JOURNAL OF COMPOSITE M AT E R I A L S
Article
Implementation of domain superposition technique for the nonlinear analysis of composite materials
Journal of Composite Materials 47(2) 243–249 ! The Author(s) 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0021998312439220 jcm.sagepub.com
Wen-Guang Jiang
Abstract A domain superposition technique for the nonlinear modeling of composite materials has been developed in this paper. Instead of explicitly modeling the conjugate matrix domain and reinforcement domain with proper matching mesh interfaces between them in the manner of a traditional finite element analysis, a domain superposition technique model uses the global domain geometry superimposed with the reinforcement phase material domain geometries. The major purpose of doing so is to avoid the difficulty of directly constructing the matrix phase material geometries, which are usually hard to build due to the presence of likely geometric degeneracies in most composite materials internal architectures. The finite element meshes for domain superposition technique analyses are much simpler to establish than those used for traditional finite element analyses. Numerical examples are given to show the validity of the proposed modeling procedures.
Keywords Composite material, finite element analysis, nonlinear analysis, domain superposition technique
Introduction Finite element analysis (FEA) has become an effective means to predict the complex response of composite materials/structures. When an attempt is made to perform a conventional FEA, one of the tough issues faced is how to deal with the topologically complex internal geometries which are inherent in composite materials. Especially in composites with high reinforcement volume fraction or with complex internal reinforcement architectures, the presence of extremely narrow multiconnected matrix regions is often encountered (the word ‘degenerated’ is used to describe this hereinafter). It is common knowledge that discretization of the material geometry into finite element (FE) mesh is a basic requirement for performing FEA. Unfortunately, it is generally very hard to establish good quality FE meshes over the degenerated regions. This is a well-known obstacle for the efficient implementation of conventional FEA techniques to assist the development of new composite materials/structures. To overcome this difficulty, a domain superposition technique (DST) which can easily overcome this difficult issue and realize fast implementation of FEA of woven
composites was proposed by the author.1 The procedure for performing linear analysis of woven composites is reported in reference 2. In the present paper, the methodology of implementing DST is further extended to render it possible to perform nonlinear analyses. Numerical results are given to demonstrate the validity of the proposed approach.
Concept of DST The concept of DST can be schematically illustrated as shown in Figure 1. In contrast to conventional FEAs which model explicitly the reinforcement domain 2 and the likely degenerated matrix domain 1 that are conjugate to each other (Figure 1(a)), DST models the reinforcement domain 2, and the global domain
¼ 1[ 2, both of which are generally School of Mechanical Engineering, Yanshan University, Qinhuangdao, PR China Corresponding author: Wen-Guang Jiang, School of Mechanical Engineering, Yanshan University, Qinhuangdao, PR China. Email:
[email protected]
244 nondegenerated (Figure 1(b) and (c)) and can be easily discretized using conventional solid elements. The implementation of DST requires two essential technical strategies. The first is the stiffness matching technique; the second is the establishment of the deformation coupling relationships between the local domain 2 and the global domains to ensure that the overlapping regions can deform together when loaded. The correct implementation of these two techniques will ensure that the model is an accurate representation of the actual composite system. In general, there are two convenient ways to realize the material matching requirement, namely, 3-region DST and 2-region DST, respectively. For 3-region DST (Figure 1(b)), three sets of geometric domains are involved in this superposition system, that is, the global geometric domain using matrix material model and the two identical duplicate reinforcement geometric domains 2 using different ‘functional materials.’ One of the duplicate domains uses ‘negative matrix material,’ and the other uses actual reinforcement phase of material property. When the domain
2 with ‘negative’ matrix material is superimposed into the global domain with matrix material, this effectively creates ‘holes’ in the global domain. The subsequential superposition of the second 2 domain with the reinforcement material into the system will be equivalent to filling the ‘holes’ with the reinforcement to accomplish the final composite material system modeled. Since the duplicate reinforcement geometric domains occupy exactly the same space, duplicate FE meshes sharing the same sets of nodal locations can be conveniently used to create the corresponding FE meshes. Furthermore, the duplicate meshes can be merged into one and consequently, a new ‘artificial
Journal of Composite Materials 47(2) material model’ must be formulated and implemented to consider the combined effect of the superposition of the ‘negative’ matrix material and the reinforcement material. This leads to an alternative way of realizing the DST, that is, the 2-region DST1,2 as illustrated in Figure 1(c). It can be shown by considering the expressions for the stiffness matrices for the superimposed elements in the 2- and 3-region DSTs that these two methods are equivalent implementations of the same technique. For nonlinear analysis, the 3-region DST is more convenient to realize than the 2-region DST.
Advantages and drawbacks of DST The DST proposed differs from traditional FEA in that the reinforcement domain and the global domain are the actually modeled geometries in DST. These two domains are generally both nondegenerated or at least have much lower degree of degeneracy than the matrix domain. They only need to be independently meshed using solid elements in DST model. Therefore the difficult-to-meet requirement of meshing the likely degenerated matrix material domain and the alignment of adjacent element faces/edges at material phase interfaces for traditional FEA is no longer an issue for DST analysis. DST can even tolerate defects in the geometric models, for instance, the hard-to-avoid interpenetration at fiber bundle crossovers when establishing textile composite geometric models. As the DST approach avoids these various obstacles, it is clearly much more straightforward in use than traditional FEA. Unlike the extended type of FE method,3 the slope discontinuity of deformation field across the material phase boundaries is not treated elaborately in the DST proposed. Thus, there exist error bands at the material
Figure 1. Schematic comparisons between conventional finite element analysis (FEA) and domain superposition technique (DST).
Jiang
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phase interfaces in current DST models. As a consequence of this, the macroscopic elastic moduli predicted by DST are slightly higher than conventional FEA when similar degree of mesh refinement is used, although this error can be reduced when the FE mesh is further refined.
Numerical techniques to implement DST Coupling technique When implementing DST, the geometric domains are all meshed with solid elements and the elements should be assigned with the correct material properties as discussed above. Then coupling equations (CEs) of degrees of freedom (DOFs) between the local meshes of domain 2 and the global mesh of domain must be established to ensure that the coincident points between the two domains can have the same displacement during deformation. This is the deformation continuity/compatibility condition of implementing DST, which must be correctly dealt with. Considering a general three-dimensional isoparametric solid element, the coordinate interpolations of a given point (x, y, z) inside the element are x¼
m X
Ni xi
y¼
i¼1
m X
N i yi
z¼
i¼1
m X
Ni zi
ð1Þ
i¼1
where xi, yi and zi are the coordinates of the m element nodes in the global Cartesian coordinate system. The interpolation functions Ni (x, Z, z) are defined in the natural coordinate system of the element, which has variables x, Z, z that each vary from 1 to þ 1. The fundamental property of the interpolation function Ni is that its value in the natural coordinate system is unity at node i and zero at all other nodes. To enhance the continuity, it is preferable that the same type of element formulation be used for all the domains. In this paper, conventional three-dimensional isoparametric solid brick elements are used for the structural discretization to establish the DST model. An important advantage of using isoparametric elements is that in the isoparametric formulation, the displacements are interpolated in the same way as the geometry; that is u¼
m X i¼1
N i ui
v¼
m X i¼1
Ni vi
w¼
m X
Ni wi
ð2Þ
i¼1
where ui, vi and wi are the displacement components of the m element nodes. To establish the coupling relationship between the two sets of FE meshes for DST, for each node of the tow meshes, n(x, y, z), a numerical procedure needs to
be developed to first find the specific global solid element inside which the reinforcement node is exactly located, and determine the local coordinates (x, Z, z) of this reinforcement nodal location within this global solid element by equation (1), and then the CEs to be established for this tow node will be in the form of the displacement interpolation functions, that is, equation (2). It should be noticed that, with these CEs enforced, the displacement fields in the superimposed regions will have exactly the same displacement at the reinforcement nodal locations only, not the whole region.
Material models To implement the DST analysis, a ‘negative matrix material’ model is needed. Compared with the normal material model, a negative material model will produce a negative state of stress of the normal material model when it is loaded under the same state of strain. For linear elastic material, the negative material model needed can be established by simply using negative stress–strain matrix of the normal material. For example, for an isotropic elastic matrix material, the stress–strain matrix for matrix phase of material of the composite material system is given by Eð1 Þ ð1 þ Þð1 2Þ 2 1 1 1 6 1 6 1 1 6 6 1 1 1 6 6 0 0 0 6 6 6 0 0 0 4 0 0 0
½Dmatrix ¼
0 0
0 0
0 12 2ð1Þ
0 0
0
12 2ð1Þ
0
0
3 0 0 7 7 7 0 7 7 0 7 7 7 0 7 5
ð3Þ
12 2ð1Þ
where E and n are the elastic modulus and Poisson ratio of the material. The corresponding negative stress–strain matrix used to create the ‘negative material model’ for the implementation of 3-region DST is simply ½Dnegative material ¼ ½Dactual matrix
ð4Þ
However, for 2-region DST analysis, a combined constitutive stress–strain matrix used for the reinforcement elements is based on the difference between that of the actual reinforcement material and the matrix material, that is ½Dreinforcement ¼ ½Dactual reinforcement ½Dactual matrix ð5Þ For the analyses of composite material system with nonlinear matrix materials, to derive the negative
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Journal of Composite Materials 47(2)
material model used to implement the 3-region DST, the same procedure for determining the state of stress of the normal matrix material can be used by adding simple command lines to alter the sign of all the six stress components at both the beginning and the end of the normal numerical procedure, that is ½negative material ¼ ½matrix
ð6Þ
The stiffness matrix of the negative material model is itself negative definite. However, when implementing DST model, the negative material domains are always superimposed onto normal material domains. The resulting final global system stiffness matrix of the superimposed model is positive defined.
Numerical examples The DST proposed has been implemented using ANSYS FE software. Test examples are given as follows to show the validity of the proposed numerical procedure.
Figure 2. Schematic of microstructure for hexagonal packing unidirectional fiber composite showing a 2-fiber RVE model and a 12-fiber RVE model.
Test example 1: linear analysis The first illustrative example is the micromechanical analysis of unidirectional fiber-reinforced composites. The microstructure of hexagonal packing fiber arrangement is shown in Figure 2. Two representative volume elements (RVEs) have been used. The first one is a 2fiber model, which is the smallest rectangular RVE for hexagonal packing system. For verification and ease of visualization purposes, a larger 12-fiber model has also been analyzed, which contains a complete pattern of hexagonal fiber packing system (see Figure 2). Example FE meshes of the 2-fiber RVE for DST analysis is given in Figure 3. As this is a generalized plane strain problem, the analysis can be carried out based on a slice geometric model. Only uniform transverse loading cases are considered here, thus, the stress/strain results are independent of the thickness direction (along the fiber), and the FE meshes used need only one element division in this direction. A translational periodic boundary condition was applied. A stress-loading procedure has been implemented to derive the equivalent elastic material constants of the composites.4 The CEs for the stress-loading periodic boundary conditions are applied to the global mesh. Nodal DOFs of the fibers are coupled to the DOFs of the global mesh using the interpolation CEs established by an efficient numerical procedure developed using FORTRAN programming language. The elastic material constants for the unidirectional composite system used in the analysis are Young’s modulus E ¼ 3.35 GPa, Poisson’s ratio n ¼ 0.35 for isotropic epoxy matrix, and E ¼ 74 GPa, n ¼ 0.2 for isotropic glass fiber. Numerical analyses have revealed that the 2-fiber model and the 12-fiber model predict exactly the same results as expected. This indicates that the boundary conditions used are implemented correctly. To validate the DST model proposed, conventional FE model has also been performed. Figure 4 shows an example conventional FE mesh. Figure 5 compares the results of the predicted
Figure 3. Example domain superposition technique (DST) finite element meshes for the analysis of the micromechanical 2-fiber hexagonal RVE model, (a) fiber mesh for 2, (b) global mesh for and (c) superimposed mesh for DST analysis.
Jiang 80 Predicted elastic moduli (GPa)
homogeneous elastic moduli using both DST and conventional FEA for various fiber volume fractions. It can be seen that the DST predictions correlate excellently well with the results of the conventional FEA. The predicted results have also shown that the hexagonal fiber packing unidirectional composite system produce effective properties showing transverse isotropy. This means that the effective elastic constants in the transverse plane are independent of the orientation angle y defined in Figure 2, and the theoretical relationship for linear elastic isotropic materials, G ¼ E/2(1 þ n), are accurately satisfied in the transverse plane.
247
Ez (DST) E z (FEA) E x (DST) E x (FEA)
60
Gxy (DST) G xy (FEA)
40
20
0
0
0.2
Test example 2: nonlinear analysis To illustrate the nonlinear analysis procedure using DST, in this example, the same geometric micromechanical model as the first test example was used again.
0.4 0.6 Fiber volume fraction
0.8
1.0
Figure 5. Comparison of predicted homogeneous elastic moduli between traditional finite element analysis (FEA) and domain superposition technique (DST; z is fiber direction).
Applied macroscopic stress σ/σo
1.6
1.2
0.8
θ=30° θ=15° θ=5° θ=0°
0.4
0 0
0.001
0.002
0.003
0.004
0.005
Predicted extension strain ε
Figure 6. Stress–strain curves predicted using 3-region domain superposition technique (DST).
Plastic collapse strength σ/σ0
1.4 DST prediction
1.2
1.1
1.0
Figure 4. Comparison conventional finite element model for 2-fiber hexagonal packing micromechanical RVE model for validating the DST.
FEM prediction [5]
1.3
0
10
20
30
Loading direction angle θ (Degree)
Figure 7. Variation of plastic collapse strength with loading direction angles.
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Figure 8. Example contour plots of the accumulated equivalent plastic strain using the 12-fiber domain superposition technique (DST) model, (a) for loading direction y ¼ 0 , and (b) for loading direction y ¼ 15 .
In this analysis, a linear elastic material was used for the fibers, but an elastic perfectly plastic material model was used for the matrix phase material. The elastic properties5 are Efiber ¼ Ematrix ¼ 100 GPa, nfiber ¼ nmatrix ¼ 0.25, and the flow stress for the matrix material is s0 ¼ 100 MPa. The volume fraction of the fibers is 56.25%. The load applied to the RVE is uniaxial state of stress in the transverse direction x’ making an angle y with the horizontal direction x (refer to Figure 2). Thus, the RVE is then in a state of generalized plane strain. The 3-region DST discussed in the section Concept of DST has been used to perform the nonlinear analysis. Typical macroscopic stress–strain curves predicted using DST are given in Figure 6. It can be seen from this figure that when the applied uniaxial stress loading s is less than the flow stress of the matrix material s0 (corresponding to extension " < 0.001), the global response is linear elastic, and all loading curves from different loading directions exactly overlap with each other. This can be further confirmed from the homogeneous single stress field responses of all the analysis results. And this also indicates that the negative matrix material model used in the 3-region DST analyses were correctly developed and implemented. However, from Figure 6, we can see that the macroscopic flow stress depends on the orientation y indicating that the effective nonlinear properties of a composite exhibiting hexagonal symmetry are not necessarily isotropic. The variation of predicted collapse strength of the composites with loading direction angle is depicted in Figure 7. The values predicted by traditional FEM5 are also given in this figure. It can be seen that the results predicted using different methods correlate very well. The contour plots of accumulated equivalent plastic strain just before plastic collapse occur for loading direction y ¼ 0 and 15 are given in Figure 8. It can be seen from these figures that for y ¼ 15 , there is a single set of parallel narrow plastic straight slip bands (strain intensive bands), which completely penetrate through the materials matrix domain. This represents the easiest plastic deformation mode for
the composite material model (refer to Figure 7). Thus, the lowest collapse strength occurs in this direction. It can be seen from loading direction y ¼ 0 case that there are two sets of parallel narrow plastic straight slip bands crossing each other at an angle of 60 . These represent the most difficult plastic deformation mode for the composite material model analyzed. This is the reason why the highest collapse strength occurs in this direction (refer to Figure 7).
Conclusion The DST proposed is an easy to implement numerical procedure for the analysis of composite materials. In this paper, the methodology has been successfully extended from linear analysis to nonlinear analysis. In principle, the DST approach could be implemented using most commercially available FE packages. Numerical examples have shown that DST analysis results correlate well with those of traditional FEAs. As the FE meshes needed for performing DST analysis are much easier to establish than those used for traditional FEAs. Once the highly automatic procedure has been developed, there will be no extra work from the user side to perform the analysis, thus, DST can be easily adopted by industry users. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Acknowledgment This research is funded by Hebei natural science research funding under grant E2010001225, Hebei research funding for returning overseas researchers under grant 20100707, and Yanshan university doctorial research funding.
Conflict of interest None declared.
Jiang References 1. Jiang WG. A computer and a method of modelling a woven composite material. WO Patent WO/2008/122, 751, 2008. 2. Jiang WG, Hallett SR and Wisnom MR. Development of domain superposition technique for the modelling of woven fabric composites. Comput Meth Appl Sci 2008; 10: 281–291. 3. Belytschko T, et al. Arbitrary discontinuities in finite elements. Int J Numer Meth Eng 2001; 50: 993–1013.
249 4. Jiang WG and Yan LJ. Implementation of stress loading repetitive unit cell finite element model. In: Proceedings of the 3rd international conference on heterogeneous material mechanics, Shanghai, 22–26 May 2011, pp.816–819. 5. Michel JC, Moulinec H and Suquet P. Effective properties of composite materials with periodic microstructure: a computational approach. Comput Meth Appl Mech Eng 1999; 172: 109–143.
