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Analysis of stress distribution near a blunt surface notch tip in an orthotropic fiber under tension
Accepted Manuscript Analysis of stress distribution near a blunt surface notch tip in an orthotropic fiber under tension Yoshiki Sugimoto, Kazuro Kageyama PII: DOI: Reference:
S0167-8442(16)30384-6 http://dx.doi.org/10.1016/j.tafmec.2017.01.004 TAFMEC 1798
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
23 November 2016 24 December 2016 13 January 2017
Please cite this article as: Y. Sugimoto, K. Kageyama, Analysis of stress distribution near a blunt surface notch tip in an orthotropic fiber under tension, Theoretical and Applied Fracture Mechanics (2017), doi: http://dx.doi.org/ 10.1016/j.tafmec.2017.01.004
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of stress distribution near a blunt surface notch tip in an orthotropic fiber under tension Yoshiki Sugimoto(*), Kazuro Kageyama
Department of Technology Management for Innovation, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Abstract The intrinsic strength of carbon fiber is determined from tensile tests of carbon fibers with a blunt surface notch using a point stress criterion. A stress distribution around the blunt surface notch tip is needed for the carbon fiber with cylindrical shape and orthotropic mechanical properties. In this study, the stress distribution around a blunt surface notch in a fiber with various orthotropy under tension were calculated using a finite element method. An effect of orthotropy on the stress distribution is investigate. To predict the stress distribution for various notch and various orthotropy, an approximate stress distribution equation was proposed.
Key words Fiber; Finite element analysis; Point Stress Criterion; Notch; Orthotropy
* Corresponding author. Tel/Fax: +81 3 5841 7767. E-mail address: [email protected]
1. Introduction Recently, the demands for carbon fiber reinforced plastics, which have high tensile strength, high tensile modulus and light weight, are increasing to reduce automobiles’ weight for reducing energy consumption [1]. Carbon fibers are, however, expensive, and cannot use for general vehicles at present stage [2]. Therefore, carbon fibers from new precursors or process are actively developed to reduce production cost [3]. The carbon fibers are produced by passing through the complicate processes: synthesize, drafting, and carbonization. Therefore, it is very difficult to achieve high tensile performance in the developing stage. It is necessary to know the potential of carbon fiber to judge whether developing should continue or not. However, the carbon fiber strength is usually determined by the structure of carbon fiber and surface defects. To develop the higher strength of carbon fiber, it is very important to separate these factors. The strength depends on the carbon fiber nano-order structure is called the intrinsic strength [4,5]. The intrinsic strength of carbon fibers is measured from tensile tests of the carbon fiber with a blunt surface notch using stress distribution around the blunt notch [4,5]. The blunt notch is different from sharp notch in point of view that the stress distribution around the notch cannot be approximated using the stress intensity factor. Therefore, the strength for carbon fiber with the blunt notch is governed by not the fracture toughness but the stress criterion. By introducing blunt notch to the carbon fiber, the stress concertation occurs and the carbon fiber breaks not at the surface defect but at the blunt notch tip. The notch tip is very smooth and free from defects, and the carbon fiber strength near the notch is governed by the strength which related to the structure of carbon fiber. The strength of notched carbon fiber has notch radius dependence and the intrinsic strength is determined using the stress near the notch tip based on the point stress criterion [6]. Therefore, the evaluation of the stress distribution near the notch tip is very important. In the case of fiber with a surface sharp notch, the stress distribution can be estimate using stress intensity factor. The correction factors for stress intensity factor of surface sharp notch of round bar are reported [7] and reviewed [8] and stress distribution can be easily estimated for the surface sharp notch. On the other hand, in the case of fiber with a surface blunt notch, the stress distribution cannot be estimated from stress intensity factors. Moreover, Carbon fiber is not isotropic materials but highly orthotropic materials [5] with cylindrical shape, and the effect of orthotropic mechanical property need to be considered. In case of the stress distribution considering orthotropic materials, a few study is reported [9]. They are, however, reported for sharp notch and limited orthotropic properties [9]. So that the effect of orthotropy on stress distribution with surface blunt notch is not studied well. Therefore, to derive stress distribution considering orthotropic mechanical properties, a finite element method (FEM) calculation is needed every time and it is very time consuming work. To measure the intrinsic strength for various carbon fiber, it is very important to produce time saving method for predicting the stress distribution around the blunt notch. In this study, the stress distribution around a blunt surface notch in a fiber with various orthotropy under tension were calculated using a FEM. An effect of orthotropy on the stress distribution is investigate. To predict the stress distribution for various notch and various orthotropy, an approximate stress distribution equation was proposed.
2. Calculation The analysis of the stress distribution around a blunt surface notch in an orthotropic fiber under tension was performed using a FEM. The pre-post processer and solver used were Abaqus CAE and Abaqus (Dassault Systèmes), respectively. Considering the geometrical symmetry of the notched fiber, one forth model was used for the analysis. The analytical model of a notched fiber is shown in Fig. 1. The fiber diameter, D, is 10 μm, and the fiber length, L, is 1 mm. ρ and a represent the notch tip radius and the notch depth, respectively. The x-, y-, and z-axis correspond to the notch depth direction, the notch front direction, and fiber axis direction. The origin is set to the center of the notch front. Fig. 2 shows the mesh density used for the model. The hexahedron element mesh was used. The angular intervals of the mesh density of the notch tip is 2.5o. The minimum mesh size was 0.5 nm at the notch front. Ex, Ey, and Ez denote the elastic moduli in x-axis, y-axis, and z-axis directions, respectively. Gxy, Gzy and Gzx denote the share moduli in xy-, yz-, and zx-planes, respectively. νzx and νzy denote the Poisson’s ratios concerning to the deformation in the x-axis and y-axis direction in response to the deformation in the z-axis direction, respectively. νxy is Poisson’s ratio in xy-plane. The mechanical property of the fiber is assumed orthotropy and isotropy in xy-plane. The orthotropic axis corresponds to the x-, y-, and z-axis. Under this assumption, Ex = Ey, Gzx = Gzy, Gxy=Ex/2(1+νxy), νzx=νzy. Therefore, orthoropic property can be determined by 5 parameters: Ex, Ez, Gzx, νzy and νxy. The orthotropic parameters, Ez/Ex and Ez/Gzx, represent the degree of orthotropy of the body, and ρ/D and a/D are the geometrical factors of the body. Considering carbon fiber properties, νzy and νxy were assumed to be 0.2 [9] and 0.4 [10], respectively, in the present calculation. Considering actual notch shape and commercial carbon fiber properties [5], the stress distribution of the fibers with various orthotropic mechanical properties and notch shapes were calculated using the values of 5 ≤ Ez/Ex ≤ 100, 5 ≤ Ez/Gzx ≤ 100, 0.02 ≤ ρ/D ≤ 0.1 and 0.1 ≤ a/D ≤ 0.5, Ez = 250 GPa. The displacement applied at one end of the fiber in the z direction was 10 μm. The force in x and y direction is free. In this calculation, liner analysis was used. Similar calculation was performed for an isotropic body with Poisson's ratio of 0.3. The linear analysis was adopted in the calculation. From the stress distribution, the stress concentration factor, α, was calculated using the following equation:
z 0,0,0 N
(1)
, where σz(x, y, z) is the tensile stress in the z direction at a position x, y and z and σN is the average tensile stress in the z direction at the fiber end.
Fig. 1 Coordinate definition and analytical model of fiber with surface notch.
Fig. 2 Mesh density model for analysis.
3. Results and discussion 3.1 Stress distribution for isotropic fiber with surface blunt notch 3.1.1 Stress distribution in notch depth direction Fig. 3 is the stress distributions in x direction, σz(x, 0, 0), with various notch depth and notch tip radius. The stress and distance are normalized by the stress at the end of the fiber, σN, and fiber diameter, D, respectively. From Fig. 3(a), it is found that σz(x, 0, 0) at the notch tip is large when the notch depth, a, is large. It is also found that notch depth influences the global stress distribution around the notch tip. From Fig. 3(b), it is found that σz(x, 0, 0) at the notch tip is large when the notch tip radius is small. σz(x, 0, 0) shows almost same behavior regardless of the notch tip radius and the size of notch tip radius influences the local stress distribution around the notch tip.
20
14
(a) a=5
= 0.6 m
15
N
(x, 0, 0)/
z(x, 0, 0)/N
a = 3 m
10
a=4
10
= 0.2
6 = 0.1 4
a=2 a=1
0 -0.02
0
= 0.4
8 = 0.3
z
a=3
5
(b)
12 = 0.5
2 0 -0.02
0.02 0.04 0.06 0.08 0.1 x/D
0
0.02 0.04 0.06 0.08 0.1 x/D
Fig. 3 Stress distributions along the x-axis of isotropic fiber with surface notch for (a) various notch depth, a, and (b) different notch tip radius, ρ.
3.1.2 Stress distribution along the notch front Fig. 4 is the stress distributions in y direction, σz(0, y, 0), with various notch depth and notch tip radius. The stress and distance are also normalized by the stress at the end of the fiber, σN, and fiber diameter, D, respectively. σz(0, y, 0) takes maximum value at y = 0 and decreases approaching to fiber surface. The sharp decrement is observed near the fiber surface and it is considered to be due to the finite size of the fiber and Poisson’s ratio. It is found that σz(0, y, 0) behavior is almost independent from the notch tip radius and the notch depth. 14
20 (a)
12
a=5
15
N
(0, y, 0)/
10
z
a=3
z
(0, y, 0)/
N
10 a=4
5
(b)
= 0.5
a=1
= 0.6 m 0
= 0.2
6 = 0.1 4
a=2
0 -0.1
= 0.4
8 = 0.3
0.1 0.2 0.3 0.4 0.5 0.6 y/D
2 0 -0.1
a = 3 m 0
0.1
0.2 y/D
0.3
0.4
0.5
Fig. 4 Stress distributions along the y-axis for (a) various notch depth, a, and (b) different notch tip radius, ρ.
3.2 Stress distribution for the orthotropic fiber with surface blunt notch 3.2.1 Effect of orthotropic parameters on stress distribution Fig. 5 is the stress distribution of orthotropic fiber, σz(x, 0, 0), with Ez/Gzx = 20 and various Ez/Ex. The notch depth, a, and the notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. Since the stress concentration at the notch tip is about 7.5 for isotropic fiber, it was confirmed that the stress concentration is greatly increased by introducing orthotropy. The stress at the notch tips shows large value, when the value of Ez/Ex is large.
Fig. 6 is the stress distribution of orthotropic fiber, σz(x, 0, 0), with Ez/Ex = 20 for various Ez/Gzx. The notch depth, a, and the notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. It indicates larger stress concentration compared to that for isotropic fiber, and it was confirmed that stress at the notch tip is large when the value of Ez/Gzx is large. Comparing Fig. 5 and Fig. 6, the stress concentration is more influenced by the value the effect of Ez/Gzx than by the value of Ez/Ex. The tensile load around the notch tip is distributed by shear load in x direction and the low shear modulus prevent stress transfer from notch tip. As a result, Ez/Gzx is shows higher stress concentration at the notch tips. 35 30
Ez/Gzx = 100
z(x, 0, 0)/N
25 Ez/Gxz = 50 Ez/Gzx = 40 Ez/Gzx = 30 Ez/Gzx = 20
20 15
Ez/Gzx = 10
10 Ez/Ex = 20
5 0 -0.02
0
0.02 0.04 0.06 0.08 0.1 x/D
Fig. 5 Stress distributions along the x-axis of orthotropic fiber with surface notch for Ez/Gzx of 20 and Ez/Ex of 10, 20, 30, 40, 50, and 100. Notch depth, a, and notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. The arrows indicate the maximum values at x/D = 0 for Ez/Ex of 10, 20, 30, 40, 50, and 100. 20 Ez/Ex = 100 Ez/Ex = 50 Ez/Ex = 40 Ez/Ex = 30 E /E = 20 z x Ez/Ex = 10
z(x, 0, 0)/N
15
10
5
0 -0.02
Ez/Gzx = 20
0
0.02 0.04 0.06 0.08 0.1 x/D
Fig. 6 Stress distributions along the x-axis of orthotropic fiber with surface notch for Ez/Gzx of 10, 20, 30, 40, 50, and 100 and Ez/Ex of 20. Notch depth, a, and notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. The arrows indicate the maximum values at x/D = 0 for Ez/Gzx of 10, 20, 30, 40, 50, and 100 3.2.2 The effect of orthotropic parameters on stress concentration Fig. 7 is the stress concentration of orthotropic fiber with a/D = 0.3 and ρ/D = 0.06. As shown in Fig. 5 and 6, the stress concentration depends on orthotropic parameters and high stress concentration is clearly observed
in case of high orthotropic parameters. In the isotropic fiber, the stress concentration is about 7.5, and in the case of Ez/Gzx = 100 stress concentration is 4 times larger than that for isotropic fiber. It is also observed that the increase of stress concentration is small when the orthotropic parameters is large. When Gzx is small, the small value change of Gzx lead the large value change of Ez/Gzx. For lager value of Ez/Gzx, Gzx are not different significantly and it leads small increase.
Fig. 7 Stress concentration factor, α, for orthotropic fiber with surface notch. Notch depth, a, and notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. 3.3 Approximated stress distribution equation for notched fiber 3.3.1 Normalized stress distribution Fig. 8 is plots of the normalized stress distribution versus x/ρ for the isotropic and orthotropic fibers with various notch depth and notch tip radius. As shown in Fig. 3 and 4, the stress distribution depends on notch shape. However, the stress near the notch tips shows same behavior regardless of the notch shape by using the normalized stress and x/ρ. The normalizing stress distribution depends on the orthotropic parameters. From this result, it is considered that the stress distribution around the notch tip can be expressed by the orthotropic parameters. 1.2
z(x, 0, 0)/z(0, 0, 0)
1.0
Isortopic(Ez/Ex = 1, Ez/Gzx = 2.6) Ez/Ex = 20, Ez/Gzx = 10
0.8
Ez/Ex = 20, Ez/Gzx = 50
0.6 0.4 0.2 0 -0.2
0
0.2
0.4 x/
0.6
0.8
1.0
Fig. 8 Plots of σz(x, 0, 0)/ σz(0, 0, 0) versus x/ρ for isotropic and orthotropic fibers for various notch shape.
An approximate equation of the normalized stress distribution in notched fiber, φ(x), can be proposed as n
x x 1 s (2) Ez 0.02 0.1 1 100, 0 x D Ex , where s and n are parameters which depends on Ez/Gzx and Ez/Ex. In the case of isotropic fiber, this value takes 2.99 for s and -0.763 for n. s and n can be approximated by following equations:
E E / E s 0.4676 z 0.6274 exp 3.904 z x 0.5956 (3) Ez / Gzx Gzx
Ez / Gzx (4). n 0.4970 1.019 exp 1.181 E / E 1 . 584 z x Fig. 9 is relation between fitting parameters using Eq. (2) and approximate parameters derived from Eq. (3) and (4). Although maximum differences of 20 % for small s and 10 % for small n are existed, this differences cause relatively small effects on stress distribution due to insensitiveness of Eq. (2) to small s and small n. It is confirmed that Eq. (3) and (4) can be relatively approximated well to predict stress distribution in the range of parameters used for calculation. 0
100
(b) n calculated from Eq. (4)
s calculated from Eq.(3)
(a) 80 60 40 20 0
0
20 40 60 80 s derived from Eq. (2)
100
-0.5
-1.0
-1.5 -1.5
-1.0 -0.5 n derived from Eq. (2)
0
Fig. 9 (a) s derived from Eq. (2) versus s calculated from Eq. (3) and (b) n derived from Eq. (2) versus n calculated from Eq. (4). Straight line indicate equal values derived from Eq. (2) and Eq. (3) or (4).
3.3.2 Approximate stress distribution equation for notched fiber The stress distribution can be express the product of the stress at the notch tip and the normalized stress distribution. Using Eq. (1) and (2), an approximate equation of the stress distribution in notched fiber around the notch can be expressed as
z x,0,0 x N (5) a E E for 0.1 0.5, 0.02 0.1, 2.6 z 100, 1 z 100 D D Gzx Ex , where α is interpolation equation of stress concentration factor which represented as follows:
Ez E a 2 a 2 a sin zx z sec C E 2 G 2 D 2 D x zx
1 2
for
0.1
(6)
a E E 0.5, 0.02 0.1, 2.6 z 100, 1 z 100 D D Gzx Ex
, where C is a correction factor which represented as 0.5882 a / D a E 0.9985 0.07217 0.1606 0.9473 0.0005525 z (7). C 1 D D Gzx Ez / Ex Ez / Gzx The stress concentration factor calculated using Eq. (6) is plotted against the values determined using the FEM in Fig. 10. This figure demonstrates that Eq. (6) well approximates the stress concentration factor within the deviation less than 4.2 % in the range of parameters used for calculation.
calculated using Eq. (6)
150
100
50
0
0
50 100 determined using FEM
150
Fig. 10 Stress concentration factor (α) calculated using interpolation equation, Eq. (6), versus that determined using FEM. Line indicates equal stress concentration factor determined using FEM and calculated using Eq. (6).
Fig. 11 shows stress distribution derived from approximate equation and FEM calculation. Although little discrepancy is observed, the stress at the notch tip and decrease behavior are correspond to FEM calculation well. The figure demonstrates that Eq. (5) well approximate the stress concentration.
8 Isortopic (Ez/Ex = 1, Ez/Gzx = 2.6)
7
15
(a)
(b) Ez/Ex = 20, Ez/Gzx = 10
3
10
z(x, 0, 0)/N
z(x, 0, 0)/N
z(x, 0, 0)/N
4
(c) Ez/Ex = 20, Ez/Gzx = 50
20
6 5
25
15 10
5
2
5
1 0 -0.2
0
0.2
0.4 0.6 x / m
0.8
1
0 -0.2
0
0.2
0.4 0.6 x / m
0.8
1
0 -0.2
0
0.2
0.4 0.6 x / m
0.8
1
Fig. 11 σz(x, 0, 0)/σN versus x for (line) approximate stress distribution equation (Eq. (5)) and (○) FEM calculation result with (Ez/Ex, Ez/Gzx) = (a) (1, 2.6) (isotoropic), (b) (20, 10), and (20, 50). Notch depth, a, and notch tip radius, ρ, are 3 μm and 0.6 μm, respectively.
4. Conclusions In this study, the stress distribution fiber with surface blunt notch under tension was analysis using FEM. From isotropic body result, notch tip radius influence only stress distribution around the notch. On the other hand, notch depth is influence global stress distribution. Stress distribution along the notch front shows maximum value at the center of the notch and decrease sharply approaching to the fiber surface. From orthotropic body result, orthotropic parameters influence the local stress distribution around the notch. The stress concentration showed large value when the orthotropic parameters are large. It was found that Ez/Gzx was more influenced to stress concentration comparing to Ez/Ex. Using FEM calculation results, approximate equation was proposed. This approximate equation can be applied to not only isotropic body but also orthotropic body (5 ≤ Ez/Ex ≤ 100, 5 ≤ Ez/Gzx ≤ 100) for various radius size of notch and for various depth of notch (0.02 ≤ ρ/D ≤ 0.1 and 0.1 ≤ a/D ≤ 0.5). The difference of the stress at the notch tips was smaller than 4.2 %. Proposed approximate equation was compared to some stress distribution, and showed good correspondence.
Acknowledgement This study was supported by New Energy and Industrial Technology Development Organization (NEDO) through the Project of Basic R&D for Carbon Fiber Innovation.
Reference [1] P. Morgan, Carbon Fibers and Their Composites, Taylor & Francis, 2005.
[2] D. A. Baker T. G. Rials, Recent advances in low-cost carbon fiber manufacture from lignin, J. Appl. Polym. Sci. 130 (2013) 713-728. [3] H. Mainka, O. Tager, E. Korner, L. Hilfert, A. Busse, F. T. Edelmann, A. S. Herrmann, Lignin – an alternative precursor for sustainable and cost-effective automotive carbon fiber, J. Mater. Res. Technol. 4 (2015) 283-296.
[4] M. Shioya, H. Inoue, Y. Sugimoto, Reduction in tensile strength of polyacrylonitrile-based carbon fibers in liquids and its application to defect analysis, Carbon 65 (2013) 63-70.
[5] Y. Sugimoto, M. Shioya, K. Kageyama, Determination of intrinsic strength of carbon fibers, Carbon 100 (2016) 208-213.
[6] J. M. Whitney, R. J. Nuismer, Stress fracture criteria for laminated composites containing stress concentrations, J. Compos. Mater. 8 (1974) 253-265.
[7] M. A. Astiz, An incompatible singular elastic element for two- and three- dimensional crack problems, Int. j. fract. 31 (1986) 105-124.
[8] J. Toribio, N. Alvarez, B. Gonzalez, J. C. Matos, A critical review of stress intensity factor solutions for surface cracks in round bars subjected to tension loading, Eng. Fail. Anal. 16 (2009) 794-809.
[9] T. Okabe, T. Motani, M. Nishikawa, M. Hashimoto, Micromechanics of failure mode in fiber reinforced plastic, J. Jpn. Soc. Compos. Mater. 35 (2009) 256-265.
[10] S. Ogiwara, Y. Imafuku, R. Yamamoto, Y. Kogo, Application of FIB technique to introduction of a notch into a carbon fiber for direct measurement of fracture toughness, J. Phys.: Conference Series 191 (2009) 012009.
Highlights
Stress distribution near a blunt surface notch for a fiber with was calculated using finite element method.
Effects of notch shape and orthotropy in mechanical properties on stress distribution were analyzed.
An approximate stress distribution equation for a fiber with a blunt surface notch was proposed.
S0167-8442(16)30384-6 http://dx.doi.org/10.1016/j.tafmec.2017.01.004 TAFMEC 1798
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
23 November 2016 24 December 2016 13 January 2017
Please cite this article as: Y. Sugimoto, K. Kageyama, Analysis of stress distribution near a blunt surface notch tip in an orthotropic fiber under tension, Theoretical and Applied Fracture Mechanics (2017), doi: http://dx.doi.org/ 10.1016/j.tafmec.2017.01.004
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of stress distribution near a blunt surface notch tip in an orthotropic fiber under tension Yoshiki Sugimoto(*), Kazuro Kageyama
Department of Technology Management for Innovation, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Abstract The intrinsic strength of carbon fiber is determined from tensile tests of carbon fibers with a blunt surface notch using a point stress criterion. A stress distribution around the blunt surface notch tip is needed for the carbon fiber with cylindrical shape and orthotropic mechanical properties. In this study, the stress distribution around a blunt surface notch in a fiber with various orthotropy under tension were calculated using a finite element method. An effect of orthotropy on the stress distribution is investigate. To predict the stress distribution for various notch and various orthotropy, an approximate stress distribution equation was proposed.
Key words Fiber; Finite element analysis; Point Stress Criterion; Notch; Orthotropy
* Corresponding author. Tel/Fax: +81 3 5841 7767. E-mail address: [email protected]
1. Introduction Recently, the demands for carbon fiber reinforced plastics, which have high tensile strength, high tensile modulus and light weight, are increasing to reduce automobiles’ weight for reducing energy consumption [1]. Carbon fibers are, however, expensive, and cannot use for general vehicles at present stage [2]. Therefore, carbon fibers from new precursors or process are actively developed to reduce production cost [3]. The carbon fibers are produced by passing through the complicate processes: synthesize, drafting, and carbonization. Therefore, it is very difficult to achieve high tensile performance in the developing stage. It is necessary to know the potential of carbon fiber to judge whether developing should continue or not. However, the carbon fiber strength is usually determined by the structure of carbon fiber and surface defects. To develop the higher strength of carbon fiber, it is very important to separate these factors. The strength depends on the carbon fiber nano-order structure is called the intrinsic strength [4,5]. The intrinsic strength of carbon fibers is measured from tensile tests of the carbon fiber with a blunt surface notch using stress distribution around the blunt notch [4,5]. The blunt notch is different from sharp notch in point of view that the stress distribution around the notch cannot be approximated using the stress intensity factor. Therefore, the strength for carbon fiber with the blunt notch is governed by not the fracture toughness but the stress criterion. By introducing blunt notch to the carbon fiber, the stress concertation occurs and the carbon fiber breaks not at the surface defect but at the blunt notch tip. The notch tip is very smooth and free from defects, and the carbon fiber strength near the notch is governed by the strength which related to the structure of carbon fiber. The strength of notched carbon fiber has notch radius dependence and the intrinsic strength is determined using the stress near the notch tip based on the point stress criterion [6]. Therefore, the evaluation of the stress distribution near the notch tip is very important. In the case of fiber with a surface sharp notch, the stress distribution can be estimate using stress intensity factor. The correction factors for stress intensity factor of surface sharp notch of round bar are reported [7] and reviewed [8] and stress distribution can be easily estimated for the surface sharp notch. On the other hand, in the case of fiber with a surface blunt notch, the stress distribution cannot be estimated from stress intensity factors. Moreover, Carbon fiber is not isotropic materials but highly orthotropic materials [5] with cylindrical shape, and the effect of orthotropic mechanical property need to be considered. In case of the stress distribution considering orthotropic materials, a few study is reported [9]. They are, however, reported for sharp notch and limited orthotropic properties [9]. So that the effect of orthotropy on stress distribution with surface blunt notch is not studied well. Therefore, to derive stress distribution considering orthotropic mechanical properties, a finite element method (FEM) calculation is needed every time and it is very time consuming work. To measure the intrinsic strength for various carbon fiber, it is very important to produce time saving method for predicting the stress distribution around the blunt notch. In this study, the stress distribution around a blunt surface notch in a fiber with various orthotropy under tension were calculated using a FEM. An effect of orthotropy on the stress distribution is investigate. To predict the stress distribution for various notch and various orthotropy, an approximate stress distribution equation was proposed.
2. Calculation The analysis of the stress distribution around a blunt surface notch in an orthotropic fiber under tension was performed using a FEM. The pre-post processer and solver used were Abaqus CAE and Abaqus (Dassault Systèmes), respectively. Considering the geometrical symmetry of the notched fiber, one forth model was used for the analysis. The analytical model of a notched fiber is shown in Fig. 1. The fiber diameter, D, is 10 μm, and the fiber length, L, is 1 mm. ρ and a represent the notch tip radius and the notch depth, respectively. The x-, y-, and z-axis correspond to the notch depth direction, the notch front direction, and fiber axis direction. The origin is set to the center of the notch front. Fig. 2 shows the mesh density used for the model. The hexahedron element mesh was used. The angular intervals of the mesh density of the notch tip is 2.5o. The minimum mesh size was 0.5 nm at the notch front. Ex, Ey, and Ez denote the elastic moduli in x-axis, y-axis, and z-axis directions, respectively. Gxy, Gzy and Gzx denote the share moduli in xy-, yz-, and zx-planes, respectively. νzx and νzy denote the Poisson’s ratios concerning to the deformation in the x-axis and y-axis direction in response to the deformation in the z-axis direction, respectively. νxy is Poisson’s ratio in xy-plane. The mechanical property of the fiber is assumed orthotropy and isotropy in xy-plane. The orthotropic axis corresponds to the x-, y-, and z-axis. Under this assumption, Ex = Ey, Gzx = Gzy, Gxy=Ex/2(1+νxy), νzx=νzy. Therefore, orthoropic property can be determined by 5 parameters: Ex, Ez, Gzx, νzy and νxy. The orthotropic parameters, Ez/Ex and Ez/Gzx, represent the degree of orthotropy of the body, and ρ/D and a/D are the geometrical factors of the body. Considering carbon fiber properties, νzy and νxy were assumed to be 0.2 [9] and 0.4 [10], respectively, in the present calculation. Considering actual notch shape and commercial carbon fiber properties [5], the stress distribution of the fibers with various orthotropic mechanical properties and notch shapes were calculated using the values of 5 ≤ Ez/Ex ≤ 100, 5 ≤ Ez/Gzx ≤ 100, 0.02 ≤ ρ/D ≤ 0.1 and 0.1 ≤ a/D ≤ 0.5, Ez = 250 GPa. The displacement applied at one end of the fiber in the z direction was 10 μm. The force in x and y direction is free. In this calculation, liner analysis was used. Similar calculation was performed for an isotropic body with Poisson's ratio of 0.3. The linear analysis was adopted in the calculation. From the stress distribution, the stress concentration factor, α, was calculated using the following equation:
z 0,0,0 N
(1)
, where σz(x, y, z) is the tensile stress in the z direction at a position x, y and z and σN is the average tensile stress in the z direction at the fiber end.
Fig. 1 Coordinate definition and analytical model of fiber with surface notch.
Fig. 2 Mesh density model for analysis.
3. Results and discussion 3.1 Stress distribution for isotropic fiber with surface blunt notch 3.1.1 Stress distribution in notch depth direction Fig. 3 is the stress distributions in x direction, σz(x, 0, 0), with various notch depth and notch tip radius. The stress and distance are normalized by the stress at the end of the fiber, σN, and fiber diameter, D, respectively. From Fig. 3(a), it is found that σz(x, 0, 0) at the notch tip is large when the notch depth, a, is large. It is also found that notch depth influences the global stress distribution around the notch tip. From Fig. 3(b), it is found that σz(x, 0, 0) at the notch tip is large when the notch tip radius is small. σz(x, 0, 0) shows almost same behavior regardless of the notch tip radius and the size of notch tip radius influences the local stress distribution around the notch tip.
20
14
(a) a=5
= 0.6 m
15
N
(x, 0, 0)/
z(x, 0, 0)/N
a = 3 m
10
a=4
10
= 0.2
6 = 0.1 4
a=2 a=1
0 -0.02
0
= 0.4
8 = 0.3
z
a=3
5
(b)
12 = 0.5
2 0 -0.02
0.02 0.04 0.06 0.08 0.1 x/D
0
0.02 0.04 0.06 0.08 0.1 x/D
Fig. 3 Stress distributions along the x-axis of isotropic fiber with surface notch for (a) various notch depth, a, and (b) different notch tip radius, ρ.
3.1.2 Stress distribution along the notch front Fig. 4 is the stress distributions in y direction, σz(0, y, 0), with various notch depth and notch tip radius. The stress and distance are also normalized by the stress at the end of the fiber, σN, and fiber diameter, D, respectively. σz(0, y, 0) takes maximum value at y = 0 and decreases approaching to fiber surface. The sharp decrement is observed near the fiber surface and it is considered to be due to the finite size of the fiber and Poisson’s ratio. It is found that σz(0, y, 0) behavior is almost independent from the notch tip radius and the notch depth. 14
20 (a)
12
a=5
15
N
(0, y, 0)/
10
z
a=3
z
(0, y, 0)/
N
10 a=4
5
(b)
= 0.5
a=1
= 0.6 m 0
= 0.2
6 = 0.1 4
a=2
0 -0.1
= 0.4
8 = 0.3
0.1 0.2 0.3 0.4 0.5 0.6 y/D
2 0 -0.1
a = 3 m 0
0.1
0.2 y/D
0.3
0.4
0.5
Fig. 4 Stress distributions along the y-axis for (a) various notch depth, a, and (b) different notch tip radius, ρ.
3.2 Stress distribution for the orthotropic fiber with surface blunt notch 3.2.1 Effect of orthotropic parameters on stress distribution Fig. 5 is the stress distribution of orthotropic fiber, σz(x, 0, 0), with Ez/Gzx = 20 and various Ez/Ex. The notch depth, a, and the notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. Since the stress concentration at the notch tip is about 7.5 for isotropic fiber, it was confirmed that the stress concentration is greatly increased by introducing orthotropy. The stress at the notch tips shows large value, when the value of Ez/Ex is large.
Fig. 6 is the stress distribution of orthotropic fiber, σz(x, 0, 0), with Ez/Ex = 20 for various Ez/Gzx. The notch depth, a, and the notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. It indicates larger stress concentration compared to that for isotropic fiber, and it was confirmed that stress at the notch tip is large when the value of Ez/Gzx is large. Comparing Fig. 5 and Fig. 6, the stress concentration is more influenced by the value the effect of Ez/Gzx than by the value of Ez/Ex. The tensile load around the notch tip is distributed by shear load in x direction and the low shear modulus prevent stress transfer from notch tip. As a result, Ez/Gzx is shows higher stress concentration at the notch tips. 35 30
Ez/Gzx = 100
z(x, 0, 0)/N
25 Ez/Gxz = 50 Ez/Gzx = 40 Ez/Gzx = 30 Ez/Gzx = 20
20 15
Ez/Gzx = 10
10 Ez/Ex = 20
5 0 -0.02
0
0.02 0.04 0.06 0.08 0.1 x/D
Fig. 5 Stress distributions along the x-axis of orthotropic fiber with surface notch for Ez/Gzx of 20 and Ez/Ex of 10, 20, 30, 40, 50, and 100. Notch depth, a, and notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. The arrows indicate the maximum values at x/D = 0 for Ez/Ex of 10, 20, 30, 40, 50, and 100. 20 Ez/Ex = 100 Ez/Ex = 50 Ez/Ex = 40 Ez/Ex = 30 E /E = 20 z x Ez/Ex = 10
z(x, 0, 0)/N
15
10
5
0 -0.02
Ez/Gzx = 20
0
0.02 0.04 0.06 0.08 0.1 x/D
Fig. 6 Stress distributions along the x-axis of orthotropic fiber with surface notch for Ez/Gzx of 10, 20, 30, 40, 50, and 100 and Ez/Ex of 20. Notch depth, a, and notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. The arrows indicate the maximum values at x/D = 0 for Ez/Gzx of 10, 20, 30, 40, 50, and 100 3.2.2 The effect of orthotropic parameters on stress concentration Fig. 7 is the stress concentration of orthotropic fiber with a/D = 0.3 and ρ/D = 0.06. As shown in Fig. 5 and 6, the stress concentration depends on orthotropic parameters and high stress concentration is clearly observed
in case of high orthotropic parameters. In the isotropic fiber, the stress concentration is about 7.5, and in the case of Ez/Gzx = 100 stress concentration is 4 times larger than that for isotropic fiber. It is also observed that the increase of stress concentration is small when the orthotropic parameters is large. When Gzx is small, the small value change of Gzx lead the large value change of Ez/Gzx. For lager value of Ez/Gzx, Gzx are not different significantly and it leads small increase.
Fig. 7 Stress concentration factor, α, for orthotropic fiber with surface notch. Notch depth, a, and notch tip radius, ρ, are 3 μm and 0.6 μm, respectively. 3.3 Approximated stress distribution equation for notched fiber 3.3.1 Normalized stress distribution Fig. 8 is plots of the normalized stress distribution versus x/ρ for the isotropic and orthotropic fibers with various notch depth and notch tip radius. As shown in Fig. 3 and 4, the stress distribution depends on notch shape. However, the stress near the notch tips shows same behavior regardless of the notch shape by using the normalized stress and x/ρ. The normalizing stress distribution depends on the orthotropic parameters. From this result, it is considered that the stress distribution around the notch tip can be expressed by the orthotropic parameters. 1.2
z(x, 0, 0)/z(0, 0, 0)
1.0
Isortopic(Ez/Ex = 1, Ez/Gzx = 2.6) Ez/Ex = 20, Ez/Gzx = 10
0.8
Ez/Ex = 20, Ez/Gzx = 50
0.6 0.4 0.2 0 -0.2
0
0.2
0.4 x/
0.6
0.8
1.0
Fig. 8 Plots of σz(x, 0, 0)/ σz(0, 0, 0) versus x/ρ for isotropic and orthotropic fibers for various notch shape.
An approximate equation of the normalized stress distribution in notched fiber, φ(x), can be proposed as n
x x 1 s (2) Ez 0.02 0.1 1 100, 0 x D Ex , where s and n are parameters which depends on Ez/Gzx and Ez/Ex. In the case of isotropic fiber, this value takes 2.99 for s and -0.763 for n. s and n can be approximated by following equations:
E E / E s 0.4676 z 0.6274 exp 3.904 z x 0.5956 (3) Ez / Gzx Gzx
Ez / Gzx (4). n 0.4970 1.019 exp 1.181 E / E 1 . 584 z x Fig. 9 is relation between fitting parameters using Eq. (2) and approximate parameters derived from Eq. (3) and (4). Although maximum differences of 20 % for small s and 10 % for small n are existed, this differences cause relatively small effects on stress distribution due to insensitiveness of Eq. (2) to small s and small n. It is confirmed that Eq. (3) and (4) can be relatively approximated well to predict stress distribution in the range of parameters used for calculation. 0
100
(b) n calculated from Eq. (4)
s calculated from Eq.(3)
(a) 80 60 40 20 0
0
20 40 60 80 s derived from Eq. (2)
100
-0.5
-1.0
-1.5 -1.5
-1.0 -0.5 n derived from Eq. (2)
0
Fig. 9 (a) s derived from Eq. (2) versus s calculated from Eq. (3) and (b) n derived from Eq. (2) versus n calculated from Eq. (4). Straight line indicate equal values derived from Eq. (2) and Eq. (3) or (4).
3.3.2 Approximate stress distribution equation for notched fiber The stress distribution can be express the product of the stress at the notch tip and the normalized stress distribution. Using Eq. (1) and (2), an approximate equation of the stress distribution in notched fiber around the notch can be expressed as
z x,0,0 x N (5) a E E for 0.1 0.5, 0.02 0.1, 2.6 z 100, 1 z 100 D D Gzx Ex , where α is interpolation equation of stress concentration factor which represented as follows:
Ez E a 2 a 2 a sin zx z sec C E 2 G 2 D 2 D x zx
1 2
for
0.1
(6)
a E E 0.5, 0.02 0.1, 2.6 z 100, 1 z 100 D D Gzx Ex
, where C is a correction factor which represented as 0.5882 a / D a E 0.9985 0.07217 0.1606 0.9473 0.0005525 z (7). C 1 D D Gzx Ez / Ex Ez / Gzx The stress concentration factor calculated using Eq. (6) is plotted against the values determined using the FEM in Fig. 10. This figure demonstrates that Eq. (6) well approximates the stress concentration factor within the deviation less than 4.2 % in the range of parameters used for calculation.
calculated using Eq. (6)
150
100
50
0
0
50 100 determined using FEM
150
Fig. 10 Stress concentration factor (α) calculated using interpolation equation, Eq. (6), versus that determined using FEM. Line indicates equal stress concentration factor determined using FEM and calculated using Eq. (6).
Fig. 11 shows stress distribution derived from approximate equation and FEM calculation. Although little discrepancy is observed, the stress at the notch tip and decrease behavior are correspond to FEM calculation well. The figure demonstrates that Eq. (5) well approximate the stress concentration.
8 Isortopic (Ez/Ex = 1, Ez/Gzx = 2.6)
7
15
(a)
(b) Ez/Ex = 20, Ez/Gzx = 10
3
10
z(x, 0, 0)/N
z(x, 0, 0)/N
z(x, 0, 0)/N
4
(c) Ez/Ex = 20, Ez/Gzx = 50
20
6 5
25
15 10
5
2
5
1 0 -0.2
0
0.2
0.4 0.6 x / m
0.8
1
0 -0.2
0
0.2
0.4 0.6 x / m
0.8
1
0 -0.2
0
0.2
0.4 0.6 x / m
0.8
1
Fig. 11 σz(x, 0, 0)/σN versus x for (line) approximate stress distribution equation (Eq. (5)) and (○) FEM calculation result with (Ez/Ex, Ez/Gzx) = (a) (1, 2.6) (isotoropic), (b) (20, 10), and (20, 50). Notch depth, a, and notch tip radius, ρ, are 3 μm and 0.6 μm, respectively.
4. Conclusions In this study, the stress distribution fiber with surface blunt notch under tension was analysis using FEM. From isotropic body result, notch tip radius influence only stress distribution around the notch. On the other hand, notch depth is influence global stress distribution. Stress distribution along the notch front shows maximum value at the center of the notch and decrease sharply approaching to the fiber surface. From orthotropic body result, orthotropic parameters influence the local stress distribution around the notch. The stress concentration showed large value when the orthotropic parameters are large. It was found that Ez/Gzx was more influenced to stress concentration comparing to Ez/Ex. Using FEM calculation results, approximate equation was proposed. This approximate equation can be applied to not only isotropic body but also orthotropic body (5 ≤ Ez/Ex ≤ 100, 5 ≤ Ez/Gzx ≤ 100) for various radius size of notch and for various depth of notch (0.02 ≤ ρ/D ≤ 0.1 and 0.1 ≤ a/D ≤ 0.5). The difference of the stress at the notch tips was smaller than 4.2 %. Proposed approximate equation was compared to some stress distribution, and showed good correspondence.
Acknowledgement This study was supported by New Energy and Industrial Technology Development Organization (NEDO) through the Project of Basic R&D for Carbon Fiber Innovation.
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Highlights
Stress distribution near a blunt surface notch for a fiber with was calculated using finite element method.
Effects of notch shape and orthotropy in mechanical properties on stress distribution were analyzed.
An approximate stress distribution equation for a fiber with a blunt surface notch was proposed.