Proceedings of the SEM Annual Conference June 1-4, 2009 Albuquerque New Mexico USA ©2009 Society for Experimental Mechanics Inc.
Biaxial failure envelopes for glass fibre reinforced composite laminates Makris A. 1, Ramault C. 1, Van Hemelrijck D.1 ,Lamkanfi E. 2, Van Paepegem W. 2 1
Vrije Universiteit Brussel, Department of Mechanics of Materials & Constructions, Brussels, Belgium 2 Ghent University, Department of Mechanical Construction and Production, Ghent, Belgium Email:
[email protected]
ABSTRACT To study the behaviour of fibre reinforced polymeric matrix composite laminates under static and cyclic in-plane complex stress states, a horizontal biaxial loading frame and a special cruciform type specimen have been developed. In the present study the cruciform specimen is biaxially loaded in its plane using four independent servo-hydraulic actuators and the quasi-static loading case is investigated. An appropriate control unit keeps the centre of the specimen at the same position during the test. For obtaining reliable biaxial failure data the design of the cruciform geometry is of paramount importance, also the accuracy of the measurement techniques has to be ensured. For that reason, finite element simulations of the cruciform specimen in combination with measurements using both strain gages and optical-numerical full field methods for strain determination over the entire biaxially loaded test zone have been carried out. Finally will be shown a comparison between failure envelopes plotted using existing failure criteria and experimental data obtained from cruciform and beam specimen’s testing. The non-linear shear stress-strain and transverse stress-strain behaviour of the lamina was taken into account. Keywords: A. Polymer-matrix composites (PMCs); B.Cruciform specimen; C. Finite element simulations (FES); D. Failure envelopes. Introduction Due to their high specific strength and stiffness, the use of fibre reinforced composite materials in aerospace, aviation, automotive and civil engineering industry as for instance wind turbine rotor blades or submersible structures has increased rapidly in recent years. In general, these composite laminates in the form of thin or moderately thick shells are developing complex stress states [1,2]. Advances by the composite materials testing community in recent years have resulted in the ability to accurately characterize the uniaxial response of fibre reinforced composite materials. On the other hand, test methods that can be used to characterize the same materials under multiaxial stress states have not been pursued as rigorously. As a result, there is little existing capability to evaluate the full multiaxial —or even the biaxial— response of composite materials, even though large demand for such information exists [3,4]. The ability to successfully model and simulate the behaviour of materials for optimum use in structural applications, depends largely on the material description (constitutive relations, failure criteria) employed in analytical formulations. The proper choice of constitutive law requires rigorous experimental characterization in order to ensure that the constitutive model adequately describes the behaviour of the material under a variety of complex loading conditions [5]. The current practice of using uniaxial test results to predict failure for multiaxial stress states seems inadequate. Numerous failure theories have been proposed in an effort to accurately predict the response of composite materials. The lack of reliable biaxial and multiaxial experimental data to validate failure theories is the critical step in the evolution and a most efficient usage of composite materials [6]. An appropriate method to perform biaxial testing of fibre reinforced composite laminates consists of applying inplane biaxial loads to cruciform specimens [7]. In order to do so, a biaxial test device was developed at the Free University of Brussels (VUB) at the Department of Mechanics of Materials and Constructions (MeMC). This device employs four servo-hydraulic actuators for testing cruciform specimens under static and cyclic loading conditions. Finite element simulations of the cruciform specimen geometry in combination with experiments using both strain gages and full field optical-numerical methods (Digital Image Correlation Technique, Electronic Speckle Pattern Interferometry Technique) for displacement measurements all over the biaxially loaded test zone have been carried out, leading to the proposal of an optimized geometry and ensure the accuracy of the measurements.
1. EXPERIMENTAL PART 1.1 Types of biaxial testing Different experimental techniques and specimens have been used to produce biaxial stress states [8, 9]. These techniques may be classified into two categories [10]: (i) tests using a single loading system and (ii) tests using two or more independent loading systems. In the first category the biaxial stress ratio depends on the specimen geometry or the loading fixture configuration; whereas in the second category it is specified by the applied load magnitudes. Examples of the first category are bending tests on cantilever beams, anticlastic bending tests of flat plates, bulge tests and tests using special fixtures. Examples of the second category are a round bar under torsion combined with bending, thin-wall tubes subjected to a combination of tension / compression and torsion or internal / external pressure, and cruciform specimens under in-plane biaxial loading. The technique with the thin-wall tube [11, 12] is the most popular one and seems to be very versatile, because it allows tests with any constant load ratio to be performed. However, it presents some inconveniences: (i) radial stress gradients may not be negligible depending on the thickness of the tube and the applied load, (ii) real construction components in fibre reinforced composite materials are often flat or gently curved and differ a lot from tubular specimens, (iii) thin-wall tubes are not easy to fabricate, (iv) obtaining a perfect alignment and load introduction is not straightforward, (v) tubular thin-walled specimens can experience various forms of elastic instability when they are subjected to circumferential or axial compression or torsion loads and (vi) tubes may exhibit changes in geometry during loading, but these effects are usually ignored when processing experimental results. The most realistic technique to create biaxial stress states consists of applying in-plane loads along two perpendicular arms of cruciform specimens. The use of hydraulic actuators represents a very versatile technique for the application of the loads. The main differences between the techniques are the use of one actuator per loading direction or two. For the techniques using one actuator per loading direction the centre of the specimen will move during a test causing side bending of the specimen. This results in undesirable non-symmetric strain field. Systems with two actuators per loading direction with a close-loop servo control using the measured loads as feedback system, allow the centre of the specimen standing still. The plane biaxial test device used at the Free University of Brussels (VUB) is of this type using four independent servo-hydraulic actuators with an appropriate control unit as will be explained in the next paragraph. 1.2 Testing configuration 1.2.1 Plane biaxial test bench for cruciform specimens The biaxial test rig, see Fig.1, developed at VUB has a capacity of 100kN in each perpendicular direction, but only in tension, limiting the experimental results to the first quadrant of the two-dimensional stress space. As no cylinders with hydrostatic bearing were used, failure or slip in one arm of the specimen will result in sudden radial forces which could seriously damage the servo-hydraulic cylinders and load cells. To prevent this, hinges were used to connect the specimen to the load cells and the servo-hydraulic cylinders to the test frame. Using four hinges in each loading direction results in an unstable situation in compression and consequently only tension loads can be applied. The stroke of the cylinders is 150mm. The loading may be static or dynamic up to a frequency of 20Hz. Each cylinder is independently controlled and any type of loading waveform, including spectral sequences of variable amplitude, can be efficiently introduced using the dedicated software and control system.
Fig.1. Plane biaxial test device for cruciform specimens.
In order to maintain the co-linearity of the loads in each direction to avoid bending moments which cause a nonsymmetric strain distribution in the bi-axially loaded zone, the centre of the specimen must remain still. An arrangement where two ends of the cruciform specimen are held fixed while the opposite two ends are loaded — as was done in the past with test devices using two actuators— is unsatisfactory in this respect. Instead, four servo-hydraulic cylinders (Fig.2a) are used with a control unit to explicitly keep the forces collinear (Fig.2b). Even when using four actuators, a small displacement might occur in a real situation and an imbalance arises in the forces. Due to the displacement in the y-direction (Fig.2c), a component Fy is added and the forces P and P’ are no longer equal. However as four load cells are used, it is possible to measure this small load difference and this is used as a control signal. P
P
Fy F
F’
P’
F
F’
P’
a) control using four cylinders b) ideal situation c) real situation Fig.2. Control unit and forces on the cruciform specimen. 1.2.2 Cruciform specimen An optimized cruciform specimen for biaxial testing should fulfil the following requirements: (i) maximization of the region of strain uniformity in the biaxial loaded zone, (ii) minimization of the value of the global shear strain ε°xy of the biaxially loaded test zone, (iii) minimization of the strain concentrations outside the test zone, (iv) specimen failure in the bi-axially loaded test zone and (v) repeatable results [13-15]. It has been proven extremely difficult to develop cruciform specimens that simultaneously fulfil all these requirements. The influence of parameters like (i) the radius of the corner fillet at the intersection of the arms, (ii) the thickness of the bi-axially loaded test zone in relation to the thickness of the uniaxially loaded arms and (iii) the geometry of the bi-axially loaded test zone, are of paramount importance. A combination of finite element simulations and experiments led to an optimized geometry [16], see Figure 3. The arm width is 25mm and the total length 250mm.
Figure 3. Dimensions of the cruciform specimen, (a) arm of the specimen, (b) adapted fillet corner radius (6.25mm), (c) milled central zone. 1.3 Material mechanical elastic properties and strengths The material used is glass fibre reinforced epoxy with a [(±45/0)4/±45] lay-up. The material system and stacking sequence are typical of wind turbine rotor blade construction. The lay-up is not symmetric and this is due to the production technique, but the maximum curvature (twist) ks that can appear under axial loading is relatively much smaller than the appearing axial strains. Cruciform specimens were machined from plates produced by LM Glasfiber, Denmark, using RTM (resin transfer moulding) technology. When cured, the [±45] plies have a
thickness of 0.61 mm and the [0] ones of 0.88 mm. This gives a total thickness of 6.57 mm for the arms of the cruciform specimen and of 3.59 mm for the biaxially loaded zone where one group of [±45/0] was milled away at each side of the specimen. An extended database of experimental static and fatigue results on beamlike glass fibre reinforced epoxy specimens with a [(±45/0)4/±45] lay-up has been set-up within the framework of the Optimat Blades project [16]. For the glass fibre reinforced composite laminate with the mentioned lay-up the average material parameter results and the coefficient of variation of about four hundred traditional beamlike tests are given below, see Table 3. Also material properties for a UD laminate were calculated using beam specimens with a [0]4 lay-up, see Table 1. The strengths of the UD material are given also in Table 2 obtained from tension/compression testing of beam coupons and from v-notched specimens testing for shear strength determination. Based on the ply data, the theoretically expected properties of the laminate could be also calculated using classical laminate theory, see Table 4. Table 1. Elastic properties obtained from beam testing of [0]4 lay-up. E1
E2
G12
ν12
[GPa]
[GPa]
[GPa]
-
average
39.10
14.44
5.39
0.294
COV [%]
2.1
0.98
1.77
0.027
Table 2. Uniaxial strengths for the UD material.
average COV [%]
XT
XC
YT
YC
S
[MPa]
[MPa]
[MPa]
[MPa]
[MPa]
766.5 4.65
686.32 5.2
53.865 4.77
165 2.93
71 2
Where XT, XC are the tensile and compressive strength longitudinal to the fibers, YT, YC are the tensile and compressive strength transverse to the fibers and S is the shear strength of the lamina. Table 3. Elastic properties obtained from beam testing of [(±45/0)4/±45] lay-up. Ex
Ey
Gxy
vxy
[GPa]
[GPa]
[GPa]
-
average
27.03
14.21
8.1
0.455
COV [%]
4.4
6
9.5
9.2
Table 4. Elastic properties of [(±45/0)4/±45] lay-up obtained from classical laminate theory calculations.
average
Ex
Ey
Gxy
vxy
[GPa]
[GPa]
[GPa]
-
28.62
16.25
8.29
0.406
1.4 Strain measurements/ experimental failure envelope In this study the biaxial failure envelope εx/εy of a glass epoxy laminate will be presented. The strain space was mainly preferred because the strain field of the zone of interest can be straightway measured or calculated by differentiating the measured displacement field. On the other hand stress calculations are not accurate because the exact loaded section of the cruciform specimen cannot be easily defined. Furthermore strain is dimensionless (it is the same in both SI and English units) and mostly constant or a linear function through the thickness of a laminate. To be able to study the symmetry of the strains and the occurring shear strains experimentally, full field methods are necessary. Strain measurements using a strain gage or extensometer are not sufficient because both give an average value of the deformation along their gauge length and sometimes fail earlier than the specimen. Due to the complexity of the specimen’s geometry different measurement techniques were applied to investigate strain concentrations and finally obtain the valid failure strain field of the zone of interest. Strain gages, Digital Image Correlation (DIC) and Electronic Speckle pattern Interferometry (ESPI) techniques were used combined or separately [17]. Digital image correlation technique (DIC) is an experimental technique, which offers the possibility to determine in-plane and out of plane displacement and deformation fields of the surface of objects under any kind of loading, based on a comparison between images taken at different load steps. Electronic speckle pattern interferometry (ESPI) is widely used in nondestructive testing (NDT). In ESPI measurement technique, the deformations are made visible as fringe patterns while an inspected sample is loaded. Strain fields obtained from different methods were compared in order to evaluate the results. Experimental strain data were also compared with data acquired from a 3D finite element model [18]. All methods showed significant qualitative and quantitative correlation see Figure 4.
Figure 4. Position of the strain gages and the first principal strain fields from DIC ESPI and FEM respectively. (uniaxial tensional load along 0° direction, 40% of the total failure load). Seven different load ratios (Fx/Fy) were applied on cruciform specimens (five different biaxial cases and two uniaxial cases) and for each ratio average four specimens were tested in tension until total failure, see Figure 5. Biaxial testing of cruciform specimens was performed using load control of the machine with a constant load speed of 5kN/min and uniaxial testing of them by displacement control with a displacement rate of 1mm/min.
Figure 5. A cruciform specimen biaxially loaded (ratio 3.85/1, Fx=48kN) very close to total failure. Failure strains obtained from beam specimens under uniaxial tension or compression loading, together with data from cruciform testing, were plotted in order to obtain the experimental failure envelope of the glass epoxy laminate. Below are presented the experimental data measured from the centre of each specimen, see Figure 6.
3.0
2.0
εy (%)
1.0
0.0 -3.0
-2.0
ratio 0/1 ratio 1.925/1 ratio 2.567/1 ratio 3.85/1 ratio 5.775/1 ratio 7.7/1 ratio 1 /0 beam ratio 0/1 beam ratio 1/0 beam ratio -1/0
-1.0
0.0
1.0
2.0
3.0
-1.0
-2.0
-3.0 εx (%)
Figure 6. Failure envelope in strain space from beam specimens testing of [(±45/0)4/±45] laminates and cruciform specimens with a loaded central zone of [(±45/0)2/±45] lay-up. 2. FAILURE THEORIES 2.1 Failure criteria for fibre reinforced polymeric matrix composites. A comprehensive study of the most popular failure criteria that exist was performed under the name of the WorldWide Failure Exercise (WWFE) [6], where 19 different failure theories were compared each other and with the existing experimental data (mainly from testing of tubular specimens). The overall strategy and procedures were outlined where each contributor describes its own theory and employs it to solve 14 challenging problems (test cases) set by the organisers. Basic conclusions considering this work were that very few failure criteria are able to predict well the behaviour of the material under all the different load cases and that there is a lack of reliable experimental data that could be compared with these theories. In this study experimental data obtained from testing of cruciform specimens under biaxial and uniaxial tension loading and beam specimens under uniaxial tension or compression loading are compared with Tsai, Puck and maximum stress failure criteria. The two first failure criteria were selected because they seemed to produce the most reliable results among all the criteria investigated from the authors of WWFE and maximum stress as a basic failure criterion for brittle materials. Below each criterion is briefly discussed.
Tsai’s criterion [19] is an empirical quadratic criterion, one of the most popular failure criteria to predict failure of composite materials, also implemented in a few commercial finite element programs. It is based on a tensoral function, see equation 1. 2nd order
4th order
6th order
Fijσ ij + Fijklσ ijσ kl + Fijklmnσ ijσ klσ mn + ......−1 ≤ 0, i, j, k, l, m, n = 1,....,3 For a plane stress case and by using the 2 2
2
nd
(1)
order of the function, equation 1 is written: 2
F11σ 1 + F22σ 2 + 2 F12σ 1σ 2 + F66σ 6 + F1σ 1 + F2σ 2 − 1 = 0
(2)
Fij are relations of the strengths of the lamina, F12 is called ‘interaction term’ and controls a lot the shape of the failure envelope. For it different values have been proposed, in the present study were used:
F12 = − F12 = −
F11 2
: Ellipsoid Parabolic Failure Surface term (EPFS).
1 F11 F12 : Hahn’s term. 2
(3) (4)
Puck’s criterion [20] is the most representative from the phenomenological theory criteria. In these ‘physical based’ criteria are investigated separately the different failure modes of the composite materials. Puck’s theory is based on Mohr’s hypothesis ‘The stresses on the fracture plane are decisive for fracture’. Puck was also the first to publish the idea in 1969 that fiber failure (FF) and inter fiber failure (IFF) should be distinguished and theoretically treated by separate and independent failure criteria. Finally maximum stress criterion expresses the physical idea that failure of a ply under multiaxial stresses occurs when the stress σij (i, j=1, 2) is equal to or exceeds the uniaxial or shear strength of the ply without taking into account any interaction between stresses. 2.2 Failure analysis code In order to be able to compare the experimental data with failure envelopes coming from various failure criteria a code was developed using the commercial software Matlab. The code is based on the classical laminate theory and on a ply by ply failure analysis of composite plates. Basic inputs are the elastic properties, the strength of the lamina and the lay up of the loaded laminate. Outputs of the program are the failure envelopes in strain (εx/εy) and stress (σx/σy) space according to various failure criteria. The algorithm is summarized below, see Figure 7.
Figure 7. Failure analysis algorithm. For the failure predictions in the present study a First Ply Failure analysis (FPF) was used, which means that for each biaxial ratio are plotted the global strains ε°x and ε°y under which the first orientation-ply of the laminate fails, according to each failure criterion. FPF is often used to determine the strength of a laminate especially when leak characteristics are investigated and even though often it is considered as a conservative tool, a comparison between it and the experimentally obtained data of total failure is interesting as it can show the degree of this ‘conservatism’. 2.3 Linear/Non-Linear behaviour of the material/comparison with experimental data Inputs for the program calculations are the mechanical elastic properties and strengths of the UD material (lamina). Experimental data showed that the evolution of the strain is linear when the UD-material is uniaxially loaded along the fibre direction, slightly non linear transverse to the fibres and highly non-linear under shear stresses.
Figure 8. Qualitative stress-strain (σ1-ε1, σ2-ε2, and σ6-ε6) curves of the glass epoxy lamina. To be able to take into account the non linear behaviour of the material are required uniaxial tension/compression and shear tests on UD laminates. From these tests are obtained the shear stress-strain curve, the transverse compression and transverse tension stress-strain curve. Each curve is divided into a number of strain intervals and for each interval the slope (tangential modulus E2, G12) is calculated, see Figure 8. By that way can be tang tang applied in the program the proper value E2 , G12 according to the strain state of each lamina of the laminate. Below, the theoretical failure envelopes, considering the non linear character of the glass epoxy lamina, together with experimental results are plotted, see Figure 9.
εx /εy - Failure Envelope Non Linear Material First Ply Failure
Tsai_Hahn_FPF Tsai_EPFS_FPF PUCK_FPF Experiments
3
Max_Stress 2
εy (%)
1 0 -3
-2
-1
0
1
2
3
-1 -2 -3 εχ (%)
Figure 9. Failure envelopes, experimentally and theoretically, considering the non-linear response of the lamina. A comparison between the experimental data and failure envelopes obtained from failure criteria considering a totally linear elastic behaviour of the glass epoxy lamina is also presented, see Figure 10.
ε x /ε y - Failure Envelope Linear Material First Ply Failure
Tsai_Hahn_FPF Tsai_EPFS_FPF PUCK_FPF Experiments Max_Stress
3 2
εy (%)
1 0 -3
-2
-1
0
1
2
3
-1 -2 -3 ε χ (% )
Figure10. Failure envelopes, experimentally and theoretically, considering linear and elastic behavior of the material. Conclusions A comparison between experimental data obtained from cruciform and beamlike specimens testing with failure envelopes coming from existing failure criteria expressed in strain space was shown. The non linear behaviour of the glass epoxy lamina under shear stresses and normal stresses perpendicular to the fibre direction has a great influence on the shape of the envelope and has to be taken into account. Furthermore Tsai’s criterion using EPFS instead of Hahn’s interaction term is closer to the rest of the predictions. An important fact is also that for the tested cases none of the predictions is ‘dangerous’ which means that for all the biaxial ratios investigated experiments showed material stronger than the predictions except from the uniaxial compression case where the accurate experimental determination of the ultimate compression strength is questionable. Acknowledgements The authors gratefully acknowledge the support for this research by the Fund for Scientific Research - Flanders (FWO) and express their gratitude to Hans Tommerup Knudsen from LM Glassfiber in Denmark for his effort in producing the cruciform specimens. References [1] Lin WP, Hu HT. Parametric study of failure stresses in fibre reinforced composite laminates subjected to biaxial tensile load. J Comp Mater 2002; 36(12): 1481-1504. [2] Susuki I. Biaxial testing of composite plates using cruciform specimens. Composites Design, Manufacture and Application, ICCM/VIII 1992; 30-39: 37A1-9. [3] Boehler JP, Demmerle S, Koss S. A new direct biaxial testing machine for anisotropic materials. Exp Mech 1994; 34(1): 1-9. [4] Welsh JS, Adams DF. Development of an electromechanical triaxial test facility for composite materials. Exp Mech 2000; 40(3): 312-320. [5] Makinde A, Thibodeau L, Neale KW. Development of an apparatus for biaxial testing using cruciform specimens. Exp Mech 1992; 32(2): 138-144. [6] Hinton MJ, Kaddour AS, Soden PD. Failure Criteria in Fibre Reinforced Polymer Composites: The World-Wide Failure Exercise, A Composite Science and Technology Compendium: Elsevier, 2004 [7] Welsh JS, Adams DF. An experimental investigation of the biaxial strength of IM6/3501-6 carbon/epoxy cross ply laminates using cruciform specimens. Compos Part A- Appl S 2002; 33(6): 829-839. [8] Makris A, Ramault C, Van Hemelrijck D, Clarke A, Williamson C, Gower M, Shaw R, Mera R, Lamkanfi E, Van Paepegem W. A review of biaxial test methods for composites, International Conference on experimental techniques, 2007.
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