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Estimation for probabilistic distribution of residual strength of sandwich structure with impact-induced damage
Renewable Energy 54 (2013) 219e226
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Estimation for probabilistic distribution of residual strength of sandwich structure with impact-induced damage Jae-Hoon Kim a, Seung-Pyo Lee b, Ji-Won Jin c, Ki-Weon Kang c, * a
High Speed Railroad System Research Center, Korea Railroad Research Institute, Uiwang, Gyeonggi-Do 437-757, Republic of Korea R&D Center, ILJIN Global, 128-5, Samsung-dong, Kangnam-go, Seoul 135-875, Republic of Korea c School of Mechanical Engineering, Kunsan National University, Kunsan, Jeonbuk 573-701, Republic of Korea b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 January 2012 Accepted 1 August 2012 Available online 24 August 2012
A new approach was developed to estimate the probabilistic distribution of the residual strength of honeycomb sandwich structures subjected to impact-induced damage. The model was derived from the assumption that the residual strength is a function of the static strength of the unimpacted sandwich structures and the impact damage of the impacted structures. The probabilistic distribution of the residual strength was calculated from the probabilistic properties (mean and variance) of the strength of the unimpacted laminates and the impact damage at a given impact energy. The model was experimentally verified by using an experimental program for a NomexÒ honeycomb core sandwich structure. The experimental program included a series of impact and three-point flexural tests to generate the residual strength of a sandwich structure. Theoretical predictions on the probabilistic distribution were in conformance with the experimental results. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Impact-induced damage Residual strength Sandwich structure Probabilistic distribution
1. Introduction Because of their superior mechanical properties, such as high specific strength and stiffness, flexible tailorabilty and improved fatigue life, composite materials have been widely used as structural components in the aerospace, marine and renewable energy industries [1]. Despite these advantages over traditional metallic materials, composite materials are susceptible to damage from foreign object impact because they lack throughthe-thickness reinforcement. Compared to metallic materials, impact events may cause entirely different types of damage to composite materials, such as delamination, matrix crack and fiber breakage [2,3]. These types of damage may behave as a discontinuity, reducing the structural strength and stiffness of the composite and leading to damage growth and premature failure of the structure at a load well below the designed load [3,4]. Many investigations have focused on the impact damage and strength reduction behaviors of damaged composite structures, mainly in conventional unidirectional laminates [3e8]. Aktas¸ et al. [3] have investigated the strength reduction behavior and
* Corresponding author.Tel.: þ82 63 469 4872; fax: þ82 63 469 4727. E-mail address: [email protected] (K.-W. Kang). 0960-1481/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.renene.2012.08.006
temperature effect of composites under compressive loading. Their experiments have shown that the compression strength after impact is affected by both the test temperature and the stacking sequence. Sánchez-Sáez et al. [4] have studied the compression after impact behavior of unidirectional and fabric carbon fiber reinforced composite laminates at low temperature. Cui et al. [5] have presented an integrated approach to the analysis of damage initiation and growth under impact and tensile loading after low velocity impact using a 3D progressive damage theory and analysis technique. Corum et al. [6] have experimentally characterized the susceptibility of various composite materials to low-energy impact loading using various test apparatuses, including pendulum, brickdrop and gas gun devices. They have investigated the damage mechanism, tension and compression after impact and reported that strength reduction was significantly affected by the constituent materials, loading conditions and stacking sequence of the test. To assess the residual properties of a sandwich structure, Schubel et al. [7] investigated the damage tolerance of sandwich structures composed of woven carbon/epoxy facesheets and a PVC foam core. Their research has shown that the major damage to the sandwich structure consisted of delamination and permanent indentation in the impacted facesheets, and the delamination in the facesheet was found to be detrimental to the load bearing capacity of the structure. Castanié et al. [8] have investigated the effects of low-velocity impacts on metal skinned sandwich
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structures and shown that the presented method agreed well with contact law [9]. From the literature survey, most prior research has focused on the strength reduction behavior of unidirectional or fabric composite plates. Most studies of sandwich structures have focused on their strength reduction post-impact behavior under compression. Although the compressive strength of the impacted structures may be compromised by delamination of the facesheet, structures of thin to medium thickness, such as wind turbine blades and the bodyshells of railway vehicles, are mainly exposed to flexural loads rather than compressive loading. In addition to their susceptibility to impact damage, composite materials have another drawback: their properties, particularly their strength, can vary widely due to their inhomogeneity and anisotropic nature, as well as the brittleness of their constituent materials [10,11]. Thus, by introducing the normal or Weibull distribution, many investigations have statistically analyzed the strength of composite materials [11e13]. However, when composite structures with impact-induced damage are loaded, their residual strength may exhibit different statistical behavior as a result of the impact damage [14e18]. Tai et al. [14] have studied the static and fatigue strength of impacted composite laminates under tensile loading. They have shown that there is significant scatter in the residual tensile strength and statistically analyzed the residual tensile strength using the Weibull distribution. Despite this investigation, their results have neither explained the difference in the variation of residual strength differs from that of unimpacted composites nor identified what factors affect the variation of strength in impacted ones. Aoki et al. [15] have evaluated the compression after impact characteristics of CFRP laminates and the combined effects of water absorption and the thermal environment. They briefly reported the variation of residual strength, but they did not explain their mechanism. Shim et al. [16] have examined the residual mechanical properties of crowfoot-weave carbon/epoxy laminates subjected to a lowvelocity impact. Their investigation has shown that there is some variation in the residual strength, but they did not mention the probabilistic characteristics of the residual strength or its mechanism. Tai et al. [17] have shown that the residual strength under tensile loading decreases with the impact energy, and its variation can be described by the two-parameter Weibull distribution. However, they did not explain the relationship between the variation of the residual strength and other parameters, such as the strength of the unimpacted laminates or the impact energy. Kang et al. [18,19] have analyzed the probabilistic properties of the residual strength of a sandwich structure under flexural loading. They introduced a random variable into Caprino’s residual strength prediction model [20], based on residue analysis, which represents the variation of the residual strength with incident impact energy. While such an approach can be fairly effective for assessing the distribution of the residual strength of impacted composite materials, the approach does not take into account the distribution of static strength in unimpacted materials, which may affect the distribution of residual strength. However, because the variation of residual strength may be affected by variations of the static strength as well as the impact damage, it is necessary to perform a probabilistic analysis considering both the variation of static strength in unimpacted materials and the impact damage in impacted materials. This paper presents a two-parameter approach to predict the residual strength distribution of composite sandwich structures subjected to low velocity impact. The approach was derived from the assumption that the residual strength can be expressed as a function of the strength of the unimpacted sandwich structure and the impact damage to the impacted structure. The model was experimentally verified using an experimental program for
Table 1 Mechanical properties of facesheet laminates. Laminates
Exx (GPa)
Eyy (GPa)
nnxy
Carbon/epoxy laminates Glass/epoxy laminates
127.53 43.90
7.85 2.74
0.34 0.31
NomexÒ honeycomb core sandwich structures. It was shown that the theoretical predictions on the probabilistic distribution conformed to the experimental results. 2. Experimental procedure 2.1. Material and specimens NomexÒ honeycomb core (Aerocell CACH 1/8-3.) is a highstrength, non-metallic honeycomb that is manufactured with aramid fiber paper, and its cell size and density are 3.2 mm and 48 kg/m3, respectively. The thickness of the core material was 10 mm and 20 mm. The facesheets were unidirectional eight-ply carbon/epoxy (TBCarbon CP200NS) and glass/epoxy (TBCarbon SGP125NS) laminates obtained from a prepreg with a thickness of approximately 0.2 mm. Their stacking sequence was [02/904/02]. The material properties of the facesheets were determined from tensile tests according to ASTM D-3039-00 [21] and are shown in Table 1. The material properties of the core material (shown in Table 2) were taken from the manufacturer’s data. Facesheet plates with a nominal fiber volume fraction of vf ¼ 50% were processed in an autoclave system according to the cure cycle recommended by the manufacturer. These plates were then bonded to the top and bottom of the honeycomb core with EA 9696(HYSOL) film adhesive. The assembly was then cured under pressure and temperature in an autoclave system, following the recommended curing cycle for the adhesive. The fabricated sandwich structures were then cut into panels with a width of 80 mm and length of 250 mm. The following notation is used for the specimens: SXY0, where X and Y indicate the facesheet type (C: carbon/epoxy, G: glass/epoxy) and core thickness (1:10 mm, 2:20 mm), respectively. 2.2. Impact and flexural tests A Dynatup model 9250HV impact testing machine (Fig. 1) was used to conduct the low velocity impact tests. The drop-weight testing machine that consists of a drop tower equipped with an impactor that has a hemispherical nose, a variable crosshead weight arrangement, high bandwidth DSP (digital signal processing) electronics, self-identifying load cells, and ImpulseÔ control and data acquisition software. The panels were roundclamped, with an opening of 76.2 mm in diameter. The radius and mass of the hemispherical impactor were 6.35 mm and 6.45 kg, respectively. After impact, the panels were inspected by scanning acoustic microscope (SONIX HS1000, 5 MHz). Threepoint flexural tests were then conducted by a universal testing machine (Instron 5581). The sandwich panels were simply supported on 25.4 mm diameter rollers with a span length of 160 mm, and the impacted surface was directed upward. The crosshead speed was 3 mm/min.
Table 2 Mechanical properties of Nomex honeycomb core. Thickness(mm) Density (kg/m3) Shear modulus (MPa) Shear strength (MPa) 10 & 20
48.0
19.3
0.58
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
221
1. Hook/Release
2. Drop Weight
3. Loadcell
4. Guide Rail
5. Impactor
6. Brake
7. Brake Control
8. Chamber (if necessary)
9. Position Control 10. Velocity Unit
1
11. Specimen
12. Pneumatic clamp
13. Emergency
14. Controller
9
14
2 3 4 5
10
11 12 13
6 7 8
Fig. 1. Illustration for instrumented impact testing machine.
3. Results and discussion 3.1. Two parameter probabilistic model for residual strength Experimental residual strength data [18e20,22,23] for composite materials indicate that the normalized residual strength sR =s0 after a low velocity impact decreases as a function of the damage caused by impact loading. Here, s0 is the original strength of the unimpacted laminates, and sR indicates the residual strength after impact loading. Hence, the residual strength can be expressed in terms of the original strength and a function of impact damage, as shown in Eq. (1).
sR ¼ s0 ,f ðci Þ
(1)
in which f(ci) is a function of the impact damage area ci and describes the strength reduction behavior. In Eq. (1), the residual strength sR can be considered to be the response variable, and it is functionally related to the original strength s0 and the impact damage area ci. Because the original strength and the damage area function ci can be assumed to be random variables [18], the residual strength should also be a random variable. The original strength and the impact damage can be treated as basic random variables as well. Because the residual strength is affected by both the residual strength and the impact damage, the joint probability density function (PDF) between the basic random variables could be used to describe the distribution of the residual strength sR, but it is practically impossible to obtain the joint PDF between the
original strength s0 and the damage area ci. It is, therefore, necessary to use an approximation method to estimate the residual strength distribution from the information about the basic variables. With the aid of a previous study [24], the following section provides a brief description of the approximation method used to identify the probabilistic characteristics of the response variable, given only limited information on the basic variables. In general, the response variable Y can be expressed by a function of the basic random variables Xi, as follows:
Y ¼ gðX1 ; X2 ; .; Xn Þ
(2)
Here, if the mean and variance of each Xi are known, the approximate mean and variance of the response variable Y can be obtained according to the following procedure [24]. Expanding the function g(X1, X2, ., Xn) in a Taylor series about the mean values, we obtain
Table 3 Flexural strength of unimpacted laminates. Flexural strength [MPa]
SC10 SC20 SG10 SG20
1
2
3
4
5
6
7
Mean
St. Dev.
66.59 39.47 56.38 42.28
59.44 38.44 55.82 41.41
65.26 42.13 54.98 41.26
77.13 39.32 52.25 42.70
79.58 43.23 55.79 43.36
69.39 43.00 57.77 43.81
63.56 37.94 56.24 44.73
68.71 40.51 55.61 42.79
7.28 2.22 1.70 1.27
222
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
1.0 Flexural Strength of unimpacted laminates
0.8 Results for SC10 Results for SC20
0.6
0.4 SC20 m=40.51 / μ=2.22
0.2
SC10 m=68.71 / μ=7.28
0.0 0
20
40
60
80
Cumulative distribution function, F
Cumulative distribution function, F
1.0
Flexural Strength of unimpacted laminates
0.8 Results for SG10 Results for SG20
0.6
SG10 m=55.61 / μ=1.70
0.4 SG20 m=42.79 / μ=1.27
0.2
0.0
100
0
Flexural Strength, σ0,i [MPa]
20
40
60
80
100
Flexural Strength, σ0,i [MPa]
a SC Panels
b SG Panels
Fig. 2. Cumulative distribution function of flexural strength of unimpacted laminates. n P vg Y ¼ g mX1 ; mX2 ; .; mXn þ xi mXi vXi i¼1
(3)
n X n v2 g 1X þ þ/ xi mXi xj mXj 2 i¼1 j¼1 vXi vXj
Here, mXi is the mean value of the basic variables Xi, and the derivatives are evaluated at mXi .
Truncating the series at the linear terms, the first-order approximate mean of Y, denoted by E(Y0 ), can be obtained as follows.
E Y ’ ¼ g mX1 ; mX2 ; .; mXn
Eq. (4) states that the first-order mean of Y can be estimated from the value of the function g, calculated at the mean values of the basic random variables.
1.2
1.2 Residual Strength for SC10
Residual Strength for SC20
1.0
Residual Strength, σ R/ σ 0
Residual Strength, σ R/ σ 0
1.0 0.8 0.6 0.4
Measured Resutls Fitted Resutls α=0.557, co=530.2
0.2
σ R ⎧ c0 ⎫ =⎨ ⎬ σ 0 ⎩ ci ⎭
α
0.0
0.8 0.6
Measured Resutls Fitted Resutls α=0.569, co=615.05
0.4
α
σ R ⎧ c0 ⎫ =⎨ ⎬ σ 0 ⎩ ci ⎭
0.2 0.0
0
500
1000
1500
2000
0
2
500
1000
1500
2000
2
Damaged area, ci (mm )
Damaged area, ci (mm )
a SC10
b SC20
1.2
1.2 Residual Strength for SG20
Residual Strength for SG10
1.0
Residual Strength, σ R/ σ 0
1.0
Residual Strength, σ R/ σ 0
(4)
0.8 0.6 0.4 Measured Resutls α Fitted Resutls σ R ⎧ c0 ⎫ α=0.681, Co=82.82 =⎨ ⎬
0.2
σ0
⎩ ci ⎭
0.0
0.8 0.6 0.4 Measured Resutls α Fitted Resutls σ R ⎧ c0 ⎫ α=1.013, Co=68.60 =⎨ ⎬
0.2
σ0
⎩ ci ⎭
0.0 0
50
100
150
200 2
250
0
50
100
150
200 2
Damaged area, ci (mm )
Damaged area, ci (mm )
c SG10
d SG20
Fig. 3. Strength reduction behavior in impacted sandwich panels.
250
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
223
Table 4 (a) Impact damage area for SC panels. (b) Impact damage area for SG panels. Ei
SC10 SC20
4.0J 8.0J 4.9J 8.9J
Damage area, ci [mm2] 1
2
3
4
5
6
7
Mean
Variance
813.1 1522.7 1017.9 1608.3
866.6 1510.9 1055.1 1576.8
873.7 1430.0 956.8 1467.3
824.8 1605.9 930.9 1591.5
885.23 1409.8 1005.8 1547.8
848.07 1583.5 967.9 1487.5
824.8 1525.0 997.3 1521.7
849.6 1512.6 990.2 1542.9
671.8 5234.8 1734.9 2841.6
(b)
Ei
Damage area, ci [mm2] 1
2
3
4
5
6
7
Mean
Variance
SG10
3.0J 8.0J 3.0J 8.0J
116.7 116.9 128.5 171.2
117.7 151.6 136.8 144.9
110.1 131.9 113.8 148.1
75.62 123.4 134.2 149.7
117.2 114.6 128.7 157.2
115.4 133.4 135.4 140.9
108.4 136.8 145.4 147.6
108.7 129.8 131.8 151.5
226.3 163.3 94.0 102.9
SG20
Similarly, the first-order variance of the response variable Var(Y0 ) can be shown to be n n X n X X Var Y ’ ¼ S2i VarðXi Þ þ Si Sj Cov Xi ; Xj for isj i¼1
i¼1
(5)
j
in which the constants Si and Sj represent the partial derivatives vg=vXi and vg=vXj , respectively, calculated at the mean values of the basic variables Xi. Also, these constants can be interpreted as
significance factors for the uncertainties in each of the corresponding random variables Xi. In general, these significance factors show the importance of the variables in the formulation [24]. If we can obtain the probabilistic properties (mean and variance) of the basic random variables, as well as the function g between the response variable and the basic variables, the probabilistic properties of the response variable can be evaluated easily according to the above procedure. Let us assume that we know the mean and variance of the basic random variables (the original strength s0 and the impact damage
1.0 Measured Results for SC10
Cumulative distribution function, F
Cumulative distribution function, F
1.0
0.8
0.6
0.4 Results at 8.0J m = 1512.8 μ = 72.35
Results at 4.0J m = 849.6 μ = 25.92
0.2
0.0 0
500
1000
1500
2000
Measured Results for SC20
0.8
0.6
0.4 Results at 8.9J m = 1542.9 μ = 53.31
Results at 4.9J m = 990.2 μ = 41.65
0.2
0.0
2500
0
2
500
1000
2000
2500
Damaged Area, ci (mm )
a SC10
b SC20
1.0
1.0 Measured Results for SG20
Cumulative distribution function, F
Cumulative distribution function, F
1500
2
Damaged Area, ci (mm )
0.8
0.6
0.4 Results at 8.0J m = 151.5 μ = 10.14
Results at 3.0J m = 131.8 μ = 9.70
0.2
0.0 0
50
100
150
200 2
Damaged Area, ci (mm )
c SG10
250
Measured Results for SG10
0.8
0.6
0.4 Results at 8.0J m = 129.8 μ = 12.79
Results at 3.0J m = 108.7 μ = 15.05
0.2
0.0 0
50
100
150
200 2
Damaged Area, ci (mm )
d SG20
Fig. 4. Cumulative distribution function for impact damage area.
250
224
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
Table 5 (a) Estimated probabilistic properties (mean and variance) for SC panels. (b) Estimated probabilistic properties (mean and variance) for SG panels. SC10 Ei
4.0J 8.0J
SC20 Measured
Estimated
Mean
Var
Mean
Var
55.43 40.14
23.75 23.77
52.85 38.33
32.18 17.55
SG10 Ei
3.0J 8.0J
Ei
Measured
Estimated
Mean
Var
Mean
Var
4.9J 8.9J
31.59 25.15
2.47 0.59
30.88 23.99
3.41 1.75
SG20 Measured
Estimated
Ei
Mean
Var
Mean
Var
39.70 25.39
6.04 1.57
38.88 26.65
13.04 2.75
3.0J 8.0J
Measured
Estimated
Mean
Var
Mean
Var
32.45 26.41
1.75 0.47
31.69 27.06
2.90 0.88
area ci). If those values are known, once the function f(ci) in Eq. (1) has been determined from the experimental data, Eqs. (4) and (5) can be easily used to find the probabilistic properties of the residual strength sR of composite materials subjected to impact loading. It is well known that strength reduction behavior can be represented as a function of impact damage according to Caprino’s approach [20], as follows.
sR ¼ s0
f ci ¼
a c0 ci
(7)
Y ¼ sR ; X1 ¼ s0 ; X2 ¼ ci
(8)
Combining Eq. (4) with Eqs. (7) and (8), one obtains a first-order approximation of the expected residual strength at a certain impact energy, as follows:
a c E s’R ¼ ms0 , 0 ci
(9)
in which ms0 indicates the mean value of the original strength. Because the original strength s0 (X1) is an inherent property of composite materials, and the impact damage area ci (X2) is caused by impact loading, it is reasonable to assume that the original
1.0
Cumulative distribution function, F
Results for SC10 at 4.0J Tests results
0.8
0.6
Measured Results mean = 0.807 standard deviation = 0.071
0.4 Simulated Results mean = 0.769 standard deviation = 0.124
0.2
0.0 0.2 0.3
0.4 0.5 0.6 0.7 0.8
Results for SC10 at 8.0J Tests results
0.8
0.6 Measured Results mean = 0.584 standard deviation = 0.071
0.4
Simulated Results mean = 0.558 standard deviation = 0.092
0.2
0.0
0.9 1.0
0.2 0.3
Normalized Residual Strength, σR,i/σ0,m
0.4 0.5 0.6 0.7 0.8
0.9 1.0
Normalized Residual Strength, σR,i/σ0,m
a SC10 : 4.0J
b SC10 : 8.0J
1.0
1.0 Results for SC20 at 4.9J Tests results
Cumulative distribution function, F
Cumulative distribution function, F
(6)
In this expression, E and Eth denote the incident impact energy and the threshold energy (which is the minimum energy required to cause a strength reduction in the structures), respectively, c0 and ci are the inherent and impact damage areas, respectively, and a is the material constant. Comparison of Eqs. (1), (2) and (6) leads to the following relationship:
1.0
Cumulative distribution function, F
a c0 ; Ei Eth ci
0.8
0.6
Measured Results mean = 0.780 standard deviation = 0.039
0.4 Simulated Results mean = 0.763 standard deviation = 0.069
0.2
0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Residual Strength, σR,i/σ0,m
c SC20 : 4.9J
Results for SC20 at 8.9J Tests results
0.8
0.6
Measured Results mean = 0.621 standard deviation = 0.019
0.4
0.2
Simulated Results mean = 0.592 standard deviation = 0.049
0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Residual Strength, σR,i/σ0,m
d SC20 : 8.9J
Fig. 5. Comparison of measured and estimated distribution of SC panels.
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
strength and the impact damage are not correlated. Therefore, the first-order variance of the residual strength can be obtained as follows from Eq. (5) [24].
Var s’R ¼ S21 s0 þ S22 ci S1 ¼
(10)
a a vg v c c0 s0 0 ¼ ¼ vX1 mX vs0 ci ci ms ms 1
S2 ¼
0
(11) 0
a as0 c0 a vg v c0 s ¼ ¼ 0 ci ci ci mc vX2 mX vci mc 2
i
(12) i
in which mci is the mean value of the impact damage at a given impact energy. From the above procedure, the probabilistic distribution of the residual strength sR of a composite material subjected to a given impact energy can be obtained from the probabilistic properties (mean and variance) of the original strength and impact damage. 3.2. Experimental verification To determine the probabilistic distribution of the residual strength in the impacted sandwich structures according to abovementioned procedure, the mean and variance of the original strength must be determined first. Therefore, seven specimens of each sandwich system were statically tested under flexural loading,
and the results are summarized in Table 3. Fig. 2 shows the cumulative distribution functions of each material, as evaluated by the normal distribution. As shown in the figure, the flexural strength data of the unimpacted laminates correlate well with the normal distribution. As shown in Eqs. (6) and (9), the relationship between the residual strength and the impact damage area must be established. To determine this relationship, impact tests were conducted in the range of approximately 2-13J for the SC and SG panels. The resulting damage was measured with a scanning acoustic microscope (SONIX HS1000, 5 MHz). Three point flexural tests were then conducted for the impacted sandwich panels. The residual strength data, normalized to the mean strength of the unimpacted sandwich panels, were plotted against the impact damage area in Fig. 3. Here, the solid marks and line represent the experimental results and the fitted results according to Caprino’s approach shown in Eq. (6) [6,20], respectively. From the figure, it is clear that the residual strength was not affected by the damage area below a threshold value. Thus, this value can be defined as the characteristic damage area c0, which is the minimum damage area that causes strength reduction in structures. Above this level, the residual strength decreases rapidly as the impact damage area increases until it approaches approximately 50% of the original strength of the unimpacted sandwich panels for both sandwich systems. According to Caprino’s approach in Eq. (6), the rate of strength reduction is described by the exponent a. To identify the probabilistic distribution of the residual strength in the impacted panels, it is also necessary to determine the 1.0
Results for SG10 at 3.0J Tests results
Cumulative distribution function, F
Cumulative distribution function, F
1.0
0.8
0.6 Measured Results mean = 0.714 standard deviation = 0.044
0.4
0.2
Simulated Results mean = 0.699 standard deviation = 0.098
0.0 0.2 0.3
0.4 0.5 0.6 0.7 0.8
Results for SG10 at 8.0J Tests results
0.8 Measured Results mean = 0.457 standard deviation = 0.023
0.6
0.4 Simulated Results mean = 0.479 standard deviation = 0.045
0.2
0.0
0.9 1.0
0.2 0.3
Normalized Residual Strength, σR,i/σ0,m
0.4 0.5 0.6 0.7 0.8
0.9 1.0
Normalized Residual Strength, σR,i/σ0,m
a SG10 : 3.0J
b SG10 : 8.0J
1.0
1.0 Results for SG20 at 3.0J Tests results
Cumulative distribution function, F
Cumulative distribution function, F
225
0.8
0.6 Measured Results mean = 0.758 standard deviation = 0.031
0.4 Simulated Results mean = 0.741 standard deviation = 0.060
0.2
0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Results for SG20 at 8.0J Tests results
0.8
0.6 Measured Results mean = 0.611 standard deviation = 0.016
0.4
0.2
Simulated Results mean = 0.632 standard deviation = 0.033
0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Residual Strength, σR,i/σ0,m
Normalized Residual Strength, σR,i/σ0,m
c SG20 : 3.0J
d SG20 : 8.0J
Fig. 6. Comparison of measured and estimated distribution of SG panels.
226
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
Table 6 (a) Significance factors Si in each random variable for SC panels. (b) Significance factors Si in each random variable for SG panels. SC10
SC20
Ei
S1
S2
Ei
S1
S2
4.0J 8.0J
0.76907 0.55778
0.03464 0.01411
4.9J 8.9J
0.76244 0.59225
0.01776 0.00885
Ei
S1
S2
Ei
S1
S2
3.0J 8.0J
0.62717 0.52418
0.32494 0.22751
3.0J 8.0J
0.72879 0.66263
0.16126 0.12750
SG10
SG20
probabilistic properties (mean and variance) of the impact damage at a given impact energy. For this measurement, four specimen sets were tested at 4.0 J and 8.0 J for SC10 and 4.9 J and 8.9 J for SC20. The impact tests for the SG panels were carried out at 3.0 J and 8.0 J for the SG10 and SG20 panels, respectively. The results are summarized in Table 4, and Fig. 4 indicates the cumulative distribution function of the impact damage area for the SC and SG panels. From the probabilistic properties (mean and variance) of the original strength and the impact damage area at a given impact energy level and the relationship between the residual strength and impact damage area of the impacted structures, it is possible to estimate the residual strength distribution for a particular incident impact energy level Ei by using Eqs. (6)e(12). At a given impact energy level, the mean and variance of the residual strength can be estimated by Eqs. (9) and (10), respectively. The predicted results are summarized in Table 5. The proposed approach can be used to predict the distribution of the residual strength at specific impact energy levels. For verification of the approach, theoretical predictions were made for given energy levels, and comparisons were made between the estimated (simulated) and experimental (measured) results for the SC and SG panels (Figs. 5 and 6, respectively). Here, the estimated and experimental results were normalized to the mean strength of the unimpacted laminates. The measured results were obtained by directly applying the normal distribution to the seven data sets of residual strength at given energy levels. In the figures, the measured results are indicated by a solid line and the estimated results as a dotted line. The estimated distributions compare well with the measured distributions for both the SC and SG panels. Because the proposed model consists of two parameters, the original strength and the impact damage area, it is of interest to determine which parameter plays the primary role in determining the uncertainty of the residual strength of the impacted structures. To identify this parameter, the significance factors S1 and S2 were calculated from Eqs. (11) and (12), respectively; their results are summarized in Table 6. The results indicate that the original strength plays the primary role in defining the uncertainty of the residual strength, and the effect of the impact damage uncertainty is relatively small. This behavior may be a result of the fact that the impact damage behaves as a discontinuity because it is a stress concentration in the structure. Although the impact damage has a considerable amount of scatter at a given impact energy, it is well known that the impact damage causes a large reduction in the strength of composite structures [20]. Therefore, the impact damage can be inferred to produce a stress concentration rather than the uncertainty in the residual strength of the impacted structure. 4. Conclusions This paper provides a two-parameter approach to the prediction of the probabilistic distribution of residual strength at a given impact energy in impacted sandwich structures. The approach was
derived from the relationship between the response variable (the residual strength) and the basic random variables (the original strength and impact damage). The distribution of the residual strength was calculated from the probabilistic properties (mean and variance) of the original strength and the impact damage at a given impact energy. By using an experimental program on NomexÒ honeycomb core sandwich structures, this approach was found to be capable of predicting the distribution of residual strength at a given impact energy level. Acknowledgment This work was financially supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No. 20124010203240) and by Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology (2011-0007012). References [1] Kang KW, Lim DM, Kim JK. Probabilistic analysis for the fatigue life of carbon/ epoxy laminates. Comp Struct 2008;85:258e64. [2] Kang KW, Kim JK. Fatigue life prediction for impacted carbon/epoxy laminates under 2-stage loading. Comp A 2006;37:1451e7. [3] Aktas¸ M, Karakuzu R, Arman Y. Compression-after impact behavior of laminated composite plates subjected to low velocity impact in high temperatures. Comp Struct 2009;89:77e82. [4] Sánchez-Sáez S, Barbero E, Navarro C. Compressive residual strength at low temperatures of composite laminates subjected to low-velocity impacts. Comp Struct 2008;85:226e32. [5] Cui HP, Wen WD, Cui HT. An integrated method for predicting damage and residual tensile strength of composite laminates under low velocity impact. Comp Struct 2009;87:456e66. [6] Corum JM, Battiste RL, Ruggles-Wrenn MB. Low-energy impact effects on candidate automotive structural composites. Comp Sci Tech 2003;63:755e69. [7] Schuble PM, Luo JJ, Daniel IM. Impact and post impact behavior of composite sandwich panels. Comp A 2007;38:1051e7. [8] Casanie B, Bouvet C, Aminanda Y, Barrau JJ, Thevenet P. Modelling of lowenergy/low-velocity impact on Nomex honeycomb sandwich structures with metallic skins. Intl J Imp Eng 2008;35:620e34. [9] Swanson RS, Jongman K. Design of sandwich structures under contact loading. Comp Struct 2003;59:403e13. [10] Tanaka S, Ichikawa M, Akita S. A probabilistic investigation of fatigue life and cumulative cycle ratio. Eng Frac Mech 1984;20:501e13. [11] Ni K, Zhang S. Fatigue reliability analysis under two-stage loading. Relia Eng System Saf 2000;68:153e8. [12] Yang JN, Jones DL. Effect of load sequence on the statistical fatigue of composites. AIAA J 1980;18:1525e31. [13] Hwang W, Han KS. Cumulative damage models and multi-stress fatigue life prediction. J Comp Mater 1989;20:125e53. [14] Tai NH, Yip MC, Lin JL. Effects of low-energy impact on the fatigue behavior of carbon/epoxy compsites. Comp Sci Tech 1998;58:1e8. [15] Aoki Y, Yamada K, Ishikawa T. Effect of hygrothermal condition on compression after impact strength of CFRP laminates. Comp Sci Tech 2008;68: 1376e84. [16] Shim VPW, Yang LM. Characterization of the residual mechanical properties of woven fabric reinforced composites after low-velocity impact. Int J Mech Sci 2005;38:647e65. [17] Tai NH, Yip MC, Tseng CM. Influences of thermal cycling and low-energy impact on the fatigue behavior of carbon/PEEK laminates. Comp B 1999;30: 849e65. [18] Kang KW, Kim HS, Kim MS, Kim JK. Strength reduction behavior of honeycomb sandwich structure subjected to low velocity impact. Mat Sci Eng 2008; A483-484:333e5. [19] Lee SP, Lee JH, Kang KW. Residual strength distribution of impacted glass/ epoxy laminates with embedded shape memory alloy at low temperatures. Comp Struct 2011;93:324e30. [20] Caprino G. Residual strength prediction of impacted CFRP laminates. J Comp Mater 1984;18:508e18. [21] ASTM. Standard test method for tensile properties of fiber resin composite, D3039eD3000; 2000. [22] Harai Y, Hamada M, Kim JK. Impact response of woven glass-fabric composites-II effect of temperature. Comp Sci Tech 1998;58(1):119e212. [23] Kim JK, Mackay DB, May YW. Drop-weight impact damage tolerance of CFRP with rubber modified epoxy matrix. Composites 1993;24:458e94. [24] Haldar A, Mahadevan S. Probability, reliability, and statistical methods in engineering design. New York: John Wiley & Sons; 2000.
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Estimation for probabilistic distribution of residual strength of sandwich structure with impact-induced damage Jae-Hoon Kim a, Seung-Pyo Lee b, Ji-Won Jin c, Ki-Weon Kang c, * a
High Speed Railroad System Research Center, Korea Railroad Research Institute, Uiwang, Gyeonggi-Do 437-757, Republic of Korea R&D Center, ILJIN Global, 128-5, Samsung-dong, Kangnam-go, Seoul 135-875, Republic of Korea c School of Mechanical Engineering, Kunsan National University, Kunsan, Jeonbuk 573-701, Republic of Korea b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 January 2012 Accepted 1 August 2012 Available online 24 August 2012
A new approach was developed to estimate the probabilistic distribution of the residual strength of honeycomb sandwich structures subjected to impact-induced damage. The model was derived from the assumption that the residual strength is a function of the static strength of the unimpacted sandwich structures and the impact damage of the impacted structures. The probabilistic distribution of the residual strength was calculated from the probabilistic properties (mean and variance) of the strength of the unimpacted laminates and the impact damage at a given impact energy. The model was experimentally verified by using an experimental program for a NomexÒ honeycomb core sandwich structure. The experimental program included a series of impact and three-point flexural tests to generate the residual strength of a sandwich structure. Theoretical predictions on the probabilistic distribution were in conformance with the experimental results. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Impact-induced damage Residual strength Sandwich structure Probabilistic distribution
1. Introduction Because of their superior mechanical properties, such as high specific strength and stiffness, flexible tailorabilty and improved fatigue life, composite materials have been widely used as structural components in the aerospace, marine and renewable energy industries [1]. Despite these advantages over traditional metallic materials, composite materials are susceptible to damage from foreign object impact because they lack throughthe-thickness reinforcement. Compared to metallic materials, impact events may cause entirely different types of damage to composite materials, such as delamination, matrix crack and fiber breakage [2,3]. These types of damage may behave as a discontinuity, reducing the structural strength and stiffness of the composite and leading to damage growth and premature failure of the structure at a load well below the designed load [3,4]. Many investigations have focused on the impact damage and strength reduction behaviors of damaged composite structures, mainly in conventional unidirectional laminates [3e8]. Aktas¸ et al. [3] have investigated the strength reduction behavior and
* Corresponding author.Tel.: þ82 63 469 4872; fax: þ82 63 469 4727. E-mail address: [email protected] (K.-W. Kang). 0960-1481/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.renene.2012.08.006
temperature effect of composites under compressive loading. Their experiments have shown that the compression strength after impact is affected by both the test temperature and the stacking sequence. Sánchez-Sáez et al. [4] have studied the compression after impact behavior of unidirectional and fabric carbon fiber reinforced composite laminates at low temperature. Cui et al. [5] have presented an integrated approach to the analysis of damage initiation and growth under impact and tensile loading after low velocity impact using a 3D progressive damage theory and analysis technique. Corum et al. [6] have experimentally characterized the susceptibility of various composite materials to low-energy impact loading using various test apparatuses, including pendulum, brickdrop and gas gun devices. They have investigated the damage mechanism, tension and compression after impact and reported that strength reduction was significantly affected by the constituent materials, loading conditions and stacking sequence of the test. To assess the residual properties of a sandwich structure, Schubel et al. [7] investigated the damage tolerance of sandwich structures composed of woven carbon/epoxy facesheets and a PVC foam core. Their research has shown that the major damage to the sandwich structure consisted of delamination and permanent indentation in the impacted facesheets, and the delamination in the facesheet was found to be detrimental to the load bearing capacity of the structure. Castanié et al. [8] have investigated the effects of low-velocity impacts on metal skinned sandwich
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structures and shown that the presented method agreed well with contact law [9]. From the literature survey, most prior research has focused on the strength reduction behavior of unidirectional or fabric composite plates. Most studies of sandwich structures have focused on their strength reduction post-impact behavior under compression. Although the compressive strength of the impacted structures may be compromised by delamination of the facesheet, structures of thin to medium thickness, such as wind turbine blades and the bodyshells of railway vehicles, are mainly exposed to flexural loads rather than compressive loading. In addition to their susceptibility to impact damage, composite materials have another drawback: their properties, particularly their strength, can vary widely due to their inhomogeneity and anisotropic nature, as well as the brittleness of their constituent materials [10,11]. Thus, by introducing the normal or Weibull distribution, many investigations have statistically analyzed the strength of composite materials [11e13]. However, when composite structures with impact-induced damage are loaded, their residual strength may exhibit different statistical behavior as a result of the impact damage [14e18]. Tai et al. [14] have studied the static and fatigue strength of impacted composite laminates under tensile loading. They have shown that there is significant scatter in the residual tensile strength and statistically analyzed the residual tensile strength using the Weibull distribution. Despite this investigation, their results have neither explained the difference in the variation of residual strength differs from that of unimpacted composites nor identified what factors affect the variation of strength in impacted ones. Aoki et al. [15] have evaluated the compression after impact characteristics of CFRP laminates and the combined effects of water absorption and the thermal environment. They briefly reported the variation of residual strength, but they did not explain their mechanism. Shim et al. [16] have examined the residual mechanical properties of crowfoot-weave carbon/epoxy laminates subjected to a lowvelocity impact. Their investigation has shown that there is some variation in the residual strength, but they did not mention the probabilistic characteristics of the residual strength or its mechanism. Tai et al. [17] have shown that the residual strength under tensile loading decreases with the impact energy, and its variation can be described by the two-parameter Weibull distribution. However, they did not explain the relationship between the variation of the residual strength and other parameters, such as the strength of the unimpacted laminates or the impact energy. Kang et al. [18,19] have analyzed the probabilistic properties of the residual strength of a sandwich structure under flexural loading. They introduced a random variable into Caprino’s residual strength prediction model [20], based on residue analysis, which represents the variation of the residual strength with incident impact energy. While such an approach can be fairly effective for assessing the distribution of the residual strength of impacted composite materials, the approach does not take into account the distribution of static strength in unimpacted materials, which may affect the distribution of residual strength. However, because the variation of residual strength may be affected by variations of the static strength as well as the impact damage, it is necessary to perform a probabilistic analysis considering both the variation of static strength in unimpacted materials and the impact damage in impacted materials. This paper presents a two-parameter approach to predict the residual strength distribution of composite sandwich structures subjected to low velocity impact. The approach was derived from the assumption that the residual strength can be expressed as a function of the strength of the unimpacted sandwich structure and the impact damage to the impacted structure. The model was experimentally verified using an experimental program for
Table 1 Mechanical properties of facesheet laminates. Laminates
Exx (GPa)
Eyy (GPa)
nnxy
Carbon/epoxy laminates Glass/epoxy laminates
127.53 43.90
7.85 2.74
0.34 0.31
NomexÒ honeycomb core sandwich structures. It was shown that the theoretical predictions on the probabilistic distribution conformed to the experimental results. 2. Experimental procedure 2.1. Material and specimens NomexÒ honeycomb core (Aerocell CACH 1/8-3.) is a highstrength, non-metallic honeycomb that is manufactured with aramid fiber paper, and its cell size and density are 3.2 mm and 48 kg/m3, respectively. The thickness of the core material was 10 mm and 20 mm. The facesheets were unidirectional eight-ply carbon/epoxy (TBCarbon CP200NS) and glass/epoxy (TBCarbon SGP125NS) laminates obtained from a prepreg with a thickness of approximately 0.2 mm. Their stacking sequence was [02/904/02]. The material properties of the facesheets were determined from tensile tests according to ASTM D-3039-00 [21] and are shown in Table 1. The material properties of the core material (shown in Table 2) were taken from the manufacturer’s data. Facesheet plates with a nominal fiber volume fraction of vf ¼ 50% were processed in an autoclave system according to the cure cycle recommended by the manufacturer. These plates were then bonded to the top and bottom of the honeycomb core with EA 9696(HYSOL) film adhesive. The assembly was then cured under pressure and temperature in an autoclave system, following the recommended curing cycle for the adhesive. The fabricated sandwich structures were then cut into panels with a width of 80 mm and length of 250 mm. The following notation is used for the specimens: SXY0, where X and Y indicate the facesheet type (C: carbon/epoxy, G: glass/epoxy) and core thickness (1:10 mm, 2:20 mm), respectively. 2.2. Impact and flexural tests A Dynatup model 9250HV impact testing machine (Fig. 1) was used to conduct the low velocity impact tests. The drop-weight testing machine that consists of a drop tower equipped with an impactor that has a hemispherical nose, a variable crosshead weight arrangement, high bandwidth DSP (digital signal processing) electronics, self-identifying load cells, and ImpulseÔ control and data acquisition software. The panels were roundclamped, with an opening of 76.2 mm in diameter. The radius and mass of the hemispherical impactor were 6.35 mm and 6.45 kg, respectively. After impact, the panels were inspected by scanning acoustic microscope (SONIX HS1000, 5 MHz). Threepoint flexural tests were then conducted by a universal testing machine (Instron 5581). The sandwich panels were simply supported on 25.4 mm diameter rollers with a span length of 160 mm, and the impacted surface was directed upward. The crosshead speed was 3 mm/min.
Table 2 Mechanical properties of Nomex honeycomb core. Thickness(mm) Density (kg/m3) Shear modulus (MPa) Shear strength (MPa) 10 & 20
48.0
19.3
0.58
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
221
1. Hook/Release
2. Drop Weight
3. Loadcell
4. Guide Rail
5. Impactor
6. Brake
7. Brake Control
8. Chamber (if necessary)
9. Position Control 10. Velocity Unit
1
11. Specimen
12. Pneumatic clamp
13. Emergency
14. Controller
9
14
2 3 4 5
10
11 12 13
6 7 8
Fig. 1. Illustration for instrumented impact testing machine.
3. Results and discussion 3.1. Two parameter probabilistic model for residual strength Experimental residual strength data [18e20,22,23] for composite materials indicate that the normalized residual strength sR =s0 after a low velocity impact decreases as a function of the damage caused by impact loading. Here, s0 is the original strength of the unimpacted laminates, and sR indicates the residual strength after impact loading. Hence, the residual strength can be expressed in terms of the original strength and a function of impact damage, as shown in Eq. (1).
sR ¼ s0 ,f ðci Þ
(1)
in which f(ci) is a function of the impact damage area ci and describes the strength reduction behavior. In Eq. (1), the residual strength sR can be considered to be the response variable, and it is functionally related to the original strength s0 and the impact damage area ci. Because the original strength and the damage area function ci can be assumed to be random variables [18], the residual strength should also be a random variable. The original strength and the impact damage can be treated as basic random variables as well. Because the residual strength is affected by both the residual strength and the impact damage, the joint probability density function (PDF) between the basic random variables could be used to describe the distribution of the residual strength sR, but it is practically impossible to obtain the joint PDF between the
original strength s0 and the damage area ci. It is, therefore, necessary to use an approximation method to estimate the residual strength distribution from the information about the basic variables. With the aid of a previous study [24], the following section provides a brief description of the approximation method used to identify the probabilistic characteristics of the response variable, given only limited information on the basic variables. In general, the response variable Y can be expressed by a function of the basic random variables Xi, as follows:
Y ¼ gðX1 ; X2 ; .; Xn Þ
(2)
Here, if the mean and variance of each Xi are known, the approximate mean and variance of the response variable Y can be obtained according to the following procedure [24]. Expanding the function g(X1, X2, ., Xn) in a Taylor series about the mean values, we obtain
Table 3 Flexural strength of unimpacted laminates. Flexural strength [MPa]
SC10 SC20 SG10 SG20
1
2
3
4
5
6
7
Mean
St. Dev.
66.59 39.47 56.38 42.28
59.44 38.44 55.82 41.41
65.26 42.13 54.98 41.26
77.13 39.32 52.25 42.70
79.58 43.23 55.79 43.36
69.39 43.00 57.77 43.81
63.56 37.94 56.24 44.73
68.71 40.51 55.61 42.79
7.28 2.22 1.70 1.27
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J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
1.0 Flexural Strength of unimpacted laminates
0.8 Results for SC10 Results for SC20
0.6
0.4 SC20 m=40.51 / μ=2.22
0.2
SC10 m=68.71 / μ=7.28
0.0 0
20
40
60
80
Cumulative distribution function, F
Cumulative distribution function, F
1.0
Flexural Strength of unimpacted laminates
0.8 Results for SG10 Results for SG20
0.6
SG10 m=55.61 / μ=1.70
0.4 SG20 m=42.79 / μ=1.27
0.2
0.0
100
0
Flexural Strength, σ0,i [MPa]
20
40
60
80
100
Flexural Strength, σ0,i [MPa]
a SC Panels
b SG Panels
Fig. 2. Cumulative distribution function of flexural strength of unimpacted laminates. n P vg Y ¼ g mX1 ; mX2 ; .; mXn þ xi mXi vXi i¼1
(3)
n X n v2 g 1X þ þ/ xi mXi xj mXj 2 i¼1 j¼1 vXi vXj
Here, mXi is the mean value of the basic variables Xi, and the derivatives are evaluated at mXi .
Truncating the series at the linear terms, the first-order approximate mean of Y, denoted by E(Y0 ), can be obtained as follows.
E Y ’ ¼ g mX1 ; mX2 ; .; mXn
Eq. (4) states that the first-order mean of Y can be estimated from the value of the function g, calculated at the mean values of the basic random variables.
1.2
1.2 Residual Strength for SC10
Residual Strength for SC20
1.0
Residual Strength, σ R/ σ 0
Residual Strength, σ R/ σ 0
1.0 0.8 0.6 0.4
Measured Resutls Fitted Resutls α=0.557, co=530.2
0.2
σ R ⎧ c0 ⎫ =⎨ ⎬ σ 0 ⎩ ci ⎭
α
0.0
0.8 0.6
Measured Resutls Fitted Resutls α=0.569, co=615.05
0.4
α
σ R ⎧ c0 ⎫ =⎨ ⎬ σ 0 ⎩ ci ⎭
0.2 0.0
0
500
1000
1500
2000
0
2
500
1000
1500
2000
2
Damaged area, ci (mm )
Damaged area, ci (mm )
a SC10
b SC20
1.2
1.2 Residual Strength for SG20
Residual Strength for SG10
1.0
Residual Strength, σ R/ σ 0
1.0
Residual Strength, σ R/ σ 0
(4)
0.8 0.6 0.4 Measured Resutls α Fitted Resutls σ R ⎧ c0 ⎫ α=0.681, Co=82.82 =⎨ ⎬
0.2
σ0
⎩ ci ⎭
0.0
0.8 0.6 0.4 Measured Resutls α Fitted Resutls σ R ⎧ c0 ⎫ α=1.013, Co=68.60 =⎨ ⎬
0.2
σ0
⎩ ci ⎭
0.0 0
50
100
150
200 2
250
0
50
100
150
200 2
Damaged area, ci (mm )
Damaged area, ci (mm )
c SG10
d SG20
Fig. 3. Strength reduction behavior in impacted sandwich panels.
250
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
223
Table 4 (a) Impact damage area for SC panels. (b) Impact damage area for SG panels. Ei
SC10 SC20
4.0J 8.0J 4.9J 8.9J
Damage area, ci [mm2] 1
2
3
4
5
6
7
Mean
Variance
813.1 1522.7 1017.9 1608.3
866.6 1510.9 1055.1 1576.8
873.7 1430.0 956.8 1467.3
824.8 1605.9 930.9 1591.5
885.23 1409.8 1005.8 1547.8
848.07 1583.5 967.9 1487.5
824.8 1525.0 997.3 1521.7
849.6 1512.6 990.2 1542.9
671.8 5234.8 1734.9 2841.6
(b)
Ei
Damage area, ci [mm2] 1
2
3
4
5
6
7
Mean
Variance
SG10
3.0J 8.0J 3.0J 8.0J
116.7 116.9 128.5 171.2
117.7 151.6 136.8 144.9
110.1 131.9 113.8 148.1
75.62 123.4 134.2 149.7
117.2 114.6 128.7 157.2
115.4 133.4 135.4 140.9
108.4 136.8 145.4 147.6
108.7 129.8 131.8 151.5
226.3 163.3 94.0 102.9
SG20
Similarly, the first-order variance of the response variable Var(Y0 ) can be shown to be n n X n X X Var Y ’ ¼ S2i VarðXi Þ þ Si Sj Cov Xi ; Xj for isj i¼1
i¼1
(5)
j
in which the constants Si and Sj represent the partial derivatives vg=vXi and vg=vXj , respectively, calculated at the mean values of the basic variables Xi. Also, these constants can be interpreted as
significance factors for the uncertainties in each of the corresponding random variables Xi. In general, these significance factors show the importance of the variables in the formulation [24]. If we can obtain the probabilistic properties (mean and variance) of the basic random variables, as well as the function g between the response variable and the basic variables, the probabilistic properties of the response variable can be evaluated easily according to the above procedure. Let us assume that we know the mean and variance of the basic random variables (the original strength s0 and the impact damage
1.0 Measured Results for SC10
Cumulative distribution function, F
Cumulative distribution function, F
1.0
0.8
0.6
0.4 Results at 8.0J m = 1512.8 μ = 72.35
Results at 4.0J m = 849.6 μ = 25.92
0.2
0.0 0
500
1000
1500
2000
Measured Results for SC20
0.8
0.6
0.4 Results at 8.9J m = 1542.9 μ = 53.31
Results at 4.9J m = 990.2 μ = 41.65
0.2
0.0
2500
0
2
500
1000
2000
2500
Damaged Area, ci (mm )
a SC10
b SC20
1.0
1.0 Measured Results for SG20
Cumulative distribution function, F
Cumulative distribution function, F
1500
2
Damaged Area, ci (mm )
0.8
0.6
0.4 Results at 8.0J m = 151.5 μ = 10.14
Results at 3.0J m = 131.8 μ = 9.70
0.2
0.0 0
50
100
150
200 2
Damaged Area, ci (mm )
c SG10
250
Measured Results for SG10
0.8
0.6
0.4 Results at 8.0J m = 129.8 μ = 12.79
Results at 3.0J m = 108.7 μ = 15.05
0.2
0.0 0
50
100
150
200 2
Damaged Area, ci (mm )
d SG20
Fig. 4. Cumulative distribution function for impact damage area.
250
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J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
Table 5 (a) Estimated probabilistic properties (mean and variance) for SC panels. (b) Estimated probabilistic properties (mean and variance) for SG panels. SC10 Ei
4.0J 8.0J
SC20 Measured
Estimated
Mean
Var
Mean
Var
55.43 40.14
23.75 23.77
52.85 38.33
32.18 17.55
SG10 Ei
3.0J 8.0J
Ei
Measured
Estimated
Mean
Var
Mean
Var
4.9J 8.9J
31.59 25.15
2.47 0.59
30.88 23.99
3.41 1.75
SG20 Measured
Estimated
Ei
Mean
Var
Mean
Var
39.70 25.39
6.04 1.57
38.88 26.65
13.04 2.75
3.0J 8.0J
Measured
Estimated
Mean
Var
Mean
Var
32.45 26.41
1.75 0.47
31.69 27.06
2.90 0.88
area ci). If those values are known, once the function f(ci) in Eq. (1) has been determined from the experimental data, Eqs. (4) and (5) can be easily used to find the probabilistic properties of the residual strength sR of composite materials subjected to impact loading. It is well known that strength reduction behavior can be represented as a function of impact damage according to Caprino’s approach [20], as follows.
sR ¼ s0
f ci ¼
a c0 ci
(7)
Y ¼ sR ; X1 ¼ s0 ; X2 ¼ ci
(8)
Combining Eq. (4) with Eqs. (7) and (8), one obtains a first-order approximation of the expected residual strength at a certain impact energy, as follows:
a c E s’R ¼ ms0 , 0 ci
(9)
in which ms0 indicates the mean value of the original strength. Because the original strength s0 (X1) is an inherent property of composite materials, and the impact damage area ci (X2) is caused by impact loading, it is reasonable to assume that the original
1.0
Cumulative distribution function, F
Results for SC10 at 4.0J Tests results
0.8
0.6
Measured Results mean = 0.807 standard deviation = 0.071
0.4 Simulated Results mean = 0.769 standard deviation = 0.124
0.2
0.0 0.2 0.3
0.4 0.5 0.6 0.7 0.8
Results for SC10 at 8.0J Tests results
0.8
0.6 Measured Results mean = 0.584 standard deviation = 0.071
0.4
Simulated Results mean = 0.558 standard deviation = 0.092
0.2
0.0
0.9 1.0
0.2 0.3
Normalized Residual Strength, σR,i/σ0,m
0.4 0.5 0.6 0.7 0.8
0.9 1.0
Normalized Residual Strength, σR,i/σ0,m
a SC10 : 4.0J
b SC10 : 8.0J
1.0
1.0 Results for SC20 at 4.9J Tests results
Cumulative distribution function, F
Cumulative distribution function, F
(6)
In this expression, E and Eth denote the incident impact energy and the threshold energy (which is the minimum energy required to cause a strength reduction in the structures), respectively, c0 and ci are the inherent and impact damage areas, respectively, and a is the material constant. Comparison of Eqs. (1), (2) and (6) leads to the following relationship:
1.0
Cumulative distribution function, F
a c0 ; Ei Eth ci
0.8
0.6
Measured Results mean = 0.780 standard deviation = 0.039
0.4 Simulated Results mean = 0.763 standard deviation = 0.069
0.2
0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Residual Strength, σR,i/σ0,m
c SC20 : 4.9J
Results for SC20 at 8.9J Tests results
0.8
0.6
Measured Results mean = 0.621 standard deviation = 0.019
0.4
0.2
Simulated Results mean = 0.592 standard deviation = 0.049
0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Residual Strength, σR,i/σ0,m
d SC20 : 8.9J
Fig. 5. Comparison of measured and estimated distribution of SC panels.
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
strength and the impact damage are not correlated. Therefore, the first-order variance of the residual strength can be obtained as follows from Eq. (5) [24].
Var s’R ¼ S21 s0 þ S22 ci S1 ¼
(10)
a a vg v c c0 s0 0 ¼ ¼ vX1 mX vs0 ci ci ms ms 1
S2 ¼
0
(11) 0
a as0 c0 a vg v c0 s ¼ ¼ 0 ci ci ci mc vX2 mX vci mc 2
i
(12) i
in which mci is the mean value of the impact damage at a given impact energy. From the above procedure, the probabilistic distribution of the residual strength sR of a composite material subjected to a given impact energy can be obtained from the probabilistic properties (mean and variance) of the original strength and impact damage. 3.2. Experimental verification To determine the probabilistic distribution of the residual strength in the impacted sandwich structures according to abovementioned procedure, the mean and variance of the original strength must be determined first. Therefore, seven specimens of each sandwich system were statically tested under flexural loading,
and the results are summarized in Table 3. Fig. 2 shows the cumulative distribution functions of each material, as evaluated by the normal distribution. As shown in the figure, the flexural strength data of the unimpacted laminates correlate well with the normal distribution. As shown in Eqs. (6) and (9), the relationship between the residual strength and the impact damage area must be established. To determine this relationship, impact tests were conducted in the range of approximately 2-13J for the SC and SG panels. The resulting damage was measured with a scanning acoustic microscope (SONIX HS1000, 5 MHz). Three point flexural tests were then conducted for the impacted sandwich panels. The residual strength data, normalized to the mean strength of the unimpacted sandwich panels, were plotted against the impact damage area in Fig. 3. Here, the solid marks and line represent the experimental results and the fitted results according to Caprino’s approach shown in Eq. (6) [6,20], respectively. From the figure, it is clear that the residual strength was not affected by the damage area below a threshold value. Thus, this value can be defined as the characteristic damage area c0, which is the minimum damage area that causes strength reduction in structures. Above this level, the residual strength decreases rapidly as the impact damage area increases until it approaches approximately 50% of the original strength of the unimpacted sandwich panels for both sandwich systems. According to Caprino’s approach in Eq. (6), the rate of strength reduction is described by the exponent a. To identify the probabilistic distribution of the residual strength in the impacted panels, it is also necessary to determine the 1.0
Results for SG10 at 3.0J Tests results
Cumulative distribution function, F
Cumulative distribution function, F
1.0
0.8
0.6 Measured Results mean = 0.714 standard deviation = 0.044
0.4
0.2
Simulated Results mean = 0.699 standard deviation = 0.098
0.0 0.2 0.3
0.4 0.5 0.6 0.7 0.8
Results for SG10 at 8.0J Tests results
0.8 Measured Results mean = 0.457 standard deviation = 0.023
0.6
0.4 Simulated Results mean = 0.479 standard deviation = 0.045
0.2
0.0
0.9 1.0
0.2 0.3
Normalized Residual Strength, σR,i/σ0,m
0.4 0.5 0.6 0.7 0.8
0.9 1.0
Normalized Residual Strength, σR,i/σ0,m
a SG10 : 3.0J
b SG10 : 8.0J
1.0
1.0 Results for SG20 at 3.0J Tests results
Cumulative distribution function, F
Cumulative distribution function, F
225
0.8
0.6 Measured Results mean = 0.758 standard deviation = 0.031
0.4 Simulated Results mean = 0.741 standard deviation = 0.060
0.2
0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Results for SG20 at 8.0J Tests results
0.8
0.6 Measured Results mean = 0.611 standard deviation = 0.016
0.4
0.2
Simulated Results mean = 0.632 standard deviation = 0.033
0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Residual Strength, σR,i/σ0,m
Normalized Residual Strength, σR,i/σ0,m
c SG20 : 3.0J
d SG20 : 8.0J
Fig. 6. Comparison of measured and estimated distribution of SG panels.
226
J.-H. Kim et al. / Renewable Energy 54 (2013) 219e226
Table 6 (a) Significance factors Si in each random variable for SC panels. (b) Significance factors Si in each random variable for SG panels. SC10
SC20
Ei
S1
S2
Ei
S1
S2
4.0J 8.0J
0.76907 0.55778
0.03464 0.01411
4.9J 8.9J
0.76244 0.59225
0.01776 0.00885
Ei
S1
S2
Ei
S1
S2
3.0J 8.0J
0.62717 0.52418
0.32494 0.22751
3.0J 8.0J
0.72879 0.66263
0.16126 0.12750
SG10
SG20
probabilistic properties (mean and variance) of the impact damage at a given impact energy. For this measurement, four specimen sets were tested at 4.0 J and 8.0 J for SC10 and 4.9 J and 8.9 J for SC20. The impact tests for the SG panels were carried out at 3.0 J and 8.0 J for the SG10 and SG20 panels, respectively. The results are summarized in Table 4, and Fig. 4 indicates the cumulative distribution function of the impact damage area for the SC and SG panels. From the probabilistic properties (mean and variance) of the original strength and the impact damage area at a given impact energy level and the relationship between the residual strength and impact damage area of the impacted structures, it is possible to estimate the residual strength distribution for a particular incident impact energy level Ei by using Eqs. (6)e(12). At a given impact energy level, the mean and variance of the residual strength can be estimated by Eqs. (9) and (10), respectively. The predicted results are summarized in Table 5. The proposed approach can be used to predict the distribution of the residual strength at specific impact energy levels. For verification of the approach, theoretical predictions were made for given energy levels, and comparisons were made between the estimated (simulated) and experimental (measured) results for the SC and SG panels (Figs. 5 and 6, respectively). Here, the estimated and experimental results were normalized to the mean strength of the unimpacted laminates. The measured results were obtained by directly applying the normal distribution to the seven data sets of residual strength at given energy levels. In the figures, the measured results are indicated by a solid line and the estimated results as a dotted line. The estimated distributions compare well with the measured distributions for both the SC and SG panels. Because the proposed model consists of two parameters, the original strength and the impact damage area, it is of interest to determine which parameter plays the primary role in determining the uncertainty of the residual strength of the impacted structures. To identify this parameter, the significance factors S1 and S2 were calculated from Eqs. (11) and (12), respectively; their results are summarized in Table 6. The results indicate that the original strength plays the primary role in defining the uncertainty of the residual strength, and the effect of the impact damage uncertainty is relatively small. This behavior may be a result of the fact that the impact damage behaves as a discontinuity because it is a stress concentration in the structure. Although the impact damage has a considerable amount of scatter at a given impact energy, it is well known that the impact damage causes a large reduction in the strength of composite structures [20]. Therefore, the impact damage can be inferred to produce a stress concentration rather than the uncertainty in the residual strength of the impacted structure. 4. Conclusions This paper provides a two-parameter approach to the prediction of the probabilistic distribution of residual strength at a given impact energy in impacted sandwich structures. The approach was
derived from the relationship between the response variable (the residual strength) and the basic random variables (the original strength and impact damage). The distribution of the residual strength was calculated from the probabilistic properties (mean and variance) of the original strength and the impact damage at a given impact energy. By using an experimental program on NomexÒ honeycomb core sandwich structures, this approach was found to be capable of predicting the distribution of residual strength at a given impact energy level. Acknowledgment This work was financially supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No. 20124010203240) and by Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology (2011-0007012). References [1] Kang KW, Lim DM, Kim JK. Probabilistic analysis for the fatigue life of carbon/ epoxy laminates. Comp Struct 2008;85:258e64. [2] Kang KW, Kim JK. Fatigue life prediction for impacted carbon/epoxy laminates under 2-stage loading. Comp A 2006;37:1451e7. [3] Aktas¸ M, Karakuzu R, Arman Y. Compression-after impact behavior of laminated composite plates subjected to low velocity impact in high temperatures. Comp Struct 2009;89:77e82. [4] Sánchez-Sáez S, Barbero E, Navarro C. Compressive residual strength at low temperatures of composite laminates subjected to low-velocity impacts. Comp Struct 2008;85:226e32. [5] Cui HP, Wen WD, Cui HT. An integrated method for predicting damage and residual tensile strength of composite laminates under low velocity impact. Comp Struct 2009;87:456e66. [6] Corum JM, Battiste RL, Ruggles-Wrenn MB. Low-energy impact effects on candidate automotive structural composites. Comp Sci Tech 2003;63:755e69. [7] Schuble PM, Luo JJ, Daniel IM. Impact and post impact behavior of composite sandwich panels. Comp A 2007;38:1051e7. [8] Casanie B, Bouvet C, Aminanda Y, Barrau JJ, Thevenet P. Modelling of lowenergy/low-velocity impact on Nomex honeycomb sandwich structures with metallic skins. Intl J Imp Eng 2008;35:620e34. [9] Swanson RS, Jongman K. Design of sandwich structures under contact loading. Comp Struct 2003;59:403e13. [10] Tanaka S, Ichikawa M, Akita S. A probabilistic investigation of fatigue life and cumulative cycle ratio. Eng Frac Mech 1984;20:501e13. [11] Ni K, Zhang S. Fatigue reliability analysis under two-stage loading. Relia Eng System Saf 2000;68:153e8. [12] Yang JN, Jones DL. Effect of load sequence on the statistical fatigue of composites. AIAA J 1980;18:1525e31. [13] Hwang W, Han KS. Cumulative damage models and multi-stress fatigue life prediction. J Comp Mater 1989;20:125e53. [14] Tai NH, Yip MC, Lin JL. Effects of low-energy impact on the fatigue behavior of carbon/epoxy compsites. Comp Sci Tech 1998;58:1e8. [15] Aoki Y, Yamada K, Ishikawa T. Effect of hygrothermal condition on compression after impact strength of CFRP laminates. Comp Sci Tech 2008;68: 1376e84. [16] Shim VPW, Yang LM. Characterization of the residual mechanical properties of woven fabric reinforced composites after low-velocity impact. Int J Mech Sci 2005;38:647e65. [17] Tai NH, Yip MC, Tseng CM. Influences of thermal cycling and low-energy impact on the fatigue behavior of carbon/PEEK laminates. Comp B 1999;30: 849e65. [18] Kang KW, Kim HS, Kim MS, Kim JK. Strength reduction behavior of honeycomb sandwich structure subjected to low velocity impact. Mat Sci Eng 2008; A483-484:333e5. [19] Lee SP, Lee JH, Kang KW. Residual strength distribution of impacted glass/ epoxy laminates with embedded shape memory alloy at low temperatures. Comp Struct 2011;93:324e30. [20] Caprino G. Residual strength prediction of impacted CFRP laminates. J Comp Mater 1984;18:508e18. [21] ASTM. Standard test method for tensile properties of fiber resin composite, D3039eD3000; 2000. [22] Harai Y, Hamada M, Kim JK. Impact response of woven glass-fabric composites-II effect of temperature. Comp Sci Tech 1998;58(1):119e212. [23] Kim JK, Mackay DB, May YW. Drop-weight impact damage tolerance of CFRP with rubber modified epoxy matrix. Composites 1993;24:458e94. [24] Haldar A, Mahadevan S. Probability, reliability, and statistical methods in engineering design. New York: John Wiley & Sons; 2000.