A SIMPLE PLASTICITY MODEL FOR PREDICTING TRANSVERSE COMPOSITE RESPONSE AND FAILURE

Transcription

1 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS A SIMPLE PLASTICITY MODEL FOR PREDICTING TRANSVERSE COMPOSITE RESPONSE AND FAILURE K.W. Gan*, M.R. Wisnom, S.R. Hallett, G. Allegri Advanced Comosites Centre for Innovation and Science, University of Bristol, University Walk, Bristol, BS8 1TR, United Kingdom. * Corresonding author (khong.wui.gan@bristol.ac.uk) Keywords: Transverse Isotroy, Plasticity, Fibre Reinforced Plastics, Failure Abstract This aer reorts on the develoment of a simle three-arameter nonlinear constitutive model for comosite materials. The model has been calibrated using two indeendent sets of exerimental data in the literature. With knowledge of the critical tensile strain at failure of the matrix constituent, it is used to redict failure loads for load cases involving through-thickness comression and interlaminar shear, in which shear strengths increase with through-thickness comressive stresses. Good correlation with the exeriments has been achieved. Introduction Comosite materials exhibit highly non-linear constitutive resonses in the matrix-dominated directions. The non-linearity in resonse can be considered generally as lastic behaviour, although hysically this might not be strictly correct. The non-linearity can as well be attributed to the effects of various damage mechanisms in the matrix. In aerosace alications, the matrices are very often olymeric materials such as eoxy resins. Very distinctive differences exist between metals and olymers. Polymers in general dislay a strong ressure deendent material behaviour. In articular, the different yield behaviour between tension and comression of the olymeric matrix together with the anisotroic nature of comosite materials recludes the use of a von-mises tye of yield criterion. The urose of this aer is to exlore the otential use of a non-linear constitutive model to redict failure initiation in comosites subjected to comlex loading. The knowledge of strains gleaned from the constitutive model can be exloited so that the matrix dominated failure initiation can be estimated with a simle maximum strain criterion. This is demonstrated with some load cases involving interlaminar shear and through-thickness comression suorted by exerimental measurements. Develoment of the constitutive model A non-linear elastolastic constitutive law requires the definition of a yield function, which essentially defines the elastic limit of a material under combined states of stress. Ideally, a yield function for a unidirectional comosite material should be formulated in the framework of transversely isotroic invariants, and be sufficiently simle to contain no more arameters than are necessary. The invariants, exressed in the conventional rincial material axis notation of a unidirectional comosite, are: I1 I I 1 I (1) I I 5 4 For most continuous fibre-reinforced comosite materials, the resonse in the fibre direction from exerimental observations is essentially linear to failure with no lastic deformation. Secondly, a yield criterion should not be deendent on the sign of shear stresses. Based on the above reasoning, the formulation of an aroriate yield function should exclude the first and the fifth invariants. A simle three-arameter lasticity yield function, f, similar to the Drucker-Prager model is roosed for this work.

2 f ( ) H( ) L 1 J( ) () where H, L and J are material constants to be determined from the exeriment data. Without loss of generality, the coefficient of the third invariant (the longitudinal shear terms) is arbitrarily set to 1, as this will facilitate the calibration rocess later. A strict adherence to the invariant formulation framework dictates L = 4H. However, due to the common lack of exerimental data on the transverse shear resonse of comosite materials and also to maintain the generality of the yield function, L is assumed to be an indeendent fitting arameter. All these material arameters are assumed to be constants and are indeendent of the stress state, i.e. the initial material orthotroy will not be affected by the subsequent lastic deformation. Since this is a feasibility study and to kee the number of arameters to a minimum, an associative lastic flow rule is adoted, i.e. the lastic strain increment, dε, can be determined from the yield function itself via: f ( ) d d () where i, j = 1,, and λ is the lastic multilier to be defined later. In order to calibrate the yield function with the exerimental uniaxial stress-strain curve, a scalar effective stress is defined: f ( ) (4) i.e. the effective stress is the same as the yield function in Equation (). From the equivalence of the increment of lastic work er unit volume, we have: dw d d (5) where d is the increment of the effective lastic strain. Substituting Equation () into Equation (5) and using the definition from Equation (4): f d d fd d (6) Therefore the lastic multilier is equal to the effective lastic strain: d d (7) By aroximating the relation between effective stress and effective lastic strain of a simle uniaxial test using a Ramberg-Osgood equation: 1 n (8) K where K and n are the curve fitting arameters, the magnitude of the lastic multilier can be known from Equation (7) as a function of the effective stress. Exerimental results for matrix-dominated resonses in a fibre-reinforced comosite very often do not indicate a well-defined yield oint. To avoid an arbitrary definition of a yield oint, the current constitutive model allows initial yielding at the moment the stresses are alied. Due to the ower law nature of the Ramberg-Osgood equation, the amount of lastic strain will be negligibly small at low alied stresses. Also in this study, an isotroic hardening is assumed, i.e. any Bauschinger effect is neglected comletely and the yield surface exands uniformly in all directions in the stress sace. Residual thermal stresses are neglected. Having the coefficient of the third invariant set to 1 in the yield function (Equation ()) allows us to establish the effective stress and effective lastic strain relation solely from a simle longitudinal shear stress and lastic shear strain (τ 1 -γ 1 ) resonse, without the need to know the values of H, L and J beforehand. In a ure longitudinal shear test, the effective stress is given by Equation (4) as: 1 (9)

3 A SIMPLE PLASTICITY MODEL FOR PREDICTING TRANSVERSE COMPOSITE RESPONSE AND FAILURE while the effective lastic strain, from the equivalence of lastic work in Equation (5) and using Equation (7), becomes: d d 1 d (10) Therefore the ure longitudinal shear stress-strain curve will act as the lasticity master curve which rovides the essential information on the magnitude of the lastic multilier as a function of the effective stress. Calibration of the constitutive model The aerosace graded carbon/eoxy IM7/855 comosite system is used here to assess the feasibility of this non-linear constitutive aroach. For the master lasticity curve, the longitudinal shear stress-strain test data was taken from [1] (Figure 1). The corresonding shear stress-strain curve can be reresented by Equation (8) with the fitting arameters K = and n = Now the arameters H, L and J remain to be determined. Parameter L requires test data on the transverse shear stress-strain (τ -γ ) resonse, which is difficult to measure and they are not readily available in the literature. The requirement that L = 4H following the invariant formulation might make the transverse shear resonse overly comliant. Therefore, as a first aroximation, the arameter L is set equal to 1, i.e. the lastic resonse of the transverse shear is assumed to be the same as that of the longitudinal shear. In fact, the exected value of L should not be much larger than 1. On the other hand, the two unknown arameters H and J require two indeendent sets of test data which involve transverse normal stress-strain resonses, such as the transverse tensile and comression tests, or the biaxial tests. They can be calibrated from the exerimental work by Koerber et al. [] who erformed off-axis comression tests on carbon/eoxy IM7/855 at six different off-axis angles: 15, 0, 45, 60, 75 and 90. In the tests, the stress comonents along the rincial material axes can be related to the off-axis alied axial stress, σ x, as: 11 x cos x sin (11) sin cos 1 x where θ is the off-axis angle. Similarly, the strain comonents along the rincial material axes are related to the off-axis alied axial strain, ε x, as: 11 x cos x sin (1) sin cos 1 x For an infinitesimal stress increment, a searation of the increment strain tensor, dε in the rincial material axes into elastic strains, dε e, and lastic strains, dε, is assumed: d d d () e The elastic stress-strain relationshi is given by the Hooke s Law for an orthotroic material. Equations (11) to () are substituted into Equations () to (10) to search for the otimal set of H and J values that best describe all the six off-axis comression exerimental curves. As there are only two arameters to be determined, this can be easily done through a brute force otimisation rocess by searching through a range of ossible H and J values that give the least squared error. This is done in MATLAB. It is found that H = 0.4 and J = 0.6 for the IM7/855 carbon/eoxy comosite. The rediction of the lasticity model using the otimised arameters is comared against the six sets of off-axis exerimental data in Figure to Figure 7. Considering the simlicity of the model with only two main arameters, reasonably accurate constitutive resonses for the comosite material are obtained. The redicted stress-strain curves for small off-axis angles (fibre dominated) are slightly underredicted, while they are over-redicted for large off-axis angles (matrix dominated). For moderate off-axis angles however, the agreement with the exerimental curves is excellent. This model is now ready for use to redict material behavior in other more comlex load cases, caturing henomena or

4 stress states within comosite structures that are unobtainable if a linear elastic analysis is used. Case study A case study is resented in this section to demonstrate the otential alication for this simle lasticity model to redict the failure load of comosite secimens subjected to through-thickness comression and interlaminar shear stress. A symmetric version of the double-notch shear test has been roosed by the authors for direct measurement of the interlaminar shear strength under moderate through-thickness comression using a biaxial test rig (Figure 8) []. In the test, through-thickness comressive stresses were imosed through a air of flat indenters with some user-secified loads. The interlaminar shear failure was designed to take lace within the three lies of 0 at the gauge section of the secimen between the notches under longitudinal tensile loading. Thickness-wise away from the 0 lies, alternating 90 and 0 lies were resent to revent the secimens from slitting longitudinally so that the interlaminar shear strength of the secimens could be tested u to high through-thickness comressive loads. The test results show that the throughthickness comressive stress can significantly increase the interlaminar shear strength. The exerimental results are shown in Figure 11. In the following, a failure criterion for matrix dominated failure is roosed. Failure due to delamination is dominantly governed by the matrix strength roerties. According to the Northwestern theory (the NU theory) develoed in [4], a shear dominated failure criterion is governed by the limiting microscoic tensile strain set by ure shear in the ly interface region. Under ure shear loading in the 1- lane, the critical rincial tensile strain at failure is one half of the shear strain at failure at 45 to the loading direction, as illustrated in Figure 9. For a biaxial σ -τ state of stress, failure is ostulated to occur when the maximum rincial tensile strain exceeds the critical rincial tensile strain at failure given by the ure shear loading. In other words, it is essentially a maximum rincial tensile strain criterion that redicts failure at the ly interface subjected to a σ -τ state of stress. Using linear elastic stress-strain relationshi, the failure criterion can be exressed in terms of the alied stresses and comosite elastic roerties as: S S G E 1 (14) where S is the interlaminar shear strength, G is the shear modulus in the 1- lane and E is the through-thickness elastic modulus of the comosite. The NU theory (Equation (14)) rediction has shown good agreement with exerimental results for interlaminar shear dominated failure in textile comosites [4] involving interlaminar shear and through-thickness comression. The failure of the symmetric double-notch shear secimens [] is also well-redicted using the NU theory. In numerical modelling, a strain-based failure criterion such as the maximum strain criterion generally loses its aeal due to the difficulty in calculating the strains in a comosite as they are usually highly non-linear with resect to the alied stresses. However, strains are fundamental quantities that can be measured directly in the exeriments, unlike derived quantities such as stresses which need the recise knowledge of the material stiffness or require that the geometry of the secimen to be sufficiently simle. In addition to that, the maximum strain criterion also has the straightforward hysical interretation that a material fractures when two material oints are searated by a critical dislacement (or strain), regardless of the comlexity of the load conditions. The critical rincial tensile strain at which the ly interface fails by interlaminar shear of IM7/855 carbon/eoxy comosite can be obtained from a simle shear test. Makeev and Seon [5] have measured the interlaminar shear stress-strain resonse for unidirectional IM7/855 using short beam shear (SBS) secimens. The measured stresses were derived using a linear elastic beam theory so only first order accuracy was obtained. However, the strain values were obtained using digital image correlation (DIC) full-field technique. Provided that the technique was well calibrated, the strain measurement thus obtained was of higher accuracy.

5 A SIMPLE PLASTICITY MODEL FOR PREDICTING TRANSVERSE COMPOSITE RESPONSE AND FAILURE In their SBS tests, the shear strain to failure was measured at about.9%, imlying that the critical maximum rincial tensile strain is 1.95% (half of the critical shear strain). Also the tensile strain to failure for the 855 matrix constituent according to the datasheet rovided by Hexcel [6] is 1.7%. Comaring the two values, it can therefore be estimated that the critical tensile strain to failure for a matrix dominated failure of IM7/855 carbon/eoxy should be around 1.8%. With the develoment of the current lasticity model, the total strains of comosite materials can be obtained more accurately than those calculated using a linear elastic analysis. It is the urose of this aer to demonstrate that a maximum rincial tensile strain criterion (the NU theory) taking into account the non-linear stress-strain relation is still alicable in a non-linear analysis. The added advantage is that the non-linear model can give a more realistic global and local comliance resonse of the comosite structure, which may alter the comlex stress distribution within a comlicated geometry. Using the authors exerimental results of the symmetric double-notch shear secimens made of IM7/855 as an examle, a 1.8% critical rincial tensile strain failure criterion is used to redict failure loads. In the initial study, the lasticity model is imlemented in MATLAB as a single element with the material roerties of IM7/855 []. For simlicity, a two-dimensional lane strain boundary condition is assumed. Using the 1.8% critical rincial tensile strain failure criterion, the results of both the linear elasticity and the lasticity models are comared against the exerimental ones in Figure 11. Although both models dislay the trend of increasing interlaminar shear failure stress with transverse comression, the lasticity aroach which incororates a nonlinear constitutive resonse gives more realistic results. The linear elastic solution way overestimates the failure load. On the other hand, the slight overestimation of the failure load in the lasticity model can be attributed to the lane strain boundary condition, which tends to induce a higher hydrostatic ressure in the model, redicting the material to be less likely to yield and failure is thus delayed to a higher load. For imrovement of the numerical results, the roosed lasticity model is imlemented in exlicit solver LS-DYNA as a user-material subroutine to simulate the actual three-dimensional stress state. Only a quarter of the symmetric double-notch shear secimen (taking advantage of the symmetry boundary conditions) is modelled, as shown in Figure 10. Note that the current lasticity model does not incororate a rogressive damage or failed element deletion caability (this could be imlemented in the future). This means that the global alied load of the model can continue to increase after the maximum strain criterion is reached, albeit at a decreasing rate due to the lasticity. To get over this, a user-defined definition of failure for the models needs to be rescribed. In this case, it is reasonable to define the failure load as the tensile load when at least 50% of the gauge section has reached 1.8% maximum rincial tensile strain. This is illustrated in Figure 11, with the region in red showing failed elements within the gauge section. The shear strength is then simly calculated in an average sense as the tensile load divided by the gauge area. The imrovement of the results can be seen in Figure 11. It aears that the NU theory (Equation 14) can be used with success for both the linear elastic [, 4] and the non-linear lasticity analyses to estimate the failure load in a biaxial shear-dominated failure. This can be exlained qualitatively by examining Equation (14). Both the resonses in the throughthickness normal direction and interlaminar shear are highly non-linear. The aroximately equal degree of non-linearity of the two cancels each other out, therefore making the NU theory alicable for both solutions. However the general comliance resonse of a comosite structure will be more accurate when the non-linearity in stress-strain relation is considered in the model. Summary A simle lasticity model for comosite materials has been roosed. It takes into account the ressure effect and is able to cature the large strain and the nonlinear behaviour of the matrix caused by yield and flow. Neglecting these effects, a linear analysis tends to overestimate the stress-strain resonse and give inaccurate stress distributions in the material, 5

6 rendering recise failure redictions roblematic, esecially when involving comlex multiaxial loading. This is demonstrated with an examle of a biaxial load case involving interlaminar shear and through-thickness comressive normal stress, using a maximum rincial tensile strain criterion. The lasticity model has been imlemented as an exlicit finite element subroutine in LS-DYNA and has roven to be a romising aroach for studying matrix-dominated failure, at the same time giving a more realistic comliance resonse of a comosite structure comared to a linear elastic model. This is esecially the case when considering comlex loading and stress states for which finite element analysis is the most suitable technique. [6] Hexcel datasheet accessed on March 01, htt:// g-data-sheets/855_us.df Figures Acknowledgement This work was suorted by a Dorothy Hodgkin Postgraduate Award. The suort of Rolls-Royce lc through the Comosites University Technology Centre of the University of Bristol is also acknowledged. Fig. 1. Non-linear shear stress vs. strain resonse for various UD comosites [1]. References [1] G.M. Vyas, S.T. Pinho Comutational Imlementation of a Novel Constitutive Model for Multidirectional Comosites. Comutational Materials Science, Vol. 51, 17-4, 01. [] H. Koerber, J. Xavier, P.P. Camanho High Strain Rate Characterisation of Unidirectional Carbon- Eoxy IM7-855 in Transverse Comression and In- Plane Shear using Digital Image Correlation. Mechanics of Materials, Vol. 4, , 010. [] K.W. Gan, S.R. Hallett, M.R. Wisnom Measurement and Modelling of Interlaminar Shear Strength Enhancement under Moderate Through- Thickness Comression. Comosites Part A, Vol. 49, 18-5, 0. [4] I.M. Daniel, J-J. Luo, P.M. Schubel, B.T. Werner Interfiber/interlaminar failure of comosites under multi-axial states of stress. Comosites Science and Technology, Vol. 69, , 009. [5] A. Makeev, G. Seon Failure redictions for carbon/eoxy tae laminates with wavy lies. Journal of Comosite Materials, Vol. 44, no. 1, 95-11, 010. Fig.. Stress-strain rediction of the lasticity model versus 15 off-axis comression test result [].

7 A SIMPLE PLASTICITY MODEL FOR PREDICTING TRANSVERSE COMPOSITE RESPONSE AND FAILURE Fig.. Stress-strain rediction of the lasticity model versus 0 off-axis comression test result []. Fig. 6. Stress-strain rediction of the lasticity model versus 75 off-axis comression test result []. Fig. 4. Stress-strain rediction of the lasticity model versus 45 off-axis comression test result []. Fig. 7. Stress-strain rediction of the lasticity model versus 90 off-axis comression test result []. Fig. 5. Stress-strain rediction of the lasticity model versus 60 off-axis comression test result []. Fig. 8. A symmetric double-notch shear (SDNS) secimen with dimensions (side and to views) []. 7

8 Fig. 9. Illustration of limiting microscoic tensile strain in the interlaminar region for shear dominated failure on the 1- lane [4]. Fig. 11. The interlaminar shear failure enveloe redicted by a D linear elastic model (single element in MATLAB), a D lastic model (single element in MATLAB) and a D lastic model (a quarter model of the actual test secimen in LS- DYNA), comared against exerimental observations. Fig. 10. (To) A quarter FE model of the actual test secimen. (Bottom) A to view of the gauge area showing more than 50% of the area (elements in red) reaching 1.8% critical rincial tensile strain as the rescribed definition of failure.