Large-deflection axisymmetric deformation of circular clamped plates with different moduli in tension and compression

International Journal of Mechanical Sciences 62 (2012) 103–110

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Large-deflection axisymmetric deformation of circular clamped plates with different moduli in tension and compression Xiao-Ting He a,b,n, Qiang Chen a, Jun-Yi Sun a,b, Zhou-Lian Zheng a,b a b

College of Civil Engineering, Chongqing University, Chongqing 400045, PR China Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400030, PR China

a r t i c l e i n f o

abstract

Article history: Received 3 July 2011 Received in revised form 30 May 2012 Accepted 8 June 2012 Available online 15 June 2012

Ambartsumyan’s bimodular model for isotropic materials deals with the principal stress state in a point, which is particularly useful in the analysis and design of structures. In this paper, based on the known flexural stiffness for a bimodular thin plate in small-deflection bending, we establish the von Ka´rma´n equations with different moduli in tension and compression and then use the perturbation method and the displacement variation method to solve the problem, respectively. The comparison shows that the perturbation solution based on the central deflection is valid. The analytical result shows that the bimodularity of the material will have an effect on the relation of load vs. deflection to a certain extent. We also investigate the yield conditions for a bimodular thin plate in large-deflection bending. It is concluded that this introduction of materials nonlinearity will eventually influence the yield stress at the edge and center of the plate, however, it does not change the yield order that when loading further, the edge of the plate will firstly yield and then the center of the plate. Moreover, during the transition from plate to membrane, the bimodular plate will gradually regress to the classical one. This work will be helpful for analyzing the mechanical behaviors of thin film materials with obvious bimodularity and with moderate thickness or hardness. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Bimodulus Tension and compression Large-deflection Thin plates Axisymmetry Yield

1. Introduction Most materials essentially exhibit different tensile and compressive strains given the same stress applied in tension and compression. These materials are known as bimodular materials [1,2]. For a better understanding of the mechanical behaviors for such bimodular materials, some useful material models are established. Overall, there are two basic material models widely used in theoretical analysis within the engineering profession. One of these models is Bert’s model [3] for orthotropic materials, which is based on the criterion of positive–negative signs in the longitudinal strain of fibers, and is therefore widely used for research on laminated composites [4–9]. Another model is Ambartsumyan’s bimodular model for isotropic materials [10,11], which is based on the criterion of positive–negative signs of principal stresses. The work presented in this paper will focus the research on the latter model. Ambartsumyan’s bimodular model asserts that Young’s modulus of elasticity depends not only on material properties, but also on the stress state of that point. Since the stress state in a point is

n Corresponding author at: College of Civil Engineering, Chongqing University, Chongqing 400045, PR China. Tel.: þ86 23 65120898; fax: þ86 23 65123511. E-mail address: [email protected] (X.-T. He).

0020-7403/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2012.06.003

generally obtained as a final result of analysis, the elastic constants involved in the governing equations are not correctly indicated beforehand. In other words, except in particularly simple problems it is not easy to estimate a priori the stress state in a point in the deformed body. In some complex problems, we have to resort to FEM based on an iterative strategy [12–14]. Generally, many researchers adopted directly iterative methods based on an incrementally evolving stiffness, including an improved algorithm in which the shear stress and shear strain are set to zero to formulate the elastic matrix [15–17], an improved algorithm keeping Poisson’s ratio constant while modifying the elastic matrix [18,19], the initial stress technique [20], the smoothing function technique [21], a new and simple analytical-iterative method for statically indeterminate structures [22] and an iterative process applying the principle of strain invariant [23]. More recently, some analytical works have been carried out, all involving simple bending components in small-deflection, such as beams, bending-compression columns and thin plates. When analyzing these bending components, many researchers adopted a simplified mechanical model on subarea in tension and compression, in which the midsurface remains unstrained under bending. Based on this simplified model, Yao and Ye firstly obtained a series of analytical solutions of bimodular columns [24] and beams [25] in the case of small-deflection. He et al. obtained the same solution via the equivalent section method

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X.-T. He et al. / International Journal of Mechanical Sciences 62 (2012) 103–110

Nomenclature Eþ, E

tensile and compressive Young’s modulus of elasticity, respectively m þ , m  tensile and compressive Poisson’s ratio, respectively a, b principal stress coordinates system ea , eb principal strain sa , sb principal stress aij ði, j ¼ 1, 2Þ elastic compliance r, y radial and circumferential coordinates, respectively t thickness of circular plate q intensity of uniformly-distributed loads a radius of circular plate t1, t2 tensile and compressive section height of plate, respectively Dn flexural stiffness of bimodular plate D þ , D  tensile and compressive components of Dn, respectively (see Eq. (3)) ur, uy, w radial, circumferential displacements and deflection at the middle plane of plate, respectively sr, sy radial and circumferential membrane stress, respectively Nr, Ny radial and circumferential membrane force per unit length, respectively er, ey radial and circumferential unit elongations at the middle plane of plate, respectively C integral constant Qr radial shear force per unit length

[26] and derived the approximate elasticity solution without plane-section assumption [27]. For a bimodular thin plate, He et al. firstly used the Kirchhoff hypothesis to judge the existence of the elastic neutral layers in small-deflection bending [28], and consequently derived a series of analytical solutions in rectangular and polar coordinate systems [28,29], respectively. However, once the plate is in large-deflection bending, the strain of midsurface cannot be ignored thus not only the flexural effect but also the membrane stress in the thin plate should be considered together. Up to now, there is few research works on this topic, especially for the analytical one. The classical von Ka´rma´n largedeflection theory for bending thin plates deals mainly with the singular modular problem. When different moduli is considered, not only the equilibrium equation but also the consistency equation, should be modified according to some practical assumptions introduced beforehand. In this paper, based on the known flexural stiffness for a bimodular thin plate in small-deflection bending [28,29], we establish the equilibrium and consistency equations in largedeflection bending. We use the perturbation method to obtain the explicit expressions for the stress and the deflection, in which the central deflection of the plate is taken as a perturbation parameter. For the comparison, we also use the displacement variation method to solve the same problem. Moreover, we analyze the influences of bimodular effect on the relation of load vs. deflection, discuss the yield conditions for the edge and center of the plate and study the transition problem from plate to membrane when considering the bimodularity of materials.

Mr radial bending moment per unit length P, Z, S, W, T1, T2 dimensionless quantities (see Eq. (15)) K dimensionless form of Dn (see Eq. (18)) E singular or average modulus of elasticity m singular or average Poisson’s ratio b ratio of different moduli (see Eq. (18)) w0 central deflection of plate Wm dimensionless form of w0, perturbation parameter a1, a3,y undetermined coefficients during perturbation (see Eq. (22)) o1(Z), o3(Z),y undetermined functions during perturbation (see Eq. (22)) f2(Z), f4(Z),y undetermined functions during perturbation (see Eq. (22)) s0r radial stretching stress at the middle plane of plate S0r ðZÞ dimensionless form of sr0 (see Eq. (36)) s00r radial bending stress on the convex surface of plate S00r ðZÞ dimensionless form of sr00 (see Eq. (36)) U total strain potential energy U1 potential corresponding to bending U2 potential corresponding to midsurface strain Am , C m undetermined coefficients s1, s2, s3 three principal stresses in yield problem sy yield stress sre, ste radial and circumferential stresses on the convex surface of plate edge, respectively src radial stress on the convex surface of plate center

It is a bilinear material model whose tangents at the origin are discontinuous. The basic assumptions of this model are as follows: (1) the body under study is continuous, homogeneous and isotropic; (2) small deformation is assumed; (3) the Young’s modulus and Poisson’s ratio of materials are E þ and m þ when the materials are in tension along certain direction and E  and m  when they are in compression; (4) when the three principal stresses are uniformly positive or uniformly negative, the three basic equations are essentially the same as those of classical theory; when the signs of the three principal stresses are different, the differential equations of equilibrium and the geometrical equations are the same as those of classical materials theory, with the exception of the physical equations; (5) m þ /E þ ¼ m  /E  , to ensure that the flexibility matrix is symmetric. The two dimensional stress–strain relation based on the Ambartsumyan–Khachatryan model for isotropic bimodulus material is expressed with respect to the principal stress coordinates (a,b) as follows:

ea ¼ a11 sa þ a12 sb , eb ¼ a21 sa þ a22 sb

where ea and eb are strain components, sa and sb are stress components in the principal coordinates, and elastic compliance aij(i, j ¼1, 2) is expressed as follows, in relation to the sign of corresponding stress components: ( ( 1=E þ ðsb 4 0Þ 1=E þ ðsa 4 0Þ a11 ¼ a22 ¼ ð2aÞ  1=E ðsb o 0Þ, 1=E ðsa o0Þ, (

2. Bimodular material model The most familiar and simplest bimodular material model for linearized elastic material was proposed by Ambartsumyan [11].

ð1Þ

a12 ¼

m þ =E þ ðsb 4 0Þ 

m =E ðsb o 0Þ, 

( a21 ¼

m þ =E þ ðsa 4 0Þ m =E ðsa o 0Þ:

ð2bÞ

It is obvious that according to the fifth basic assumption mentioned above, a12 ¼a21 holds true, which ensures the flexibility

X.-T. He et al. / International Journal of Mechanical Sciences 62 (2012) 103–110

matrix is symmetric and makes it possible to apply a numerically iterative technology based on FEM. It must be noted that since the stress state of the point in question is unknown in advance, we have to begin with a single modulus problem, thus gaining the initial stress state to form a corresponding elasticity matrix for each element. This method is only available for the numerically iterative technology based on FEM. For the analytical solution of such a problem, however, we will find that, since the physical equations originally built on the principal stress direction a and b are rebuilt on general coordinates x and y via coordinates conversion, many nonlinear items concerning the principal stress and its direction cosine are generated in the physical equations. Therefore, it is very difficult to solve such a complicated problem, analytically. In the next section, we will take a bimodular circular plate in large-deflection axisymmetric deformation as our studied object, in which the radial and circumferential stresses coincide with the principal stresses, thus establish von Ka´rma´n equations considering different moduli in tension and compression. We will focus the derivation on the modification for Ka´rma´n equations introduced by different moduli of materials.

105

3.2. Consistency equation Now, let us consider the consistency relation between the deflection and the membrane stress of the plate. If let radial displacement at the middle plane of the plate be ur, the radial and circumferential unit elongations at the middle plane of the plate be, er and ey, respectively, the geometrical equation in largedeflection axisymmetric bending should be  2 9 = r er ¼ du þ 12 dw dr dr : ð5Þ ur ; ey ¼ r Let the radial and circumferential membrane stress be, sr and sy, respectively; and the radial and circumferential membrane force per unit length be, Nr and Ny, respectively, where Nr ¼tsr and Ny ¼tsy. Note that under large-deflection axisymmetric bending, any point at the middle plane of the plate is in tension along two perpendicular directions. Therefore, based on the bimodular materials model founded on the principal direction, we have the stress–strain relation as follows: 9 er ¼ E1þ ðsr m þ sy Þ ¼ E 1þ t ðNr m þ N y Þ = : ð6Þ ey ¼ E1þ ðsy m þ sr Þ ¼ E 1þ t ðNy m þ N r Þ ; After considering the in-plane equilibrium relation d ðrN r Þ, dr

3. Ka´rma´n equations considering different moduli

Ny ¼

3.1. Equilibrium equation

we obtain the consistency equation via Eqs. (5)–(7)  2 2 d Nr dN r E þ t dw r2 þ þ3r ¼ 0: 2 2 dr dr dr

In polar coordinate systems, let the radial and circumferential coordinates of any point be r and y, respectively. A bimodular circular plate with thickness t is under the action of the normal uniformly-distributed loads q. If the small-deflection axisymmetric bending is considered, the governing differential equation of the neutral layer may be expressed in terms of the deflection w(r) of any point as r4w(r)¼q/Dn, where Dn is the flexural stiffness of the bimodular circular plate in small-deflection bending and is given by [28,29] Dn ¼ D þ þ D ¼

E þ t 31 3½1ðm

þ Þ2 

þ

E t 32 3½1ðm Þ2 

,

ð3Þ

where t1 and t2 are the tensile and compressive section height of the plate, respectively, and they has been determined in the earlier work presented by He et al. [28,29]. The determination of t1 and t2 is based completely on t, E þ , E  , m þ and m  , which indicates that Dn depends on the plate thickness and the mechanical parameters of the bimodular material. It is known that when a thin circular plate is under largedeflection bending, the external load is equilibrated not only by the bending effect but also by the membrane stress of the plate. Therefore, the differential equation of equilibrium in smalldeflection should be modified as Dn

      1d d 1d dw 1d dw r r  rNr ¼ q, r dr dr r dr dr r dr dr

ð4Þ

where Nr ¼tsr and sr is the radial membrane stress of the plate. It is obvious that the first term on the left side of Eq. (4) represents the bending effect and the second one represents the stretching effect. We note that comparing with the classical equilibrium equation with singular modulus, the difference introduced by bimodularity of materials is only embodied in the flexural stiffness; the detailed derivation of Eq. (4) may be obtained from any general theory of plates and shells. It is not necessary to discuss this problem here.

ð7Þ

ð8Þ

Therefore, under the known boundary conditions, Eqs. (4) and (8) may be used for the solution of the two basic unknown function, w¼w(r) and Nr ¼Nr(r). Taking the central deflection of the plate as a perturbation parameter, Chien [30] successfully used the perturbation method to solve the classical Ka´rma´n equations with singular modulus and the analytical expressions obtained by Chien is very close to the experimental results. Next, we will use the same method to solve Eqs. (4) and (8). Due to the introduction of different moduli and the modification for Ka´rma´n equations, however, the perturbation process will change correspondingly.

4. Application of perturbation method ´rma ´n equations 4.1. Simplification and non-dimensionalization of Ka Let us consider such a problem that a bimodular circular plate with radius a and peripheral clamped is under the action of uniformly distributed loads q. Due to the constant q, Eq. (4) may be simplified as    d 1d dw dw q C Dn r Nr ¼ rþ , ð9Þ dr r dr dr dr 2 r where C is an integral constant. In an axisymmetric bending problem, the radial shear force per unit length may be expressed as Qr ¼  Dnd(r2w)/dr (see Ref. [29]), thus Eq. (9) may be written as Q r N r

dw q C ¼ rþ : dr 2 r

ð10Þ

Due to the axisymmetry, at the center of the plate, the boundary conditions should satisfy dw ¼ 0, dr

Q r ¼ 0,

at r ¼ 0:

ð11Þ

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X.-T. He et al. / International Journal of Mechanical Sciences 62 (2012) 103–110

Thus, C¼0. Eq. (4) becomes lastly ! 3 2 1 d w 1 dw dw q n d w ¼ r: D þ  Nr r dr 2 r 2 dr dr 2 dr 3

ð12Þ

Moreover, at the peripheral of the plate, the corresponding boundary conditions should be dw ¼ 0, dr

w¼

r

dN r þ ð1m þ ÞN r ¼ 0, dr

at r ¼ a,

ð13Þ

where the original boundary condition ur ¼0 at r ¼a is transformed into the condition expressed in terms of Nr. At the center of the plate, the corresponding boundary conditions are dw and Nr are the finite values, dr

at r ¼ 0:

ð14Þ

We introduce the following dimensionless quantities P¼

qa4 , E þ t4

Z ¼ 1

r2 , a2

N r a2 , E þ t3

S¼

W¼

w , t

T1 ¼

t1 , t

T2 ¼

t2 , t ð15Þ

Eqs. (12) and (8) become, respectively 

 2  d dW 3ð1 þ bÞ 3ð1 þ bÞ dW P S ð1ZÞ ¼ 2 dZ 16K 4K dZ dZ

ð16Þ

and   2  1 dW 2 d

ð1 Z ÞS þ ¼ 0, 2 dZ dZ2

ð17Þ

where K¼ E¼

ð1 þ bÞT 31 1ðm

þ

þ Þ2

E þ þ E , 2

ð1bÞT 32  Þ2

1ðm

¼

3Dn , Et3

b¼

E þ E , E þ þE

E þ ¼ ð1 þ bÞE

ð18Þ

and E is an average modulus. The boundary conditions, Eqs. (13) and (14), become, respectively W¼

dW ¼ 0, dZ

2ð1ZÞ

dS ð1m þ ÞS ¼ 0, dZ

at Z ¼ 0

ð19Þ

which should satisfy the boundary conditions 9 = o1 ¼ 0, ddoZ1 ¼ 0, at Z ¼ 0,

o1 ¼ 1,

do1 dZ

is the finite value,

at Z ¼ 1: ;

ð24Þ

The solution of Eq. (23) under the condition Eq. (24) is

a1 ¼

4K , 1þb

o1 ðZÞ ¼ Z2 ,

ð25Þ

which is exactly the solution in the case of small-deflection after the comparison. The differential equation used for the solution of f2 may be obtained from the coefficient of W 2m in Eq. (17)   2  1 do1 2 d

ð1 Z Þf ¼ 0: ð26Þ þ 2 2 dZ dZ2 The boundary conditions may be obtained from the coefficient of W 2m in Eqs. (19) and (20) 9 2 dfdZ2 ð1m þ Þf 2 ¼ 0, at Z ¼ 0 = ð27Þ f 2 is the finite value, at Z ¼ 1: ; Using the known o1(Z), the solution of Eq. (26) under the condition Eq. (27) is   1 2 f 2 ðZÞ ¼ þ Z þ Z2 þ Z3 : ð28Þ þ 6 1m Similarly, the differential equation used for the solution of a3 and o3 may be obtained from the coefficient of W 3m in Eq. (16)  2  d do3 1þb 3ð1 þ bÞ do1 f2  2 ð1ZÞ a3  , ð29Þ ¼ 4K dZ K dZ dZ which should satisfy the boundary conditions 9 = o3 ¼ 0, ddoZ3 ¼ 0, at Z ¼ 0,

o3 ¼ 0,

do3 dZ

is the finite value,

at Z ¼ 1: ;

ð30Þ

Using the known o1(Z) and f2(Z), we have the solution of Eq. (29) under the condition Eq. (30)

and dW and S are the finite values, dZ

The differential equation used for the solution of a1 and o1 may be obtained from the coefficient of Wm in Eq. (16)  2  d do1 1þb  2 ð1ZÞ a1 , ð23Þ ¼ dZ K dZ

at Z ¼ 1:

ð20Þ

a3 ¼

17373m þ , 360ð1m þ Þ

o3 ðZÞ ¼

  1þb 8343m þ 2 Z þ 23Z3 þ 8Z4 þ2Z5 : ð1ZÞ þ 1440K 1m

ð31Þ Similarly, the differential equation used for the solution of f4 may be obtained from the coefficient of W 4m in Eq. (17)

4.2. Perturbation solution

2

Now, we introduce the following perturbation parameter w w0 , ð21Þ ¼ W m ¼ ðWÞZ ¼ 1 ¼ t r¼0 t where w0 is the center deflection of the plate, that is, the maximum deflection. It is obvious that P ¼P(Wm), W¼W(Wm, Z) and S ¼S(Wm, Z), thus P, W and S may be expressed in the terms of power series of Wm 9 3 5 3 > > 16 P ¼ a1 W m þ a3 W m þ a5 W m þ    , = W ¼ o1 ðZÞW m þ o3 ðZÞW 3m þ o5 ðZÞW 5m þ    , : ð22Þ > > 2 4 6 ; S ¼ f ðZÞW þf ðZÞW þf ðZÞW þ    , 2

m

4

m

6

m

Obviously, there are no even terms with respect to Wm in the expansion of P and W, and no odd terms in the expansion of S, because they are all eliminated during the perturbation.

 do1 do3 d

ð1ZÞf 4 þ ¼ 0: dZ dZ dZ2 The corresponding boundary conditions are 9 2 dfdZ4 ð1m þ Þf 4 ¼ 0, at Z ¼ 0 = f 4 is the finite value,

at Z ¼ 1: ;

ð32Þ

ð33Þ

Using the known o1(Z) and o3(Z), we also have the solution of Eq. (32) under the condition Eq. (33) " 1þb 160104m þ 8052m þ f 4 ðZÞ ¼ þ ðZ þ Z2 þ Z3 Þ 2 þ 30240K 1m þ ð1m Þ  501249m þ 4 5 6 7  Z 123 Z 39 Z 9 Z : ð34Þ 1m þ Thus, a1, o1(Z), f2(Z), a3, o3(Z), f4(Z),y may be determined in such a procedure. When E þ ¼E  and m þ ¼ m  , we may have b ¼0

X.-T. He et al. / International Journal of Mechanical Sciences 62 (2012) 103–110

and K ¼1/[4(1  m2)] via Eq. (18), thus a1, o1(Z), f2(Z), a3, o3(Z) and f4(Z) may regress to the corresponding expressions of the classical problem. Substituting the determined a1 and a3 into the first expression of Eq. (22) and considering Eq. (15), we may obtain the relation of load vs. deflection as follows: 3qa4 4K w0 17373m þ w0 3 þ , þ 4 ¼ 1þb t 360ð1m þ Þ t 16E t

ð35Þ

which may regress to the corresponding expression of the classical problem when E þ ¼E  and m þ ¼ m  . Moreover, the first term in the right side of Eq. (35) exactly corresponds to the solution of a bimodular plate in small-deflection bending, i.e. w0 ¼qa4/64Dn. 4.3. Stress analysis For the purpose of studying the stress of each point in the plate, we need to derive the stretching stress from the membrane forces and the bending stress from the flexure of the plate. Let S0r ðZÞ be the dimensionless quantity of the radial stretching stress at the middle plane of the plate s0r ; let S00r ðZÞ be the dimensionless quantity of the bending stress on the convex surface s00r , that is

S0r ðZÞ ¼

s0r a2 , E þ t2

From

the

S00r ðZÞ ¼

third

s00r a2 : E þ t2

ð36Þ

expression

of

Eq.

(22),

we

may

take

S0r ðZÞ  f 2 ðZÞW 2m þf 4 ðZÞW 4m . At the peripheral of the plate, the radial stretching stress is   W 2m 1 þ b 2013m þ 2 1þ S0r ð0Þ ¼ Wm þ þ 3ð1m Þ 1260K 1m

ð37Þ

and at the center of the plate, we have " # W 2m 1 þ b 33ðm þ Þ2 74m þ þ 34 2 þ ð53 S0r ð1Þ ¼ m Þ W m : 1m þ 6ð1m þ Þ 630K

The radial bending stress on the convex surface is 12M r t1 , t3

ð39Þ

where t1 is the tensile section height of the plate and Mr is the radial bending moment per unit length and is given by Eq. (41) in Ref. [29]. Thus, from Eqs. (36) and (39), we have ( ) 2  24t 1 þ 00  nd W þ  dW Sr ðZÞ ¼ þ 4 D ð1 þ m Þ þD ð1 þ m Þ 2ð1ZÞD : dZ dZ2 E t ð40Þ From the second expression of Eq. (20), we may take WðZÞ  o1 ðZÞW m þ o3 ðZÞW 3m . At the peripheral of the plate, the radial bending stress is   96t Dn 1 þ b 8343m þ 2 S00r ð0Þ ¼ þ1 4 W m 1 þ W ð41Þ m 1440K 1m þ E t and at the center of the plate, we have

S00r ð1Þ ¼

  48t 1 ½D þ ð1 þ m þ Þ þ D ð1 þ m Þ 1 þ b 2919m þ 2 W m 1 Wm : þ 4 þ 720K 1m E t ð42Þ þ



þ



5.1. Strain potential energy The total strain potential energy stored in the bimodular plate in large-deflection bending, U, will include the potential corresponding to bending, U1, and the potential corresponding to midsurface strain, U2, that is U ¼ U1 þ U2 :

ð43Þ

In an axisymmetric problem, U1 stored in a bimodular thin plate may be written as 2 3 !2  2 Z 2 2 d w 1 dw dw d w þ 4r 5dr þ þ2m þ U 1 ¼ pD r dr dr dr 2 dr 2 2 3 !2  2 Z 2 2 d w 1 dw dw d w   4r 5dr, ð44Þ þ pD þ þ 2m r dr dr dr 2 dr 2 and U2 may be expressed in terms of the stress and strain at the middle plane of the plate as Z ð45Þ U 2 ¼ pt ðsr er þ sy ey Þrdr: Using the geometrical and physical equations in large-deflection bending, as shown in Eqs. (5) and (6), we may have 8  2 #2 Z <" pE þ t dur 1 dw U2 ¼ þ 2 dr 1ðm þ Þ2 : dr "  2 #) u2r 1 dw þ ur dur þ 2 þ 2m rdr: ð46Þ þ 2 dr r dr r

Since there is no circumferential displacement uy in an axisymmetric problem, the displacement components for each point on the midsurface may be taken as X X ur ¼ Am ður Þm , w ¼ C m wm , ð47Þ m

m

where (ur)m and wm are only the function of r and satisfy the boundary conditions of displacement. The equations determining the coefficients Am and Cm may be obtained from Ritz method as follows: Z @U @U ¼ 0, ¼ 2p qwm rdr: ð48Þ @Am @C m For the comparison with the result in Section 4.2, we still study a bimodular clamped circular plate with radius a is under the action of uniformly-distributed loads q. The formulas for the radial displacement and deflection may be taken as   r r r r2 ur ¼ 1 A0 þA1 þ A2 2 þ    a a a a #)   "    2 2 2 2 r r r2 C 0 þ C 1 1 2 þ C 2 1 2 þ    , ð49Þ w ¼ 1 2 a a a which satisfy the boundary conditions

3

When E ¼E and m ¼ m , we have b ¼0, t1 ¼t/2, D ¼Et / [12(1  m2)] and K ¼1/[4(1  m2)], Eqs. (37), (38), (41) and (42) may regress to the corresponding expressions in the classical problem with singular modulus. The results obtained above are helpful for the analysis of yield problem of the plate, especially while the bimodular effect of materials is more obvious. n

5. Application of variation method

5.2. Ritz method ð38Þ

s00r ¼

107

ur ¼ w ¼

dw ¼ 0, dr

at r ¼ a

ð50Þ

and also satisfy the axisymmetric conditions ur ¼

dw ¼ 0, dr

at r ¼ 0:

ð51Þ

108

X.-T. He et al. / International Journal of Mechanical Sciences 62 (2012) 103–110

8

Now, we take only two undetermined coefficients in the first expression and one undetermined coefficient in the second expression in Eq. (49), respectively, that is r r r , ur ¼ A0 þ A1 1 a a a



w ¼ C 0 1

7

 2 2

r a2

:

Like a classical variation method, we obtain the following relation after a series of computation 2:592 E þ t qa4 C0 þ C 30 ¼ , n 2 64D 1ðm þ Þ 64Dn

5 P 4

ð53Þ

3

which may be used for the determination of C0, thus A0 and A1 may also be determined. Note that C0 is exactly the center deflection w0 of the plate, thus Eq. (53) may be written as w0 þ

2:592 E þ t qa4 w30 ¼ : n 2 þ 64D 1ðm Þ 64Dn

1 0

0

0.1

0.2

0.3

0.4

0.5 Wm

0.6

0.7

0.8

0.9

1

Fig. 1. Relation of load vs. deflection for m þ ¼ 0.3 and b ¼1/3, in which the dashed line is for variation solution (Eq. (55)) and the solid line is for perturbation solution (Eq. (56)).

6.1. Comparisons of two solutions Eq. (54) presents the relation between the load and the central deflection via variation method. After considering the dimensionless relations shown in Eq. (18), Eq. (54) becomes

14 12

ð55Þ

10

Moreover, the same relation obtained via perturbation method is shown in Eq. (35) in Section 4.2 and its dimensionless form is 3 4K 17373m þ 3 P¼ W : Wm þ 16 1þb 360ð1m þ Þ m

variation solution

2

ð54Þ

6. Results and discussions

3 4K 0:486 P¼ Wm þ W 3m : 16 1þb 1ðm þ Þ2

perturbation solution

6

ð52Þ

ð56Þ

We note that the only difference between Eqs. (55) and (56) lies in the factor for the term W 3m . For any given values of m þ and b , from Eq. (18) and m þ /E þ ¼ m  /E  , we may have m  ¼ m þ (1 b)/(1þ b) and the dimensionless T1 and T2, thus K is also determined via Eq. (18). For the values of m þ ¼0.3 and b ¼1/3, we plot the variation curves of the load and central deflection, as shown in Fig. 1. It is easily seen that the solution obtained via displacement variation method is very close to that obtained via perturbation method, which indicates that the perturbation method based on the central deflection is valid. 6.2. Influence of bimodular effect on load vs. deflection We use the perturbation solution to analyze the influences of different moduli on the deflection. The variation curves when m þ ¼0.3 and b are taken as 1/3, 0 1/3, respectively, are plotted in Fig. 2, where b ¼1/3 corresponds to the case E þ ¼2E  , which may be seen from Eq. (18); b ¼0 to the case E þ ¼E  and b ¼ 1/3 to the case E  ¼2E þ . It is seen that from Fig. 2 (1) Comparing with the case of singular modulus E þ ¼E  , for the same magnitude load, the central deflection of the plate will increase when E þ ¼2E  and will decrease when E  ¼2E þ , which indicates that if the tensile modulus of the materials decreases further, the central deflection of the plate will become smaller. (2) Comparing with the case of singular modulus E þ ¼E  , for the same magnitude central deflection, the external load applied on the plate will increase when E  ¼2E þ and will decrease when E þ ¼2E  , which indicates that if the tensile modulus of

E − = 2E +

8 P

E+ = E−

6 4

E + = 2E −

2 0

0

0.1

0.2

0.3

0.4

0.5 Wm

0.6

0.7

0.8

0.9

1

Fig. 2. Relation of load vs. deflection for m þ ¼ 0.3 when E þ /E  are taken as 1/2, 1, and 2, respectively, in which the dotted lines are for the small-deflection solutions and the solid lines are for the large-deflection solutions.

the materials increases further, the applied load will become smaller. (3) For all the three cases, when the central deflection is less than 0.3, the linear results from the small-deflection solution are very close to the nonlinear results from the large-deflection ones, which indicate that when the central deflection is very small, the cubic term in the large-deflection solution is negligible.

6.3. Yield conditions The yield problem of the plate may be analyzed via the stress expressions obtained in Section 4.3. If let the three principal stresses be s1, s2 and s3, respectively, we may take the following

X.-T. He et al. / International Journal of Mechanical Sciences 62 (2012) 103–110

yield conditions 2

103 2

2

2 y,

ðs1 s2 Þ þ ðs2 s3 Þ þðs3 s1 Þ ¼ 2s

ð57Þ

edge yield

where sy is yield stress. At the edge of the plate, the radial displacement, ur ¼r(Ny  m þ Nr)/E þ t, is equal to zero, thus we may have

s1 ¼ sre , s2 ¼ ste ¼ m þ sre , s3 5 sre ,

sy m þ sy sre ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ste ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 1m þ þ ðm þ Þ 1m þ þðm þ Þ2

ð59Þ

thus, the yield conditions on the convex surface of the plate edge is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sy a2 1m þ þ ðm þ Þ2 ðS0r ð0Þ þ S00r ð0ÞÞ: ð60Þ þ 2 ¼ E t Substituting Eqs. (37) and (41) into Eq. (60), the yield conditions at the edge becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sy a ¼ 1m þ þ ðm þ Þ2 W m Eþ" t2 # 32KT 1 1 T 1 ð8343m þ Þ 2 ð1 þ bÞð2013m þ Þ 3 W W þ þ þ W m m m : 1þb 3ð1m þ Þ 45ð1m þ Þ 3780Kð1m þ Þ2 2

ð61Þ Theoretically, we may also derive the yield conditions at the center of the plate. The principal stresses should be equal at the center, thus

s1 ¼ s2 ¼ src , s3 {src ,

102

ð58Þ

where sre and ste are the radial and circumferential stresses on the convex surface of the plate edge, respectively. Neglecting s3, from Eqs. (57) and (58) we may solve

ð62Þ

where src is the radial stress on the convex surface of the plate center. Similarly, neglecting s3, from Eqs. (57) and (58) we may solve

src ¼ sy ,

109

ð63Þ

Substituting Eqs. (38) and (42) into Eq. (63), the yield conditions at the center becomes    sy a2 Wm 1b 1m þ 3 3 þ þ 2 ¼ 6ð1m þ Þ 96T 1 T 1 þ 1 þ b 1m T 2 þð53m ÞW m E t   2T 1 ð2919m þ Þ 1 þ b 3 1b 3 T þ T W 2m  15K 1m þ 1 1m 2 ) 1þ b 33ðm þ Þ2 74m þ þ34 3 ð64Þ  Wm 1m þ 630K Using logarithm coordinates, we plot the variation curves for edge and center yield when m þ ¼0.3 and b are taken as  1/3, 0 and 1/3 (or E þ /E  are taken as 1/2, 1, and 2), respectively, as shown in Fig. 3. It is seen from Fig. 3 that (1) Comparing with the case of singular modulus E þ ¼E  , for the same magnitude yield stress, the central deflection of the plate will increase when E þ ¼2E  and will decrease when E  ¼2E þ , which indicates that if the tensile modulus of the materials decreases further, the central deflection of the plate will become smaller. (2) Comparing with the case of singular modulus E þ ¼E  , for the same magnitude central deflection, the yield stress of the plate will increase when E  ¼2E þ and will decrease when E þ ¼2E  , which indicates that if the tensile modulus of the materials increases further, the yield stress will become smaller. (3) However, like the case of singular modulus E þ ¼E  , the introduction of different moduli does not change the rule that when the circular plate is loaded further, the edge of the plate will start to yield and then the center of the plate.

y a2 E +t 2

101 E - = 2E + 100

E+ = EE + = 2E -

10-1 10-1

center yield 100 w0/t

101

Fig. 3. Yield condition for m þ ¼0.3 when E þ /E  are taken as 1/2, 1, and 2, respectively, in which the dashed lines are for the center yield and the solid lines are for the edge yield.

It is also interesting to study the transition problem from plate to membrane when considering the bimodularity of materials. If the bimodular plate is very flexible and the deflection of the plate increases further, the plate will behave like a membrane, which implies the bending effect will disappear and the stretching effect becomes dominant, as discussed in the analysis of clamped punch-loaded blister test [31]. In such a so-called bimodular membrane problem, however, the discussion on different moduli will be of little significance. Due to the fact that the modulus of elasticity and Poisson’s ratio in this stress state are uniformly taken as the tensile ones according to Ambartsumyan’s bimodular material model, the problem will eventually regress to a classical one with singular modulus.

7. Concluding remarks In this paper, we establish the von Ka´rma´n equations with different moduli in tension and compression for a thin circular plate in large-deflection bending and use the perturbation method and the displacement variation method to solve the problem, respectively. We compare the two results obtained in different ways, analyze the influences of bimodular effect on the relation of load and central deflection, study the yield conditions for the edge and center of the plate and discuss the transition from plate to membrane when considering different moduli. The following main conclusions can be drawn. (1) The comparison between the perturbation result and the variation one shows that the perturbation solution based on the central deflection of the plate, is valid. (2) The bimodularity of the material will have an effect on the relation of load vs. deflection to a certain extent, which should be considered in the analysis and design of bimodular thin plates. (3) The introduction of different moduli will eventually influence the yield stress at the edge and center of the plate, however, this introduction of materials nonlinearity does not change the yield order that the edge of the plate will firstly yield and then the center of the plate. (4) During the transition from plate to membrane, the bimodular plate will gradually regress to the classical one with singular modulus.

110

X.-T. He et al. / International Journal of Mechanical Sciences 62 (2012) 103–110

This work will be helpful for analyzing the mechanical responses of some thin-plate-like structures in large-deflection bending, especially for studying the mechanical behaviors of thin film materials with obvious bimodularity and with moderate thickness or hardness (in this case, the stretching and bending effects should be considered together) [32,33]. For example, the information about the adhesion strength of a film-substrate system can generally be obtained by analyzing experimental data from pressurized blister tests; however, most of existent research works are based on the singular modulus elasticity theory and in most cases only stretching effect is considered. But for some coating films such as with certain thickness or hardness and also with relatively obvious bimodularity (some polymer films, for example), it is appropriate that the large-deflection theory for thin plates with different moduli in tension and compression should be adopted in the analysis of experimental data. The same problems mentioned above may be found in the analysis of experimental data from pressurized bulge tests for diaphragms of instruments (some diaphragm is made by organic glass with obvious bimodular effect [11], for example). The work presented here is only a theoretical study and we will extend it to the above-mentioned application fields in our further works.

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