Symplectic system analysis for finite sector plates of viscoelastic media

International Journal of Engineering Science 79 (2014) 30–43

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Symplectic system analysis for finite sector plates of viscoelastic media Weixiang Zhang ⇑, Yang Bai, Jianwei Wang, Li Chen School of Civil Engineering and Architecture, Henan University of Technology, Zhengzhou 450052, China

a r t i c l e

i n f o

Article history: Received 12 December 2013 Received in revised form 17 February 2014 Accepted 28 February 2014 Available online 25 March 2014 Keywords: Symplectic method Viscoelastic Laplace integral transform Eigenvector

a b s t r a c t The symplectic method is applied to boundary condition problems of finite viscoelastic solids in a sector domain. On the basis of the state space formalism and the use of the Laplace integral transform, the general eigensolution of the governing equations are obtained in the polar coordinate system. Since the eigenvectors are expressed in concise analytical forms, the adjoint symplectic relation of the Laplace domain is generalized to the time domain. Therefore, the particular solution and boundary condition problems can be discussed directly in the eigenvector space of the time domain with the use of the eigenvector expansion method. Numerical examples show a good agreement between the numerical results and the exact analytical ones. Thus, the correctness and accuracy of the proposed method are well guaranteed. Ó 2014 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

4.

5.

6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The fundamental problem and the symplectic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General eigensolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Nonzero-eigenvalue eigenvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Zero-eigenvalue eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Lateral boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Circular boundary conditions at q = a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Circular boundary conditions at q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Circular boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Lateral boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author. Tel./fax: +86 371 67789202. E-mail addresses: [email protected] (W. Zhang), [email protected] (Y. Bai), [email protected] (J. Wang), [email protected] (L. Chen). http://dx.doi.org/10.1016/j.ijengsci.2014.02.034 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.

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1. Introduction In the modern engineering designs, the knowledge in material behavior is necessary to obtain predictive numerical simulations. This is particular true for viscoelastic solids and structures. Viscoelastic materials have more complicated properties as they display both elastic and viscous type response at different strain rates, which is usually defined as viscoelastic property (Atanackovica, Pilipovicb, & Zorica, 2011; Fernández, Rodríguez, Lamela, & Fernández-Canteli, 2011; Hirsekorn, Petitjean, & Deramecourt, 2011; Johlitz, Diebels, & Possart, 2012; Yun & Richard Kim, 2011). In recent years, one can find in the literature a great amount of computational techniques to solve viscoelastic problems (Altenbacha, Eremeyeva, & Morozov, 2012; Bhattacharjee, Swamy, & Daniel, 2012; Loktev, 2012; Tscharnuter, Gastl, & Pinter, 2012). Shariyat (2011) proposed a higher-order global–local theory based on the double-superposition concept for free vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates subjected to thermomechanical loads. Minster and Novák (2011) discussed the time-dependent recovery process in a local part of a quasi-homogeneous and quasi-isotropic viscoelastic hemisphere after microindentation unloading. Based on the time–temperature superposition principle with vertical shift as well as horizontal shift, Nakada, Miyano, Cai, and Kasamori (2011) studied the prediction of long-term viscoelastic behavior of amorphous resin at a temperature below the glass transition temperature from measuring the short-term viscoelastic behavior at elevated temperatures. However, compared with elastic materials, it is very difficult to achieve analytical solutions due to the time-dependent property, and therefore numerical methods are inevitably taken into account. One of the most popular approaches is the finite element method, which has been widely applied to viscoelastic analyses (Kroon, 2011; Sasso, Palmieri, & Amodio, 2011). Levin and Markov (2011) employed a self-consistent scheme named the effective field method for the calculation of the velocities and quality factors of elastic waves propagating in double-porosity media, and analyzed the effective viscoelastic properties of the matrix with the primary small-scale pores. Cao and Chen (2012) investigated the indentation of viscoelastic composites, and performed a theoretical analysis which is followed by finite element analysis. The other important approach is the boundary element method. Compared with the finite element method, this method can reduce the dimension of the problem and provide an attractive idea for the viscoelastic research. Zhang, Rong, and Li (2010) discussed crack problems in linear viscoelastic materials by generalizing the Heaviside function to represent the displacement discontinuity across the crack surface. Based on differential constitutive relations, Mesquita and Coda (2007) provided the important algebraic equations and presented a method for the treatment of two dimensional coupling problems between the finite element method and the boundary element method by discussing Kelvin and Boltzmann models. It should be pointed out that the Laplace transform is an effective method for viscoelasticity since the problem can be transformed from the time domain to Laplace domain (Chowdhuri & Xia, 2012). The mathematical integral transform method is applied in most of the above mentioned researches. However, it is difficult and important to solve Laplace inverse transforms. A lot of inverse transforms cannot be solved analytically, therefore the numerical methods of Laplace inverse transformation had to be developed and applied (Syngellakis, 2003). Traditional analysis mostly applied displacement or force method for the discussion of governing equations. These methods belong to Lagrange formulation which may consequentially lead high order partial differential equations. Taking original variables (displacements) and their dual variables as the basic variables, Zhong (2004) developed the symplectic approach for elasticity problems on the basis of the mathematical theory of symplectic geometry. In contrast to the well-known semi-inverse method, the symplectic method takes original variables and their dual variables as the basic variables in the discussion of the governing equations (Lim, Lü, Xiang, & Yao, 2009; Xu, Leung, Gu, Yang, & Zheng, 2008). Since the difficulty of solving high-order differential equations in the traditional methods, such as the semi-inverse method, is overcome, the symplectic approach has gained much attention in recent years and has been applied successfully into various branches of applied mechanics (Lim, 2010; Lim, Cui, & Yao, 2007; Zhao & Chen, 2009). However, this method cannot be applied directly into viscoelasticity for its energy non-conservation property. Noticing that, by using the Laplace integral transform, viscoelastic constitutive equations can be transformed into a set of

z

θ O

2α

ρ y

Fig. 1. The coordinate system of the plain sector domain.

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W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

corresponding elastic ones, and the symplectic method can be introduced into viscoelastic researches (Zhang & Xu, 2012). In this paper, the symplectic method is further extended to the deformation and stress analysis of finite viscoelastic materials in a sector domain under certain boundary conditions. In the polar coordinate system, the symplectic system method can also be applied by introducing a new axis n = ln q and using the Laplace transform. With the direct method, all the general solutions of the governing solutions including zero eigenvectors and non-zero eigenvectors are obtained in analytical form. Using this method, solutions of problems of non-homogeneous equations and various boundary conditions can be obtained numerically by applying the eigenvector expansion method. The polar coordinate symplectic system provides a convenient for method for solving viscoelastic problems in sector domains, especially for analyses related with viscoelastic crack. 2. The fundamental problem and the symplectic system A homogeneous isotropic media of a plain sector domain, shown in Fig. 1, is considered in the polar coordinate system (q, h). The q-axis is along the radial direction, and the origin is located at the center of the sector. The Cartesian coordinates ðy; zÞ are used as a referential system. For plane stress problems, the strain–displacement relations are

@u @q u 1 @v ¼ þ q q @h @v 1 1 @u ¼  vþ @q q q @h

eq ¼ eh eqh

ð1Þ

v are displacements along q and h directions, respectively. The equilibrium equations are @ rq 1 @ rqh 1 þ þ ðrq  rh Þ þ hq ¼ 0 @ q q @h q @ rqh 1 @ rh 2 þ þ r þ hh ¼ 0 @q q @h q qh

in which u and

ð2Þ

where hq and hh are body forces. The constitutive equations for linear viscoelastic materials can be expressed in an integral form as

Z

t

@eij ðr; sÞ ds @s Z t @ e ðr; tÞ rmm ðr; tÞ ¼ 3 Kðt  sÞ mm ds @s 0 sij ðr; tÞþ ¼ 2

lðt  sÞ

0

ð3Þ

where r is the position vector, l(t) and K(t) are the relaxation functions of shear modulus and bulk modulus. rmm and emm are the volumetric stress and strain, and sij and eij are the deviatoric components of the stress and strain tensors

1 sij ¼ rij  dij rmm ; 3

1 eij ¼ eij  dij emm 3

ð4Þ

The viscoelastic stress–stain relations can be transformed into a set of corresponding elastic ones using the Laplace transformation. As a result, the transformed constitutive equations are

 ðsÞeij ðr; sÞ sij ðr; sÞ ¼ 2l

ð5Þ

r mm ðr; sÞ ¼ 3KðsÞemm ðr; sÞ

E1

1

E2

η2

En

ηn

Fig. 2. The generalized Maxwell type model.

W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

33

where a bar over a variable designates its Laplace transform, and s is the transform parameter. The relaxation modulus of EðsÞ  ðsÞ in the Laplace domain can accordingly be expressed as and Poisson’s ratio v

EðsÞ ¼

 ðsÞKðsÞ 9l ;  ðsÞ 3KðsÞ þ l

tðsÞ ¼

 ðsÞ 3KðsÞ  2l :  2lðsÞ þ 6KðsÞ

ð6Þ

In the differential approach, viscoelastic constitutive models are constructed by various combinations of linear springs and dashpots. One of the most popular models is the generalized Maxwell model shown in Fig. 2. Its Young’s modulus can be written as

EðtÞ ¼

X X Ei eEi t=gi or EðsÞ ¼ i

i

Ei : s þ Ei =gi

ð7Þ

For many cases, we can assume that the viscoelastic materials behave the same in dilatation as in shear, which results in a constant Poisson’s ratio t:

tðsÞ ¼ tðtÞ ¼ t:

ð8Þ

In practical implementations, this assumption is made only for simplicity and could reduce the complexity of the mathematical derivations. According to the variational principle, we get a Z a

Z d

a

r q

0

     i  r h  h 1 @u  @u @ v @ v r 1h 2 þ qh  q þ2r  2h þð1þ tÞs 2qh  tr  hr q u  hq  v r  h qdqdh ¼ 0 þs  2r u þ  þ @q q @h @ q q q @h E

ð9Þ

where a is the radius, and 2a is the boundary angles of the sectorial domain. In the polar coordinate system, the symplectic method can also be applied. However, a new axis must be introduced:

n ¼ ln q

ð10Þ

Under the new coordinates (n, h), Eq. (9) is rewritten as

Z

a

Z

ln a

d a

Ldndh ¼ 0

ð11Þ

0

where

2

3 2 3 qr q Sq 6 7 6  7 4 Sh 5 ¼ 4 qrq 5; qr qh Sq h

ð12Þ

L is a Lagrangian function

!   i  @ v @ v_ @u 1h   e2n v h  _ þ h  þ Sh u  2S2q þ 2S2h þ ð1 þ tÞS2qh  tSh Sr  e2n u þ Sq h L ¼ Sq u  v þ q h @h @h @n E

ð13Þ

in which over dot on a variable donates its differentiation to coordinate n. To obtain the fundamental governing equations in the Hamiltonian system, we write the displacement variables in vector form:

Q ¼

   u

ð14Þ

v

Then, its corresponding dual vector is written as

" # " #  þ @@hv Þ þ 2u _ 2tðu Sq @L E  ¼ P¼ ¼ 2ð1  t2 Þ ð1  tÞðv_  v þ @ uÞ _ Sqh @Q @h

ð15Þ

 The Hamiltonian function can be established by using the mutually dual vectors Q and P:

  ; PÞ  ¼P  T Q  L Q ; Q_ HðQ

ð16Þ

Applying the variational method,

dH _ Q ¼ ; dP

dH P_ ¼  dQ

ð17Þ

one has

  þh _ ¼ Hw w

h iT  ¼ 0 T , h  ¼ Q TP in which w

ð18Þ 0

q h

h h

T

2n

e

is the vector of body forces per unit area, and H is the operator matrix

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W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

2

t

1t2 2E

@ t @h

6 6 @ 1 6 H ¼ 6 @h @ 6 E E @h 4 @ @ E @h E @h

0

3

7 t7  1þ2 E 7 7 @ 7 5 @h 1

0

t t @h@

ð19Þ

The other stress component is

  @ v h ¼ E u þ þ tSq Sh ¼ q r @h

ð20Þ

The corresponding lateral boundary conditions are



tSq þ E u þ @@hv ¼ qr h qr qh ¼ qr qh

ðh ¼ aÞ:

ð21Þ

The complete solutions of Eq. (18) includes be two parts: the general eigensolutions and a particular solution. To derive the general solutions, we usually suppose the lateral boundary of the sector is homogeneous, that is



tSq þ E u þ @@hv ¼ 0 ðh ¼ aÞ: qr qh ¼ 0

ð22Þ

The homogeneous assumption is only to derive the general solutions of the governing equations. For non-homogeneous case, the satisfaction approach will be discussed in detail in Section 4. 3. General eigensolutions Since the governing equations (18) belong to linear Hamiltonian system, the method of separation of variables can be applied. So, we can suppose the general solutions satisfy

 hÞ ¼ v  ðnÞu  ðhÞ wðn;

ð23Þ

 ¼ 0, we get Substituting Eq. (23) into Eq. (18), and letting the non-homogeneous term h

v ðnÞ ¼ ejn

ð24Þ

 ðhÞ ¼ ju  ðhÞ Hu

ð25Þ

and

 ðhÞ is its corresponding eigenvector: where j is an eigenevalue, and u

 ¼ u

   q 1 ; q 2 ; p 1 ; p 2 T ¼ ½q  p

ð26Þ

Eqs. (25) are ordinary differential equations about the coordinate h. The characteristic equation along h direction is

2

ðj þ tÞ

kt

ð1  t2 Þ=E

k

1j

0

E

Ek

Ek

Ek2

tj kt

6 6 det 6 4

0

3

2ð1 þ tÞ=E 7 7 7¼0 5 k

ð27Þ

1  j

Its solutions are

k1;2 ¼ ð1 þ jÞi;

k3;4 ¼ ð1  jÞi

ð28Þ

3.1. Nonzero-eigenvalue eigenvectors Since four different characteristic values are obtained from Eq. (28), the general solution can be expressed as

2

A1 cosð1 þ jÞh þ B1 sinð1 þ jÞh þ C 1 cosð1  jÞh þ D1 sinð1  jÞh

3

6 A sinð1 þ jÞh þ B cosð1 þ jÞh þ C sinð1  jÞh þ D cosð1  jÞh 7 2 2 2 2 7 7: 4 A3 cosð1 þ jÞh þ B3 sinð1 þ jÞh þ C 3 cosð1  jÞh þ D3 sinð1  jÞh 5

 ¼6 u 6

A4 sinð1 þ jÞh þ B4 cosð1 þ jÞh þ C 4 sinð1  jÞh þ D4 cosð1  jÞh

ð29Þ

W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

35

Not all of the constants are independent. Substituting Eq. (29) into Eq. (25), we get twelve linear equations about the constants. Thus, only four constants are independent. For brevity, we can divide the solutions into two groups: symmetric solutions and anti-symmetric ones. As a result, they are expressed as

2

c1 a11 cosð1 þ jÞh þ c2 a12 cosð1  jÞh

3

6 c a sinð1 þ jÞh þ c a sinð1  jÞh 7 1 21 2 22 7 7 4 c1 a31 cosð1 þ jÞh þ c2 a32 cosð1  jÞh 5

s ¼ 6 u 6

ð30Þ

c1 a41 sinð1 þ jÞh þ c2 a42 sinð1  jÞh and

2

c3 a13 sinð1 þ jÞh þ c4 a14 sinð1  jÞh

3

6 c a cosð1 þ jÞh þ c a cosð1  jÞh 7 3 23 4 24 7 7 4 c3 a33 sinð1 þ jÞh þ c4 a34 sinð1  jÞh 5

a ¼ 6 u 6

ð31Þ

c3 a43 cosð1 þ jÞh þ c4 a44 cosð1  jÞh respectively, in which the coefficients are: a11 = a12 = a13 = a14 = a21 = a23 = 1, a22 = a24 = (t  j  jt  3)/ (3  t  j  jt), a31 ¼ a33 ¼ a41 ¼ a43 ¼ 2Ej=ð1 þ tÞ, a32 ¼ a34 ¼ Ejð3  jÞ=ð3  t  j  jtÞ; a42 ¼ a44 ¼ Ejð1  jÞ= ð3  t  j  jtÞ: To determine the unknown eigenvalues, substituting homogeneous lateral boundary conditions (22) into the solutions (30) and (31), we get



ðjt  3 þ t þ jÞ cosð1 þ jÞa ð1 þ jÞð1 þ tÞ cosð1  jÞa ðjt  3 þ t þ jÞ sinð1 þ jÞa



ð1  jÞð1 þ tÞ sinð1  jÞa

c1



c2

¼0

ð32Þ

¼0

ð33Þ

for the symmetric solutions, and



ðjt  3 þ t þ jÞ sinð1 þ jÞa

ð1 þ jÞð1 þ tÞ sinð1  jÞa

ðjt  3 þ t þ jÞ cosð1 þ jÞa ð1  jÞð1 þ tÞ cosð1  jÞa



c3 c4



for the anti-symmetric ones. Due to the existence of non-zero eigenvectors, the integral constants c1c2 or c3c4 cannot be zeros simultaneously. Thus the coefficient determinants of Eqs. (32) and (33) must be zeros, Accordingly, the eigenvalue equations are established. The anti-symmetric non-zero-eigenvalue solutions of j = ±1 have two important physical interpretations. One is

 ¼ ½ 0 1 0 0 T u

ð34Þ

for j = 1, which is the solution for the rigid body rotation about the origin. For j = 1, the solution is

3 2 sinð2hÞ 6 ð1 þ tÞ cosð2aÞ þ ð1  tÞ cosð2hÞ 7 7  ¼6 u 7 6 5 4 2E sinð2hÞ 2

ð35Þ

E cosð2aÞ þ E cosð2hÞ Its physical explanation is shown in Fig. 3, representing solution acting by a concentrated bending moment at the origin. On the Bases of the property of the Hamiltonian operator matrix H, we define the integral product as

 1 ; J; u  2i ¼ hu

Z

a

a

 T1 Ju  2 dh u

ð36Þ

z

M O

y

Fig. 3. Physical explanation of the eigenvector of j = 1.

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W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

 1 and u  2 are arbitrary eigenvectors, and J is a unit rotational matrix. Since the eigenvalues come in pairs, they can be where u divided into two groups:

jðj aÞ : ReðjÞ > 0; ReðjÞ ¼ 0; ImðjÞ < 0 ðbÞ ða Þ jðbÞ : jj ¼ jj j

ðj ¼ 1; 2; . . .Þ

ð37Þ

The eigenvectors can satisfy the adjoint symplectic relationships:

D D

E

D

E

 iðaÞ ; J; u  ðbÞ  ðbÞ  ðaÞ ¼ dij u ¼ u j i ; J; uj E

D

E

 iðaÞ ; J; u  jðaÞ ¼ u  ðbÞ  ðbÞ ¼ 0 u i ; J; uj

ð38Þ

Noticing that both the zero eigenvectors and non-zero ones are expressed in concise analytical forms, we can discuss the problem in the time domain directly simply by using the inverse Laplace transform. The time domain integral product is defined as





ui ; J; uj ¼

where

⁄

Z

a

a

uTi  Juj dh

ð39Þ

denotes the conventional convolution. The corresponding adjoint symplectic relations in the time domain are













ðbÞ ða Þ uiðaÞ ; J; uðbÞ ¼  ui ; J; uj ¼ dij dðtÞ j





ðbÞ uiðaÞ ; J; ujðaÞ ¼ uðbÞ ¼0 i ; J; uj

ð40Þ

According to the adjoint symplectic relations, any solution vector can be described by the combinations of the eigensolutions:

¼ w

1 X

n ðnÞu n ðnÞu  ðnaÞ þ b  ðbÞ a n



ð41Þ

n¼1

or

w¼

1 X

an ðnÞ  uðnaÞ þ bn ðnÞ  unðbÞ



ð42Þ

n¼1

D E D E   ðaÞ n ðnÞ ¼  w;  J; u  J; u n ðnÞ ¼ w;  ðbÞ  ðnaÞ , an ðnÞ ¼ w; J; uðbÞ ;b and bn ðnÞ ¼  w; J; un . Since the eigenvalues come in where a n n pairs, we can regard the maximal negative eigenvalue as the first order of the eigenvalues. To describe the local effect of nonzero eigenvectors, we consider the generalized Maxwell viscoelastic model including a single spring-dashpot analog. The moduli of the materials are selected as: E1 = 4E, g1 = g, E1/g1 = 0.01 and t = 0.3 (so do the following discussions). Figs. 4 exhibit the displacement components of the first four symmetric eigenvectors. From their expressions (30), we notice that the displacements do not including the variable t, thus they only change with coordinate h. In Fig. 5, stress components of the first symmetric eigenvectors are exhibited. These figures reflect the time-dependent property of viscoelastic materials. In fact the higher order eigenvectors correspond with more complex curves, and therefore they should be employed in the numerical calculations for complex boundary condition problems. 3.2. Zero-eigenvalue eigenvectors Eq. (25) has various orders of principal vectors in the Jordan normal form for j = 0. Each principal vector has a physical interpretation. The corresponding governing equation is

 ðhÞ ¼ 0 Hu

ð43Þ

By solving Eq. (43) directly, it is not difficult to find the fundamental solutions which satisfy the lateral boundary conditions (22). Zero eigenvectors are:

 1ð0Þ u

3 2  q1 ¼ cos h 7 6q 6 2 ¼  sin h 7 ¼6 7; 1 ¼ 0 5 4 p 2 ¼ 0 p

3 1 ¼ sin h q 7 6q 6 2 ¼ cos h 7 ¼6 7: 1 ¼ 0 5 4 p 2 ¼ 0 p 2

 ð0Þ u 2

ð44Þ

 ð0Þ ¼ u  ð0Þ  ð0Þ  ð0Þ The final solutions corresponding to zero eigenvectors (44) are w 1 1 ; w2 ¼ u2 . Their geometrical interpretations are translations along y and z coordinates. Besides the fundamental solutions, there are Jordan form solutions which are determined by ð1Þ

j Hu

 jð0Þ ¼u

ðj ¼ 1; 2Þ

ð45Þ

W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

(a)

37

25 κ=4.0593+1.9520i κ=10.2457+2.7780i κ=16.3142+3.2078i κ=22.3512+3.5021i

20 15 10

q

1

5 0 −5 −10 −15 −20 −0.5

(b)

−0.25

5

0 θ

0.25

0.5

κ=4.0593+1.9520i κ=10.2457+2.7780i κ=16.3142+3.2078i κ=22.3512+3.5021i

4 3

q

2

2 1 0 −1 −2 −3 −0.5

0 θ

0.5

Fig. 4. Displacement components of the first four eigenvectors (a) displacement component q1. (b) Displacement component q2.

From Eq. (45), two Jordan form solution can be obtained:

3 ð1  tÞh sin h 6 ð1  tÞh cos h  ð1 þ tÞ sin h 7 7 6 ¼6 7 5 4 2E cos h 2

 ð1Þ u 1

ð46Þ

0 2

ðt  1Þh cos h

3

6 ð1  tÞh sin h þ ð1 þ tÞ cos h 7 7 7 5 4 2E sin h

6  ð1Þ u 2 ¼ 6

ð47Þ

0 As Fig. 6 shows, they are the solutions of concentrated forces at the origin along the vertical and horizontal directions, respectively. 4. Boundary conditions Generally, boundary conditions may be displacement conditions or stress conditions, and they also can be the mixed conditions of displacement and stress. In the symplectic system, the boundary conditions just correspond to the fundamental variables. Therefore, whether the boundaries are given by displacements or stresses, this technique can be applied. For sector domain problems, boundary conditions includes lateral conditions (h = ±a) and circular conditions (q = 0, a).

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W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

Fig. 5. Stress components of the first eigenvector: (a) Stress component p1/E. (b) Stress component p2/E.

z

z

FN

O Fs

y

O

y

 ð1Þ  ð1Þ Fig. 6. Physical interpretations of the Jordon form solution u 1 and u2 .

4.1. Lateral boundary conditions All the general solutions can satisfy the homogeneous lateral boundary condition (22). For non-homogeneous cases, we can apply the variable substitution method to transform into homogeneous ones. For example, the boundary conditions are given as:

r h ¼ r þh ; r qh ¼ r þqh ðh ¼ aÞ r h ¼ r h ; r qh ¼ r qh ðh ¼ aÞ To apply the variable substitution method, we express Eq. (48) by the fundamental dual variables

ð48Þ

W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

  @ v þ þ tSq ¼ Sþh ; Sqh ¼ Sþqh E u @h   @ v þ þ tSq ¼ Sh ; Sqh ¼ Sqh E u @h

39

ðh ¼ aÞ ð49Þ ðh ¼ aÞ

Considering Eq. (49) and the governing equations in the Hamiltonian system (18), we introduce new variables

 ¼ w  w   w h

  ¼ 1 0 0 ðh þ aÞSþ þ ðh  aÞS where w h h 2at homogeneous boundary conditions:

  @ v   þ þ tSq ¼ 0; E u @h

ð50Þ

iT

tðh þ aÞSþqh þ 2at2 ðh  aÞSqh . Obviously, these new variables can satisfy

Sqh ¼ 0 ðh ¼ aÞ

ð51Þ

However, the governing equation (18) is changed into:

  þ h _  ¼ Hw w

ð52Þ

  ¼ Hw  Now, the problem to be solved is to find a particular solution of Eq.    w _  þ h. where the non-homogeneous term is h (52). Since the method of this research involves only operations with the radial and polar variables, we discuss the problems in the following chapters in the time domain directly without using the inverse Laplace transform. According to the adjoint symplectic orthogonal relation between the eigenvectors and the completeness of the eigensolution space, the non-homogeneous term and the particular solution can be developed as 

h ¼

X

hn ðnÞ  uðnaÞ þ hn ðnÞ  uðbÞ n



ð53Þ



ð54Þ

n

and

wg ¼

X

pn ðnÞ  uðnaÞ þ pn ðnÞ  uðbÞ n

n

    ðaÞ  ðbÞ ; hn ðnÞ ¼  h ; J; un , and the coefficients p±n are to be determined. Considering Eq. respectively, where hn ðnÞ ¼ h ; J; u n (52), we get relations about the coefficients

p_ n ðnÞ ¼ jn pn ðnÞ þ hn ðnÞ; p_ n ðnÞ ¼ jn pn ðnÞ þ hn ðnÞ

ð55Þ

one can easily derive the solutions by solving the ordinary differential equations directly:

pn ðnÞ ¼ ejn n pn ðnÞ ¼ e

Z 0

jn n

n

hn ðsÞejn s ds

Z

ð56Þ

n

hn ðsÞe

jn s

ds

0

4.2. Circular boundary conditions at q = a For this case, the boundary conditions on can be displacement or stress conditions. As an example, we suppose the circular boundary conditions are given as

P ¼ P0 ðaÞ ðq ¼ aÞ

ð57Þ

The final solution can be rewritten by

w¼

X

 an  uðnaÞ ejn n þ an  unðbÞ ejn n

ð58Þ

n

Accordingly, Eq. (57) is expressed as

P0 ðaÞ ¼

X X jn ln a an  pðnaÞ ejn ln a þ an  pðbÞ n e n

ðq ¼ aÞ

ð59Þ

n

where an and an are coefficients to be determined. Based on the adjoint symplectic relationships, one get equations about the coefficients:

Z

a

a

ðaÞ

P0  qj dh ¼

Z X an  ejn ln a n

h

h

ðaÞ

pðnaÞ  qj dh þ

Z X an  ejn ln a n

a

a

ða Þ

pðbÞ ðq ¼ aÞ n  qj dh

ð60Þ

in which j = 1, 2, . . . , n. As an approximation, we usually take first N orders of the eigenvectors in numerical calculations, and the independent property of these equations can be guaranteed by the adjoint symplectic relationships.

40

W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

4.3. Circular boundary conditions at q = 0 To deal with this kind of boundary conditions, we introduce a least-square method. Suppose displacements at q = 0 are given as

u ¼ uðhÞ

v ¼ v ðhÞ

ðq ¼ 0Þ

ð61Þ

The complete solution is

~¼ w

N X

an uðnaÞ þ anþ1 uðbÞ n



ð62Þ

n¼1

The error of the solution (60) on the boundary can be described as

D¼

Z

a

a

2

k1 þ k22 dh

ð63Þ

~ ðhÞ  uðhÞ; k2 ¼ v ~ ðhÞ  v ðhÞ. The boundary conditions cannot be satisfied exactly but in a least-square sense, where k1 ¼ u

@D ¼ 0 ðj ¼ 1; 2; . . . ; 2NÞ @aj

ð64Þ

So, another set of linear equations to determine the coefficients is obtained. 5. Numerical examples 5.1. Circular boundary conditions Suppose the boundary conditions are given as

u¼v ¼0 3

rq ¼ sin h; rqh ¼ h

ðq ¼ 0Þ 3

ð65Þ

ðq ¼ aÞ:

Ra Ra The geometric parameter a is selected to be p/4. It can be verified that a rq ¼ a rqh ¼ 0. Figs. 7 and 8 show that the boundary effects only appear near the boundary where the stresses are applied, and fall off at a rapid rate along the coordinate q. The results just well explained the famous Saint–Venant principle about boundary effect. The figures also exhibits a good agreement between the numerical results and the exact analytical ones at q = a, which demonstrates the correctness and accuracy of the proposed method. 5.2. Lateral boundary conditions Consider a viscoelastic circular containing a radial crack shown in Fig. 9 (h = 0, 2p). Suppose the boundary conditions are:

0.6

ρ/a=1 ρ/a=0.9 ρ/a=0.8 ρ/a=0.7 Exact result

0.5 0.4

σρ

0.3 0.2 0.1 0 −0.1 −1

−0.5

0

θ

0.5

Fig. 7. The normal stress rq vs. coordinate h.

1

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W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

0.5

σρ θ

ρ/a=1 ρ/a=0.9 ρ/a=0.8 ρ/a=0.7 Exact result 0

−0.5 −1

−0.5

0

0.5

θ

1

Fig. 8. The shear stress rqh vs. coordinate h.

crack

o

Fig. 9. The computational model.

3

0.95

0.98

2.5

θ

2

1 1.1

1.5 1.3 1.05

0

1.02

1.01

1.03

1 0.5

0.99

0.95

1

0.99 0.98

0.2

0.4

0.6

ρ/a Fig. 10. Stress rh/P distribution.

0.8

1

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W. Zhang et al. / International Journal of Engineering Science 79 (2014) 30–43

3 2.5

1

0.5 0.3

θ

2 0.1

1.5 1 0.5 0

0.02

0.1

0.002

0.02

0.02 0.1

0.2

0.4

0.6

0.8

1

ρ/a Fig. 11. Stress rq/P distribution.

u ¼ v ¼ 0 ðq ¼ 0Þ Sq ¼ P

ðh ¼ 0; 2pÞ

ð66Þ

On the basis of the symplectic theory of the present research, we firstly transform the lateral boundary conditions (h = 0, 2p) into homogeneous ones using the variable substitution method. Then, according to Eqs. (53)–(56) in Section 4.2, a particular solution of the governing equations is obtained analytically. Lastly, considering circular boundary conditions ðq ¼ 0; aÞ, algebra equations about the coefficients in the eigenvector expansion are established, and the numerical results are obtained. The results show that the shear stress is much smaller than the other stresses. Thus only normal stresses are considered. Figs. 10 and 11 give the distributions of Sh and Sq, respectively. It can be seen from the figures that stress concentrations appear near the clamped end (q = 0) due to the displacement constraints, and the concentration effects decrease with the distance from the boundary. The calculation also explains that the proposed symplectic approach is very convenient for engineering problems in sector domain related with boundary conditions, especially for problems related with viscoelastic crack.

6. Conclusion With the use of the polar coordinate system and the Laplace transform, the symplectic method can be applied to a two-dimensional sector viscoelastic domain by introducing a new axis n = ln r. Using the method of separation of variables, all the general solutions including zero-eigenvalue eigenvectors and nonzero-eigenvalue eigenvectors are obtained analytically. Accordingly, the solution of the problems boundary conditions and the non-homogeneous governing equations can be described by linear combination of the eigenvectors since the eigenvectors construct a complete solution space. The polar coordinate symplectic method proposed in this paper provides a convenient approach for solving sectoral problems, especially for analyses related with elastic and viscoelastic crack in engineering.

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