Scaling effects of composite laminates under out-of-plane loading

Composites Part A 116 (2019) 1–12

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Scaling effects of composite laminates under out-of-plane loading A. Wagih a b

a,b,⁎

a

a

, P. Maimí , N. Blanco , E.V. González

T

a

AMADE, Polytechnic School, University of Girona, M. Aurèlia Capmany 61, 17003 Girona, Spain Mechanical Design and Production Dept., Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Sharkia, Egypt

A R T I C LE I N FO

A B S T R A C T

Keywords: A. Polymer-matrix composites (PMCs) B. Impact behaviour C. Analytical modelling D. Failure

The scaling effects of composite laminates under quasi-static indentation and low-velocity impact tests are studied with the aim of reducing the experimental cost of low-velocity impact tests. First, an analytical model is proposed using some equations available in the simplified analytical models in the literature and other equations derived in this work to predict the quasi-static indentation response during the elastic and delamination regime up to fiber failure. The model is able to predict the load-displacement response and delamination area. Finally, the effect of scaling the geometrical parameters such as in-plane dimension and shape, thickness of the laminate, thickness of the plies and indenter radius is analyzed. This scaling approach is developed based on the presented analytical model and the experimental results available in the literature. As a result, it is possible to predict the response of large structures under impact loads by testing small coupons under static indentation test.

1. Introduction The use of composite laminates by the aeronautic industry has increased significantly in the last years. The use of these materials by the largest civil aircraft companies is a real example of this increase. Additionally, the automotive industry has recently started to use this type of material in the structure of the new generation of electric cars to reduce weight and, therefore, energy consumption. Among the different types of composite materials, carbon fiber reinforced polymers (CFRP), especially those with epoxy matrix, are the most important and common composite laminates due to their high specific mechanical properties [1–3]. In spite of these advantages, one of the main disadvantages of these materials is the low resistance against impact, which may affect their load capacity performance [1]. To cope with this problem, composite structures are generally over-designed with a high safety factor [4], which results in material waste and weight increase. The first studies on low-velocity impact of composite materials appeared during the 70s, such as the work introduced by Kelkar et al. [5], Bostaf et al. [6] and Caprino et al. [7], in which the parameters governing the low-velocity impact test have been defined. Also, Cantwell et al. [8–10] anlaysed the influence of different parameters (e.g. laminate thickness, stacking sequence, impact velocity, and impactor shape) on the damage extension in CFRP laminates. In the 90s, more studies were done not only on the impact behaviour of composite laminates but also, on the residual strength after impact [11–20], which is used to define design allowables in the aeronautic industry. Apart from these

⁎

works, it is important to mention the work of Kwon et al. [12], in which a large experimental campaign on low-velocity impact and quasi-static indentation (QSI) tests was carried out. The effects of impactor radius, in-plane dimensions, and mismatch angle were studied. In the last two decades, most of the experimental works related to low velocity impacts were used to validate numerical models [21–26] and analytical models [27–30] to predict damage behavior. In order to get meaningful results from impact tests, a large test campaign should be carried out including the analysis of all the parameters that affect this test. Apart from the fact that such a campaign would be really expensive and time consuming, the test itself is complex and there are a large number of related standards, which makes difficult to obtain a true agreement between researchers. Moreover, low-velocity impact tests do not allow observing the succession and evolution of the degradation mechanisms within the plate since it can only be inspected upon the completion of impact testing. Fortunately, and according to the limits established by Swanson [31], Bucinell et al. [32], Olsson [29] and Yigit and Christoforou [33], low-velocity impacts caused by large masses can be treated as a static indentation problem because the impact duration is much longer than the time required by the propagating waves to travel from the impact site to the supports or free edges [29]. According to Swanson [31], the problem can be considered as quasi-static if the impactor mass is more than ten times larger than the lumped mass of the plate. For this reason, some researchers use static indentation tests to qualitatively elucidate the damage induced during low-velocity impact events [12,34–38].

Corresponding author at: AMADE, Polytechnic School, University of Girona, M. Aurèlia Capmany 61, 17003 Girona, Spain. E-mail address: [email protected] (A. Wagih).

https://doi.org/10.1016/j.compositesa.2018.10.001 Received 29 May 2018; Received in revised form 6 September 2018; Accepted 2 October 2018 Available online 10 October 2018 1359-835X/ © 2018 Elsevier Ltd. All rights reserved.

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termed stage II. However, in other cases this load drop is not present and the damage threshold load is equal to the load after the damage initiation (Fth = Fdn ) [37]. The third stage is related to the delamination growth. During this stage, stable delaminations grow between the interfaces of the laminate. The final stage is defined as the fiber breakage stage which corresponds to a large load drop in the load displacement curve.

Since these tests are static tests, they can be easily interrupted at different stages to observe the damage evolution within the plate. Moreover, they are easier to be carried out, require simpler test devices and less number of specimens. Scaling is an important aspect of every physical theory [39] and scaling laws are commonly used in many physical and engineering problems. A historical review of scaling was introduced by Bažant et al. [39]. Moreover, understanding scaling laws allows to extrapolate testing results to large scale real structures. The scaling technique has two important advantages: (i) reduce the experimental cost by testing a specimen with smaller in-plane and out-of-plane dimensions and (ii) increase the accuracy of analytical models by considering an experimental result as an input of the model. Although several researchers [40–48] have paid attention to scaling of composite laminates under inplane loading, applying this idea for specimens under out-of-plane loading is a complex task even when considering only a linear elastic response. Morton [49] applied the classical scaling laws for the elastic behaviour (undamaged) of transversely impacted carbon-fiber beams. Qian et al. [50] developed scaling laws for the elastic response (undamaged) of impacted composite laminates. In their study, an experimental analysis was carried out by increasing the initial in-plane size by a factor of five to validate their laws. They also considered that the damage process in composite laminates is too complex so the damage part of the response cannot be scaled with analytical laws. Sankar [51] provided a method for nondimensionalizing the impact problem. In this study, the impact problem of a plate was defined by five dimensionless parameters. Although good agreement was found between these laws and the experimental results, their use is limited to predict the elastic response and the maximum impact force. Liu et al. [52] have studied the size effects on impact experiments. In their study, the influence of the laminate thickness and the in-plane dimension was studied. They concluded that the thickness effect is much more significant than the inplane dimensional effect. Recently, Abisset et al. [53] have done an attempt to give some preliminary interpretations based on simplified analytical models of the main scaling effects by observing experimental Quasi-Static Indentations (QSI). Having a look to the parameters that affect the QSI test, the in-plane dimension is the most effective parameter to be scaled to reduce the size of the tested specimens reducing testing and computational costs. From this short review it can be concluded that the available scaling laws are only valid to scale the elastic response of the impact test and the maximum impact force. However, there is a lack of knowledge in the scaling of the damage threshold load, the response of the damaged laminates and the damage size. The present work is an effort in this direction. In this paper, a scaling approach based on a simplified analytical model and experimental results available in the literature is presented for scaling quasi-isotropic composite laminate specimens under out-ofplane loading. The analytical model used in this study is developed using some equations already available in the literature in combination with others proposed in this work. So, the aim of this work can be summarized as: analytical prediction of QSI response and the use of a small scale QSI test on a quasi-isotropic composite laminate to predict the response of large scale QSI and low-velocity impact tests.

2.1. Elastic response Considering the simplified mass structural model, Fig. 1 (c), the plate response of a composite laminate of thickness h and circular inplane dimension of radius R indented by hemispherical indenter of radius r, can be defined by an approximate solution as [55]:

F = Ku + Km u3

(1)

where K is the linear part of the stiffness (including the bending and shear stiffness) and Km is the membrane stiffness. The linear part of the stiffness can be calculated by the combination of the bending and the shear compliances which can be expressed as:

K = (Kb−1 + K s−1)−1

(2)

h3

h

where Kb = kb R2 , K s = k s h and Km = k m 2 , being kb and k m two conR stants that depend on the elastic properties, the geometry, and the boundary conditions, but not on the specimen size. Table 1 shows the expression for computing kb and k m for different boundary conditions [56]. k s depends on the material properties and the geometry and can be computed as: −1

ks =

4πGxz Ex ⎛⎜ 4 + log ⎛ R ⎞ ⎞⎟ 3 Ex−4νxz Gxz ⎝ 3 ⎝ rc ⎠ ⎠ ⎜

⎟

(3)

where Ex , Gxz and νxz are the elastic properties of the quasi-isotropic laminate (subscripts x and z refer to the in-plane and out-of-plane directions, respectively) and rc is the contact radius between the impactor and the plate [56]. With the aim of simplifying the calculation, rc is considered as a constant value as suggested by Shivakumar et al. [56]. For rectangular or square laminate plates with in-plane dimensions β × γ , the value of R can be computed as βγ / π [57]. The presence of n equally distributed circular delaminations of radius a produces a decrease of bending stiffness. In this case the bending stiffness can be expressed as [58]:

Kb−1 =

R2 a2 a2 (n + 1)2 1 R2 n (n + 2) a2 ⎞ − + = 3⎛ + 3 3 3 kb h kbc h kbc h h ⎝ kb kbc ⎠ ⎜

⎟

(4)

where kbc is the bending stiffness of a circular clamped plate. The first term represents the compliance of the plate of in-plane radius R without delamination, the second term represents the compliance of a circular plate of radius a without delamination and the last term represents the compliance of a circular plate of radius a with n equally distributed through the thickness circular delaminations. Consequently, by substituting Eqs. (2) and (4) in Eq. (1), the elastic load-displacement response assuming n delaminations of radius a can be expressed as:

2. Quasi-static indentation model

−1

The plate response or load displacement response (F-u) of a composite laminate subjected to quasi-static indentation load can be defined by four different stages as shown in Fig. 1 (a) and represented by a simplified schematic drawing as shown in Fig. 1 (b) [36,37]. The first stage is the elastic response of the laminate which corresponds to the bending and contact responses. It is well known that the bending response depends on the in-plane dimensions. However, the contact response is independent of the in-plane dimensions [54]. At a certain load (Fth ) damage onset is reached. In most laminates, it corresponds to a load drop in the load displacement response until Fdn [36], this is

F=

h3 ⎛ 1 n (n + 2) a2 1 h2 ⎞ h + + u + k m 2 u3 2 2 R ⎝ kb kbc R k s R2 ⎠ R ⎜

⎟

(5)

2.2. Delamination growth The load displacement curve when delaminations are growing can be determined by means of linear elastic fracture mechanics. The elastic energy (U) can be expressed as the area under the unloading curve and can be defined as: 2

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Fig. 1. Representation of the mechanical response and delamination area of a QSI test (a) the associated damage mechanisms (b) and a simplified mechanical model (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) −1

3

U=

∫ Fdu = 2hR2 ⎛ k1b ⎜

⎝

+

n (n + 2) a2 1 h2 ⎞ 2 k h + u + m 2 u4 2 kbc R k s R2 ⎠ 4R

Fdn/(2nG IIc ) . The response of Eqs. (5) and (9) is equivalent to the model presented by Olsson [28]. A key point in the proposed model to define the load Fdn and the delamination area is to define the number of the delaminated interfaces, n.

⎟

(6)

−1 ∂U

Applying linear elastic fracture mechanics, G = 2πan ∂a = G IIc , the relation between the applied displacement and delamination radius during crack growth is governed by:

u = Fdn

R2 ⎛ 1 n (n + 2) a2 1 h2 ⎞ + + h3 ⎝ kb kbc R2 k s R2 ⎠ ⎜

2.3. Onset of delamination

⎟

(7) During a QSI or impact test the load required for the damage threshold (Fth ) is usually larger than the load after damage initiation (Fdn ), promoting an unstable behavior with a sudden load drop. According to Shivakumar and Elber [61] delaminations are initiated if the transverse shear stress is larger than the critical shear strength and these delaminations grow if the energy release rate is larger than the mode II fracture toughness of the interface. The strength criterion to define the onset of cracking or delamination for hemispherical indenters is formulated as [57,62]:

where

Fdn =

2πG IIc kbc h3 n+2

(8)

where G IIc is the mode II interlaminar fracture toughness [28,59,60]. So, taking into account Eqs. (5) and (7), the load-displacement response when delaminations propagate can be expressed as:

F = Fdn + k m

h 3 u R2

(9)

FS =

Eq. (5) defines the elastic response of the plate and it is valid for no delamination and constant radius delamination (black lines in Fig. 1(a)). When delaminations grow, the load displacement is defined by Eq. (9) (blue line in Fig. 1(a)). Eq. (7) defines the maximum plate displacement for a given delamination size and G IIc value. Therefore, it is possible to define the projected delaminated area for a specific plate displacement as:

πa2 =

Fdn 1 1 h2 ⎞ R2 u−⎛ + Fdn 2nG IIc ⎝ kb k s R2 ⎠ h3 ⎜

κτs3 π 3h3r E

(11)

where τs is the interlaminar shear strength of the laminate and E is the 1

1 − ν2

1−ν2

contact modulus which can be computed as E = E 1 + E 2 , where 1 2 subscripts 1 and 2 refer to the indenter and the laminate, respectively. κ is a constant that depends on the stress distribution considered, parabolic: κ = 16/9 [57] or constant κ = 6 [62]. The process of delamination grow is defined by means of linear elastic fracture mechanics. The growth of delaminations approximately follows the sequence: a single delamination starts to grow in the middle of the laminate dividing it into two sublaminates. After this delamination, two delaminations grow at the middle of the two resulting

⎟

(10)

where < •> are the Macauley brackets. According to this, the delamination area increases proportionally to the displacement with a slope of

Table 1 Determination of the kb and k m constants for different boundary conditions [56]. In the table B C refers to boundary condition, C refers to clamped support, S refers to simply supported, M refers to movable support and I refers to immovable support. x is the in-plane direction. BC

kb

km

C-M

4πEx 3(1 − ν x2)

191πEx 648

C-I

4πEx 3(1 − ν x2)

πEx 353 − 191ν x 1 − νx 648

S-M

4πEx 3(1 − ν x )(3 + ν x )

S-I

4πEx 3(1 − ν x )(3 + ν x )

πEx (3 + ν x )4

SM km +

3

(

191 (1 648

+ νx ) 4 +

2πEx (1 − ν x )(3 + ν x )4

(

1 (1 8

41 (1 27

+ νx

+ νx )3 +

)4

32 (1 9

+ (1 + νx

)3

+ νx ) 2 +

+ 4(1 + νx

40 (1 9

)2

+ νx ) +

8 3

) )

+ 8(1 + νx ) + 8

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3.1. Thickness effect

sublaminates and this phenomenon is repeated until Fdn is reached. Applying Eq. (8) with n = 1 the load required to growth the first delamination is defined as [58,63]:

Fd1 =

2πG IIc kbc h3 3

In general, a specific thickness of a laminate can be achieved by the repetition of a group of plies, or sublaminate, with specific orientations and keeping the thickness of the individual plies constant, sublaminate repetition, or by grouping or clustering different plies with the same orientation and increasing the ply thickness. Such a procedure is commonly referred as ply clustering. In the first case, sublaminate clustering, the ply thickness (tp ) is kept constant while the number of interfaces for delamination (N = h/ tp−2 , for symmetric laminates) is increased. In the second case, ply clustering, the ply thickness is increased but the number of interfaces for delamination remains constant. The values of the damage threshold load (Fth ) and the load after damage initiation (Fdn ) of the laminates with different thickness and different ply thicknesses used to generate the figures in this section are summarized in Table 3.

(12)

It must be pointed out that the indenter radius must influence the energy release rate somehow. The numerical models of Shivakumar and Elber [61] showed that the energy release rate associated to delaminations smaller than the contact radius is very small. This suggests that a certain level of shear stress is necessary to produce matrix and interface damage large enough for delamination growth. Therefore, a criterion for delamination threshold can be expressed as: (13)

Fth = max{Fd1, FS}

3.1.1. Damage threshold load (Fth ) The damage threshold load (Fth ) defines the maximum elastic load before delamination initiation. This load is defined in Eq. (13) as the maximum of two criterion, a strength criterion (Eq. (11)) and a fracture mechanics criterion (Eq. (12)). According to both conditions the threshold load scales with laminate thickness as Fth ∝ h1.5. In Fig. 2(a) several experimental results from bibliography (summarized in Table 2) confirm this scaling law, this trend is also validated by the experimental observations reported by Schoeppner and Abrate [64]. It must be pointed out that the results presented in Fig. 2(a) are from the same material and the thickness is increased by means of sublaminate scaling (constant tp ). In Fig. 2(b) the effect of ply thickness is shown for different values of the ply thickness (tp ) keeping constant laminate thickness (h). As it can be seen in the figure, the damage threshold load decreases with respect to the ply thickness.

2.4. Energy balance The elastic energy is defined by means of Eq. (6). In the elastic region, when the displacement value is smaller than uth = Fth/ K (with the stiffness defined in Eq. (2)), the total energy of the system is elastic and determined by Eq. (6) with a = 0 . After the threshold displacement is reached the total energy is the contribution of the elastic energy (UE ), the dissipated energy by delamination (UD ) and in the case, Fth > Fdn , some energy that is assumed to be dissipated as elastic waves (UK ). The dissipated energy UD , can be computed by multiplying Eq. (10) by nG IIc while UK can be computed as the energy released in the unstable drop of loads between Fth and Fdn . Therefore, the total energy can be expressed as: UT = UE + UD + UK , where:

Fdn k h u + m 2 u4 , 2 4R 2 Fth−Fdn = 2K

UE =

UD =

Fdn F ⎛u− dn ⎞ = πa2nG IIc 2 ⎝ K ⎠

and UK

3.1.2. Load after damage initiation (Fdn ) and number of delaminated interfaces The variation of the load after the damage initiation load (Fdn ) is characterized by the growth of n equally distributed through-thethickness circular delaminations. Based on LEFM analysis (Eq. (8)), this load is proportional to the laminate thickness and the number of delaminated interfaces according to Fdn ∝ h3/(n + 2) . In Fig. 3 the effect of laminate thickness on Fdn is shown. In all the cases the laminate thickness is increased by means of sublaminate scaling, i.e. the number of interfaces is increased while tp is kept constant. The experimental results presented show a linear trend in a bilogarithmic plot with a slope close to 1.5, this suggests a null influence of the number of interfaces available to delamination growth, i.e. n is constant and independent on N. The projected delamination area of all interfaces is usually quite circular in quasi-isotropic laminates. When each delaminated interface is analyzed, it can be observed that the shape of delamination is nearly elliptical, with an aspect ratio of approximately 1/3 and oriented according to the directions defined by the fibers in the bottom and top plies of the interface. In Fig. 4 the C-scan of a statically indented quasiisotropic laminate by Wagih et al. [36] is presented. While the projected delamination area is circular, the shape of each interface is oriented in the direction of the fiber. Due to this experimental evidence,Olsson [27,28] suggested that the number of equivalent delaminations is about one third of the interfaces between plies: n = N /3. When considering this hypothesis the scaling law for delamination growth is expressed as: Fdn ∝ h 3tp/(1 + 4tp/ h) . In Fig. 3(b) some experimental results of Fig. 3(a) are reproduced again. The dotted lines correspond to the hypothesis n = N /3, while the solid lines correspond to n constant and independent on N. For the experimental results E2 from Evci [65] the best possible fit is defined while for the results G1 [66] the material properties Hexply AS4/8552 of Table 4 are used. By considering the clamped bending stiffness of Table 1

(14)

3. Scaling effects Because of the complexity of the impact problem, experimental data are needed to determine the extent of impact damage in particular material systems and structural geometries. Scaling of impact is of particular concern in the design of large composite structures, where the cost and the difficulty of full-scale impact testing is, in general, prohibitive. The emphasis in this section is on establishing scaling laws for QSI and impact tests. As a preliminary step for scaling, it is mandatory to identify the geometrical parameters that influence the QSI and low-velocity impact response. The geometrical parameters that affect the response are the in-plane shape and size, the laminate and ply thicknesses and the indenter radius. In QSI and low-velocity impact tests, the response can be summarized as: the initial stiffness, the damage threshold load (Fth ), the load after damage initiation (Fdn ) and the damage size (πa2 ) (see Fig. 1(a)). In the following subsections the effect of each of the aforementioned geometrical parameters on each part of the QSI response will be studied to obtain a clear picture of the scaling laws involved. For a better understanding of the influence of the aforementioned geometrical parameters (specimen in-plane dimension, laminate and lamina thicknesses and indenter radius) different experimental results available in the literature are analyzed to highlight the influence of each parameter. Table 2 summarizes the experimental tests used in the present study. In the case of G1, G2, S1, W1 and W2, the tests campaigns were carried out by the authors of this work after the material was fully characterized. 4

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Table 2 Summary of the parameters studied in experimental tests in the literature. In the table, for case K3, π/8 stands for [0/22.5/45/67.5/90/ −67.5/ −45/ −22.5]2s and LVI refers to low-velocity impact test. Label

Ref.

Staking sequence

Test

R (mm)

h (mm)

r (mm)

tp (mm)

A1 A2 E1 E2 G1 G2 G3 K1 K2 K3 S1 S2 W1 W2 Y1 Y2

[53] [53] [65] [65] [66] [26] [73] [12] [12] [12] [70] [72] [36] [37] [71] [71]

[45/0/ −45/90]ns [45/0/ −45/90]ns Woven GFRP Unidirectional GFRP [45/0/ −45/90]ns [45m /0m/ −45m /90m]ns [02/902/(45/ −45)2]s [0/45/90/ −45]4s [0/90]8s Lam. π/8 [45/0/ −45/90]3s Woven [45/0/ −45/90]3s [(45/ −45)/(0/90)]ns [0/90]ns [0/90]ns

QSI QSI LVI LVI LVI LVI LVI LVI and QSI LVI and QSI LVI and QSI LVI LVI QSI QSI LVI LVI

150 × 100 150 × 100 100 × 100 100 × 100 150 × 100 150 × 100 45 and 150 × 100 25.4, 38.1 and 50.8 25.4, 38.1 and 50.8 25.4, 38.1 and 50.8 150 × 100 25 and 50 25 150 × 100 25, 50, 75 and 100 50

2 and 4 4 2.1, 4, 5.9 and 8.1 2.2, 4, 6.1 and 8 3, 5.8 and 8.8 5.8 2.75 4 4 4 4.4 3.3, 6.6, 9.8 and 12.4 4.4 4.5 1.8, 2.5, 2.7 and 3.6 2.7 and 3.6

8 8 5 5 8 8 6.35 3.175 and 12.7 3.175 and 12.7 3.175 and 12.7 8 5 6.35 8 5 2.3, 5, 6, 7.2 and 9.6

0.125 0.125 and 0.25 ≈ 0.59–0.7 ≈ 0.43–0.45 0.181 0.181, 0.36 and 0.724 0.172 0.125 0.125 0.125 0.184 0.66 0.184 0.08 and 0.16 0.225 0.225

In Fig. 5(a) the response of the impacted specimens G1 with different laminate thicknesses (h = 3, 5.8 and 8.8 mm) and constant ply thickness (t = 0.181 mm) are shown together with the model predictions considering; a constant number of delaminated interfaces (n = 5) and n = N /3. The response considering an increasing number of delaminated interfaces (n = N /3) tends to underpredict the load to delamination growth when the laminate thickness increases. The set of experiments G2 in Table 2 have the same laminate thickness (h = 5.8 mm) with three different ply thicknesses: tp = 0.181, 0.36 and 0.724 mm. The experimental results with the model predictions are represented in Fig. 5(b). For ply thicknesses of tp = 0.181 and 0.36 mm, the global load-displacement response is almost the same except for the threshold load. For the thicker ply (tp = 0.724 mm) the response after damage onset follows almost a linear response that does not fit to the model presented. This laminate has only eight plies which are not enough to fulfill the hypothesis of the model, furthermore they present extensive matrix cracking and the delamination shape is no longer circular but they are driven by the presence of the matrix cracking. For the ply thicknesses of tp = 0.181 and 0.36 mm, Fdn is independent of the ply thickness and the number of available interfaces for delamination (N). The same conclusions can be obtained by means of the experimental results presented by Abisset et al. [53] and Wagih et al. [37] (A2 and W2 in Table 2).

Table 3 The damage threshold load (Fth ) and the load after damage initiation (Fdn ) of the laminates with different thickness and different ply thicknesses. Label

h or tp (mm)

Fth (N)

Fdn (N)

G1 [66]

h = 3.0 h = 5.8 h = 8.8 h = 2.0 h = 4.0 h = 1.8 h = 2.5 h = 2.7 h = 3.6 h = 3.3 h = 6.6 h = 9.8 h = 6.6 h = 9.7 h = 12.4 h = 2.1 h=4 h = 5.9 h = 8.1 h = 2.2 h = 4.0 h = 6.1 h = 8.0 tp = 0.181

3800 10000 18870 1700 4750 1424 2109 2383 3644 1469 4659 8904 4295 8044 12816 576 1930 3644 5590 530 1543 2835 4056 9910

2600 5100 9521 1636 3930 – – – – – – – – – – 544 1705 3420 4813 489 1470 2698 3966 –

tp = 0.36 tp = 0.725

7442

–

5983

–

A2 [53]

tp = 0.125

4830

–

3920

–

W2 [37]

tp = 0.25 tp = 0.08

5100

–

tp = 0.16

4180

–

A1 [53] Y1-All [71]

S2-R25 [72]

S2-R50 [72]

E1 [65]

E2 [65]

G2 [26]

3.1.3. Projected delamination area According to the model, the projected delamination area increases linearly with the applied displacement as defined in Eq. (10). In Fig. 6, the projected delamination area for test results G1 and G2 are represented considering two different values for the delaminated interfaces: n = 5 and n = N /3. The effect of the laminate thickness is shown in Fig. 6(a), the constant number of delaminations n = 5 shows a very good correlation with the experimental results while the hypothesis n = N /3 tends to underpredict the response. The delamination area for the thinner laminate are collected beyond the region of delamination growth and there are significative fiber damage. It is valuable noting that according to Eq. (10) and the experimental results shown in Fig. 6(a), the slope of the delaminated area-displacement curve (slope of the black solid line for laminate with thickness 5.8 mm and the slope of the blue solid line for laminate with thickness 8.8 mm) scales with the laminate thickness according to h1.5 . In Fig. 6(b), the effect of ply thickness is shown. For tp = 0.181 and 0.36 mm, the delamination area is quite similar (a little smaller for the thinner ply). For the thicker ply the delamination area has no longer a circular shape and the model described is not applicable because the main dominant damage mechanism is matrix cracking, which dissipates the largest portion of

(kbc = 226.1 GPa) and n = 5, the prediction is significantly better than considering n = N /3 = 4.7, 10 and 15.3 from the thinner to the thicker. To clarify the scaling of number of delaminated interfaces the load displacement presented by González et al. [26,66] and summarized as G1 and G2 in Table 2 are represented in Fig. 5. All experiments are LVI tests performed according the ASTM D7136 [67] specifications. The specimen with dimensions 150 × 100 mm is placed over a flat support with a 125 × 75 mm rectangular cut-out, with corresponds to an equivalent radius of R = 125 × 75/ π = 54.63 mm. The shear stiffness defined by Eq. (3) and the bending stiffness of Table 1 for simply supported plate are ks ≈ 6.6 GPa (for rc = r/2 ) and kb = 90.29 GPa, respectively. The determination of the membrane stiffness is approximated as km = 23.79(2R/ γ )2 = 54.6 GPa, where γ = 75 mm. 5

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Fig. 2. Effect of laminate thickness and lamina thickness on the damage threshold load. The data points correspond to the experimental results of the tests performed by Abisset et al. [53], Evci [65], González [66], González et al. [26], Sutherland and Soares [72], Yang and Cantwell [71] and Wagih et al. [37] and summarised in Table 3 and the lines are the linear fitting of this data. The legend includes the code of the test, the in-plane dimension and the slope of the fitting lines in (a) and the code of the test and the slope of the fitting lines in (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the presented model correlates well with the experimental results for TP and UTP laminates considering n = 5. Again, the response considering an increasing number of delaminated interfaces (n = N /3) tends to under predict the load for both laminates. The same observation is found for the delamination area prediction. The model considering n = N /3 under-predicts the response with an error of 166% and 290% for TP and UTP laminates, respectively, at a displacement equal to 2.9 mm. However, at the same displacement level, the error between the model predictions considering n = 5 and the experimental results is 22% and 20% for TP and UTP laminates. Fig. 8(a) shows the variation of the total and the dissipated energy for laminates TP and UTP under QSI [37] and LVI tests [68]. The figure reflects a better correlation between the experimental results and the presented model prediction considering n = 5 than considering n = N /3. In their study, Soto et al. [68] carried out a detailed finite element simulation using continuum damage models and cohesive elements in all the interfaces to accurately capture both the intralaminar and interlaminar damage in the material. According to their results, at a displacement of u = 2.5 mm the total energy is ET = 9.37 J,

energy. Again, considering n = 5 the prediction of the delaminated area for the thinner plies, tp = 0.181 and 0.36 mm, is better than n = N /3. The maximum error between the model prediction considering n = 5 for the laminate with tp = 0.181 is equal to 8%. This error slightly increases reaching 17% for the laminate with larger ply thickness, tp = 0.36 . However, the error in the prediction of the model considering n = N /3 for the laminate with tp = 0.181 is 190%. This error is decreased to 20% for the laminate with tp = 0.36. In Fig. 7, the load and the delamination area as a function of the applied displacement is presented from the results presented by Wagih et al. [37] (W2 in Table 2) for laminates with two different ply thicknesses, 0.08 mm and 0.16 mm referred as UTP and TP. The material is a TeXtreme plain weave with 20 mm wide yarn fabrics, manufactured by Oxeon AB and the number of plies for the TP and UTP ply thicknesses are 28 and 56 plies, respectively. The specimen, which dimensions are 150 × 100 mm, is placed over a flat support with a 125 × 75 mm rectangular cut-out. The figure also includes the experimental results of the LVI tests carried out by Soto et al. [68] impacting the UTP laminate at 20 J. As shown in Fig. 7(a), the load-displacement curve predicted by

Fig. 3. Effect of laminate thickness on the load after damage initiation. The data points in (a) correspond to the experimental results of the tests performed by Abisset et al. [53], González [66] and Yang and Cantwell [71] and summarised in Table 3 and the lines are the linear fitting of this data. The legend includes the code of the test and the slope of the fitting lines. The results in (b) are the experimental results of G1 and E2 tests and the prediction of the proposed model with two different values of n. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 6

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Fig. 4. C-scan projection of static indentation delaminations and true shape of delamination and adjusted 1/3 ellipse in interfaces 7 and 13 [36]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3.2. In-plane dimension (R)

Table 4 Elastic and fracture properties of the ply. Material

Ref.

E11 (GPa)

E22 (GPa)

G12 (GPa)

ν12

Hexply AS4/ 8552 TeXtreme AS4D/TC350

[26]

128

7.6

4.4

0.35

0.79

[74] [70]

69.1 135.4

69.1 9.3

4.0 5.3

0.03 0.32

1.09 1.17

The influence of the in-plane dimension on the damage threshold load is shown in Fig. 9(a). As shown in the figure, the experimental data can be fitted with almost horizontal lines in a bi-logarithmic plot with very low scatter. From this figure, it is clear that the in-plane dimension has no influence on the damage threshold. Similar observations were reported by Olsson [28] and Olsson et al. [69]. Fig. 9(b) shows the influence of the in-plane dimension on the load after damage initiation, Fdn . As shown in the figure, the influence of the in-plane dimension can be also neglected since the slope for all the tested specimens is almost zero. It can be also observed in Fig. 9 that there is no effect of the inplane shape on Fd1 and Fdn . In fact, the figure summarises the experimental data for two different in-plane shapes, circular and rectangular, with similar results. The experimental results S1 and W1 in Table 2 were performed by Sebaey et al. [70] and Wagih et al. [36], respectively. Both tests were carried out with the same material (AS4D/TC350, see Table 4) and the same stacking sequence. Sebaey et al. [70] impacted the specimens following the ASTM D7136 [67] specifications while Wagih et al. [36] indented the specimens quasi-statically using the small circular fixture. In Fig. 10, the load and delamination area are represented as a function of the applied displacement for both tests considering n = 5 and n = N /3. Again, the response considering n = N /3 tends to under

G IIc (N/ mm)

the energy dissipated by delamination is 2.45 J and the associated to intralaminar damage is 0.38 J. In Fig. 8(b) the energy dissipated by each interface is schematically shown. The top and bottom one sixth of the interfaces dissipate a very small amount of energy by delamination (most of interlaminar energy). Between them there are only twelve significantly delaminated interfaces (22% of the total number of interfaces) that dissipate 82% of the energy. The most delaminated interface is situated at three quarter of the laminate thickness and dissipates 0.37 J. In the same figure (with dotted lines) the model prediction, Eq. (14), with n = 5 delaminated interfaces is also shown. In this case each interface dissipates 0.41 J.

Fig. 5. Effect of laminate and lamina thickness on the load displacement curve. The data points correspond to the experimental results of González et al. [26,66] summarized as G1 and G2 in Table 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 7

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Fig. 6. Effect of laminate and lamina thickness on the projected delamination area. The data points correspond to the experimental results of González et al. [26,66] summarized as G1 and G2 in Table 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

results of Yang and Cantwell [71] in Table 6 are of special interest because five indenter radii are considered. The damage threshold is plotted in Fig. 11(a) in a bi-logarithmic scale. The dash lines have a slope equal to zero and fit the damage threshold load for small indenter radii. The solid lines have a slope equal to 1/2 and fit the damage threshold loads for indenter radii larger than 4 mm. This transition radius, 4 mm, is defined by equaling Eqs. (11) and (12). The influence of the indenter radius on the load after damage initiation, Fdn , is shown in Fig. 11(b). From the figure, the influence of the indenter radius can be neglected since the slope of the linear fitting is close to zero and also because by the Saint Venant’s principle, when the delamination radius is large enough the impactor radius does not influence the energy release rate due to delamination. According to Eq. (10), there is no influence of the indenter radius on the projected delamination area. The experimental results presented by Kwon et al. [12] confirm the fact that the indenter radius has no influence on the delamination area. In their results two different indenter radii, 3.175 and 12.7 mm, were considered, and the projected delamination area was almost equal for both radii.

predict the load and the delamination area for both tests.

3.3. Indenter radius (r) The damage threshold expressed with Eq. (13) is a combination of the load required to reach shear strength (Eq. (11)) and to grow a middle delamination according to fracture mechanics (Eq. (12)). The threshold load according to the shear criterion scales with indenter radius as r 1/2 , while according to fracture mechanics it does not depend on r. For sufficiently small indenter radii, despite the shear stress is larger than the material shear strength and the shear criterion is fulfilled, the energy release rate is not large enough for crack growth. Therefore, the fracture mechanics criterion becomes dominant and the damage threshold load is independent of the indenter radius. On the other hand, for larger indenter radii, the load is distributed in a larger zone and the external load to nucleate shear cracks is larger than the energy release rate for these cracks to grow. Therefore, for large indenter radii, the strength criterion becomes dominant and the damage threshold load varies with the indenter radius. Experimental results from Kwon and Sankar [12] (K1, K2 and K3 in Table 5), Wagih et al. [36] and Sebaey et al. [70] (W1 and S1 in Table 6) with two indenter radii show an increment of damage threshold load with respect the indenter radius. The experimental

4. Conclusions The scaling of elastic properties with respect to specimen size are

Fig. 7. Effect of the ply thickness. The data points correspond to the experimental results of Wagih et al. [37] summarized as W2 in Table 2 for QSI results and experimental results of Soto et al. [68] for LVI tests. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 8

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Fig. 8. Energy balance and delaminated interfaces for TeXtreme material. In (b) the solid lines represent the delaminations positions presented by Soto et al. [68] and the dash lines represent the presented model prediction with n = 5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Effect of the in-plane dimension on the damage threshold load and the load after the damage initiation. The data points correspond to the experimental results of the tests performed by Ghelli and Minak [73] and Kwon and Sankar [12] and summarised in Table 5 and the lines are the linear fitting of this data. The legends include the code of the test, the indenter radius and the slope of the fitting lines in each case. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Effect of in-plane size (R) on load-displacement response and delamination area. The data points correspond to the experimental results of the tests performed by Wagih et al. [36] and Sebaey et al. [70]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 9

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Table 5 The damage threshold load (Fth ) and the load after damage initiation (Fdn ) of the laminates with in-plane size. Label

In-plane size (mm)

Fth (N)

Fdn (N)

R = 25.4 R = 38.1 R = 50.4 R = 25.4 R = 38.1 R = 50.4 R = 25.4 R = 38.1 R = 50.4 R = 25.4 R = 38.1 R = 50.4 R = 25.4 R = 38.1 R = 50.4 R = 25.4 R = 38.1 R = 50.4 150× 100 R = 45

2981 2894 2887 3833 3854 3748 2681 2914 2874 3821 4114 3941 2407 2561 2368 2774 2987 2941 3540 3620

2401 2314 2320 2780 2787 2834 2430 2487 2280 2741 2861 2827 2330 2414 2334 2347 2861 2854 3122 3156

Table 7 Summary of scaling of geometrical parameters during out-of-plane loading.

Fth K1-r3.175 [12]

K1-r12.7 [12]

K2-r3.175 [12]

K2-r12.7 [12]

K3-r3.175 [12]

K3-r12.7 [12]

G3 [73]

Y2-h2.7 [71]

Y2-h3.6 [71]

W1 [36] S1 [70]

In-plane size (mm)

Fth (N)

Fdn (N)

r = 2.3 r = 5.0 r = 6.0 r = 7.2 r = 9.6 r = 2.3 r = 5.0 r = 6.0 r = 7.2 r = 9.6 r = 6.35 r = 8.0

1888 2039 2183 2517 2880 2961 3195 3378 3711 4636 5605 6020

– – – – – – – – – – 4100 4055

In-plane size, R

Indenter size, r

∝ h1.5

–

– for small r

Fdn

∝ h1.5

–

∝ r 1/2 for large r –

A

∝ h1.5

–

–

growth load (Fdn ). Both loads are independent on the in-plane size (R) and scale with the laminate thickness as h1.5 , as predicted by the analytical expressions presented in Section 2. Based on the experimental results included in Table 2, an increment of ply thickness reduces the damage threshold load, but it does not influence the load after damage initiation. The indenter radius does not influence the load after damage initiation, but it has an influence on the damage threshold load. According to the analytical models, the load required to reach the shear strength of the material scales with the indenter radius according to r 1/2 . The experimental results considered show that for indenter radii large enough, the damage threshold load increases with the indenter radius. For small indenter radii, the damage threshold load becomes constant. A summary of the scaling relations between the different parameters is presented in Table 7. Experimental results show that delaminations grow in almost all interfaces with mismatch angles. However, these delaminations are of different size and with elliptical shapes with the major length in the direction of the fibers of the adjacent plies. The analytical model defined in Section 2 requires the definition of n equivalent circular delaminations. This definition influences the propagation load (Fdn ) and the delamination area. After analysing the great variety of experimental results reported in Table 2, it can be concluded that values of n between 3 and 6 result in good fitting of the load after damage initiation and delamination area independently of the number of interfaces in the laminate. However, more experimental results have to be performed on different materials with the aid of modern inspection techniques such as X-ray computed tomography to enable implementing a robust relation for the equivalent number of delaminated interfaces, n. The projected delamination area predicted by the presented analytical model increases linearly with the applied displacement and correlates well with the experimental results. The slope of the linear relation between delamination area and applied displacement scales only with laminate

Table 6 The damage threshold load (Fth ) and the load after damage initiation (Fdn ) of the laminates with in-plane size. Label

Laminate thickness, h

well defined by elasticity theory: the bending stiffness scales as h3/ R2 and the membrane stiffness scales as h/ R2 . The delamination response is defined by means of two loads, the damage threshold load (Fth ) and the

Fig. 11. Effect of indenter radius on the damage threshold load and the load after damage initiation. In (b), the legends include the code of the test and the slope of the fitting lines in each case. The data points correspond to the experimental results of the tests performed by Kwon and Sankar [12], Yang and Cantwell [71], Sebaey et al. [70] and Wagih et al. [36] and summarised in Tables 5 and 6. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 10

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thickness according to h1.5 . The model and the results presented justify the use of small in-plane and out-of-plane coupons under static indentation tests to evaluate the response of large impacted structures.

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