Axi-symmetrical deflection and buckling of circular porous-cellular plate

ARTICLE IN PRESS

Thin-Walled Structures 46 (2008) 333–337 www.elsevier.com/locate/tws

Axi-symmetrical deflection and buckling of circular porous-cellular plate E. Magnucka-Blandzi Institute of Mathematics, Poznan´ University of Technology, ul. Piotrowo 3a, 60-965 Poznan´, Poland Received 10 March 2006; received in revised form 11 June 2007; accepted 11 June 2007 Available online 25 October 2007

Abstract The main goal of this paper is a solution of the problem of buckling and deflection. A circular porous plate with simply supported edge under radial uniform compression and uniformly distributed load (pressure) is considered. Mechanical properties of the isotropic porous material vary across the thickness of the plate. Middle plane of the plate is its symmetry plane. A field of displacements (geometric model of nonlinear hypothesis) is described. The principle of stationarity of the total potential energy allowed to get a system of differential equations that govern the plate stability. A critical load and a deflection are determined. The results obtained for porous plates are compared to homogeneous circular plates. r 2007 Elsevier Ltd. All rights reserved. Keywords: Circular plate; Elastic buckling; Porous-cellular metal

1. Introduction Problem of deflection and buckling of the plates is described in many works and monographs. Some of them deal with the classical (Kirchhoff) theory which is not adequate in providing accurate buckling. This is due to the effect of transverse shear strains. Shear deformation theories provide accurate solutions compared to the classical theory. During the last several years this problem has been developed by many authors. The simplest, widely used approach for modelling plates made of non-homogeneous material is using Kirchhoff–Love’a hypothesis for describing the displacement field and taking into account modified forms of stiffness coefficients. For example, this way of modelling was used by Ambartsumian [1] in his monographs. The first shear deformation theory is presented by Vinson [2]. Later, many hypotheses, which include shearing, have been formulated. One of the monograph devoted to this problem is the work of Wang et al. [3], where authors presented not only their own solutions but also a review of E-mail address: [email protected] 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.06.006

previous attempts to model beams and plates. A comparison of theories used for modelling compressed and bent multilayered composite plates is presented in Chattopadhyay [4]. Banhart [5] provided a comprehensive description of various manufacturing processes of metal foams and porous metallic structures. Structural and functional applications to various industrial sectors are discussed. Instead, porous plates and beams with varying properties were described by Malinowski and Magnucki [6], Magnucki and Stasiewicz [7,8] and Magnucki [9] where also a nonlinear hypothesis was assumed. Porous-cellular materials exist in the nature, for example, structure of bones cross-section. Taking into account these structures, many searchers describe and manufacture similar structures. Suresh and Mortensen [10] presented detailed technology of these graded materials and its mechanical properties. Presented circular porous plate is a generalization of sandwich structure. These circular plates, for example as flat baffle plate are used in water filter or horizontal cylindrical pressure vessels. Flat baffle plates of vessels are loaded by pressure and radial compression. So this paper is concerned with the problem of deflection and buckling of the plate.

ARTICLE IN PRESS E. Magnucka-Blandzi / Thin-Walled Structures 46 (2008) 333–337

334

2. Displacements of a porous plate

-h/2

This work is divided into two parts. Both of them are concerned with a circular porous plate with simply supported edge. But in the first one, the plate under uniformly distributed load is described and in the second one, the plate under radial uniform compression. Mechanical properties of the material vary through the thickness of the plate. Minimal value of Young’s modulus occurs in the middle surface of the plate and maximal values at its top and bottom surfaces. For such a case, the Euler–Bernoulli or Timoshenko plate theories do not correctly determine displacements of the plate’s cross-section. Wang [3] discussed in details the effect of non-dilatational strain of middle layers on bending of plates subject to various load cases. A porous plate (Fig. 1) is a generalized sandwich plate. The material is of continuous mechanical properties. The top and the bottom plate surfaces are made of nonporous material, while maximal porosity of the material occurs in the middle surfaces of the plate. The degree of porosity varies in normal direction. This plate is described in polar (cylindrical) coordinate system with the z-axis in the normal direction. The moduli of elasticities and mass density vary continuously too, as follows: EðzÞ ¼ E 1 ½1  e0 cosðpzÞ; RðzÞ ¼ R1 ½1  em cosðpzÞ,

GðzÞ ¼ G 1 ½1  e0 cosðpzÞ, ð1Þ

where, e0 is the coefficient of plate porosity, e0 ¼ 1  E 0 =E 1 , E 0 , E 1 the Young’s moduli at z ¼ 0 and z ¼ h=2, respectively, G0 , G 1 the shear moduli for z ¼ 0 and z ¼ h=2, respectively, G j the relationship between the moduli of elasticy for j ¼ 0; 1, G j ¼ E j =½2ð1 þ nÞ, n the Poisson’s ratio (constant for the entire plate), em the dimensionless parameter of mass density, em ¼ 1  R0 =R1 , R0 , R1 the mass density for z ¼ 0 and z ¼ h=2, respectively, z the dimensionless coordinate, z ¼ z=h, h the thickness of the plate. Choi and Lakes [11] presented mechanical properties for porous materials. Taking into account the results of investigations of this paper the relation between dimensionless parameter of mass density em ¼ 1  R0 =R1 and dimensionless parameter of the porosity pffiffiffiffiffiffiffiffiffiffiffiffiof ffi the metal foam e0 is defined as follows: em ¼ 1  1  e0 .

-h/2

E1

E0

0

h/2 z Fig. 1. Scheme of porous plate.

E (z)

E1

0

r

z h/2

z w (r)

z

u (r,z)

∂w ∂r

r

r∼ Fig. 2. Scheme of a deformation of a plane cross-section of the beam—the nonlinear hypothesis.

The physical model of deformation of a plane crosssection of the plate (the nonlinear hypothesis) is shown in Fig. 2. The cross-section, being initially planar surface, becomes curved after the deformation. The surface perpendicularly intersects the top and the bottom surfaces of the plate. This geometric model is analogous to the broken-line hypothesis applied to three-layered structures. A field of displacements in any cross-section is assumed in the following form:  dw 1  ½c ðrÞ sinðpzÞ uðr; zÞ ¼  h z dr p 1  þ c2 ðrÞ sinð2pzÞcos2 ðpzÞ , wðr; zÞ ¼ wðr; 0Þ ¼ wðrÞ,

(2)

where uðr; zÞ is the longitudinal displacement along the raxis, wðrÞ the deflection (displacement along the z-axis), c1 ðrÞ, c2 ðrÞ the dimensionless functions of displacements. The strains are linear and components of the strain field describing the geometric relationships are defined as follows:  2  qu d w 1 dc1 ¼ h z 2  er ¼ sinðpzÞ qr dr p dr  dc2 2 þ sinð2pzÞcos ðpzÞ , dr ej ¼

  u 1 dw 1 1 ¼ h z  c sinðpzÞ r r dr p r 1  1 2 þ c2 sinð2pzÞcos ðpzÞ , r

ARTICLE IN PRESS E. Magnucka-Blandzi / Thin-Walled Structures 46 (2008) 333–337

grz ¼

qu dw þ ¼ c1 cosðpzÞ þ c2 ½cosð2pzÞ þ cosð4pzÞ, qz dr

(3)

where er is the normal strain along the r-axis, ej is the circular strain and grz is the shear strain. The physical relationships, according to Hooke’s law, are EðzÞ ðer þ nej Þ; 1  n2 trz ¼ GðzÞgrz . sr ¼

sj ¼

EðzÞ ðej þ ner Þ, 1  n2

c2 c4 c6

ð4Þ

Moduli of elasticy (1) occurring here are variable and depend on the z-coordinate. 3. Equations of stability The above stated problem includes three unknown functions: wðrÞ, c1 ðrÞ and c2 ðrÞ. Hence, three equations are necessary to solve this problem. They may be formulated on the grounds of the principle of stationarity of total potential energy of the plate under pressure and compressive force. dðU e  W Þ ¼ 0,

(5)

where U e is the energy of elastic strain Z R Z 1=2 U e ¼ ph rðsr er þ sj ej þ trz grz Þ dz dr. 0

c0

c8

335

  p2  8 3 8  e0 , ¼16 e0 ; c1 ¼ 2 p3 p p     3 3 128 2 4  e0 ; c3 ¼ 2 3  e0 , ¼ 2 p 4 75p p p   1 32 3 8  e0 ; c5 ¼ 3  e0 , ¼ 2 p 5p 2 p     1 32 1 15 512  15e0 ; c7 ¼ 2  e0 , ¼ 10 p p 8 105p   832 e0 . ¼2 3 105p

Boundary conditions for r ¼ 0 or r ¼ R are formulated in the form: dw ¼ 0, wðRÞ ¼ 0; M r ðRÞ ¼ 0; c1 ð0Þ ¼ c2 ð0Þ ¼ 0; dr r¼0 (7) where

  d M r ¼ D c0 LðwÞ  c1 Lðc1 Þ  c2 Lðc2 Þ , dr df n þ f. Lðf Þ ¼ dr r

4. Axi-symmetrical deflection of the plate

1=2

W is the work which follows from the compressive force and pressure Z R  2 Z R dw W ¼ 2p prwðrÞ dr þ pN r r dr, dr 0 0 R is the radius of the plate, R is the mass density of the plate, N r is the compressive force, p is the pressure. A system of three equations of stability for the porous plate under compression and pressure is formulated in the following form:       dw d2 w 1 dw D c0 R þ  c1 Rðc1 Þ  c2 Rðc2 Þ ¼ p  N r , dr dr2 r dr    d 1d dw r c1  c3 c1  c4 c2 dr r dr dr 1n þ 2 ðc5 c1 þ c6 c2 Þ ¼ 0, h    d 1d dw r c2  c4 c1  c7 c2 dr r dr dr 1n þ 2 ðc6 c1 þ c8 c2 Þ ¼ 0, h where    1d d 1d r ðrf Þ ; Rðf Þ ¼ r dr dr r dr

E 1 h3 D¼ , 12ð1  n2 Þ

In this part of the paper only deflection is described. So, N r ¼ 0, p ¼ p0 ¼ const are assumed. Therefore, the first equation of system (6) can be written in the form:    d 1d dw Q r c0  c1 c1 ðrÞ  c2 c2 ðrÞ (8) ¼ , dr r dr dr D where shear force    Z 1=2 2 e0 8e0 QðrÞ ¼ h  c ðrÞ , trz dz ¼ G1 h c1 ðrÞ  p 2 15p 2 1=2 Z 1 pr dr. QðrÞ ¼ r Based on Eq. (8) and assuming that dimensionless functions c1 ðrÞ, c2 ðrÞ are linear, system (6) can be reduced to the system of two equations in the form:

ð6Þ

c5 c 1 þ c6 c 2 ¼ 

c 1 h2 p r, 2Dc0 ð1  nÞ 0

c6 c 1 þ c8 c 2 ¼ 

c 2 h2 p r. 2Dc0 ð1  nÞ 0

Therefore, c1 ðrÞ, c2 ðrÞ have the form: c1 ðrÞ ¼

c 2 c 6  c 1 c 8 h2 p0 r, c5 c8  c26 1  n 2c0 D

c2 ðrÞ ¼

c 1 c 6  c 2 c 5 h2 p0 r. c5 c8  c26 1  n 2c0 D

ð9Þ

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Substitution of the above two functions (9) into Eq. (8) gives the deflection  4 3ð1 þ nÞ 1 r 3þn 2 2 2 2 r R  wðrÞ ¼ c9 ðR  r Þ þ 2 32c 1þn D 2 c0 E 1 h 0  15 þ n 4 R þ p0 , 21 þ n where c9 ¼

c21 c8 þ c22 c5  2c1 c2 c6 . c5 c8  c26

The maximal values of the normal stress are in the middle of the plate (r ¼ 0) and can be written in the form: ( "  2 # 1  e0 cosðpzÞ 1 þ n c9 3 R sr ¼ z 6 þ ð3 þ nÞ c0 1  n c0 4 h  6 1 þ n c2 c6  c1 c8 c1 c6  c2 c5 þ sinðpzÞ þ 2 p 1  n c5 c8  c6 c5 c8  c26 #)  sinð2pzÞ cos2 ðpzÞ

p0 .

1 Below two examples for the following porous plate: Rh ¼ 10 , n ¼ 0:3, p0 ¼ 1 MPa, e0 ¼ 0 and 0.9 were considered. Graphs of the maximal stresses varying on the depth of the circular plate are shown in Fig.3.

σr

The last problem considered in this paper is buckling. Now, N r ¼ const., p ¼ 0 are supposed. The system of differential equations (6) may be approximately solved with the use of Galerkin’s method. Hence, three unknown functions satisfying boundary conditions are assumed:   2 þ n r 2 1 þ n r 3 wðrÞ ¼ wa 1  3 þ2 , 4þn R 4þn R   2 þ n r  1 þ n r 2  , c1 ðrÞ ¼ 6ca1 4þn R 4þn R   2 þ n r  1 þ n r 2  . ð10Þ c2 ðrÞ ¼ 6ca2 4þn R 4þn R Substituting the above three functions (10) into Eqs. (6) and using Galerkin’s method yields a system of three equations in the form:   R2 c0  c9 N r wa  c1 Rca1  c2 Rca2 ¼ 0, 15D c1 wa  c11 Rca1  c12 Rca2 ¼ 0, c2 wa  c12 Rca1  c13 Rca2 ¼ 0,

ð11Þ

where n2 þ 8n þ 22 ð1  nÞR2 n2 þ 7n þ 16 ; c10 ¼ , ðn þ 1Þðn þ 5Þ 15h2 ðn þ 1Þðn þ 4Þ ¼ c3 þ c5 c10 ; c12 ¼ c4 þ c6 c10 , ¼ c7 þ c8 c10 .

c9 ¼ c11 c13

From the second and third equations of system (11) the ca1 and ca2 functions may be calculated e wa ; c ¼ c e wa , (12) ca1 ¼ c a1 a2 a2 R R where e ¼ c1 c13  c2 c12 ; c e ¼ c2 c11  c1 c12 . c a1 a2 2 c13 c11  c12 c13 c11  c212

184.45 e0 = 0.9 100 e0 = 0

0.5

-0.5

5. Buckling of the plate

Substituting these functions (12) into the first equation of the system (11) yields the critical load

-100

N cr ¼

15D e  c2 c e Þ. ðc0  c1 c a1 a2 c9 R2

(13)

In particular, the critical load for the homogeneous plate (e0 ¼ 0) was considered by Volmir [12]. Dimensionless critical loads Ncr for different coefficients of porosity for the following plates: E 1 ¼ 2:05  105 MPa, n ¼ 0:3, Rh ¼ 1002400 are presented in Table 1. Hence, influence

-184.45

Fig. 3. Maximal stresses. Table 1 Dimensionless critical loads

100 200 300 400

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4.2195 4.2200 4.2200 4.2201

4.0669 4.0673 4.0674 4.0674

3.9142 3.9146 3.9147 3.9147

3.7615 3.7619 3.7620 3.7620

3.6088 3.6092 3.6093 3.6093

3.4561 3.4565 3.4566 3.4567

3.3034 3.3039 3.3040 3.3040

3.1506 3.1512 3.1513 3.1513

2.9978 2.9985 2.9986 2.9986

2.8448 2.8457 2.8459 2.8459

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and buckling of the porous plates include the classical solutions for homogeneous plates which are presented in references.

4.22

NCr

References

2.85

0.1

0.2

0.3

0.4

0.5 e0

0.6

0.7

0.8

0.9

Fig. 4. Dimensionless critical loads.

of porosity on the critical load shown in Fig. 4 can be written in the form: Ncr ¼ 1:527e0 þ 4:22, where Ncr ¼ NcrðR2 =DÞ. 6. Conclusions The porous-cellular circular plate is a generalization of a sandwich or multi-layer plates. The hypotheses of a plane cross-section deformation for a porous-cellular plate includes a transverse shear deformable effect. Distribution of normal stress along the thickness of the plate is nonlinear. Whereas for homogeneous plate, this distribution is linear. The critical load linearly decrease with increase porosity of the plate. Both solutions for deflection

[1] Ambartsumian SA. Theory of anisotrpic plates. Moscow: Nauka; 1987. [in Russian]. [2] Vinson JR. The behavior of sandwich structures of isotropic and composite materials. Lancaster: Technomic Publishing Company Inc.; 1999. [3] Wang CM, Reddy JN, Lee KH. Shear deformable beams and plates. Amsterdam, Lousanne, New York, Oxford, Shannon, Singapore, Tokyo: Elsevier; 2000. [4] Chattopadhyay A, Gu H. Exact elasticity solution for buckling of composite laminates. Compos. Struct. 1996;34(3):291–9. [5] Banhart J. Manufacture, characterisation and application of cellular metals and metal foams. Prog. Mater. Sci. 2001;46:559–632. [6] Malinowski M, Magnucki K. Buckling of an isotropic porous cylindrical shell. In: Topping BHV, editor. in: Proceedings of the tenth international conference on civil, structural and environmental engineering computing. Civil-Computer Press Stirling, Scotland Paper 53; 2005. p. 1–10. [7] Magnucki K, Stasiewicz P. Elastic bending of an isotropic porous beam. Int. J. Appl. Mech. Eng. 2004;9(2):351–60. [8] Magnucki K, Stasiewicz P. Elastic buckling of a porous beam. J. Theoret. Appl. Mech. 2004;42(4):859–68. [9] Magnucki K. Differential equations of multilayer isotropic beams. In: The fourth international conference on tools for mathematical modelling, Saint-Petersburg; June (2003). [10] Suresh S, Mortensen A. Fundamentals of functionally graded materials. London, UK: IOM Communications Ltd; 1998. [11] Choi JB, Lakes RS. Analysis of elastic modulus of conventional foams and of re-entrant foam materials with a negative Poisson’s ratio. Int. J. Mech. Sci. 1995;37(1):51–9. [12] Volmir AS. Stability of deformation systems. Nauka; Moscow: 1967. [in Russian].