Determination of the 3D failure envelope of a composite based on a modified Arcan test device

Composite Structures 131 (2015) 585–593

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Determination of the 3D failure envelope of a composite based on a modified Arcan test device L. Alfonso, A. Uguen, C. Badulescu, J.-Y. Cognard, T. Bonnemains, E. Lolive, N. Carrere ⇑ Laboratoire brestois de mécanique et des systèmes, ENSTA Bretagne/Université de Brest/ENIB/UEB, ENSTA Bretagne, 2 rue François Verny, 29806 Brest Cedex 09, France

a r t i c l e

i n f o

Article history: Available online 20 June 2015 Keywords: Composite laminate Failure Strength analysis Out-of-plane Experimental Model

a b s t r a c t This paper describes a 3D failure criterion identified through Arcan tests, to analyze the behavior of a laminate composite subjected to out-of-plane loadings. The proposed criterion is based on the Hashin’s hypothesis and the interactions between tensile and shear out-of-plane loadings are taken into account. The out-of-plane stresses generated in the composite subjected to an Arcan test are studied using 3D Finite Element calculations in order to determine the stack sequence influence. Using different angles of the loading and different stacking sequences allows the ply to be subjected to complex 3D stress state. Using the experimental results and an inverse identification procedure, it is possible to identify the out-of-plane failure envelope. It is shown that a quadratic failure envelope, which takes into account a decrease of the apparent shear strength in the presence of out-of-plane tensile stress, permits the model to describe in a correct manner the experimental results. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In the last few years, the use of composite laminates and their assemblies has drastically increased in almost all engineering applications, e.g. automotive, aerospace, medical prosthetics and sport devices. In these applications complex 3D loadings are often generated. It could be the case due to the shape of the specimen (for instance curved beam specimen subjected to bending), due to the thickness or to the edge effects [1]. Thus, the analysis of the failure of composite laminates under out-of-plane loadings is necessary in order to ensure the design requirements. It has been shown that a coupled strength and toughness initiation criterion [2] must be used to attain this goal. This approach has recently been used to model the failure of open-hole specimens or the initiation of delamination from the edges [1]. Different tests exist in the literature to identify the toughness: Double Cantilever Beam for the mode I [3], End-Notched Flexure for the mode II [4] and Mixed Mode Bending test for the mixed-mode (mode I + mode II) [5]. These tests make it possible to determine the toughness as a function of the mode mixity and thus to identify a propagation law such as the power-law [6] or the semi-empirical model proposed in [7] (usually known as BK model). ⇑ Corresponding author. E-mail address: [email protected] (N. Carrere). http://dx.doi.org/10.1016/j.compstruct.2015.06.029 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.

If the methodology to determine the evolution of the toughness as a function of the mode mixity is now well established, it is not the case for the strength. Indeed, some standards exist in the literature to identify the out-of-plane strength. For example, the four-point bending test on curved beam for the tensile out-of-plane strength [8–10] and the 3-point bending test on short beam for the out-of-plane shear strength [11]. A complete review of the different tests can be found in [12]. However, there is no consensus among researchers regarding the identification of the out-of-plane strengths under multiaxial loadings. Some authors propose to use a cylindrical specimen subjected to tensile and torsional loadings. The main drawback lies in the manufacturing of the specimens and their representativity as compared with the final application [12]. Other authors propose to use the Arcan test device to subject the butterfly-shaped specimen to multiaxial loadings (with a given ratio of tensile/shear loadings) [13,14]. The main drawback of this approach lies on the manufacturing of the specimen machined from high thickness composite plates that could lead to a high scattering of the results [15]. To overcome this difficulty, Cognard and co-workers [16] have developed a modified Arcan test device. It is based on the use of specimen manufactured from thin composite plates bonded to metallic substrates. It has already been shown that this method allows us to obtain reliable results [17]. In these previous works, the results were presented in terms of macroscopic failure envelope of the laminates. This is the reason why this article is aimed at developing a methodology

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to identify the failure criterion of a UD fiber reinforced polymer matrix composite. 2. Experimental and numerical background 2.1. Material and experimental setup The material under investigation is a carbon/epoxy material whose elastic properties are give in Table 1. The out-of-plane properties are investigated thanks to the modified Arcan test device developed a few years ago by Cognard and co-workers [16]. The principle of this device is given in Fig. 1(a). The specimen is constituted of a composite plate bonded to two metallic substrates (see Fig. 1(b)). The bonded surface measured 50 mm in length and 9.5 mm in width. This specimen is fixed to the Arcan device which allows the analysis of the influence of a wide range of tensile/compression-shear proportional loads using a classic tensile testing machine. By choosing the angle between the normal axis to the specimen and the loading direction, it is possible to subject the specimen to: an out-of-plane tensile loading (with c ¼ 0 ), an out-of-plane shear loading (with c ¼ 90 ), combined out-of-plane tensile/shear loading (for 0 < c < 90 ) and combined out-of-plane compressive/shear loading (for 90 < c). It

Table 1 Properties of the M55J/M18 UD ply measured by CNES (the elastic properties are assumed to be transverse isotropic). Young modulus (GPa) Poisson’s ratio Shear modulus (GPa) Thickness ply (mm)

E1 = 315 E2 = E3 = 6.75 v 12 ¼ v 13 ¼ 0:3 v 23 ¼ 0:4 G12 ¼ G13 ¼ 4:5 G23 ¼ 2:4 0.13

has been shown that the geometry of the specimen and the fixture system of the Arcan device have a great influence on the stress distribution and could lead to stress concentration in the adhesive and in the composite plate. Consequently, both the specimen geometry and the set fixation must be optimized in order to reduce these stress concentration. The current specimen geometry used in this study has been proposed by Cognard et al. [17] and avoids the stress concentration in order to obtain reliable results (thanks to the grooves machined in the specimen see Fig. 1(c)). In the present study, the substrates are made of aluminum (Young modulus Eal ¼ 75 GPa, Poisson’s ratio v al ¼ 0:3). In order to induce failure in the composite, it is necessary for the adhesive used to bond the composite to the substrates to have a strength greater than the out-of-plane strength of the composite. An epoxy Huntsman™ AralditeÒ 420 A/B adhesive has been used to join the aluminium and the composite (Ead ¼ 2 GPa;v ad ¼ 0:3). Different stacking sequences with 8 plies in the composite thickness have been studied (see Table 2). 2.2. Model In order to identify the out-of-plane strength an inverse identification procedure is used. It involves applying on a Finite Element Table 2 List of stacking sequences investigated in the present study. Composite reference

Stacking sequence

A0 A10 A20 A30 A90

UD [0 4 ]s [þ10 þ 10  10  10 ]s [þ20 þ 20  20  20 ]s [þ30 þ 30  30  30 ]s [90 4 ]s

Fig. 1. Principle of the Arcan test (a). Specimen used to test the out-of-plane properties of the composite (b). Detail of the substrate design to reduce the stress concentrations close to the free edge (c).

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model the load corresponding to the failure determined thanks to the experimental test. Only half of the Arcan specimen is modeled with node by node anti-symmetric conditions between the left and the right parts of the bottom face of the composite (see Fig. 2 and Eq. (1)).

(

i;j;r U i;j;l x ¼ U x i;j;r U i;j;l z ¼ U z

ð1Þ

where U xi;j;l and U zi;j;l are respectively the displacement in the x direction and the z direction of the node i; j in the left part of the bottom surface. The displacement in the x direction and the z direction of the node i; j in the right part of the bottom surface are noted U i;j;r x

and U zi;j;r . The displacement in the x and z direction of the nodes of the central line of the bottom face are null. The loads are applied on a Reference Point that is linked to the top surface of the substrates by kinematic couplings. Due to the possible stress concentration near the edges of the substrate at the adhesive/composite interface, a refined mesh is required in these zones. In order to ensure accurate results, 5 elements are used in the thickness of adhesive leading to a size around 20 lm. The size of the elements in the plane ðO; ~ x; ~ yÞ is around 100 lm near the edges and a bias technique is used to increase the size of the elements far from the edges. Finally, in the thickness of the composite, the minimum size of the element (located at the interface with the adhesive) is equal to 20 lm. A

Fig. 2. Half geometry of the Arcan specimen and boundary conditions used in the FE model.

Bias

Bias

Bias

Bias Fig. 3. Typical mesh used in the Finite Element analysis of the Arcan test on a composite plate.

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Fig. 4. Tension and compression failure modes under Hashin’s hypotheses: fiber mode (a), in-plane inter-fiber mode (b) and out-of-plane inter-fiber mode (c).

bias technique is used to gradually increase the size of the element in the thickness of the composite, the maximum size being equal to the thickness of one ply (in this case around 100 lm). A typical mesh is shown in Fig. 3. The elements used are 8-node linear brick elements with reduced integration and hourglass control. The Hashin’s hypotheses [18] are assumed in order to distinguish three failure modes in the UD ply: the fiber mode, the in-plane interfiber mode and the out-of-plane interfiber mode (Fig. 4). Three failure criteria are compared in this article. The first one is the classical maximal stress criterion (see Eq. (1))

8 > < Z c < r33 < Z t jr13 j < SR13 > : jr23 j < SR23

ð2Þ

where Z t and Z c are respectively the tensile and compressive peel strengths, SR13 is the out-of-plane shear strength in the fiber direction and SR23 is the out-of-plane shear strength in the transverse direction. The second criterion is a classical quadratic failure criterion (generalization in 3D of the Hashin’s criterion [18]) 2 f 33

 ¼

r33 Zt

2 þ

s13 SR13

!2 þ

s23 SR23

!2 ¼1

ð3Þ

The final criterion is a quadratic failure criterion with two shape parameters p13 and p23 that represent an increase of the apparent shear strength (reinforcement) in presence of out-of-plane compressive stress or a decrease of the apparent shear strength (weakening) in presence of out-of-plane tensile stress 2

f 33 ¼



r33 Zt

2 þ

r13 SR13 ð1  p13 r33 Þ

!2 þ

r23 SR23 ð1  p23 r33 Þ

!2 ¼1 ð4Þ

This criterion [19] is a 3D generalization of the in-plane criterion proposed in [20]. Under in-plane loadings, it has been shown that   a transverse compression stress r 22 increases the load transfer capability of the fiber/matrix and reduces the micro-damages due to shear loadings ðr12 Þ. On the contrary, a transverse tensile stress  þ r22 increases the micro-damages and reduces the interfiber strengths of the composite [20]. Here, a similar hypothesis is made to take into account the interactions between out-of-plane tensile stress and out-of-plane shear stresses; it is represented by the ð1  p13 r33 Þ and ð1  p23 r33 Þ terms in Eq. (4). These two shape parameters represent at the mesoscopic scale some mechanisms observed on a microscopic scale. Failure under out-of-plane shear is due to the onset of microcracks parallel to the plane of the plies. These microcracks percolate leading to the out-of-plane failure. An out-of-plane tensile stress furthers the percolation of micro-damages and reduces the out-of-plane shear strengths. On

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the contrary, an out-plane compressive stress reduces the percolation of the microcracks leading to an increase of the out-of-plane shear strengths. These criteria necessitate the identification of the out-of-plane

z is the direction perpendicular tp the plane of (a)). The direction ~ the composite plate. The plane (O; ~ x; ~ y) is the plane of the composite plate. Finally, the coordinate system associated to the fiber orientation is noted (~ 1; ~ 2; ~ 3) where ~ 1 is the fiber direction and ~ 3 is the

strength Z t ; SR13 and SR23 . Two more parameters are necessary for the last criterion with reinforcement (parameters p13 and p23 ). In order to achieve this goal, the Arcan test device is used. However, a Finite Element analysis is necessary to determine the stress state in the composite as a function of the stacking sequence and of the angle c that define the loading direction (see Fig. 1).

z see direction perpendicular to the plane of the composite (~ 3 ¼~ Fig. 5b). The bonded surface is such that x 2 ½25 mm;25 mm and y 2 ½4:25 mm; 4:25 mm.

3. Stress analysis in the composite plate during an Arcan test The Arcan device used in this study allows loading a bonded assembly with mixed tensile/shear stresses using a simple tensile testing machine. There are three different coordinate systems in this test. The coordinate system linked to the tensile machine is v ) (see Fig. 1(a), ~ v is the loading direction). The coordinate u; ~ noted (~ x; ~ y; ~ z) (see Fig. 5 system associated to the specimen is referenced (~

Plane of symmetry

Line where the results are plotted

Fig. 8. Location of the line where the results are post-treated.

Fig. 5. Coordinate systems in an Arcan device. Machine coordinate system and specimen coordinate system (a) and ply coordinate system (b).

Plane B-B Plane A-A

Fig. 6. Definition of the planes (A–A) and (B–B) used to plot the results.

0,6 0,4 0,2 0

-30

-20

-10

1 0,8 0,6 0,4 0,2 0

Normalized stress

Normalized stress

1 0,8

0

X(mm)

Fig. 7. Out-of-plane tensile stress

10

20

30

-30

-20

-10

0

X(mm)

10

20

30

rzz in each of the plies in the planes A–A (a) and B–B (b) for an Arcan test with c ¼ 0 .

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Modifying the angle of the loading (c) modifies the ratio between tensile and shear stresses in the specimen:  For c ¼ 0 , the composite plate is subjected to out-of-plane x; ~ y; ~ z) coordinate system. tensile stress rzz in the (~  For c ¼ 90 , the composite plate is subjected to out-of-plane x; ~ y; ~ z) coordinate system. An angle shear stress rxz in the (~ 0 < c < 90 will apply a combination of tensile/shear stresses (rzz ; rxz ).

Fig. 9. Stress r33 ; r23 and c ¼ 90 ((d)–(f)).

 Naturally, it could be possible to have compression/shear loadings when 90 < c < 180 , but in this study only tensile/shear loads are investigated. In order to identify a failure criterion, it is necessary to evaluate the stress state in the coordinate system associated to the ply (the orientation of the plies in the composite laminate is defined by the angle h see Fig. 5(c)). The five stack sequences have been modeled under three different Arcan loadings: tensile (c ¼ 0 ), tensile/shear (c ¼ 45 ) and

r13 (in the coordinate system associated to the fiber direction) along the line defined in Fig. 8 for Arcan tests performed at c ¼ 0 ((a)–(c)) and

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shear (c ¼ 90 ). All stresses are also normalized to the maximal out-of-plane stress. For the sake of simplicity, only the results corresponding to the tensile (c ¼ 0 ) and the shear loading (c ¼ 90 ) are presented. The results obtained for the tensile/shear loading (c ¼ 45 ) are a combination of the two other loading cases. The out-of-plane stress rzz0 in each of the plies in the planes A–A (middle of the specimen) and B–B (edge of the metallic substrates) (see Fig. 6) are plotted in Fig. 7 for an Arcan test with c ¼ 0 for the laminate A30. The results presented in Fig. 7 show that there is no stress concentration near the edges of the specimen (x ¼ 25 mm and x ¼ 25 mm) in the plane A–A. The stress is greater in the middle of the specimen (plane A–A) than near the border of the specimen (plane B–B). These results are consistent with the numerical study of the effect of the beaks on the stress distribution in a specimen subjected to an Arcan test. The stresses are homogeneous in the thickness of the composite in the plane A–A. This is the reason why, in the following, the results will be plotted in the ply located in the plane of symmetry of the composite along the line defined by the plane A–A in this ply (see Fig. 8). The Fig. 9 present the evolution of the stress state for the two elementary loadings along the x0 axis (x 2 ½25 mm; 25 mm) along the line defined in Fig. 8. The results presented in Fig. 9 show that: 1. For all the laminates, for a loading angle c ¼ 0 , the plies are mainly subjected to an out-of-plane stress r33 . 2. For A0 laminate: the coordinate system associated to the loading direction is the same as the coordinate system associated to the material directions. For a loading angle c ¼ 90 the plies are subjected to an out-of-plane shear stress r13 . An angle 0 < c < 90 will apply a combination of tensile/shear stresses (r33 ; r13 ).

3. For A90 laminate: the coordinate system associated to the loadz; ~ 1 ¼~ y and ~ x). For a loading angle ing direction is such (~ 3 ¼~ 2 ¼~  c ¼ 90 the plies are subjected to an out-of-plane shear stress r23 . An angle 0 < c < 90 will apply a combination of tensile/shear stresses (r33 ; r23 ). 4. For the other laminate, for a loading angle c ¼ 90 the plies are subjected to a combination of the out-of-plane shear stresses r23 and r13 . An angle such 0 < c < 90 will apply a combination of tensile/shear stresses (r33 ; r23 and r13 ). The results presented in this section show that for each laminate and loading conditions, the stresses are maximum in the middle of the specimen (x ¼ 0 mm, y ¼ 0 mm in the central ply: see Fig. 8). It means that failure will be initiated in this zone when the stress state is such as the failure criterion is fulfilled. Fig. 10 summarizes the Arcan tests that must be performed to identify the complete 3D envelop. Arcan tests on 0 laminate permit the identification of the (r33 ; r13 ) plane of the failure envelope (see Fig. 10). Arcan tests on 90 laminate permit the identification of the (r33 ; r23 ) plane of the failure envelope (see Fig. 10). Arcan tests on ½hns with 0 < h < 90 permit the identification the other parts of the failure envelope (see Fig. 10).

4. Identification of a failure criterion Six specimens for each stack sequence defined in Table 2 have been tested under three different out-of-plane loadings (tensile,tensile/shear and shear). Fig. 11 shows the post-failure micrography of the A0, A90 and A30 specimens subjected to an out-of-plane tensile loading. The failure is due to delamination in the middle plane of the composite plate for all the stacking sequences and the three different out-of-plane loadings (tensile,tensile/shear and shear). The local stresses (rF33 ; rF13 ; rF23 ) at failure are calculated thanks to the Finite Element model presented in Section 2.2. As explained in the previous section, for the load corresponding to the failure, the stress state in the middle of the specimen (x ¼ 0 on the line defined in Fig. 8) fulfills the failure criterion f:

for F ¼ F R f 33 ðrF33 ðMÞ; rF13 ðMÞ; rF23 ðMÞÞ ¼ 1

Fig. 10. Part of the failure envelope that could be identified as a function of the angle h of the ½hns laminate and of the angle c of the loading direction.

591

ð5Þ

where F is the load applied on the Reference Point defined in Fig. 2, FR the load corresponding to the failure and M the point in the middle of the specimen (defined by x ¼ 0 on the line shown in Fig. 8) where the stresses have been shown to be maximum. This stress state at failure has been calculated for each laminate and each type of loading. The results are shown in Fig. 12. It is worth mentioning that some calculations have been performed using elastic property modified of ±10% around their mean values. The effect of the

Fig. 11. Specimen subjected to out-of-plane tensile loading after failure (a) A0 specimen, (b) A90 specimen and (c) A30 specimen.

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Fig. 12. Failure envelope in the plane ðr33 ; r13 Þ (a), ðr33 ; r23 Þ (b) and ðr13 ; r23 Þ (c). (d) is the 3D representation of the failure criterion.

uncertainties of the elastic properties (see Table 1) on the determination of the strength remains low (less than 1%). The failure envelope is compared to the three failure criteria defined in Section 2.2. The parameters of the failure criterion (i.e. the strength (Z t ; SR13 and SR23 ) involved in the three criteria and the shape parameters (p13 and p23 ) involved in the quadratic failure criterion with reinforcement) are identified in order to fit as well as possible all the experimental results. The results show that the maximum stress criterion overestimates drastically the strength for multiaxial loadings because it neglects the coupling between the stresses. No weakening is observed in the plane (r33 ; r13 ) (the shape parameter p23 could be chosen equal to 0). In the plane (r33 ; r23 ) the quadratic criterion overestimates the strength under multiaxial loadings. The failure criterion with the weakening permits to reproduce in a better manner the experimental results (parameter p13 – 0). 5. Conclusions and discussion The out-of-plane failure criterion of a laminated composite has been investigated in this paper thanks to the use of a modified Arcan test device. A 3D Finite Element model has been developed in order to determine the stress state in the plies. It has been shown that by varying the angle of the loading it is possible to subject the composite plate to combined rzz ; rxz loading. The stress state in the plies in the material axis system is a function of the stacking sequence. Indeed, the rzz ; rxz loading subjected to the composite plate leads to.

 in the 0 plies to r33 ; r23 stresses in the material axis system,  in the 90 plies to r33 ; r13 stresses in the material axis system,  in the h plies (with h – ð0 or 90 Þ) to combined r33 ; r13 ; r23 stresses in the material axis system. Some tests have been performed on a Carbon/Epoxy material with different stacking sequences. Using an inverse identification procedure based on elastic Finite Element calculations, the stress state at failure has been determined for each test. These results allow us to plot the 3D out-of-plane failure envelope of the composite. The failure envelope has been compared to three failure criteria already proposed in the literature. It has been shown that the maximum stress criterion drastically overestimates the strength under combined stresses. A quadratic failure criterion permits to obtain better results even though a reinforcement is observed in the plane (r33 ; r23 ). It is worth mentioning that the results presented in this paper validate the use of a quadratic criterion to describe out-of-plane failure. Only three tests are necessary to identify this criterion (four point bending test for the out-of-plane tensile strength Z t and ILSS tests on two stacking sequence for the out-plane shear strengths SR13 and SR23 ). Tests on other composite materials (glass epoxy and others carbon epoxy materials) are necessary to generalize these results. It will be also interesting to compare the strengths identified using the modified Arcan test device and those obtained using the classical tests. Moreover, it could be interesting to study the effect of the non-linear behavior of the composite and of the adhesive on the stress distribution as proposed in [21].

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