Stresses around polygonal hole in an infinite laminated composite plate

European Journal of Mechanics A/Solids 54 (2015) 44e52

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Stresses around polygonal hole in an infinite laminated composite plate Dharmendra S. Sharma Department of Mechanical Engineering, Faculty of Technology and Engineering, The M S University of Baroda, Vadodara, 390001, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 November 2014 Accepted 3 June 2015 Available online 17 June 2015

A General solution for determining the stress field around polygonal hole in a laminated composite infinite plate subjected to arbitrary biaxial loading, with layers of arbitrary fiber orientations and stacking sequence, is obtained using complex variable approach. The effect of material parameters, hole geometry, fiber orientation angle and loading pattern, on stress field around hole is studied. © 2015 Elsevier Masson SAS. All rights reserved.

Keywords: Composites Polygonal holes Stress concentration

1. Introduction The isotropic and anisotropic plates are widely used in aerospace, mechanical and civil engineering structures. In engineering structures different type of holes and openings are made, in order to satisfy certain service requirements. These openings and cutouts result in strength degradation of the structural members. In order to predict the strength of the structure, it is essential to study the effect of hole geometry and types of loading on stress distribution around the openings. Muskhelishvili's (1963) complex variable approach is useful and handy tool to study two dimensional problems of theory of elasticity. For the solution of stress distribution around holes in plates (isotropic as well as anisotropic) Muskhelishvili's (1963) complex variable approach has been used extensively (Savin (1961), Lekhnitskii (1963), Ukadgaonker and Kakhandki (2005), Sharma (2012), etc). In an infinite isotropic plate (plate size is atleast 10 times greater than the size of the hole), the stress concentration is affected by geometrical parameters like hole shape, corner radius at vertex and orientation of hole, and the magnitude and direction of the applied load vectors (Savin (1961), Lekhnitskii (1963), Sharma (2012), Ukadgaonker and Awasare (1993, 1994), Gao (1996), Rezaeepazhand and Jafari (2010), Batista (2011), etc). The stress concentration around holes in an infinite anisotropic plate is not only affected by geometrical parameters and loading

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.euromechsol.2015.06.004 0997-7538/© 2015 Elsevier Masson SAS. All rights reserved.

pattern, but also affected by anisotropy of the plate material (Ukadgaonker and Kakhandki (2005), Jong (1981), Ukadgaonker and Rao (1990), Daoust and Hoa (1991), Sharma (2011), Sharma et al. (2014) etc). A laminated composite plate is made of orthotropic lamina stacked together. The orthotropy of the lamina and their stacking pattern decides anisotropy of the plate. The hole shape considered above are having finite radius at the corners. It may be possible in some of the cases that the corner radius is zero (singular point). In such circumstances one has to evaluate stress intensity factors. Sharma and Dave (2015) presented the stress intensity factors for hypocycloidal hole in anisotropic plate. The highly stressed zone around the hole may be potential zone for crack initiation and propagation. The crack propagation problem has been addressed effectively by Gentilini et al. (2004), Piva et al. (2005, 2006) in orthotropic media. The stresses around triangular hole (Ukadgaonker and Rao (1990), Daoust and Hoa (1991), Sharma (2011)), square and rectangular hole (Jong (1981), Rao et al. (2010)) and irregular shaped hole (Ukadgaonker and Kakhandki (2005)) in infinite composite plate is available in the literature. The stress distribution around regular polygonal hole and irregular polygonal hole in isotropic infinite plate has been presented by Sharma (2012) and Batista (2011), respectively. In this paper, generalized solution is obtained using complex variable approach for an infinite laminate with polygonal holes. The shape of the hole considered here are triangle, square, pentagon, hexagon, heptagon and octagon with rounded corners. The effect of corner radius, material property, loading angle, fiber orientation and stacking sequence on stress pattern is studied.

D.S. Sharma / European Journal of Mechanics A/Solids 54 (2015) 44e52

2. Complex variable formulations

D1 D2 D3 D4 U ¼ 0;

The thin anisotropic plate is assumed to be loaded in such a way that resultants lies in XOY plane (refer Fig. 1). The stresses on the top and bottom surface of the plate as well as sz, txz and tyz are zero everywhere within the plate. The mean values of strains along thickness (h) of the plate can be represented by generalized Hooke's law.

Dk1 ðk1 ¼ 1; 2; 3; 4Þ ¼

45

v v  sk1 ; vy vx

(5)

where sk1(k1 ¼1,2,3,4) are the roots of the characteristic equation presented below:

a11 s4  2a16 s3 þ ð2a12 þ a66 Þs2  2a26 s þ a22 ¼ 0:

(6)

The roots of Eq. (6) can be written as

εx ¼ a11 sx þ a12 sy þ a16 txy ; εy ¼ a12 sx þ a22 sy þ a26 txy ; gxy ¼ a16 sx þ a26 sy þ a66 txy ;

(1)

¼ a2  iU2 ðU1 > 0; U2 > 0Þ:

where

Uðx; yÞ ¼

The mean value of stresses can be written as

sx ¼

1 h

Z

sx dz; sy ¼ 2h

1 h

Z

h 2

sy dz; txy ¼ h2

1 h

Z

txy dz:

(2)

h2

(3)

Substituting Eq. (1) and Eq. (3) in strain-displacement compatibility condition.

a22

þ

v 2 εy vx2

¼

v2 gxy vxvy ,

k1 ¼1

  Fk1 x þ sk1 y ;

(8)

The analytic functions fðz1 Þ andj(z2), and their conjugates are given by

h 2

v2 U v2 U v2 U : ; sy ¼ 2 ; txy ¼  2 vxvy vy vx

v2 εx vy2

4 P

Uðx; yÞ ¼ F1 ðz1 Þ þ F2 ðz2 Þ þ F1 ðz1 Þ þ F2 ðz2 Þ:

In the absence of body forces, the stress components can be written in terms of Airy's stress function (U) as follows:

sx ¼

(7)

On integrating Eq. (5), the Airy's stress function U(x,y) can be represented as

sx,sy,txy ¼ mean value of stresses along thickness, aij ¼ compliance co-efficient.

h 2

s1 ¼ a1 þ iU1 ; s2 ¼ a2 þ iU2 ; s3 ¼ a1  iU1 ; s4

dF1 dF2 dF1 dF2 ¼ fðz1 Þ; ¼ jðz2 Þ; ¼ fðz1 Þ; ¼ jðz2 Þ: dz1 dz2 dz1 dz2

(9)

By substituting analytic functions from Eq. (9) into Eq. (8), and finally Eq. (8) into Eq. (3), the stress components in terms of fðz1 Þ andj(z2) can be obtained as

i h sx ¼ 2
(10)

sy ¼ 2
the following equation is obtained 3. Mapping function The area external to a given hole, in Z-plane is mapped conformably to the area outside the unit circle in z-plane using following mapping function.

v4 U v4 U v4 U v4 U v4 U 2a þa Þ 2a þa þð2a 26 12 66 16 11 vx4 vx3 vy vx2 vy2 vxvy3 vy4

¼ 0: (4)

z ¼ uðzÞ ¼ Rz þ Tk Rzð1knÞ ; k  Y

Symbolically, above Eq. (4) can be written in terms of four linear differential operators (Lekhnitskii (1963)) as where, Tk ¼

PN

 ðj  1Þn  2

j¼1

k¼1

nk ð1knÞðk!Þ

!

(11)

;

n ¼ number of sides of polygon, R ¼ hole size constant, z ¼ reiq(r,q are the coordinates in z-plane). For the given polygonal hole, number of terms (k) decides the corner radius. The corner radius can be calculated from the following formula

r1 ¼

Fig. 1. Arbitrary biaxial loading.

Rð1  ðkn  1ÞTk Þ2 1 þ ðkn  1Þ2 Tk

:

(12)

The normalized radius is defined asr ¼ r1/R. It can be very easily found that a polygon will be inscribed in a circle having diameter equal to 2R(1þTk). For anisotropic materials, the deformations undergo affine transformation. Hence, the mapping function (Eq. (11)) is modified by introducing complex parameterssj.

46

D.S. Sharma / European Journal of Mechanics A/Solids 54 (2015) 44e52

2 zj ¼ uj ðzÞ ¼

6 N X   R6 6 aj z1 þ bj z þ aj zð1knÞ þ bj zð1knÞ 26 4 k¼1

1 3 k Y ððj  1Þn  2ÞA7 7 j¼1 7; k n ð1  knÞðk!Þ 7 5

where aj ¼ (1þisj),bj ¼ (1isj),(j ¼ 1,2).

4. Stress functions for single hole under remote loading The anisotropic plate containing a polygonal hole is subjected to ∞ ∞ remotely applied load (s∞ x0 ,sy0 and tx0 y0 ) at the outer edges of the plate as shown in Fig. 1. These stresses can be written in terms of Gao's (1996) arbitrary biaxial loading conditions as follows: (Refer Fig. 1) ∞ ∞ s∞ x0 ¼ bs; sy0 ¼ s; tx0 y0 ¼ 0;

(14)

where b is bi-axial loading factor (b ¼ 0 and b ¼ 1 means uni-axial and equi-biaxial loading conditions, respectively). By applying transformation of axis, boundary conditions about XOY can be written explicitly as shown below: ∞ s∞ x þ sy ¼ s½1 þ b;

∞ ∞ ið2gpÞ ½1  b: s∞ y  sx þ 2itxy ¼ se

(15)

The stresses at infinity can be represented in terms of stress functions f1 ðz1 Þ andj(z2) as follows:

  h  i ∞ 2 0 2 0 s∞ x þsy ¼2
(13)

41 ðz1 Þ ¼ ðBÞz1 ;

(17)

  j1 ðz2 Þ ¼ B* þ iC * z2 :

(18)

By substituting first derivative of f1 ðz1 Þ andj(z2) in Eq. (16), one can find constants B, B* and C*. The presence of a hole alters the stress field in the infinite plate. Thus, f1 ðz1 Þ andj(z2) alone can not define the stresses around hole. In fact, they can not produce traction free boundary at the hole contour. In order to get traction free boundary, following boundary conditions are to be applied.

  f10 ¼  f1 ðz1 Þ þ j1 ðz2 Þ þ f1 ðz1 Þ þ j1 ðz2 Þ ;   f20 ¼  s1 f1 ðz1 Þ þ s2 j1 ðz2 Þ þ s1 f1 ðz1 Þ þ s2 j1 ðz2 Þ :

(19)

The stress functions for the plate with a hole (in the absence of remote load) are obtained using these boundary conditionsðf10 ; f20 Þ into Schwarz formula:

f0 ðzÞ ¼

j0 ðzÞ ¼

i 4pðs1  s2 Þ i 4pðs1  s2 Þ

Z 

 t þ z dt þ l1 ; tz t

(20)

Z   t þ z dt þ l2 ; s1 f10  f20 tz t

(21)

g

s2 f10  f20

g

where g is the boundary of the unit circle in z-plane, t is the value of z on the hole boundary, l1 and l2 are imaginary constants which will have no contribution towards stress field and may be dropped hereon. Substituting boundary conditions into Eqs. (20) and (21), f0 ðzÞ

Fig. 2. Stress distribution around triangular hole (corner radius, r ¼ 0.0476, load angle, g ¼ 0 ) for different plate material.

D.S. Sharma / European Journal of Mechanics A/Solids 54 (2015) 44e52

47

Fig. 3. Stress distribution around triangular hole (corner radius, r ¼ 0.0476) in isotropic steel plate.

Fig. 4. Stress pattern for isotropic plate under equi-biaxial loading (b ¼ 1) condition. Table 1 Effect of number of sides of a polygon and number of terms in mapping function on normalized tangential stress.

n n n n n n n n n n

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

3 4 5 6 7 8 10 20 40 ∞

k¼4

k¼3

k¼2

k¼1

22.88 15.52 12.17 10.18 8.852 7.901 6.616 4.215 3.088 2.00

18.9 12.41 9.644 8.059 7.024 6.296 5.328 3.568 2.763 2.00

14.72 9.273 7.153 5.999 5.27 4.768 4.117 2.974 2.468 2.00

9.998 6.0 4.666 3.999 3.599 3.333 3.0 2.444 2.212 2.00

and j0(z) takes the following form

i f0 ðzÞ ¼ 4pðs1  s2 Þ 



Z 

  ðs2  s1 Þf1 ðz1 Þ þ s2  s1 f1 ðz1 Þ

g

þ s2  s2 j1 ðz2 Þ

 t þ z dt ; tz t (22)

48

D.S. Sharma / European Journal of Mechanics A/Solids 54 (2015) 44e52

Fig. 5. Stress distribution around different polygonal holes-Graphite/epoxy plate subjected to uniaxial load (b ¼ 0) at infinity.

Fig. 6. Stress distribution around different polygonal holes-Graphite/epoxy plate subjected to equi-biaxial load (b ¼ 1) at infinity.

D.S. Sharma / European Journal of Mechanics A/Solids 54 (2015) 44e52

j0 ðzÞ ¼

Z 

i 4pðs1  s2 Þ 

þ s1  s2



  ðs1  s2 Þj1 ðz2 Þ þ s1  s1 f1 ðz1 Þ

Where,

g

 t þ z dt : j1 ðz2 Þ tz t (23)

Evaluating the integrals, we get

f0 ðzÞ ¼

j0 ðzÞ ¼

      i ðs2 s1 ÞI1 þ s2 s1 I2 þ s2 s2 I4 ; 4pðs1 s2 Þ

i 4pðs1 s2 Þ

     s1 s1 I2 þðs1 s2 ÞI3 þ s1 s2 I4 ;

I10 I20 I30 I40

¼ 2BRpi½a1 z2 þ b1 Tk ð1  knÞzðknÞ ; ¼ 2BRpi½a1 Tk ð1  knÞzðknÞ  b1 z2 ; ¼ 2ðB* þ iC * ÞRpi½a2 z2 þ b2 Tk ð1  knÞzðknÞ ; ¼ 2ðB*  iC * ÞRpi½a2 Tk ð1  knÞzðknÞ  b2 z2 :

The stress components in Cartesian co-ordinate given in Eqs.26e28, can be represented in orthogonal curvilinear coordinate system by means of the following relations:

(24)

sq þ sr ¼ sx þ sy ;   sq  sr þ 2itrq ¼ sy  sx þ 2itxy e2ia :

(29)

(25)

where, I1 I2 I3 I4

49

¼ 2BRpi[a1z1þb1Tkz(1kn)], ¼ 2BRpi[a1Tkz(1kn)þb1z1], ¼ 2(B*þiC*)Rpi[a2z1þb2Tkz(1kn)], ¼ 2(B*iC*)Rpi[a2Tkz(1kn)þb2z1].

By superimposing, Eqs. 17 and 24 and Eqs. 18 and 25, the final form of the stress functions can be obtained. The stress components can be written as follows:

5. Results and discussion The generalized solution obtained above is coded and numerical results are presented for different materials. The compliance coefficient, aij are obtained from generalized Hooke's law (Daniel and Ishai (2009)). Using compliance co-efficient, aij the value of complex parameters of anisotropy s1 and s2 are found by solving characteristic equation (Eq. (6)). Based on the different values of ‘n’, various polygonal shapes (using Eq. (11)) are obtained. The corner radius (Eq. (12)) at the vertices of the polygon depends upon the number of terms (k) used in the mapping function. As the number of terms increases, the radius of curvature reduces at the corner. The corner radius

      1 i ðs2  s1 ÞI10 þ s2  s1 I20 þ s2  s2 I40 4 2@    A   sx ¼s∞ x þ 2
      1 i ðs2  s1 ÞI10 þ s2  s1 I20 þ s2  s2 I40 ∞ 4 @   A   sy ¼sy þ 2
(26)

20

      1 i ðs2  s1 ÞI10 þ s2  s1 I20 þ s2  s2 I40 4 @   A   ¼t∞ xy  2
(27)

2 0

txy

(28)

50

D.S. Sharma / European Journal of Mechanics A/Solids 54 (2015) 44e52

Fig. 7. Comparision of stresses in x and y-direction around blunt square hole for EGlass/epoxy material.

has considerable effect on the stress concentration near the vertices. The stress is a point function and varies as we go around the hole boundary. Fig. 2 shows the stress distribution around the triangular hole for different materials (corner radius, r ¼ 0.0476). The hole geometry and material parameters are taken same as Daoust and Hoa (1991), and Ukadgaonker and Rao (1990), for sake of comparison. Fig. 2 can be compared with Fig. 6 (pp. 127) of Daoust and Hoa (1991) and Fig. 3 (pp. 178) of Ukadgaonker and Rao (1990). The normalized tangential stress variation around triangular hole (r ¼ 0.0476) in isotropic plate for load angles, g ¼ 0 and 90 (Load

angles measured from þve x-direction) can be seen from the Fig. 3. The results are found to be in very good agreement with Savin (1961), Daoust and Hoa (1991) and Ukadgaonker and Rao (1990). Fig. 4 shows stress pattern for isotropic plate under equi-biaxial loading condition. The effect of number of vertices (n) and number of terms in the mapping function (k) can be observed in Fig. 4. As number of terms (k) increases, the stress at a vertex for the given loading condition increases. This is because the radii at the vertices approaches zero. With the increase in the number of sides of the polygon (n) for given number of terms (k), the geometry approaches to become a circle (because the polygon is having vertex with finite radius). The stress value accordingly approaches to that of a circular hole. This behavior can be observed from Table 1 for equi-biaxial loading. The normalized tangential stress at all the vertex of the particular hole is same for isotropic media under biaxial loading. The anisotropy alters this behavior. The stacking sequence, fiber orientation and material properties have great effect on stress field around polygonal holes. Figs. 5 and 6 describes stress fields around different polygonal cutouts in graphite/epoxy (0/90) and graphite/epoxy (45/45), under uniaxial (b ¼ 0) and equi-biaxial (b ¼ 1) loading conditions, respectively. The normalized corner radius for triangular, square, pentagonal, hexagonal, heptagonal and octagonal hole are respectively 0.0476, 0.0117, 0.0317, 0.0843, 0.0261 and 0.0481. All hole shapes under consideration are symmetrical about x-axis (with one of the vertices on it) and therefore, for the given stacking sequence and fiber orientation, the stress fields are also symmetrical about it, under uni-axial and biaxial loading. As shown in Fig. 5, the maximum stress concentration is found higher in graphite/epoxy (0/90) plates than graphite/epoxy (45/45), when the load is applied normal to the x-axis. The Fig. 7 shows the comparison of stresses (sx,sy) for square hole for E-Glass/epoxy (E1 ¼ 41 GPa, E2 ¼ 10.4 GPa, G12 ¼ 4.3 GPa and n12 ¼ 0.28). The results are also obtained using commercial finite element software (ANSYS) and are found to be in close agreement. Graphite/epoxy and carbon/epoxy lamina are considered to study the effect of fiber orientation angle. For every fiber orientation angle (F), the maximum normalized tangential stress is obtained for uni-directional load applied along y-axis and plotted as shown in Fig. 8. The minimum and maximum values of stress are found at fiber angle 0 and 90 , respectively. However, there are few peaks and valleys in between due to presence of vertex of polygonal geometry. The normalized corner radius for triangular, square, pentagonal, hexagonal, heptagonal and octagonal hole are 0.0031, 0.0117, 0.0317, 0.0843, 0.0261 and 0.0481 respectively. The maximum normalized tangential stress corresponding to each load angle (g ¼ 0 to 90 ) are presented for Graphite/epoxy (0/ 90), Graphite/epoxy (45/45) and isotropic steel in Fig. 9. When the load vector is perpendicular to the line joining centre of polygon and one of vertices, the maximum normalized tangential stress is found higher compared to other load angles, for isotropic materials. This behavior is not observed in the anisotropic plates under consideration. The orthotropy of each lamina and their stacking over one another affect stress pattern greatly, but definitely a proper stacking sequence corresponding to a particular hole geometry and loading angle can be tailored to minimize the stress concentration. 6. Conclusions A general solution is presented for determining the stress pattern around polygonal holes in laminated composite infinite plate using complex variable approach. The computer implementation of the present solution is easier and faster. The general

D.S. Sharma / European Journal of Mechanics A/Solids 54 (2015) 44e52

Fig. 8. Maximum (sq =s) corresponding to different fiber angle (F) for (A). Graphite/epoxy and (B). Carbon/epoxy.

Fig. 9. Maximum normalized tangential stress (sq =s) corresponding to different load angle, g.

51

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D.S. Sharma / European Journal of Mechanics A/Solids 54 (2015) 44e52

form of mapping function and the arbitrary loading condition facilitates consideration of different polygonal hole shapes and loading conditions. The loading angle, fiber orientation, stacking sequence, material parameters and hole geometry has significant effect on stress field and failure strength. References Batista, M., 2011. On the stress concentration around a hole in an infinite plate subject to a uniform load at infinity. Int. J. Mech. Sci. 53, 254e261. Daniel, I.M., Ishai, O., 2009. Engineering Mechanics of Composite Materials, second Indian edition. Oxford university press. Daoust, J., Hoa, S.V., 1991. An analytical solution for anisotropic plates containing triangular holes. Compos. Struct. 19, 107e130. Gao, X.L., 1996. A general solution of an infinite elastic plate with an elliptic hole under biaxial loading. Press. Vessels Pip. 67, 95e104. Gentilini, C., Piva, A., Viola, E., 2004. On crack propagation in orthotropic media for degenerate states. Eur. J. Mech. A/Solids 23, 247e258. Jong, T.D., 1981. Stresses around rectangular holes in orthotropic plates. J. Compos. Mater. 15 (7), 311e328. Lekhnitskii, S.G., 1963. Theory of Elasticity of an Anisotropic Elastic Body. San Francisco- Holden-Day Inc. Muskhelishvili, N.I., 1963. Some Basic Problems of the Mathematical Theory of Elasticity, second ed. P.Noordhooff Ltd. Piva, A., Tornabene, F., Viola, E., 2006. Crack propagation in a four- parameter piezoelectric medium. Eur. J. Mech. A/Solids 25, 230e249.

Piva, A., Viola, E., Tornabene, F., 2005. Crack propagation in an orthotropic medium with coupled elastodynamic properties. Mech. Res. Commun. 32, 153e159. Rao, D.K.N., Ramesh, B.M., Reddy, K.R.N., Sunil, D., 2010. Stress around square and rectangular cut-outs in symmetric laminates. Compos. Struct. 92, 2845e2859. Rezaeepazhand, J., Jafari, M., 2010. Stress concentration in metallic plates with special shaped cut-out. Int. J. Mech. Sci. 52, 96e102. Savin, G.N., 1961. Stress Concentration Around Holes. Pergamon Press. Sharma, D.S., 2011. Stress distribution around circular/elliptical/triangular holes in infinite composite plate. Eng. Lett. 20 (1), 1e9. Sharma, D.S., 2012. Stress distribution around polygonal holes. Int. J. Mech. Sci. 65, 115e124. Sharma, D.S., Dave, J.M., 2015. Stress intensity factors for hypocycloidal hole with cusp in infinite orthotropic plate. Theor. Appl. Fract. Mech. 75, 44e52. Sharma, D.S., Patel, N.P., Trivedi, R.R., 2014. Optimum design of laminates containing an elliptical hole. Int. J. Mech. Sci. 85, 76e87. Ukadgaonker, V.G., Awasare, P.J., 1993. A novel method of stress analysis of an infinite plate with triangular hole with uniform loading at infinity. J. Inst. Eng. (India) 73, 312e317. Ukadgaonker, V.G., Awasare, P.J., 1994. A novel method of stress analysis of an infinite plate with rectangular hole with rounded corners with uniform loading at infinity. Indian J. Eng. Mater. Sci. 74, 17e25. Ukadgaonker, V.G., Kakhandki, V., 2005. Stress analysis for an orthotropic plate with an irregular shaped hole for different in-plane loading conditionsdpart 1. Compos. Struct. 70, 255e274. Ukadgaonker, V.G., Rao, D.K.N., 1990. Stress distribution around triangular holes in anisotropic plates. Compos. Struct. 45, 171e183.