mesh density in numerical models should be done with care. Mode II cohesivezone lengths are generally much longer than ones of mode I zone, and similar mode II interfacial strength reductions(as done for the DCB test) can result in significant inaccuracies. For a thorough discussion on the relation betweencohesive zone length and finite element mesh density, we refer to [7].
4.3. MMB test
The MMB specimen, see Figure 9c, is modeled with isotropic material to make a fair comparison with analyticalsolutions [35] which are valid for isotropic materials only. The properties for the isotropic material are E = 150 GPaand ν = 0.25. The properties for the cohesive elements are GIc = 0.352 N/mm, GIIc = 1.45 N/mm and τ01 = 80 MPa,τ03 = 60 MPa. The interface stiffness is k = 106 N/mm3. We use the mixed mode cohesive law given in [25] withη = 1.56. In order to prevent interpenetration of the two arms, in addition to cohesive elements, contact elements areplaced along the initial crack. The loads applied are P1 = 2Pc/L and P2 = P (2c+ L)/L, where L is the beam length,c is the lever arm length, and P is the applied load. From these relationships, it is clear that the applied loads P1 andP2 are proportional i.e., P2/P1 = (2c+L)/L. We choose c = 43.72 mm so that the mixed-mode ratio GI/GII is unity.The external force vector is therefore f ext = λ[1,−2.1436]T (a unit force was assigned to P1) in which the variable loadscale λ is solved together with the nodal displacements using the energy based arc-length method.
16
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beam theoryQ4, 2x400 elemsQ4, 4x400 elemsQ4, 6x400 elems
(a) Q4 elements
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(b) quadratic Bezier elements
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(c) cubic-quadratic Bezier elements
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P [N
]
displacement u [mm]
beam theoryB52, 4x30 elemsB52, 4x50 elems
(d) quartic-linear Bezier elements
Figure 14: Load-displacement response of the ENF test.
17
Figure 15: ENF test: delamination stages with contour plot of the tranverse stress σyy. A magnification factor of twowas adopted.
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Figure 16: ENF test: efficient performance of the energy based arc-length control. A precision of 10−6 was used forthe nonlinear solver.
18
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]
displacement u [mm]
beam theoryB43, 80 MPaB43, 60 MPaB43, 40 MPaB43, 20 MPa
Figure 17: ENF test: effect of artificially reducing the interface strength.
Figure 18 plots the load-displacement responses of the beam in which Figure 24a show the results obtained with2×200 Q4 and 2×400 Q4 (1604 nodes, 400 linear interface elements, hence 800 Newton-Cotes integration points) meshesand Figure 24b gives the results obtained with 2×50 and 2×100 B4×2 meshes. With coarse meshes, responses withspurious oscillations are observed. By using the cubic-linear B4×2 elements, a coarse mesh of 2×100 elements consistingof 1204 nodes and 100 cubic Bezier interface elements (400 Newton-Cotes integration points) can be used. Comparingto the fine mesh of bilinear elements–2×400 Q4, there are less nodes and less integration points, thus less computationalresources. We also investigate the performance of quartic-linear elements (B5×2 elements) and the result is plotted inFigure 18c. It is clear that a very coarse mesh of 2×70 elements (1124 nodes, 70 quartic Bezier interface elements, 350Newton-Cotes points) can be used. Although using B5×2 elements does not significantly reduce the computationalexpense compared to cubic-linear elements, the fact that less cohesive integration points are needed can be beneficiallyexploited in a multiscale model for thin layers [20]. Different delamination stages of the MMB is given in Figure 19.
The authors in [13] developed a discrete cohesive zone model (DCZM) in which nonlinear springs replace thecontinuum interface elements. They claimed that DCZM is an efficient tool for delamination analysis and wrote in theconclusion part of their article ”All the computations were carried out on a SunBlade-100 machine with one processor(UNIX environment). For a model with 400 elements, the CPU time ranges from 93 to 101 s for the DCB (120 steps),385 to 399 s for the ENF (240 steps and contact), and 188 to 205 s for the MMB (300 steps). This is a drastic reductionin computation time compared to other CZM implementations that would take on the order of hours for correspondingsimulations when the CZM was implemented with ABAQUS user subroutine UMAT.”. In order to make a comparison,the MMB test is analysed again with two meshes–one of 2×200 B4×2 elements and one of 2×200 B9 elements. Only50 steps are needed as shown in Figure 20 and the elapsed time was therefore only 11 seconds on a personal computer(also UNIX system) with one processor used and a RAM of 4 GB. It is obvious from Figures 20a-20c that the firsteight steps were solved using a load control with a constant load increment. At step 9, the code automatically switchedto the arc-length control and this control remains since then. It should be noted that the CPU time for the meshconsisting of 2×70 B5×2 elements (140 solid elements) was only about 6 seconds. The comparison with [13] is simply areference due to different machines used, implementation aspects and the adopted solvers for the system of equations.
4.4. Multiple delamination of a composite specimen
This example addresses a multiple mixed-mode delamination analysis of a composite specimen. The geometry andloading is given in Figure 21. This problem was studied several times see e.g., [37, 5, 12, 10]. There are two initial
19
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]displacement u1 [mm]
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(b) B4x2 elements
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P1
[N]
displacement u1 [mm]
beam theory2x50 B522x70 B52
2x100 B52
(c) B5x2 elements
Figure 18: Load-displacement response of the MMB test.
20
Figure 19: MMB test: delamination stages with contour plot of the transverse stress σyy. A magnification factor oftwo was used.
cracks–the first initial crack on the left-hand of the specimen is placed along the mid plane and the second initial crackis positioned 20 mm on the right of and two plies below the first crack. In the numerical model, a plane strain stateis assumed and material properties, which are taken from [5], is given in Table 2. The interface stiffness is k = 105
N/mm3. Contact elements are placed along the second initial crack to avoid interpenetration there. Without doing so,an extra local limit point that is not immediately obvious from the experimental results was observed [5].
E11 E22 G12 ν12 = ν13 = ν23
115 GPa 8.5 GPa 4.5 GPa 0.29
GIc GIIc τ01 τ03 µ
0.33 N/mm 0.8 N/mm 7.0 MPa 3.3 MPa 2.0
Table 2: Multiple delamination analysis: material properties.
Analyses were performed with standard eight-noded quadrilateral elements (Q8), biquadratic Bezier elements (B9)and cubic-quadratic Bezier elements (B4×3). Meshes of 5×360 Q8 elements (7214 nodes, 1800 solid elements, 420interface/contact elements), 3×360 B9 elements (6129 nodes, 1080 solid elements and 420 interface/contact elements)and 3×240 B4×3 elements (6129 nodes, 720 solid elements, 280 interface/contact elements) are utilized. The deformedconfiguration obtained with Q8 elements is presented in Figure 22. The response of the specimen in terms of thereaction and two times the displacement at the top left corner is plotted in Figure 23. Excellent agreement with theresult reported in [12] can be observed. Excellent performance of the energy based arc-length control is illustratedin Figure 24 for Q8 elements (other elements behave in a similar manner). The analysis was commenced with a loadcontrol of constant increment of 0.2 N. At step 9, the program switched to arc-length control automatically based ona sudden increase in the number of iterations per step. The arc-length control was used to the end of the analysis.Note that the entire equilibrium curve was traced with no diverged step. Only 100 load steps are needed and the totalnumber of iteration is approximately 600. The CPU time is 62 seconds. In [12], the double line search (DLS) algorithm
21
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Figure 20: MMB test: efficient performance of the energy based arc-length control. A precision of 10−6 was used forthe nonlinear solver.
22
12040 2020
interface elements
contact elementsinitial cracks
10 layers (1.325)
2 layers (0.265)12 layers (1.59)
width=20
u
−u
Figure 21: Multiple delamination analysis: geometry and loading.
was used and it requires 278 steps and 2109 iterations for the same problem (cf. Table IX, page 1037 in the referredpaper).
Figure 22: Multiple delamination analysis: deformed finite element mesh.
5. Conclusion
We have for the first time presented a high order finite element formulation that uses the Bernstein polynomialsfor the delamination analysis of composites. A pre-processing tool was presented along with its open source code. Thetool is able to duplicate nodes and generate interface elements for two/three dimensional meshes of Lagrange elementsand Bezier elements. Computer implementation of the simple-yet-efficient global energy-based arc-length method wasalso provided to facilitate the incorporation into existing FE codes. The numerical examples demonstrated that thecombination of the proposed Bezier elements and the energy-based arc-length control results in an efficient means forsolving mode II and mixed-mode I-II problems. Typical mode II and mixed-mode benchmark problems were solvedwithin a dozen of seconds on a personal computer. A mixed-mode problem with multiple delamination was solved in acouple of minutes. For mode I problems such as the double cantilever beam for which the cohesive zone length is toosmall, the proposed elements do not bring any significant advantages. It is our plan to test the proposed formulationwith other cohesive laws rather than the bilinear law. We anticipate a higher robustness and more efficiency when highorder Bezier elements are used with smooth cohesive laws.
Future work can be (1) extension of the proposed formulation to three dimensional problems, (2) to impact loadingswhere the non-negativity of the Bernstein basis provides positive mass matrices [38] and (3) to NURBS elements inorder to reduce the number of degrees of freedom. The last deserves an explanation as follows. Consider a square platefrom which a horizontal weak interface divides the plate in two equal halves is to be modeled. Let us assume thatquadratic elements are utilized. Starting with the coarsest mesh shown in Figure 25a, via an order elevation process(analog to p-refinement in FEM), one obtains a mesh of higher order (Figure 25b). Then, using knot insertion (analog
23
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Figure 23: Multiple delamination analysis: load-displacement response.
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Figure 24: Multiple delamination analysis: good performance of the energy based arc-length control.
24
to h-refinement in FEM), a refined mesh is obtained, see Figure 25c. Thanks to the sharing of control points/nodesbetween neighbouring elements, there are 20 nodes instead of 25 nodes for a mesh of 2×2 bi-quadratic Bezier or Lagrangeelements. Moreover, along the horizontal direction, there is Cp−1 continuity. In order to have a C0 continuity at theweak interface, knot insertion is realized again but this time for the vertical direction only (Figure 25d). Finally, nodedoubling and interface elements generation leads to the mesh shown in Figure 25e. This technique can be implementedin LS-DYNA–a commercial FE package that supports NURBS-based finite elements. This is a work in progress andwe are going to report our findings in a forthcoming paper.
Acknowledgements
The authors would like to express the gratitude towards Drs. Erik Jan Lingen and Martijn Stroeven at the DynaflowResearch Group, Houtsingel 95, 2719 EB Zoetermeer, The Netherlands for providing us the numerical toolkit jem/jive.
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