Jensen, H.M.

Mikkelsen, Lars Pilgaard

Sørensen, K.D.

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017On the simulation of kink bands in fiber reinforced compositesSørensen, K.D.; Mikkelsen, Lars Pilgaard; Jensen, H.M.Published in:Interface design of polymer matrix composites - mechanics, chemistry, modelling and manufacturing.ProceedingsPublication date:2007Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Sørensen, K. D., Mikkelsen, L. P., & Jensen, H. M. (2007). On the simulation of kink bands in fiber reinforcedcomposites. In B. F. Sørensen, L. P. Mikkelsen, H. Lilholt, S. Goutianos, & F. S. Abdul-Mahdi (Eds.), Interfacedesign of polymer matrix composites - mechanics, chemistry, modelling and manufacturing. Proceedings (pp.281-288). Roskilde: Risø National Laboratory.  (Risoe International Symposium on Materials Science.Proceedings; No. 28).ON THE SIMULATION OF KINK BANDS IN FIBERREINFORCED COMPOSITESKim D. Sørensen∗, Lars P. Mikkelsen∗∗and Henrik M. Jensen∗∗ Department of Civil Engineering, Aalborg University,9000 Aalborg, Denmark∗∗ Materials Research Department, Risø,4000 Roskilde, DenmarkABSTRACTSimulations of kink band formation in fiber reinforced composites are carried out using thecommercial finite element program ABAQUS. A smeared-out, plane constitutive model forfiber reinforced materials is implemented as a user subroutine, and effects of fiber misalign-ment on elastic and plastic deformation are studied under plane strain conditions.Keywords: Kink band, constitutive model, ABAQUS, fiber misalignment1. INTRODUCTIONFailure by kinking in fiber reinforced composites has been the subject of a number of recentinvestigations. It has been found that the compressive failure strength is considerably lowerthan the tensile strength, typically in the order of 50 to 60 percent for carbon fiber composites(Kyriakides et al., 1995). It has also been found that the compressive strength is governed byplastic yielding in the matrix (Budiansky, 1983), and futhermore, that small misalignmentsof the fibers have a large influence on the compressive strength, see Kyriakides et al. (1995).Several investigations of compression of a fiber reinforced material under the assumptionof perfectly aligned fibers have predicted bifurcations stresses much higher than resultsobtained from experiments. Liu, Fleck and Sutcliffe (2004) include the effect of randomwaviness of the fibers using a Cosserat smeared-out finite element model.In Kyriakides et al. (1995) the problem of predicting compressive strength for fiber rein-forced materials is approached through an idealized model composite, with individual dis-cretization of fiber and matrix material, and this two-dimensional michromechanical modelwas later (Kyriakides et al.,1998) extended to a three dimensional model. In these investi-gations the post-buckling respone is also studied, and it is shown that deformation localizesinto well-defined bands of bent fibers.281Proceedings of the 28th Risø International Symposium on Materials Science: Interface Design of Polymer Matrix Composites – Mechanics, Chemistry, Modelling and Manufacturing Editors: B. F. Sørensen, L. P. Mikkelsen, H. Lilholt, S. Goutianos, F. S. Abdul-Mahdi Risø National Laboratory, Roskilde, DenmarkSørensen and MikkelsenKink band formation in fiber reinforced materials was investigatated in Christoffersen andJensen (1996) and in Jensen and Christoffersen (1997), where a plane constitutive model forperfectly bonded layered materials was introduced. It was found that this model containsessentially the same information in one point of the material as a complete finite elementdiscretization of a representative volume element, like the model introduced in Kyriakideset al. (1995), and furthermore, that the critical stress is highly influenced by fiber volumefraction and the constitutive behavior of the constituents. Another conclusion is that thekink stress is reduced by taking non-linearity of the fibers into account compared to thepredicted critical stress assuming linear-elastic fibers.In the present study, a smeared-out plane constitutive model, as formulated by Christof-fersen and Jensen (1996), is implemented as a user subroutine in the finite element programABAQUS. Effects of fiber misalignment for elastic and plastic response are studied, andqualitatively compared to earlier results.2. IMPLEMENTATION OF THE CONSTITUTIVE MODELFor a detailed description of the constitutive model used in these kinkband simulations, seeJensen and Christoffersen (1997) and Christoffersen and Jensen (1996).2.1 The constitutive model.The general relations between rates of Cauchy stress and the strain rates in three dimensionsare given byσ˙ij = Lijklkl (1)where Lijkl are the elastic-plastic tangent moduli. The strain rates are given byij =12(vi,j + vj,i) (2)with vi,j denoting the components of the gradient of velocity components vi.The two-dimensional constitutive equations implemented in ABAQUS are given in the forms˙ij = Cijklvl,k, i, j, k, l ∈ {1, 2} (3)where s˙ij are components of the nominal stress rate, vi,j are components of the gradientof velocity components vi, and Cijkl are components of the tensor of nominal moduli, andfrom this formulation Lijkl is found by the relationLijkl = Cijkl +12δilσkj +12δikσlj +12σilδkj −12σikδlj, i, j, k, l ∈ {1, 2} (4)by futhermore using the assumption τij ≈ σij, with τij being components of Kirchoff stress.The components of the tensor of nominal moduli Cijkl are found from a mixture of fiberand matrix properties.The constitutive model is based on three assumptions:• Material lines parallel with the fibers are subject to a common stretching and rotation282Simulation of kink bands• Planes parallel with the fibers transmit identical tractions• The material of the constituents is elastic or elastic-plasticThe first of these assumptions correspond to a Voigt estimate for effective material propertieswhereas the second corresponds to a Reuss estimate. The third assumption is a standardspecification of time independant materials.The expressions for Cijkl, the components of the tensor of nominal moduli, can be foundin Christoffersen and Jensen (1996).2.2 The behavior of the constituents.For both constituents we have the relationss˙cij = Ccijklvcl,k, (5)similar to equation 3. Furthermore, both materials are assumed to be characterized by apower-law hardening with isotropical hardening. The power-law is given by ={σE, σ ≤ σyσyE[1n(σσy)n− 1n+ 1], σ > σy(6)where σy is the yield stress of the material and n the hardening parameter. From thisrelation the tangentmodulus Et is found, and subsequently the values of Lcijkl.In the kink-band simulations in the present paper, yielding only occurs in the matrix mate-rial, and thus the fibers are treated as linear elastic.2.3 ABAQUS implementation.The user subroutine UMAT in ABAQUS is used to implement the material behavior of thecomposite. In this routine, stresses in fibers and matrix are updated within each increment,and the elastic-plastic moduli are calculated by mixture of the properties of the constituents.The moduli Lijkl are determined from equation 4 and this value is returned to ABAQUSfrom UMAT. ABAQUS uses an updated Lagrangian formulation.The model is implemented as a plane strain model, but can also be formulated for planestress situations. In the present study, the fiber volume fraction is assumed to remainconstant throughout the deformation.Further details regarding the implementation of a constitutive model in ABAQUS can befound in Dunne and Petrinic (2005).3. RESULTSIn this section, results for simulation of kink-band formation, and the response of com-pression of a single element, are presented. In the simulations, a rectangular block of fiberreinforced material, as shown in Fig. 1, is analyzed. The block is loaded under plane strainconditions, and has the dimensions L = 10 and H = 3 with the fiber direction outside the283Sørensen and MikkelsenX1X2βHLφbFig. 1: Kink band geometrykink band coinciding with the x1-axis. Furthermore, the displacements u1 and u2 satisfythe boundary conditionsu1 = 0 on x1 = −L2(7)u2 = 0 on (x1, x2) =(−L2,−H2)(8)The width of the imperfection is b and β is the angle of the kink band. Inside the kink bandthe fibers are assumed to be at an angle φ, and this imperfection is given by the expression:φ (x1, x2) =12φm[cos(2pi cos βb(x1 + x2 tan β))+ 1](9)where φm is the value of the angle in the middle of the kink-band. In all simulations, thevalues b = 2 and β = 5◦ are used.The material parameters used in these simulations are found in Kyriakides et al. (1995),with superscripts f and m denoting fiber and matrix respectively:νf = 0.263, νm = 0.356,σfyEf= 0.019,σmyEm= 0.013 (10)In all simulations, a fiber volume fraction of cf = 0.6 is being used. The hardeningparameters are chosen to be nm = 4.5 and nf = 2.5, so the power-law curve closelyresembles the Ramberg-Osgood curve used in Jensen and Christoffersen (1997), and resultsare obtained for Ef/Em = 35 and Ef/Em = 100.The dotted green curve in Fig. 2 shows the material response under compression of anelement of the fiber reinforced material. In Fig. 2, the plastic deformation of the matrixmaterial is suppressed. The material response is modeled using one 4-node element with2× 2 Gauss integration points. The fibers are given a small initial homogenous inclinationof 1◦. During the compressive deformation the fiber inclination will increase, resulting ina lower overall stiffness of the composite material. Nevertheless, for a stiffness ratio of thefiber/matrix system given by Ef/Em = 35 the behavior is rather linear, whereas a stiffnessratio of Ef/Em = 100 will result in a much more pronounced non-linear material response.However, even for this stiffness ratio the material will not experience a material softeningbehavior. Therefore, a localized deformation state in the rectangular block (Fig. 1) is notexpected and is not found. Instead of a localized deformation state, an imperfection insen-sitive overall Euler buckling mode is developed as shown in Fig. 3. The red curve in Fig. 2284Simulation of kink bands 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0  0.005  0.01  0.015  0.02F/(AE m)DL/LEf/Em = 35, Kink-bandEf/Em = 35, ElementFig. 2: Elastic deformation(Ave. Crit.: 75%)SDV15-7.583e-05+1.397e-03+2.871e-03+4.344e-03+5.817e-03+7.290e-03+8.764e-03+1.024e-02+1.171e-02+1.318e-02+1.466e-02+1.613e-02+1.760e-02Fig. 3: Contourplot of imperfectionshows the corresponding load versus shortening curve for a rectangular block, indicating amaximum load carrying capacity of approximately F/ (AEm) = 0.42. In addition to the de-formed mesh, Fig. 3 shows a contourplot of the initial prescribed misalignment of the fibers,with a maximum value of φm = 0.0176 rad at the center of the kink band, corresponding toan imperfection of φm = 1◦.In Fig. 4, the load-displacement curves for one single element, Fig. 4a, and for a block ofmaterial, Fig. 4b, are given for the stiffness ratio of the fiber/matrix system Ef/Em =35. In contrast to Fig. 2, plastic yielding occurs in the following simulation in the matrixmaterial. During compressive loading it can be seen from Fig. 4a, that the smeared outmodel show extensive material softening and actually also snap-back behavior. Consequentlythe deformation state in the rectangular block localizes into a kink-band as shown in Fig.5. The load-displacement curves are linear until a kink stress is reached and for sufficientlysmall angles of fiber misalignment, the phenomenon of snap-back occurs. From Fig. 4 itcan be seen that the critical stress is very sensitive to the initial fiber-misalignment. Forinstance, for the kink-band formation in the block of material the critical stress almostdoubles when the maximum angle of fiber-misalignment φm is reduced from 5◦ to 2◦.A contour plot of the effective plastic strains in the matrix material during kink band285Sørensen and Mikkelsen 0 0.05 0.1 0.15 0.2 0.25 0  0.002  0.004  0.006  0.008  0.01  0.012  0.014F/(AE m)DL/Lφm = 1φm = 2φm = 5φm = 8φm = 12 0 0.05 0.1 0.15 0.2 0.25 0  0.002  0.004  0.006  0.008  0.01  0.012  0.014F/(AE m)DL/Lφm = 2φm = 5φm = 8φm = 12Fig. 4: Plastic deformation Ef/Em = 35(Ave. Crit.: 75%)SDV3-2.564e-02+3.972e-02+1.051e-01+1.704e-01+2.358e-01+3.012e-01+3.665e-01+4.319e-01+4.973e-01+5.626e-01+6.280e-01+6.933e-01+7.587e-01Fig. 5: Contour plot of effective plastic strain in the matrix material 0 0.05 0.1 0.15 0.2 0.25 0.3 0  0.001  0.002  0.003  0.004  0.005  0.006  0.007  0.008F/(AE m)DL/Lφm = 1φm = 2φm = 5φm = 8φm = 12 0 0.05 0.1 0.15 0.2 0.25 0.3 0  0.001  0.002  0.003  0.004  0.005  0.006  0.007  0.008F/(AE m)DL/Lφm = 2φm = 5φm = 8φm = 12Fig. 6: Plastic deformation Ef/Em = 100formation is shown in Fig. 5.The load-displacement curves for one single element and for a block of material, are shownin Fig. 6 for the stiffness ratio of the fiber/matrix system Ef/Em = 100. These curvesdemonstrate a behavior similar to the one where Ef/Em = 35. The implementationof the constitutive model does not include a material length-scale, and consequently thesolutions show strong mesh dependency. This is illustrated in Fig. 7 which shows theload-displacement curves using four different meshes. All curves are found using the fibermisalignment variation (9) with φm = 8◦. Similar to what is obtained by e.g. (Pamin, 1994)a more brittle post-localization behavior are obtained when the mesh is refined. From Fig. 8it can be seen that the kink-band formation occurs in one row of elements. Not only the286Simulation of kink bandspost-necking behavior, but also the load carrying capacity is seen to be influenced by themesh size. 0 0.02 0.04 0.06 0.08 0.1 0  0.001  0.002  0.003  0.004  0.005F/(AE m)DL/L3*106*2012*4030*100Fig. 7: Load versus deflection curve for 4 different meshesFig. 8: Kink band mesh dependency4. DISCUSSIONA plane constitutive model for fiber reinforced composites is implemented in ABAQUS as auser subroutine. It is demonstrated that the model qualitatively produces similar behavioras a micro-mechanical model, with regards to kink band development.The implementationof the model in ABAQUS does not include a material length-scale and therefore the kink-band will show strong mesh dependency. The implementation of the smeared-out compositematerial model in a finite element model has some immediately future applications, forinstance kink-band development in more complex structures, such as a plate with a holesubject to compression. Other possible application is simulation of competing compressivefailure mechanisms such as buckling and kink-band development in large fiber compositestructures.ACKNOWLEDGEMENTThe work has been financial supported by the Danish Technical Research Counsil throughthe project ’Interface Design of Composite Materials’.287Sørensen and MikkelsenREFERENCESBudiansky, B. (1993). Micromechanics. Computers & Structures 16, 3-12.Christoffersen, J., and Jensen, H.M. (1996). Kinkband analysis accounting for the mi-crostructure of fiber reinforced materials. Mechanics of Materials 24, 305-315Dunne, F., and Petrinic, N. (2005). Introduction to Computational Plasticity. OxfordUniversity Press.Hsu, S.-Y., Vogler, T.J., and Kyriakides, S. (1998). Compressive Strength Predictions forFiber Composites. Journal of Applied Mechanics 65Jensen, H.M., and Christoffersen, J. (1997). Kink Band Formation in Fiber ReinforcedMaterials. J. Mech. Phys. Solids 45, 1121-1136.Kyriakides, S., Arseculeratne, R., Perry, E. J. and Liechti, K. M. (1995). On the compressivefailure of fiber reinforced composites. Int. J. Solids Struct. 32, 689-738.Liu, D., Fleck, N. A., and Sutcliffe, M. P. F. (2004). Compressive Strength of Fibre Com-posites with Random Fibre Waviness. J. Mech. Phys. Solids 52, 1481-1505.Pamin, J.K. (1994). Gradient-dependent plasticity in numerical simulation of localizationphenomena. PhD-thesis, Delft University Press, The Netherlands.Pane, I. and Jensen, H.M. (2004) Plane strain bifurcation and its relation to kinkbandformation in layered materials. European Journal of Mechanics A/Solids 23, 359-371288

On the simulation of kink bands in fiber reinforced composites

Simulations of kink band formation in fiber reinforced composites are carried out using the commercial finite element program ABAQUS. A smeared-out, plane constitutive model for fiber reinforced materials is implemented as a user subroutine, and effects of fiber misalignment on elastic and plastic deformation are studied under plane strain conditions.Simulations of kink band formation in fiber reinforced composites are carried out using the commercial finite element program ABAQUS. A smeared-out, plane constitutive model for fiber reinforced materials is implemented as a user subroutine, and effects of fiber misalignment on elastic and plastic deformation are studied under plane strain conditions

On the simulation of kink bands in fiber reinforced composites

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