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27
УДК 539.3
Анализ прочности хаотично-армированных волокнами слоистых композитов с изменчивостью макромеханических свойств
S. Li1, Z.Y. Ma2
1 Северо-Китайский технологический университет, Пекин, 100144, Китай 2 Шанхайский университет, Шанхай, 200444, Китай
Механические свойства слоистых композитов, армированных хаотично-направленными волокнами, имеют случайный характер, что вызывает трудности при проектировании безопасных конструкций. В статье проведен анализ чувствительности свойств хаотично-армированных слоистых композитов к схеме укладки слоев при различных нагрузках на основе усовершенствованной теории Puck. Усовершенствованная теория Puck с учетом in situ влияния прочности, которая также учитывает влияние как отдельного слоя, так и соседних слоев, позволяет точно предсказывать начало и конец разрушения. Рассмотрено поведение трех видов композитов: композиты [450/0°/-450/90°]2s при одноосном растяжении и композиты [0°/90°/90°/0°]s при одноосном и двухосном растяжении. Показано, что разрушающие нагрузки подчиняются нормальному или логнормальному распределению при логнор-мальном распределении свойств материала, имеющих случайную природу вследствие хаотичного распределения волокон. Согласно результатам анализа чувствительности, продольная прочность однонаправленного слоя на растяжение оказывает наибольшее влияние на окончательное разрушение слоистых композитов и является положительной по отношению к конечным разрушающим нагрузкам. Однако влияние отдельных свойств материала на начало разрушения отличается ввиду различных механизмов разрушения слоистых композитов в зависимости от схемы укладки слоев и условий нагружения. Полученные результаты дают представление о влиянии свойств материалов на разрушение слоистых композитов для обеспечения безопасности конструкций.
Ключевые слова: армированный волокнами слоистый композит, разрушение, in situ влияние прочности, изменчивость, чувствительность
DOI 10.24412/1683-805X-2021-2-83-90
Strength analysis of multidirectional fiber-reinforced composite laminates with uncertainty in macromechanical properties
S. Li1 and Z.Y. Ma2
1 School of Mechanical and Materials Engineering, North China University of Technology, Beijing, 100144, PR China
2 Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200444, PR China
Mechanical properties of multidirectional fiber-reinforced composite laminates are dispersive and random, and these uncertainties cause difficulties in safety design of structures. So, it is necessary to analyze the sensitivity of uncertain material properties on failure of structures. In this paper, sensitivity analysis of uncertainties for composite laminates with different stacking sequences and under various loadings is performed by the improved Puck's theory and the importance measurement analysis method. The improved Puck's theory with the in situ strength effect which considers the influence of both the lamina itself and its neighboring laminae can predict the initial and final failure accurately. Three examples ([45°/0°/-45°/90°]2s laminates under uniaxial tension, [0°/90°2/0°]s laminates under uniaxial tension and biaxial tension) are examined, and results show that the failure loads obey normal or log-normal distribution when the distribution of uncertain material properties is log-normal. Results of sensitivity analysis show that the longitudinal tensile strength of a unidirectional lamina has the greatest influence on the final failure for laminated composites, and is positive relative to final failure loads. As for initial failure, the influence of each material property on failure is different because of different failure mechanisms of composite laminates with different stacking sequences and under various loadings. The results provide an in-depth understanding of the influence of material properties on failure of composite laminates for safety design.
Keywords: fiber-reinforced composite laminate, failure, in situ strength effect, uncertainty, sensitivity
© Li S., Ma Z.Y., 2021
1. Introduction
Laminated composite as a special engineering structure is applied extensively in many fields, including aerospace, ship and energy industry, etc. However, there are always uncertainties in the structure resulting from defects [1], manufacturing process [2] and experimental measurement. These uncertainties cause the dispersive and random material properties and geometries of the laminated composite within all scales [3-6], and then cause uncertain mechanical behaviors such as failure strengths of the structure. Therefore, the analysis of the uncertainties has been a necessary task to design composite structures and guarantee structural safety.
Currently, many research activities are dedicated to the analysis for the effect of material, geometrical and environmental uncertainties on composite structures within all scales. Cederbaum et al. [7] studied in-plane failure probability of composite laminates with considering random strength parameters of unidirectional lamina at the macroscale. Zhou et al. [8] studied the influence of material uncertainties on static strength of quartz/epoxy composite structures under out-of-plane loadings. Zhou et al. [5] studied the reliability of composite structures accounting for uncertainties at both micro- and macroscales, such as material and ply angles. Omairey et al. [6] studied the effect of microscale uncertainties on the macro-scale elastic properties of fibre-reinforced composites. Bogdanor et al. [9] studied the influence of random geometries on notched tensile strength. Khiat et al. [3] analysed uncertainties of the unidirectional lamina strength due to variation of environmental condition. However, to the authors knowledge, the influence of uncertain material properties on initial and final failure strengths of a fibre-reinforced composite laminate under uniaxial and biaxial loadings is yet to be investigated, especially when the in situ strength effect of the laminated composites is considered.
In order to analyse the influence of uncertainties on composite structures during structural optimization designing, the sensitivity analysis combined with mechanical models is used [6, 10, 11]. The sensitivity analysis includes local sensitivity analysis and global sensitivity analysis. Local sensitivity analysis only considers the influence of uncertain input variables on output variables near a certain input point, but global sensitivity analysis works in the whole distribution area of the input variables. In order to analyse the influence of uncertainties on failure of composites, the failure model is used as the corresponding mechanical model. As to failure models,
most of them for multidirectional fiber-reinforced composite laminates are macroscopic strength theories. Among these theories, stress analysis and failure criteria are of great importance [12, 13]. For stress analysis, the linear classical laminate theory (CLT) is used widely as a constitutive model [12], and other factors such as nonlinear effect and residual stress are also considered in some failure theories. As to failure criteria [14-16], there are two world wide failure exercises namely WWFE-I and WWFE-II [17, 18] to evaluate the accuracy of them. The Puck theory which accounts for different physical damage mechanisms resulting from composite anisotropy and in-homogeneity behaves well, and it becomes one of the first ten theories with high accuracy. However, there is a large difference between the predicted initial failure strength and experiments [19]. In order to solve this problem, the improved Puck's theory with in situ strength effect is proposed by Wang's research group [20]. It is confirmed that the improved Puck's theory is accurate enough to predict the initial and final failure strengths of the composite laminates under uniaxial and biaxial tension [20], and to study the fatigue behavior of the laminates [12]. Due to excellent performance of the improved Puck's theory, it will be used in this paper to predict the failure strengths which are preliminary data for sensitivity analysis.
The main goal of this paper is to analyse the influence of uncertain material properties on failure strengths of multidirectional fiber-reinforced composite laminates accurately.
2. The improved Puck's theory with the in situ strength effect
The 2D model of a composite laminate under tensile loadings is shown in Fig. 1. In order to predict the initial and final failure strengths of the above model, the improved Puck's theory which has been
11 ■ t i
—
-— o
U I 1
Fig. 1. The model of a composite laminate under tensile loadings
verified accurate enough [20] is used here. In this failure theory, the influence of both the thickness of lamina itself and the ply angles of adjacent laminae is considered. The improved Puck's theory contains the fiber failures and interfiber failures, and there is a brief review of these two failure types.
The fiber failures denoting final failure have two different modes defined as [20]
2 , ,, E1
— V
X 12 X
> o,
f12
-m,
Of
= 1,
Jfi
X t
— V
X b < 0,
12
X c
f12
Ei
m
of
-fi
X t
(1)
+ (10 Y12)2 =1,
where E1, v12 are the longitudinal modulus and the Poisson ratio of the lamina, Ef1, vf12 are the longitudinal modulus and the Poisson ratio of the fiber, Xt, Xc are the tensile and compressive strengths of the lamina in the fiber direction, c1, c2 are the longitudinal and transverse normal stresses of the lamina, y12 is the in-plane shear strain of the lamina, mf is the 'stress magnification effect' because of the mismatch between the moduli of fibers and matrix.
As to interfiber failures, there are three different failure modes defined as [20] mode A:
Y (
42
SI
S12 7
1 - P-Y Y
i Y
S i
S12 7
2
Y1
VJ t 7
P(+) -O2.
"PHI Si S12
= 1 -
1D
, 02 > 0,
mode B:
^2 + ( Pi ||)o2)2 + Pi ||)o2 = . SI =
S12
o2 < 0and 0 <
(2)
On
42
R
ii
12c
mode C:
(
Y ( ~ \
12
2(1 + Pj-L)S/
+ 02
Y
c
= 1 -
—Oo
1D
o2 < 0and 0 <
12
On
<
12c
R
ii
where Y/, S12I are the transverse tensile and in-plane shear strength of the lamina embedded in the laminate, Yc is the transverse compressive strength of the
isolated lamina, c1/c1D denotes the degradation of the fracture resistance, p(+:l||, p(-)i||, p(-)n are constants interrelated with the material of the laminate. The expressions for all these parameters have been shown in detail in [20]. The interfiber failure criteria include initial, intermediate and final failures. If the initial or intermediate failure occurs, the stiffness {E2, G12, ^12} is gradually reduced by a degradation factor n, i.e., mode A:
{E2, G^ vi2 } ^ fr^ nv12 }, (3)
mode B and mode C:
{^ v12 } ^ {^ nG^ v12 },
where n = 1/fE (fE > 1) and fE is the left hand terms of (2). The final failure will occur when tan9Cfp > is satisfied, where 0Cfp is the angle of the fracture plane in mode C and ^ is the coefficient of friction.
3. Sensitivity analysis of uncertainties
In this section, the method based on nonlinear regression is used to analyse the importance measurement of the uncertain material properties, and it can indicate the contribution of input variables to response variance briefly and accurately [21]. A brief introduction of this method is shown next.
There are some independent input variables expressed as
X = [X1, *2,..., Xm ], (4)
where m is the total number of the input variables. The corresponding response variables are
Y = f (X) = f (X1, X2,..., Xm), f: Mm ^ M. (5) The method of importance measurement analysis based on variance is divided into three steps [21].
Step 1: For the input variable xi (i = 1, 2, ..., m), N random samples (x1i, x2i, ..., xNi,) are generated based on the input variable's probability density function and are arranged in a column. (N* m)-dimensional sample matrix for input variables is
X =
41
21
42
v22
4m
2m
Nm
(6)
and (N * m)-dimensional matrix for the square of input samples is
-2 „2 „2
X2 =
11 x21
N1
12 x22
N 2
1m x2m
xNm
(7)
D
Step 2: Substituting the random input variables into corresponding files (the improved Puck's theory mentioned in Sect. 2) leads to output variables. (N x 1)-dimensional matrix for response variables is
y1 f ( X11 x12, . ., X1m )
Y = y2 = f ( X21: X22, . ., X2m ) . (8)
_ yN _ _ f ( XN1' XN2, . .., XNm )_
The mean value y of response variables is
yi + J2 + - + ym
y =-
N
(9)
and the total variance of response variables is 1 N
V=NTïg<y-y)2 (10)
Step 3: In order to estimate the contribution of one input variable x, to response variance, the regression of response variables expressed by x, and x,2 is used here, i.e.,
y, = a, +Pixi + y,x,2 +e, i = 1,2,..., m, (11) where i represents the order of variables and is not summed, a,, p, and y, are regression coefficients, e is error. The contribution of xi to response variance is estimated by
V =-
1 I ( j - y)2
N -1
j=1
where
y(i) = « i + ( ,x,i + Y ,x 2
(12)
(13)
j' " j"
ex', (3' and y' can be obtained by the least squares method. The effect index which represents the total contribution rate of the input variable x' itself to response variance is obtained as
S, = x100%. i V
(14)
4. Results and discussion
In this section, we apply the above analysis of failure prediction and uncertainties to multidirectional fiber-reinforced composite laminates. All laminates are with the same material type, and the corresponding properties of the unidirectional lamina are listed in Table 1. It is assumed that uncertain material properties obey log-normal distribution, and the corresponding standard deviation is shown in Table 1. In the following parts, the influence of uncertain material properties on the failure of laminates with arbitrary stacking sequences and under various loadings is analysed.
The first example is a quasi-isotropic [45°/0°/ -45°/90°]2s laminate under uniaxial tension (oy/ax = 0/1). Due to uncertain material properties, the initial and final failure are also random, and the corresponding distribution of failure strength determined by the improved Puck's theory with the in situ strength effect is shown in Fig. 2. Figure 2, a shows the distribution of initial failure loads, and it is normal or lognormal. Final failure distribution is also normal or log-normal shown in Fig. 2, b. After failure loads obtained, the effect indices of random material properties to initial failure and final failure are determined and shown in Fig. 3. Figure 3, a shows that transverse tensile strength, longitudinal modulus and inplane shear strength have great influence on initial failure loads, and results of the improved Puck's theory show that the first damage occurs in outer 45° laminae. Figure 3, b indicates that longitudinal tensile strength has a decisive effect on final failure of the [45°/0°/-45°/90°]2s laminate under uniaxial tension. After sensitivity analysis of material, the influence of important factors on failure tendency is shown in Fig. 4. Figure 4, a shows that initial failure loads increase with increasing Yt, E1 and ^12, but with
Table 1. Material properties [22]
Random variables Mean value Standard deviation Distribution type
Longitudinal modulus E1, GPa 39.042 1.032 Log-normal
Transverse modulus E2, GPa 14.077 0.325
In-plane shear modulus G12, GPa 4.239 0.099
In-plane Poisson ratio v12 0.291 0.027
Longitudinal tensile strength Xt, MPa 776.500 36.143
Longitudinal compressive strength Xc, MPa 521.820 16.501
Transverse tensile strength Yt, MPa 53.950 2.576
Transverse compressive strength Yc, MPa 165.000 4.849
In-plane shear strength S12, MPa 56.080 1.119
Fig. 2. The distribution of failure loads for a [45°/0o/-45o/90o]2s laminate under uniaxial tension (oy/ox = 0/1). Initial failure (a); final failure (b). Frequency (1), normal law (2), log-normal law (3) (color online)
H A'l 13.78% |a_
■ G12 26.8%
□ v12
X i
□ 1
v 40.89% n/ {^■4.1%
™7C ™ 1 \ 1^7.13%
mSl2 \ 7% 0.31%
5.6% UL Others 0.18%
□ G12
□ V12 / \
□ ^t /
!
[ 1 Yt \ 93.68% /
msl2
Fig. 3. The effect indices of random material properties to the failure loads for a [45°/0°/-45°/90°]2s laminate under uniaxial tension (oy/ox = 0/1). Initial failure (a); final failure (b) (color online)
Fig. 4. The influence of material properties on the tendency of failure strength for a [45°/0°/-45°/90°]2s laminate under uniaxial tension (oy/ox = 0/1). Initial failure (a), final failure (b) (color online)
decreasing G12, v12 and E2. Figure 4, b shows that final failure loads increase with the increase of both X and Yt.
The second example is an orthotropic [0°/90°/90°/ 0°]s laminate under uniaxial tension (ay/ax=0/1). Figure 5 shows the distribution of initial and final failure loads, and the distribution is almost normal or log-normal which is the same with the distribution law of quasi-isotropic laminates. Figure 6, a shows that Yt, E1 and E2 have great influence on initial failure loads, and results of the improved Puck's theory indicate that matrix tensile damage occurs succes-
sively in 90° laminae, outer 0° laminae and inner laminae. Figure 7, a shows that initial failure loads increase with the increase of Yt and E1, but with decreasing E2. As to final failure, Xt is crucial and positive relative to final failure loads which are shown in Figs. 6, b and 7, b, respectively.
The third example is an orthotropic [0°/90°/90°/ 0°]s laminate under biaxial tension (ay/ax = 2/1). Figure 8 shows that the distribution of failure loads is normal or log-normal which is the same with the distribution type for other two examples. As to initial failure, the results of the improved Puck's the-
Fig. 5. The distribution of failure loads for a [0°/90°/90°//0°]s laminate under uniaxial tension (cy/cx = 0/1). Initial failure (a), final failure (b). Frequency (1), normal law (2), log-normal law (3) (color online)
H/'M |a_ 13.69%
1 1E2
□ G12 /m X
□ v12 I
□^t I JKK^*- 0.85%
m
— 11 ■ w
«12 ^ 1
75.2%
Fig. 6. The total effect indices of random material properties to the failure loads for a [0°/90°/90°//0° tension (oy/ox = 0/1). Initial failure (a), final failure (b)
laminate under uniaxial
Fig. 7. The influence of material properties on the tendency of failure strength for a [0°/90°/90°/0° sion (oy/ox = 0/1). Initial failure (a); final failure (b) (color online)
laminate under uniaxial ten-
ory show that matrix tensile damage occurs successively in outer 0° laminae, 90° laminae and inner 0° laminae, which is different from the failure data of [0°/90°/9070°]s laminates under uniaxial tension. Figures 9, a and 10, a show that Yt, E1 and E2 have great influence on initial failure, and initial failure loads increase with increasing Yt and E1 but with decreasing E2. This is almost the same with the influence of material on initial failure loads of the [0°/90°/90°/0°]s laminate under uniaxial tension. However, the results in Figs. 9, a and 6, a show that
Yt has greater influence on laminates under biaxial tension than uniaxial tension. Figures 9, b and 10, b show that Xt is important to final failure loads, and final failure loads increase with increasing Xt.
5. Conclusions
In this paper, sensitivities of uncertain material properties for multidirectional fiber-reinforced com posite laminates with different stacking sequences and under various loadings are analysed. In order to
Fig. 8. The distribution of failure loads for a [0°/90°/90°/0°]s failure (b). Frequency (1), normal law (2), log-normal law (3)
failure
laminate under biaxial tension (ay/ax = 2/1). Initial failure (a), final
Fig. 9. The total effect indices of random material properties to the failure loads for a [0°/90°/90°/0°]s laminate under biaxial tension (Cy/ox = 2/1). Initial failure (a), final failure (b) (color online)
Fig. 10. The influence of material properties on the tendency of failure strength for a [0°/90°/90°/0° sion (oy/ox = 2/1). Initial failure (a), final failure (b) (color online)
laminate under biaxial ten-
solve this problem accurately, the improved Puck's theory considering the in situ strength effect is combined with the importance measurement analysis here. Results show that the failure loads obey normal or log-normal distribution. The results of sensitivity analysis show that longitudinal tensile strength has the greatest influence on the final failure, and is positive relative to final failure loads. As to initial failure of laminates with different stacking sequences and under various loadings, the influence of each material property is different because of different failure mechanisms of composites.
Funding
The authors thank the support of the National Key Research and Development Program of China (Grant No. 2020YFF0304905) and the Scientific Research Foundation for Scholars in NCUT (1100513 60002).
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Received 19.10.2020, revised 07.04.2021, accepted 08.04.2021
Сведения об авторах
Shu Li, North China University of Technology, PR China, lishu@ncut.edu.cn Zhaoyang Ma, Shanghai University, PR China, mazhaoyang@shu.edu.cn
