epoxy tubes under combined static loading. Part II: Validation of FEA progressive damage model

Composites Science and Technology 69 (2009) 2248–2255

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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech

Mechanical behavior of glass/epoxy tubes under combined static loading. Part II: Validation of FEA progressive damage model Alexandros E. Antoniou a, Christoph Kensche b, Theodore P. Philippidis a,* a b

Department of Mechanical Engineering and Aeronautics, University of Patras, P.O. Box 1401, GR 26504 Panepistimioupolis, Rio, Greece Hexion Specialty Chemicals Stuttgart GmbH, Am Ostkai 21/22, D-70327 Stuttgart, Germany

a r t i c l e

i n f o

Article history: Received 10 December 2008 Received in revised form 4 June 2009 Accepted 12 June 2009 Available online 18 June 2009 Keywords: B. Non-linear behavior B. Stress–strain curves C. Failure criterion C. Finite element analysis (FEA) C. Damage mechanics

a b s t r a c t Experimental results from a series of biaxial static tests of E-Glass/Epoxy tubular specimens [±45]2, were compared successfully with numerical predictions from thick shell FE calculations. Stress analysis was performed in a progressive damage sense consisting of layer piece-wise linear elastic behavior, simulating lamina anisotropic non-linear constitutive equations, failure mode-dependent criteria and property degradation strategies. The effect of accurate modeling of non-linear shear stress–strain response, dependent on the plane stress field developed, was proved of great importance for the numerical FEA predictions, concerning macroscopic stress–strain response. Ultimate load prediction was influenced more decisively when degradation strategies for the compressive strength along the fiber direction were considered. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In the first part of this work [1], experimental results were presented from a test series on [±45]2 E-Glass/Epoxy tubes under complex loading consisting of axial force, either tensile or compressive and torsion. Tests were conducted in the frame of a research project [2] aiming, amongst other, in simulating complex stress fields encountered in large wind turbine rotor blades. The purpose was twofold; derive design allowable static and fatigue strength values and validate a progressive damage FEA routine developed in parallel. Nevertheless, in this research only static test results were considered and further discussed along with the respective numerical predictions. The numerical tool developed by Philippidis and Antoniou [3] was validated by comparing with experimental results from prismatic coupons of various stacking sequences under axial tensile or compressive, load. Test data, consisting of detailed stress–strain curves, failure patterns and ultimate load values were corroborated satisfactorily by the numerical predictions. The progressive damage FEA routine composed of non-linear material constitutive equations, gradual property degradation and appropriate failure criteria distinguishing between the various failure modes of the building UD ply was implemented in a thick shell formulation of a commercial FE code.

* Corresponding author. Tel.: +30 2610 969450/997235; fax: +30 2610 969417. E-mail address: [email protected] (T.P. Philippidis). 0266-3538/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2009.06.011

Verification of failure theories and progressive damage models by comparing with experimental data from biaxial tube tests was initiated as early as in 1970 by Lantz and Foye [4]. Recently, the World Wide Failure Exercise (WWFE) [5–8] contributed in the assessment of the most well-known failure theories against common experimental results. All model predictions diverged from biaxial experimental data provided by Gl/Ep tubular specimen testing. Discrepancies launched innovative research studies to circumvent the drawbacks. Laurin et al. [9] implementing Hashin failure criteria, introduced a viscoelastic material progressive damage model to predict laminate behavior under different loading rate. Numerical calculations were corroborated satisfactorily by the experimental data especially for tube tests, where most of all other theories in WWFE had difficulties. Knops and Bögle [10] have succeeded in obtaining experimental degradation functions for shear, G12 and transverse Young modulus E2. Their analytical calculations, based on an inhouse developed Classical Lamination Theory code and Puck failure theory enhanced with post-failure assumptions, were in agreement with the WWFE tube test data, predicting accurately the failure load. Despite the plethora of material models comprising both failure theory and property degradation strategy for strength prediction of multilayer laminates, it was decided [2] to implement Puck material model [11] in a FEA user routine for design of composites. The choice was implied by the actual standards [12] and certification rules for wind turbine rotor blades [13], while also WWFE ranking was very favourable (top three selections). As an alternative to this

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60 Exp. Theoretical

-0.025

ðnÞ

E2t E1

de1 þ

m12 E2t ðnÞ de2 1  EE2t1 m212

m m12 E2t E2t ðnÞ ðnÞ dr ¼ de1 þ de2 1  EE2t1 m212 1  EE2t1 m212 1

2 12

ðnÞ 2 ðnÞ

ðnÞ

dr6 ¼ G12t de6

ð2Þ

ðnÞ i

and de are the strain increments in the principal lamina directions. In the above equations, E1 and m12, elastic modulus in the fiber direction and major Poisson ratio respectively, were considered constant throughout the static loading up to failure and equal to 37,950 MPa and 0.278, respectively. To describe tangential transverse Young, E2t, and shear, G12t, modulus evolution, the following non-linear relations established by Richard and Blacklock [17] were implemented by Philippidis and Antoniou [3]:

  n2 n1 þ1 2 dr2 r2 ¼ E o2 1  ; de2 ro2   n6 n1 þ1 6 r6 ¼ Go12 1 

E2t ¼

ro6

G12t ¼

-60

-180

ε2

Fig. 1. Transverse stress–strain response of glass/epoxy UD ply [3].

Table 1 Elastic properties of the E-Glass/Epoxy ply. Eo2 (MPa)

Go12 (MPa)

ðTÞ

15,035

–

ðCÞ

15,262 –

– 5500

roi (MPa) 75 188 67

n2

n6

3

–

2.18 –

– 1.3

Ply failure stresses (in-plane) and strains were given in Table 2 and 3. As usually, symbols X, Y and S denoted strength in the fiber direction, transversely and in-plane shear respectively, while subscripts T and C stand for tension and compression. Indices 1 and 2 in Table 3 denote the fiber and the transverse in-plane direction. 2.2. Material models

ð1Þ

where

E1

0.005

-150

2.1. Lamina mechanical properties

ðnÞ

0 0.000

-0.005

-120

E2t G12t

dr1 ¼

-0.010

-90

E2t

ðn1Þ ðnÞ rðnÞ ¼ ri þ dri ; i ¼ 1; 2; 6 and ðnÞ ¼ load step i

-0.015

-30

2. Constitutive models

All material properties used in this work were reported in optiDAT [2], a public access database. More details on test methods, data reduction and analysis can be found in Refs. [15,16]. To account for non-linear mechanical behavior, incremental stress– strain analysis in the ply level was performed, retaining the validity of the generalized Hooke law for each individual interval (load step). Stress at the nth load increment is given by:

-0.020

30

σ 2 [MPa]

model, mainly with respect to the different post-failure analysis, i.e. ply discount instead of progressive degradation, a theory by Lessard and Shokrieh [14] was also integrated. The actual realization of Puck material model, i.e. non-linear incremental stress–strain relations instead of secant modulus approach [11], implied some modifications in the failure conditions to improve performance and thus, a third new model was also developed [3]. In the present work, the numerical tool developed by Philippidis and Antoniou [3] was further validated by extensive comparison with experimental data from E-Glass/Epoxy tube testing under combined axial and torsion loads. Modified versions of compressive fiber failure criteria were implemented in the algorithm correlating non-linear shear modulus reduction and lamina compressive strength degradation. Processing of the experimental strain data measured in the outer tube surface, presented in the first part of this work [1], revealed interesting trends of the interaction of transverse to the fiber normal stress and shear strain and the associated influence on nonlinear shear stress–strain response. Several such non-linear ply shear properties were implemented in the constitutive material models, investigating their impact on FEA predictions for macroscopic stress–strain response of tubular specimens.

dr6 de6 ð3Þ

The parameters Eo2 ; ro2 ; n2 for the transverse to the fiber elastic behavior differentiate between tension and compression, see also Fig. 1. Numerical values for all the above constants derived through non-linear regression on the experimental data were summarized in Table 1.

Mechanical response of a composite laminate was simulated by implementing a FE procedure with progressive damage mechanics. Numerical predictions were based on a combination of incremental stress–strain analysis, non-linear material properties (experimentally obtained), failure mode-dependent damage onset conditions and associated property degradation strategy. The assembly of the above forms the material constitutive model. Three alternatives as proposed by Philippidis and Antoniou [3] and extensively compared with experimental data from statically loaded moderately thick prismatic coupons, were further validated in this work. A brief description follows. Material model A implements Puck failure criteria [11] and associated progressive stiffness degradation rules. For 2D stress analysis five intra-ply failure modes were assumed, Table 4. Besides failure stress symbols already defined in Section 2.1, all other parameters displayed in the failure conditions of Table 4 are kept with the same symbols used by Puck and Shürmann [11], while the implemented values for the present numerical investigation were summarized in Table 5. Since tangent elasticity instead of secant was formulated in the present FEM routine to describe material non-linearity, some modifications were performed on property degradation strategy. Table 2 Strength properties (MPa). XT

XC

YT

YC

S

776.5

686.0

54.0

167.0

56.0

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A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2248–2255

Table 3 Strains at failure (%).

Table 6 Progressive stiffness degradation model.

e1T

e1C

e2T

e2C

eu6

2.09

1.42

0.42

2.00

3.40

Failure mode ðiþ1Þ

E1

ðiþ1Þ E2 ðiþ1Þ G12 ðiþ1Þ 12

FFT or FFC

More specifically, the degradation factor, g 6 1, multiplying inplane engineering elastic constants, simulating that way growing damage was defined by:

1  gr g¼  n þ gr e2 1 þ c efail  1

m

where e2 stands for the transverse to the fibers strain at the current load step while efail 2 accounts for strain at the failure onset point. The term gr specifies the residual stiffness value after damage (matrix cracking) accumulation reaches its saturation level. A value of gr equal to zero was assumed herein for all failure modes. The parameters c and n of Eq. (4) were tuned in Ref. [3] to achieve optimal stress–strain prediction for a particular multidirectional material system under tensile loading, where matrix cracks accumulation dominates material performance. The obtained parameters for the FEA model were found equal to c = 5 and n = 3. Since then, in all remaining calculations these values were considered to be constant for all laminate lay-ups, coupon geometry and loading conditions. Stiffness degradation scenario, depending on the failure mode as defined by Puck and Shürmann [11] was summarized in Table 6, assuming that calculations take place at the ith load increment. FF stands for fiber fracture either in tension (T) or compression (C) while three distinct modes for intra-ply matrix cracking (or interfibre-fracture) are foreseen by Puck, depending on the stress state. Mode A corresponds to tensile transverse stress, r2 > 0. Modes B and C occur for r2 < 0 and differentiate according to the r2/r6 ratio; higher values trigger mode C. The exponent (fail) designates the respective parameter value at failure. Multiplying by the numerical factor of 1010, equivalent to zeroing the respective modulus was introduced to maintain numerical stability. Model B comprised Lessard and Shokrieh [14] limit theory, Table 7, suggesting five possible failure modes. The degradation module implemented in the numerical procedure was based on the concept of sudden zeroing upon failure onset (ply discount) [14], instead of progressive stiffness degradation, see Table 8.

 E2t

¼ 10

10

 G12t

¼ 10

10

IFFA

IFFB

G12

ðfailÞ

m

ðfailÞ 12

ðiÞ ðiÞE2t ðiÞ ðiÞG12t

g g

ðiþ1Þ

¼ gðiÞG12t

ðiþ1Þ

¼ 1010  E2t

E2

IFFC

ðfailÞ

¼ 10

10

ðiþ1Þ E2 ¼ ðiþ1Þ G12 ¼

ð4Þ

2

ðfailÞ

¼ 1010  E1

ðiþ1Þ G12

ðiÞ

ðfailÞ

¼ 10

10

ðfailÞ

 G12t

Limit conditions for tensile or compressive IFF damage modes shown in Table 7 are in their original forms [14] where linear stress–strain response in the transverse to the fiber direction was assumed. For the material system investigated in this work however, the respective behavior was markedly non-linear as exposed in Section 2.1 and therefore, it could be of interest to investigate the effect of non-linearity in the first term as well of the tensile and compressive IFF failure modes of the above criteria. Such an endeavor will lead however to different failure criteria and will be undertaken in the future. Model C is almost the same with A, i.e. Puck IFF criteria and associated gradual stiffness degradation rules implementing however different conditions for fiber breakage:



r1

2

 2

e6 6 1; r1 P 0 eu6  2  2 r1 e6 C ¼ þ u 6 1; r1 6 0 fEðFFÞ XC e6 T fEðFFÞ ¼

XT

þ

ð5Þ ð6Þ

The second term in the above equations is the ratio of instant shear strain, i.e. at the current load step, and its allowable value, as obtained from ISO [±45]S shear tests. The inclusion of these terms in the above equations was implied by the poor performance of Model A for multidirectional laminates without fibers in the loading directions, i.e. matrix dominated failures. In such cases, IFF damage modes are predicted as initial failure and due to the subsequent stiffness degradation and stress redistribution, only

Table 4 Puck failure criteria under plane stress. Type of failure

Failure mode

Fiber fracture (FF)

Tensile

Limit condition h   i T ¼ X1T r1 þ EEf11 v f 12 mrf  v 12 r2 6 1 fEðFFÞ C fEðFFÞ

compr. Inter-fiber-fracture (IFF)

Mode A tensile

A fEðIFFÞ

Mode B compr.

B fEðIFFÞ

Mode C compr.

C fEðIFFÞ

Condition for validity ½. . . P 0

h   i   ¼ X1C  r1 þ EEf11 v f 12 mrf  v 12 r2  þ ð10e6 Þ2 6 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  2ffi  4

r 2  ðþÞ ðþÞ r2 6 ¼ þ 1  p?jj YST þ p?jj rS2 þ 0:9f EðFFÞ 6 1 S YT ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  4

2   ðÞ 2 þ 0:9f EðFFÞ 6 1 ¼ 1S r6 þ p?jj r2 þ pðÞ ?jj r2 " 2   #   4 2 r6 YC ¼ þ rY C2 61 ðÞ ðr2 Þ þ 0:9f EðFFÞ 2ð1þp?? ÞS

½. . . 6 0

r2 P 0 r2 < 0   RA 0 6 rr26  6 jr?? 6c j r2 < 0   0 6 rr62  6 jrA6c j R??

Table 5 Parameters of Puck failure criteria. ðþÞ

ðÞ

Ef1 (MPa)

vf1

m rf

p?k

p?k

RA??

72,450

0.22

1.3

0.3

0.25

YC ðÞ 2ð1þp?? Þ

ðÞ

p?? qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðÞ 1 1 þ 2p?k YSC  1 2

r6c

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðÞ S 1 þ 2p??

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A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2248–2255 Table 7 Lessard and Shokrieh failure criteria for plane stress conditions. Type of failure

Failure mode

Fiber fracture (FF)

Tensile

Limit condition  2 R e6 r de 6 6 T ¼ rX T1 þ R0eu 61 fEðFFÞ 6

 2

C ¼ rX C1 fEðFFÞ

compr.

Inter-fiber fracture (IFF)

0

6

 2

T ¼ rY T2 fEðIFFÞ

Tensile

 2

C ¼ rY C2 fEðIFFÞ

compr.

R0e6

Table 8 ‘Ply discount’ degradation model. Failure mode ðiþ1Þ

¼ 1010  E1

ðiþ1Þ

¼ 1010  E2t

Fiber tensile failure or fiber compressive failure

E2

ðiþ1Þ G12 ðiþ1Þ 12

m

Fiber–matrix shearing

Matrix tensile failure or matrix compressive failure

ðfailÞ ðfailÞ

10

¼ 10

10

¼ 10

ðfailÞ

 G12t m

ðfailÞ 12

ðiþ1Þ

¼ 1010  G12t

E2

ðiþ1Þ

¼ 1010  E2t

G12

¼ 1010  G12t

G12

ðiþ1Þ

ðfailÞ

ðfailÞ ðfailÞ

r6 de6 r6 de6

r1 6 0

61

r2 P 0

R e6 r6 de6 þ R0eu 61

r2 6 0

þ R0eu

6

0

6

0

stresses in the fiber direction keep increasing along with matrix dominated strains. The use of Eqs. (5) and (6) remedies the situation and although final failure is assigned to fiber damage mode it should be rather considered as a shearing failure. Material highly non-linear shear behavior suggested the use of shear strain ratio instead of shear stress improving the resolution offered in calculating failure effort from a loading step to another, especially when approaching ultimate shear stress and strain. In validating the progressive damage models considered by Philippidis and Antoniou [3] through extensive comparisons of numerical predictions with experimental data from axial tests on prismatic coupons of various stacking sequences, it was shown that mechanical property degradation due to incipient failure could be limited to stiffness reduction solely. No strength degradation was deemed necessary and this was corroborated by the close agreement of theoretical predictions and experimental data. Failure modes were restricted to 2D in-plane patterns for the glass/ epoxy coupons tested and this was also valid for the compressive load cases where no delaminations were observed at least up or close to the maximum test load. Nevertheless, this was not the case in testing the laminated tubes where delamination and local instability effects such as layer buckling affected the laminate strength. This was more pronounced in cases where the outermost layer, [+45], was compressed along the fibers while simultaneously tensile stresses were acting in the transverse direction. An example of the failure pattern is shown in the left picture of Fig. 5 of the accompanying part I [1] where the outermost layer of a tubular specimen is clearly seen to buckle after an extensive delaminated area was formed. None of the aforementioned failure modes could be predicted by the static solution of the shell model due to the constitutive assumptions for the building ply. Instead, a combined 3D static analysis to simulate accurately the through-the-thickness stress

r1 P 0 r1 6 0

61

 2 R e6 r de 6 6 SC ¼ rX C1 þ R0eu 61 fEðIFFÞ

Fiber–matrix shearing

E1

r 6 d e6

Condition for validity

r6 de6

r6 de6

field and a non-linear buckling analysis to reproduce the instability mode would be advisable. Nevertheless, to overcome the inherent difficulty of the proposed methodology to model accurately delamination and buckling, the degradation of static compressive strength along the fibers was adopted as one of the potential mechanisms contributing to the initiation and spread of delaminations. The formulation used was that of Liu and Tsai [18]; a non-linear degradation rule for compressive strength as a result of fiber micro-buckling triggered by in-plane shear modulus reduction. Their approach, due originally to Rosen, see e.g. [19], was expressed by:

X 0C ¼ X C



G12t Go12

q ð7Þ

X 0C being the actual compressive strength of the ply in the fiber direction, used along with the relevant limit conditions, e.g. Eq. (6) while G12t, Go12 the tangent and initial shear moduli. The above equation is valid from the beginning of the loading procedure however its effect is activated as shear non-linearity sets in and even more as failure is encountered reducing further the G12t value according to the degradation strategies. The exponent q was given a value of 0.8 for the present investigation after a trial and error procedure resulting in the best tube strength predictions. 2.3. FEA model The material models considered were implemented in SHELL181 of ANSYS, allowing for user defined constitutive equations through a compatible FORTRAN routine [20]. The thick shell element used is formulated in terms of Mindlin–Reissner shear deformation theory. Single tube geometry, consisting of 1504 elements and 9216 DOF was used for all loading cases, adopting a cylindrical coordinate system to define the boundary conditions, see Fig. 2. All displacements in the radial, hoop and axial directions were restricted in the gripping area attached to the fixed part of the test rig. Also rotations about the hoop and axial directions were constrained. At the other tube end, attached to the moving head of the test rig, radial displacements were constrained. Forces in the axial and hoop directions were applied to simulate tensile or compressive axial load and torsion respectively. Forces were evenly distributed in all nodes of the tab area to reduce numerical artifacts. Independent rotational degrees of freedom about the radial axis, hr, compatible with the formulation of the shell element implemented, were restricted along specimen length, avoiding ANSYS default penalty method that relates them with in-plane displacement components [21]. Tube laminate was modeled as a [±45]2 lay-up, without simulating ply overlap. Coupon thickness fluctuations were taken into account by assuming an average lamina thickness of 0.3 mm.

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A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2248–2255 Table 9 Initial failure predictions, Model B.

Fig. 2. FE model of cylindrical specimen.

3. Results and discussion 3.1. FE failure locus prediction The progressive damage models described in Section 2.2, implemented in the FEA model of Section 2.3 were used to predict ultimate loads at failure of the cylindrical specimens under the various loading conditions shown in Table 2 of Part I [1]. Numerical results were compared to the experimental data in the effective axialshear stress plane, see Fig. 3; theoretical predictions for initial failure were also provided. It is worthwhile mentioning that the term ‘‘initial failure” has the same meaning as ‘‘First Ply Failure” and in the numerical procedure of this work denoted that some layer of some element in the FE model has failed by some specific mode. Subsequently, numerous distinct failure events occur to some layers of the same or different elements of the structure by various failure modes; all these could be termed ‘‘intermediate failures”. Ultimate laminate load, corresponding to ‘‘final failure” or ‘‘Last Ply Failure” was defined by the FE model itself as at a certain load step calculated strain increments were extremely disproportional to the previous strain history. The three FE model implementations were as initially introduced by Philippidis and Antoniou [3] with material properties presented in Section 2.1. Initial failure predictions according to Models A and C indicated for all tubes the same tensile matrix cracking mode (IFFA); since both models implement the same IFF

Tube #

Mode

Ply

01 02 03 06 07 08 09 11 12 13 14 15

IFFT IFFT IFFT IFFT IFFT IFFT IFFT IFFT IFFT IFFC IFFC IFFT

[45] [+45] [+45] [45] [45] [+45] [+45] [+45] [+45] [45] [45] [45]

limit conditions, their predictions coincide. Reminding that the inner layer had a fiber orientation of [45], the lay-up towards the outer surface reads as [45/+45/45/+45]. Then, for all tubes subjected to positive torque, pure or combined with axial load, i.e. #6, 7, 8, 14, 15, failure under IFFA was initiated at the [45] plies developing tensile transverse normal stresses and compressive ones along the fibers. Reversing the applied torque sign, caused failure initiation in the [+45] plies of tubes lying in the 3rd and 4th quadrants of (Nx/h  Ns/h) space; the stress field was similar, i.e. tensile r2 and compressive r1. Initial failure predictions according to Model B, although of similar trends with those previously discussed, presented nevertheless some differentiations. Results for each tube test were presented in Table 9. IFFT and IFFC are in accordance to the matrix failure modes of Table 7. FPF predictions are close to final tube failure when the axial load is compressive while ultimate load failure envelopes of Fig. 3 are optimistic especially in cases with increasing contribution of torsional loading. Discrepancies between test data and numerical predictions for ultimate loads were mostly driven by the incompatibility of FE constitutive modeling and observed failure modes, see Fig. 5 of Part I [1]. As explained in Section 2.2, this was alleviated by implementing in the original models the progressive deterioration of compressive strength in the fiber direction of the building ply. Numerical predictions of all three models with the additional implementation of Rosen concept were plotted in Fig. 4 along with the respective experimental data. Due to higher compressive stresses along the fiber direction, initial failure envelopes were contracted in the 2nd and 3rd quadrants. Model B predictions were

600

Exp.

600

FPF A & C

Exp.

400

FPF A & C

400

Ns /h [MPa]

14

LPF B

200

15

06

07 08

LPF C

0

-150 13

-100

-50

12

0 -200

50

03 02

100

11

150

09

200

Ns /h [MPa]

FPF B

01

LPF A FPF B

01

LPF A

14

-150 13

15

-100

200

-50 12

0

LPF B

06 07

LPF C

08

0

50

-200 03 02

100 11

150

200

09

-400

-400

-600

Nx/h [MPa]

-600

Nx /h [MPa]

Fig. 3. FEA predictions vs. experimental data.

Fig. 4. FEA predictions featuring degradation of longitudinal compressive strength vs. experimental data.

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A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2248–2255

[+45] or [45] under certain external constant biaxial loading ratio is under a different r2/|r6| stress ratio. Furthermore, this ratio in each ply may also present slight variations due to non-linearity. Nevertheless, for this numerical exercise, it was proved that using the exact in-plane shear stress–strain response for each ply had not a major impact on results derived by just using the appropriate material behavior of the outer [+45] ply for that of opposite sign as well. Results from the calculations were presented in Fig. 5, only for Model C to avoid multiple curve overlap. The same trends were also observed for the other two model implementations as well. Nevertheless, as it was also seen in Fig. 4, Model C enhanced with Rosen’s compressive strength degradation concept performs better in the average than the other two and it is suggested by the authors for future use. Concerning initial failure envelopes, FPF0 is similar to that of Fig. 3 derived using Model C in its original form while in FPF3 Rosen’s approach as well as proper experimental shear stress–strain behavior for each tube were implemented. Deviations were observed only in the 2nd and 3rd quadrants although failure modes remained the same for all tubes. The contour labeled C0 (dotted thin line) is in fact the same with that of Model C in Fig. 3 and corresponds to the initial model implementation [3] with properties from Section 2.1. The locus labeled C1 (solid thin line) was derived by performing the FE calculations for each tube at different biaxial loading ratio implementing experimental in-plane shear stress–strain response, as in Fig. 8 of Part I [1]. There is some improvement over the prediction performance of case C0 but this is not as drastic as for the case that degradation of compressive residual strength in the fiber direction is also taken into account. This is done in two steps; for the derivation of locus C2 (dashed thick line) the shear property implemented for all tubes is the same, that of Table 1. In fact, curve C2 is the same with that of Model C in Fig. 4. Finally, in deriving contour C3 (thick solid line), both effects were modeled. In-plane shear response was different for each tube and at the same time degradation of residual compressive strength was taken into account. According to the Rosen model, this degradation is also dependent on the in-plane shear non-linearity, see Eq. (7). The final result for ultimate load prediction by Model C, i.e. contour C3, is considered satisfactory. Nevertheless, discrepancies from predictions derived by modeling just the residual compressive strength reduction, contour C2, are limited for biaxial loading cases where important longitudinal compressive and in-plane shear stress are induced in the plies. However, for pure torsion test,

600 Exp. FPF0

400

FPF3

01

Ns/h [MPa]

mostly affected as a result of compressive strength reduction shifting to matrix shearing failure for tubes #12 to #15. Failure modes remained unchanged for Models A and C. The agreement of FE results and experimental data of Fig. 4 is very good, supporting the reasoning exposed in Section 2.2 and highlighting the effect of compressive strength degradation as a potential trigger mechanism for local instability and delamination onset. Predictions of all material models seem to be drastically affected by the implementation of Rosen degradation concept although Model B performs less successfully than the other two, A and C, based on Puck inter-fiber failure theory and degradation strategy. Nevertheless, this should be attributed to the employed stiffness degradation strategy, i.e. zeroing the shear modulus after initial failure, thus abruptly reducing the value of the compressive strength along the fibers in contrast to the progressive stiffness degradation law implemented in Models A and C. Final failure modes predicted by all three models are more or less in agreement; when loading consists of positive torque, the [45] plies fail under compressive FF and at the final loading steps failure occurs at the [+45] plies by IFF compressive modes. The opposite is valid under negative torque, i.e. the [+45] plies fail under compressive FF and the [45] under compressive IFF. Especially for Models A and C, where Puck matrix failure criteria consider two distinct compressive modes, compressive IFFB failure was predicted in the first quadrant of (Nx/h  Ns/h) space while IFFC prevailed in all other cases. Although they predict same modes of final failure, for load combinations in the first quadrant again, tensile FF is superimposed on IFFB at the [+45] plies according to Model C. The observed deviation of predictions for tube 08 and the associated bulging of failure envelope as per Model A are mostly due to the realization of the progressive algorithm routine. The external [+45] ply fails initially by IFFA while tensile stresses are acting transverse to the fibers. Upon compressive FF at the [45] layer underneath and zeroing of all stiffness at this layer, stress redistribution causes sign reversal at the transverse normal stresses at the [+45] plies which finally fail under compressive mode B. In carrying the compressive transverse stresses, although IFFA failure has preceded, the [+45] ply assumes transverse modulus of elasticity of intact material. This helps in increasing the load carrying capacity of the laminate. FF criteria implemented in Model C, Eqs. (5) and (6) contribute to alleviate this artifact. Tubes loaded under pure torque conform to the above cited final failure descriptions. Positive torque leads to catastrophic IFFC according to Models A and C at the outer [+45] ply while compressive fiber failure was predicted under negative torque; the [45] ply underneath failing with IFFC. Experimental evidence of these failure modes was given in Fig. 5 of [1]. Induced burst type matrix cracks at the outer [+45] layer (IFFC) are clearly seen near the gripped areas of the tube at right. On the left picture, the [45] ply underneath the outer [+45] fails by mode C and this promotes delamination of the exterior layer which further fails by local instability. This is again observed near the grips where the IFFC cracks form an angle of 90° with the fiber direction of the outer ply. Thus, although delamination cannot be strictly predicted by the actually implemented 2D stress and failure analysis, the predicted IFF modes by Models A and C contribute in understanding the mechanisms of the induced failures. The FE results presented in Figs. 3 and 4 were derived using the non-linear shear stress–strain material behavior expressed by Eq. (3) and the numerical values of Table 1. However, in light of the shear behavior dependence on transverse normal stress, presented in Section 4.2 of Part I [1], it was further investigated what the effect would be on ultimate load prediction of using the correct each time shear stress–strain material response, as it was already shown in Fig. 8 of [1]. Caution should be given to the fact that each ply

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there is no shear stress in the plies and thus the effect of the behavior on the final failure prediction is negligible. The predicted failure patterns remain the same as previously discussed.

Axial strain, ex, along the cylinder axis and hoop strain, eh, equivalent to ey with respect to the local laminate coordinate system, see Fig. 3 of Part I [1], were measured for all tests performed under various loading conditions. Detailed comparison of stress– strain curves measured experimentally was performed with respective FEA ones. Results from all FE models and loading conditions suggest an excellent agreement in general between numerical and experimental data. Due to space limitations, only some typical loading conditions were considered with Model C results, implementing Rosen assumption for the degradation of compressive strength. Cylindrical specimens #06 and #11 were subjected to biaxial load consisting of axial tensile force and torsion of opposite sign with the same absolute value of k ratio; see Table 2 of [1]. Experimental results for effective axial normal stress Nx/h vs. axial, ex and hoop, eh, strains were plotted in Fig. 6. Numerical predictions derived initially using FE Model C implementation with in-plane shear stress–strain response as defined by Eq. (3) and Table 1 elastic properties, obtained by testing ISO [±45]S coupons [16], were also included for comparison in the same figure. While the experimental stress–strain data from the two tubes clearly demonstrate a distinct mechanical response, the opposite is observed for the numerical results which overlap in the entire strain range of interest and yield in addition the same ultimate load prediction. Initial failure predictions were also shown as horizontal line marks. The corresponding damage modes were as discussed in the previous section. Furthermore, these FEA data lie in-between the two experimental responses, albeit closer to that for tube #06. Nevertheless, for the initial linear part, up to 30 MPa approximately, both experimental behavior and FE predictions are in good agreement. Discrepancies emerge as non-linearity, in the macroscopic laminate behavior, sets in. The reason for this distinct mechanical response of the tubes under biaxial loading when torsion is reversed was already discussed and analyzed in Sections 3 and 4.2 of [1]. Application of torque in the thin-walled tube generates only normal stresses in the principal coordinate system of each ply. In fact, these stresses are of opposite sign for the [+45] and the [45] layers and as they

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superimpose to the respective stresses generated by the tensile force result finally in different in-plane stress fields in each ply. Further, by reversing the torque sign, plies with fibers of different orientation exchange their stress fields. Then, the stress ratio r2/|r6| in each ply, thus also in the outermost [+45], is different for the loading cases of tube #06 and #11 respectively. As it was clearly shown in Fig. 8 of Part I [1], the inplane shear stress–strain response, especially the non-linear part, is drastically affected by the simultaneous application of transverse normal stress. Therefore, the tube mechanical response is altered. Improved numerical results for macroscopic stress–strain tube response were obtained when replacing the in-plane shear mechanical properties in the FE models with stress–strain curves measured in the outermost layer of tubular specimens during the static tests, i.e. those displayed in Fig. 8 of [1]. FE results were plotted again with the test data for comparison purposes in Fig. 7. As it is seen this time the agreement between the two sets of data is very good. A similar example for tubes #13 and #14, loaded by compressive force and torsion, is presented in Figs. 8 and 9. Again, the implementation of accurate non-linear shear stress–strain response has a major impact in the FEA predictions. According to the aforementioned discussion and the analysis presented in Section 3 of [1], subsequent layers should possess dissimilar shear properties since different r2/|r6| ratios were

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of a ply can improve strength predictions. A non-linear version of the Rosen model relating compressive strength reduction with that of the in-plane shear modulus was implemented for this purpose.

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Research was funded in part by the European Commission in the framework of the research programmes ‘‘Reliable Optimal Use of Materials for Wind Turbine Rotor Blades” (OPTIMAT BLADES), Contract No.: ENK6-CT-2001-00552 and ‘‘Integrated Wind Turbine Design (UPWIND), Contract No.: 019945, SES6. Partial funding was also provided by the General Secretariat of Research and Technology (GSRT) of the Greek Ministry of Development, Contract Nos. F.K. 6660 and C037.

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Fig. 9. Effect of accurate in-plane shear non-linear elastic behavior in numerical predictions for the macroscopic mechanical response of tubes #13 and 14.

developed. Implementation of different in-plane shear response for each ply, depending on the r2/|r6| ratio, resulted in slight improvement, as e.g. in the case of tubes #13 and 14 of Fig. 9, or had no effect, in most numerical predictions with respect to the experimental data, for all tube tests, deteriorating slightly however the predictions for tubes #06 and #11 are shown in Fig. 7. 4. Conclusions An exhaustive comparison of experimental results from biaxial loading tests of tubular specimens and FEA numerical predictions was performed to validate three material constitutive models, featuring progressive damage algorithms, implemented in a commercial shell element formulation. Model C numerical predictions were slightly closer to the experimental data, especially with concern to ultimate load values. It is based on Puck IFF failure functions although modified fiber failure criteria were implemented in the FE user routine. In addition, degradation of fiber compressive strength in a damage progressive sense was also considered. Numerical simulation provided accurate results for ultimate loads and very good agreement with detailed strain gauge data when the effect of transverse normal stress on in-plane shear ply response was taken into account. It has to be clarified that this was done indirectly, by using appropriate experimental data, and not by proposing and implementing a constitutive model fit to purpose. In addition, the companion effect, i.e. variation of the transverse elastic modulus due to superimposed in-plane shear, was not taken into account due to lack of test data. Further research on this issue, both theoretical and experimental by testing tubes with fibers oriented only in the hoop direction under biaxial loading is highly recommended. It was further evidenced for the laminated tube considered in this work that in case of failure mechanisms such as elastic instability and delamination onset, incompatible with stress–strain fields derived by shell constitutive formulation, adoption of progressive degradation of compressive strength in the fiber direction

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