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Failure of orthotropic plates containing a circular opening
Composite Structures 46 (1999) 53±57
Technical Note
Failure of orthotropic plates containing a circular opening H.A. Whitworth *, H. Mahase Department of Mechanical Engineering, College of Engineering, Architecture and Computer Sciences, Howard University, Washington, DC 20059, USA
Abstract Failure of composite plates containing a circular opening is investigated and results are given for plates subjected to uniaxial loading. Stresses and strains in the composite were analyzed using the Lekhnitskii anisotropic elasticity theory. A failure criterion based on the strain energy density function is presented for predicting the externally applied load that will initiate failure and to determine the location of the failure at the opening. Typical results are presented for boron/epoxy, graphite/epoxy and glass/epoxy composites. Ó 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction
2. Failure analysis
Extensive acceptance and use of composite structures require con®dence with their load carrying capacity. However, in many applications cutouts of various shapes and sizes must be considered. To assure the serviceability of advanced composite structures, one must assess the eect of these stress raisers on the strength degradation of the composite laminate. However, many factors such as accurate stress analysis and reliable failure criteria in¯uence the prediction of initial failure [1]. This makes the stress analysis more complex and failure more dicult to predict and analyze. The problem of the stress distribution around a circular hole in a composite plate has been treated by several investigators [2±6]. In anisotropic plates containing openings, the failure will take place not as a result of stress concentration but, rather, as a result of interaction of various stress components [3]. Thus, it is necessary to employ a failure criterion for orthotropic materials in order to predict the failure at the opening in terms of the applied loading. In this paper, the approach of Lekhnitskii in conjunction with a failure theory proposed in Ref. [7] is used to predict the location of failure and the externally applied loads that will initiate failure at the opening of a composite plate subjected to uniaxial tension.
The coordinate system for an orthotropic plate subjected to uniaxial loading is illustrated in Fig. 1. If the plate is stretched at an angle / to the ®ber direction, the circumferential stress on the boundary of the hole is given by Lekhnitskii [8]
*
Corresponding author.
Eh cos2 / l1 l2 ÿ n sin2 / l1 l2 cos2 h E1 1 n cos2 / l1 l2 sin2 / sin2 h ÿ n 1 n ÿ l1 l2 sin / cos / sin h cos hg
rh p/
1
or rh p/
Eh f h; E1
2
where f h represents the term in curly brackets, h is the angle measured counter clockwise from the ®ber direction, E1 is the stiness in the ®ber direction and l1 , and l2 are roots of the characteristic equation l4 1 2m12 2 1 ÿ l 0: 3 E1 El G12 E2 In Eq. (1), the modulus Eh in a direction tangent to the opening can be obtained from 1 sin4 h 1 2m12 cos4 h ÿ sin2 h cos2 h 4 E1 Eh E1 G12 E2 and the roots of the characteristic equation are related to the engineering constants by r E1 ; 5 l1 l 2 ÿ E2
0263-8223/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 4 2 - 2
54
H.A. Whitworth, H. Mahase / Composite Structures 46 (1999) 53±57
related to the circumferential stress through transformation as r1 rh sin2 h;
r2 rh cos2 h;
s12 rh sin h cos h:
8
Thus, from Eqs. (2), (7) and (8), the failure can be expressed as
f 2 h sin4 h f 2 h cos4 h XL2 XT2 f 2 h sin2 h cos2 h p/2 1:
1 m12 m21 ÿ XL2 XT2 S2
9
The minimum value of p/ will give the external stress that will produce failure at the opening, while the value of h at which p/ is a minimum will give the location of the failure. 3. Numerical results
Fig. 1. Composite lamina subjected to uniaxial loading.
s r E1 E1 n ÿi l1 l2 ÿ 2m12 2 : G12 E2
6
Once the circumferential stress is computed, an appropriate failure criterion can be applied to compute the strength and location of failure. A failure criterion proposed in Ref. [7] will be used to evaluate the composite failure stress, p/ . This failure criterion can be expressed as 2 2 r1 r2 s12 2 m12 m21 1; ÿ r1 r2 XL XT S XL2 XT2 7 where XL , XT and S are the familiar lamina strengths and m12 and m21 the Poisson's ratio. The stresses r1 , r2 , and s12 , associated with the principal axes of the plate are
Using the material properties shown in Table 1 [9], the stress distribution around a circular hole in a lamina is evaluated for graphite/epoxy, boron/epoxy and glass epoxy composites. In Eq. (1) / represents the orientation of the ®ber angle to the load axis, and h the location around the hole measured from the ®ber direction. However, in plotting the data, this equation is transformed so that the x-axis becomes the reference axis for the measurements. Fig. 2 displays typical results from this analysis. Here it can be observed that the material properties strongly in¯uence the maximum value of the circumferential stress and the location at which the maximum stress occurs. Numerical results for the strength analysis of these composites are summarized in Table 2 and displayed in Fig. 3. For comparison, prediction associated with the Azzi±Tsai±Hill failure theory is also shown in Fig. 3. As can be observed, the failure strength is also strongly in¯uenced by the ®ber orientation. In addition, Table 2 indicates that the location of maximum stress concentration does not coincide with the location of failure. Similar results were also obtained by Greszczuk [3] for boron/epoxy composites.
Table 1 Mechanical properties for various composite materials [9] Property
Glass/epoxy
Boron/epoxy
Graphite/epoxy
Young's modulus in ®ber direction, E1 (GPa) Young's modulus in transverse direction, E2 (GPa) Shear modulus, G12 (GPa) Major Poisson's ratio, t12 Minor Poisson's ratio, t21 Strength in the ®ber direction, XL (MPa) Strength in the transverse direction, XT (MPa) Shear strength, S (MPa)
53.8 17.9 8.96 0.25 0.08 1.03 ´ 103 27.58 41.37
207 20.7 6.90 0.3 0.03 1.38 ´ 103 82.78 124.11
207 5.17 2.59 0.25 0.00625 1.03 ´ 103 41.37 68.95
H.A. Whitworth, H. Mahase / Composite Structures 46 (1999) 53±57
55
Fig. 2. Stress concentration around a circular hole for a 30 laminae.
Comparison of stiness ratios E1 =E2 show that unlike the stress concentration results, the material with the highest stiness ratio does not necessarily have the highest failure load. For example, from Fig. 3 it can be observed that the boron/epoxy lamina sustained higher failure loads than the graphite/epoxy lamina. However, when this data is normalized relative to the failure stress associated with the 0 ®ber angle (Fig. 4), the response of all three composites appears to be quite similar. Fig. 3
also indicates good agreement between the present theory and the Azzi±Tsai±Hill theory. 4. Conclusion The Lekhnitskii anisotropic theory in conjunction with a failure theory proposed in Ref. [7] is used to evaluate failure in composite plates with circular
Table 2 Failure stress results for various composite systems Fiber angle /
Failure stress (MPa)
Location of failure h
Maximum stress concentration
Location of maximum stress concentration h
(a) Boron/epoxy 0 30 45 60 90
170.8 71.0 45.9 34.9 28.5
82 58 64 72 90
7.03 5.63 4.19 2.72 2.91
90 115 126 135 90
(b) Glass/epoxy 0 30 45 60 90
48.7 19.9 14.5 11.7 9.9
67 70 73 78 90
4.08 3.61 3.15 2.80 2.77
90 105 109 101 90
(c) Graphite/epoxy 0 30 45 60 90
99.3 41.2 26.5 20.1 16.4
87 61 66 75 90
10.62 8.35 6.04 3.72 2.52
90 117 129 140 90
56
H.A. Whitworth, H. Mahase / Composite Structures 46 (1999) 53±57
Fig. 3. Failure strength predictions as a function of ®ber orientation.
Fig. 4. Normalized failure strength as a function of ®ber orientation.
openings. From this investigation it was observed that the maximum stress concentration and the failure stress are reduced as the ®ber angle deviates from the load
direction. It was also observed that the material with the higher stiness ratio also had the higher stress concentration factor. Additionally, the predicted location of
H.A. Whitworth, H. Mahase / Composite Structures 46 (1999) 53±57
failure does not coincide with the location of maximum stress concentration. References [1] Herakovich CT, Nagarkar A, O'Brien DA. Failure Analysis of Composite Laminates with Free Edges, Modern Developments in Composite Materials and Structures, Vinson JR, (Ed.), ASME 1979:53±66. [2] Lo KH, Wu EM, Konishi DY. Failure Strength of Notched Composite Laminates. Journal of Composite Materials, 1983;17:384±98. [3] Greszczuk LB. Stress Concentration and Failure Criteria for Orthotropic and Anisotropic Plates with Circular Openings. ASTM STP 1972;497:363±81.
57
[4] Lakshminarayana HV. Stress distribution around a semi-circular edge-notch in a ®nite size laminated composite plate under uniaxial tension. Journal of Composite Materials 1983;17:357±67. [5] Lin CC, Ko CC. Stress and strength analysis of ®nite composite laminates with elliptical holes. Journal of Composite Materials 1988;22:373±85. [6] Chang KY, Liu S, Chang FK. Damage tolerance of laminated composites containing an open hole and subjected to tensile loading. Journal of Composite Materials 1990;5:274±301. [7] Whitworth HA, Yin SW. A failure criterion for laminated ®berreinforced composites. Composites Engineering 1991;1(2):61±7. [8] Lekhnitskii SG. Theory of Elasticity of an Anisotropic Body, Holden-Day, Inc., San Francisco, California, 1963. [9] Jones RM. Mechanics of Composite Materials. New York: McGraw-Hill, 1975.
Technical Note
Failure of orthotropic plates containing a circular opening H.A. Whitworth *, H. Mahase Department of Mechanical Engineering, College of Engineering, Architecture and Computer Sciences, Howard University, Washington, DC 20059, USA
Abstract Failure of composite plates containing a circular opening is investigated and results are given for plates subjected to uniaxial loading. Stresses and strains in the composite were analyzed using the Lekhnitskii anisotropic elasticity theory. A failure criterion based on the strain energy density function is presented for predicting the externally applied load that will initiate failure and to determine the location of the failure at the opening. Typical results are presented for boron/epoxy, graphite/epoxy and glass/epoxy composites. Ó 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction
2. Failure analysis
Extensive acceptance and use of composite structures require con®dence with their load carrying capacity. However, in many applications cutouts of various shapes and sizes must be considered. To assure the serviceability of advanced composite structures, one must assess the eect of these stress raisers on the strength degradation of the composite laminate. However, many factors such as accurate stress analysis and reliable failure criteria in¯uence the prediction of initial failure [1]. This makes the stress analysis more complex and failure more dicult to predict and analyze. The problem of the stress distribution around a circular hole in a composite plate has been treated by several investigators [2±6]. In anisotropic plates containing openings, the failure will take place not as a result of stress concentration but, rather, as a result of interaction of various stress components [3]. Thus, it is necessary to employ a failure criterion for orthotropic materials in order to predict the failure at the opening in terms of the applied loading. In this paper, the approach of Lekhnitskii in conjunction with a failure theory proposed in Ref. [7] is used to predict the location of failure and the externally applied loads that will initiate failure at the opening of a composite plate subjected to uniaxial tension.
The coordinate system for an orthotropic plate subjected to uniaxial loading is illustrated in Fig. 1. If the plate is stretched at an angle / to the ®ber direction, the circumferential stress on the boundary of the hole is given by Lekhnitskii [8]
*
Corresponding author.
Eh cos2 / l1 l2 ÿ n sin2 / l1 l2 cos2 h E1 1 n cos2 / l1 l2 sin2 / sin2 h ÿ n 1 n ÿ l1 l2 sin / cos / sin h cos hg
rh p/
1
or rh p/
Eh f h; E1
2
where f h represents the term in curly brackets, h is the angle measured counter clockwise from the ®ber direction, E1 is the stiness in the ®ber direction and l1 , and l2 are roots of the characteristic equation l4 1 2m12 2 1 ÿ l 0: 3 E1 El G12 E2 In Eq. (1), the modulus Eh in a direction tangent to the opening can be obtained from 1 sin4 h 1 2m12 cos4 h ÿ sin2 h cos2 h 4 E1 Eh E1 G12 E2 and the roots of the characteristic equation are related to the engineering constants by r E1 ; 5 l1 l 2 ÿ E2
0263-8223/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 4 2 - 2
54
H.A. Whitworth, H. Mahase / Composite Structures 46 (1999) 53±57
related to the circumferential stress through transformation as r1 rh sin2 h;
r2 rh cos2 h;
s12 rh sin h cos h:
8
Thus, from Eqs. (2), (7) and (8), the failure can be expressed as
f 2 h sin4 h f 2 h cos4 h XL2 XT2 f 2 h sin2 h cos2 h p/2 1:
1 m12 m21 ÿ XL2 XT2 S2
9
The minimum value of p/ will give the external stress that will produce failure at the opening, while the value of h at which p/ is a minimum will give the location of the failure. 3. Numerical results
Fig. 1. Composite lamina subjected to uniaxial loading.
s r E1 E1 n ÿi l1 l2 ÿ 2m12 2 : G12 E2
6
Once the circumferential stress is computed, an appropriate failure criterion can be applied to compute the strength and location of failure. A failure criterion proposed in Ref. [7] will be used to evaluate the composite failure stress, p/ . This failure criterion can be expressed as 2 2 r1 r2 s12 2 m12 m21 1; ÿ r1 r2 XL XT S XL2 XT2 7 where XL , XT and S are the familiar lamina strengths and m12 and m21 the Poisson's ratio. The stresses r1 , r2 , and s12 , associated with the principal axes of the plate are
Using the material properties shown in Table 1 [9], the stress distribution around a circular hole in a lamina is evaluated for graphite/epoxy, boron/epoxy and glass epoxy composites. In Eq. (1) / represents the orientation of the ®ber angle to the load axis, and h the location around the hole measured from the ®ber direction. However, in plotting the data, this equation is transformed so that the x-axis becomes the reference axis for the measurements. Fig. 2 displays typical results from this analysis. Here it can be observed that the material properties strongly in¯uence the maximum value of the circumferential stress and the location at which the maximum stress occurs. Numerical results for the strength analysis of these composites are summarized in Table 2 and displayed in Fig. 3. For comparison, prediction associated with the Azzi±Tsai±Hill failure theory is also shown in Fig. 3. As can be observed, the failure strength is also strongly in¯uenced by the ®ber orientation. In addition, Table 2 indicates that the location of maximum stress concentration does not coincide with the location of failure. Similar results were also obtained by Greszczuk [3] for boron/epoxy composites.
Table 1 Mechanical properties for various composite materials [9] Property
Glass/epoxy
Boron/epoxy
Graphite/epoxy
Young's modulus in ®ber direction, E1 (GPa) Young's modulus in transverse direction, E2 (GPa) Shear modulus, G12 (GPa) Major Poisson's ratio, t12 Minor Poisson's ratio, t21 Strength in the ®ber direction, XL (MPa) Strength in the transverse direction, XT (MPa) Shear strength, S (MPa)
53.8 17.9 8.96 0.25 0.08 1.03 ´ 103 27.58 41.37
207 20.7 6.90 0.3 0.03 1.38 ´ 103 82.78 124.11
207 5.17 2.59 0.25 0.00625 1.03 ´ 103 41.37 68.95
H.A. Whitworth, H. Mahase / Composite Structures 46 (1999) 53±57
55
Fig. 2. Stress concentration around a circular hole for a 30 laminae.
Comparison of stiness ratios E1 =E2 show that unlike the stress concentration results, the material with the highest stiness ratio does not necessarily have the highest failure load. For example, from Fig. 3 it can be observed that the boron/epoxy lamina sustained higher failure loads than the graphite/epoxy lamina. However, when this data is normalized relative to the failure stress associated with the 0 ®ber angle (Fig. 4), the response of all three composites appears to be quite similar. Fig. 3
also indicates good agreement between the present theory and the Azzi±Tsai±Hill theory. 4. Conclusion The Lekhnitskii anisotropic theory in conjunction with a failure theory proposed in Ref. [7] is used to evaluate failure in composite plates with circular
Table 2 Failure stress results for various composite systems Fiber angle /
Failure stress (MPa)
Location of failure h
Maximum stress concentration
Location of maximum stress concentration h
(a) Boron/epoxy 0 30 45 60 90
170.8 71.0 45.9 34.9 28.5
82 58 64 72 90
7.03 5.63 4.19 2.72 2.91
90 115 126 135 90
(b) Glass/epoxy 0 30 45 60 90
48.7 19.9 14.5 11.7 9.9
67 70 73 78 90
4.08 3.61 3.15 2.80 2.77
90 105 109 101 90
(c) Graphite/epoxy 0 30 45 60 90
99.3 41.2 26.5 20.1 16.4
87 61 66 75 90
10.62 8.35 6.04 3.72 2.52
90 117 129 140 90
56
H.A. Whitworth, H. Mahase / Composite Structures 46 (1999) 53±57
Fig. 3. Failure strength predictions as a function of ®ber orientation.
Fig. 4. Normalized failure strength as a function of ®ber orientation.
openings. From this investigation it was observed that the maximum stress concentration and the failure stress are reduced as the ®ber angle deviates from the load
direction. It was also observed that the material with the higher stiness ratio also had the higher stress concentration factor. Additionally, the predicted location of
H.A. Whitworth, H. Mahase / Composite Structures 46 (1999) 53±57
failure does not coincide with the location of maximum stress concentration. References [1] Herakovich CT, Nagarkar A, O'Brien DA. Failure Analysis of Composite Laminates with Free Edges, Modern Developments in Composite Materials and Structures, Vinson JR, (Ed.), ASME 1979:53±66. [2] Lo KH, Wu EM, Konishi DY. Failure Strength of Notched Composite Laminates. Journal of Composite Materials, 1983;17:384±98. [3] Greszczuk LB. Stress Concentration and Failure Criteria for Orthotropic and Anisotropic Plates with Circular Openings. ASTM STP 1972;497:363±81.
57
[4] Lakshminarayana HV. Stress distribution around a semi-circular edge-notch in a ®nite size laminated composite plate under uniaxial tension. Journal of Composite Materials 1983;17:357±67. [5] Lin CC, Ko CC. Stress and strength analysis of ®nite composite laminates with elliptical holes. Journal of Composite Materials 1988;22:373±85. [6] Chang KY, Liu S, Chang FK. Damage tolerance of laminated composites containing an open hole and subjected to tensile loading. Journal of Composite Materials 1990;5:274±301. [7] Whitworth HA, Yin SW. A failure criterion for laminated ®berreinforced composites. Composites Engineering 1991;1(2):61±7. [8] Lekhnitskii SG. Theory of Elasticity of an Anisotropic Body, Holden-Day, Inc., San Francisco, California, 1963. [9] Jones RM. Mechanics of Composite Materials. New York: McGraw-Hill, 1975.