BEM formulation for plane orthotropic bodies – a modification for exterior regions and its proof

Ŕ periodica polytechnica

Civil Engineering 51/2 (2007) 23–35 doi: 10.3311/pp.ci.2007-2.05 web: http://www.pp.bme.hu/ci c Periodica Polytechnica 2007 RESEARCH ARTICLE

BEM formulation for plane orthotropic bodies – a modification for exterior regions and its proof

GyörgySzeidl/JuditDudra

Received 2007-11-07

Abstract

Assuming linear displacements and constant strains and stresses at infinity, our aim is to reformulate the equations of the direct boundary element method for plane problems of elas- ticity. We shall consider a body made of orthotropic material.

The equations we have reformulated make possible to attack plane problems on exterior regions without replacing the region in question by a bounded one.

Keywords

Boundary element method · Somigliana relations · direct method·orthotropic body·plane problems·exterior regions

Acknowledgement

The support provided by the Hungarian National Research Foundation within the framework of the project OTKA T046834 is gratefully acknowledged.

György Szeidl

Department of Mechanics, University of Miskolc„ H-3515 Miskolc- Egyetemváros„ Hungary

e-mail: Gyorgy.SZEIDL@uni-miskolc.hu

Judit Dudra

Department of Mechanics, University of Miskolc„ H-3515 Miskolc- Egyetemváros„ Hungary

e-mail: judit.dudra@gmail.com

1 Introduction

As is well known a large literature studies plane problems for orthotropic bodies, including for instance [9], [8], [7] as well as the books [1, 2] and the references therein. However, as ex- plained by Schiavone [10], the standard formulation for exterior regions has the disadvantage that it is impossible to prescribe a constant stress state at infinity.

The reason is that an assumption about the far field pattern of the displacements is needed in order to establish an appro- priate Betti formula and to prove uniqueness and existence for the exterior Dirichlet and Neuman problems. Unfortunately, this assumption excludes those problems from the theory for which the displacements are linear while the strains and stresses are constant at infinity.

To make progress on plane problems with such displace- ments, we note that if the direct formulation reproduces this dis- placement field, then the resulting strain and stress conditions must also be constant at infinity. Consequently, plane problems for the exterior regions can be attacked without replacing the re- gion by a bounded one. The work in [11] and [12] presents such direct formulations by assuming constant strains and stresses at infinity for an isotropic body. For exterior regions [11] refor- mulates the classical approach to plane problems. For the same class of problems but in a dual formulation [12] sets up the equa- tions of the direct method in terms of stress functions of order one.

The present paper is an attempt to clarify how the formula- tion changes if we apply the ideas presented in paper [11] to orthotropic bodies.

2 Preliminaries

Throughout this paperx1andx2are rectangular Cartesian co- ordinates, referred to an originO. Greek subscripts are assumed to have the range (1,2), summation over repeated subscripts is implied. The doubly connected exterior region under considera- tion is denoted byAeand is bounded by the contourL_o. We stip- ulate that the contour admits a nonsingular parametrization in terms of its arc lengths. The outer normal is denoted byn_π. In accordance with the notations introduced,δ_κλis the Kronecker

symbol,∂α stands for the derivatives taken with respect to x_α and3κλis the permutation symbol. Assuming plane problems letu_κ,e_κλ andt_κλ be the displacement field and the in plane components of strain and stress, respectively. For orthotropic bodies the material constants are denoted bys₁₁,s₁₂ =s₂₁,s₂₂ ands₆₆.

For homogenous and orthotropic material the plane problem of classical elasticity is governed by the kinematic equations

e_ρλ= 1

2(∂_ρu_λ+∂_λu_ρ), (1) Hooke’s law

t11=c11e11+c12e22, t₂₂=c₁₂e₁₁+c₂₂e₂₂, t₁₂=t₂₁=2c₆₆e₁₂,

(2)

where

c₁₁ =s22

d , c₁₂=c₂₁= −s12

d , c₂₂= s11

d , c66 = 1

s₆₆, d =s11s22−s₁₂² ; (3) and the equilibrium equations

t_ρλ∂λ+b_ρ =0 (4) which should be complemented with appropriate boundary con- ditions not detailed here since they play no role in the present investigations. The basic equation foru_λtakes the form

D_ρλu_λ+b_ρ =0, (5a) where the differential operatorD_ρλhas the form

D_ρλ

=

"

c₁₁∂₁²+c₆₆∂₂² (c₁₂+c₆₆) ∂1∂2

(c₂₁+c₆₆) ∂2∂1 c₂₂∂₂²+c₆₆∂₁²

#

. (5b)

Let Q(ξ1, ξ2) and M(x1,x2) be two points in the plane (the source point and the field point). We shall assume temporar- ily that the point Q is fixed. The distance between Q and M is R, the position vector of M relative to Q isr_κ. The small circle as a subscript (for instanceM_◦or Q_◦) indicates that the corresponding points, i.e.,QorM are taken on the contour.

It is obvious that

r_α(M,Q)=x_α(M)−ξ_α(Q)=x_α−ξ_α. (6) Let us introduce the following notations

λ1+λ2=(2s12+s66) /s22, (7) λ1λ2=s₁₁/s₂₂, (8) A_α =s12−λαs22, (9) ρ_α²=λ_αr₁²+r₂², (10)

D= 1

2π (λ1−λ2)s22 . (11)

For our later considerations we note that Eqs. (7) and (8) imply

λ1,2= 2s₂₁+s₆₆ 2s₂₂ ±

s

(2s₂₁+s₆₆

2s₂₂ )²−s₁₁

s₂₂. (12) The well-known singular fundamental solutions for the basic Eq. (5a) [7, 9] are given by the formulas

U₁₁(M,Q)=D p

λ1A²₂lnρ1−p

λ2A²₁lnρ2

, U₁₂(M,Q)=D A₁A₂arctan

√λ1−√ λ2

r1r2

√λ1√

λ2r₁²+r₂² , U₂₁(M,Q)=U₁₂(M,Q) ,

U22(M,Q)= −D A²₁lnρ1

√λ1

− A²₂lnρ2

√λ2

!

(13)

and

T11(M,Q)=D

"√ λ2A₁ ρ₂² −

√λ1A₂ ρ₁²

#

(r1n1+r2n2) ,

T12(M,Q)=D ( √

λ1A₁ ρ₁² −

√λ2A₂ ρ₂²

! r1n2−

√1 λ1

A₁ ρ₁² − 1

√λ2

A₂ ρ₂²

! r₂n₁

) ,

T₂₁(M,Q)=D ( λ1

√λ1A2

ρ²₁ −λ2

√λ2A1

ρ₂²

! r₁n₂−

√λ1A2

ρ₁² −

√λ2A1

ρ₂²

! r₂n₁

) ,

T22(M,Q)=D

"√ λ1A₁ ρ₁² −

√λ2A₂ ρ₂²

#

(r1n1+r2n2) , (14) where

u_λ(M)=U_λκ(M,Q)e_κ(Q), andt_λ(M)=T_λκ(M,Q)e_κ(Q) are the displacement vector and stress vector on a line element with a normaln_λ=n_λ(M)to it at the pointM due to the force e_κ=e_κ(Q)atQ.

3 Basic formulas for exterior regions

Fig. 1 depicts a triple connected region A⁰_e bounded by the contoursL_o, L_εand the circleL_R with radius_eRand center at O. HereL_εis the contour of the neighborhoodA_εofQwith ra- diusR_εwhile_eRis sufficiently large so that the region bounded byL_Rcovers bothL₀, andL_ε. If_eR → ∞andR_ε → 0then clearlyA⁰_e→ A_e.

Letu_κ(M)andg_κ(M)be sufficiently smooth – continuously differentiable at least twice – but otherwise arbitrary displace- ment fields on Ae. The stresses obtained from these displace- ment fields are denoted byt_λκ

u_ρ(M)

andt_λκ g_ρ(M)

respec- tively.

A eR en

Ae

O x1

x2

Q

L_R

’ R L

L

o

ds ds

ds

Fig. 1.

Equation

Z A⁰_e

u_λ(M)

µ^MD_λσg_σ(M)

−g_λ(M)

µ^MD_λσu_σ(M)

dA_M = I

Lo

u_λ(M_◦)t_λκ g_ρ(M_◦)

n_κ(M_◦)−g_λ(M_◦)t_λκ u_ρ(M_◦)

n_κ(M_◦) dsM_◦+ I

L_ε

u_λ(M_◦)t_λκ g_ρ(M_◦)

n_κ(M_◦)−g_λ(M_◦)t_λκ u_ρ(M_◦)

n_κ(M_◦) ds_M◦+ I

L_R

u_λ(M_◦)t_λκ g_ρ(M_◦)

n_κ(M_◦)−g_λ(M_◦)t_λκ u_ρ(M_◦)

n_κ(M_◦) ds_M◦,

(15)

in which M over a letter denotes derivatives taken with respect to the point coordinates M andn_κ(M_◦)is the outward normal, is the primal Somigliana identity applied to the triple connected regionA⁰_e.

Letg_λ(Q)=U_λκ(M,Q)e_κ(Q), which is a non singular elas- tic state of the plane in A⁰_e. We regardu_λ(M)as a different elastic state in the regionA_e. Further we assume thatu_λ(M)has the far field pattern (asymptotic behaviour)

˜

u_κ(M)=c_κ+ε3ρκx_ρω+e_κβ(∞)x_β (16) asx_βor equivalentlyMtends to infinity. Herec_κis a translation, ωis a rotation in finite,c_κ+ε3ρκx_ρωis the corresponding rigid body motion,e_κβ(∞)is a constant strain tensor at infinity and e_κβ(∞)x_β is the corresponding displacement field.

The stresses induced by the strainse_κβ(∞)can be obtained by the Hooke law:

t₁₁(∞)=c₁₁e₁₁(∞)+c₁₂e₂₂(∞) , t₂₂(∞)=c₁₂e₁₁(∞)+c₂₂e₂₂(∞) , t12(∞)=t21(∞)=2c66e12(∞) ,

(17)

Substituting the above quantities into the Somigliana identity,

we obtain

Z A⁰_e

u_λ(M)

µD^M_λσU_σκ(M,Q)

−

µD^M_λσu_σ(M)

U_λκ(M,Q)

dA_Me_κ(Q)=

= I

Lo

[u_λ(M_◦)T_λκ(M_◦,Q)−t_λ(M_◦)U_λκ(M_◦,Q)]ds_M◦e_κ(Q)+

I

L_ε[u_λ(M_◦)T_λκ(M_◦,Q)−t_λ(M_◦)U_λκ(M_◦,Q)]dsM_◦e_κ(Q) I

L_R[u_λ(M_◦)T_λκ(M_◦,Q)−t_λ(M_◦)U_λκ(M_◦,Q)]ds_M◦e_κ(Q) (18)

sincet_λκ

u_ρ(M_◦)

n_κ(M_◦) = t_λ(M_◦)is the stress on the con- tour and obviously

t_λκ

g_ρ(M_◦)

n_κ(M_◦)=T_λκ(M_◦,Q)e_κ(Q).

In the sequel we shall assume that there are no body forces.

This assumption has no effect on the result we will obtain.

It is clear that one can omite_κ(Q). Regarding the equation obtained by omittinge_κ(Q), our goal is to compute its limit as R_ε→0andeR→ ∞. As is well known, the left hand side van- ishes under the conditions detailed above(see [1] ), and hence

I

Lo

· · · + lim

R_ε→0

I

L_ε· · · = u_κ(Q)+

I

L_o

[u_λ(M)T^o _λκ(M_◦,Q)−t_λ(M_◦)U_λκ(M_◦,Q)]dsM_◦. Consequently

u_κ(Q)= lim

eR→∞

I

L_R

[t_λ(M_◦)U_λκ(M_◦,Q)−u_λ(M_◦)T_λκ(M_◦,Q)]ds_M◦+ I

L_o

[t_λ(M_◦)U_λκ(M_◦,Q)−u_λ(M_◦)T_λκ(M_◦,Q)]ds_M◦. (19) In order to establish the first Somigliana formula for the exterior region, we need to find the limit of the first integral on the right hand side.

4 Somigliana formulas modified for exterior regions In this section our main objective is to prove that

I_κ= lim

eR→∞

I

L_R

[t_λ(M_◦)U_λκ(M_◦,Q)−u_λ(M_◦)T_λκ(M_◦,Q)]ds_M◦ = c_κ+ε3ρκξ_ρω+e_κβ(∞)ξ_β = ˜u_κ(Q) . (20) Using

x_β(M^o)=eR n_β(M^o)

we substitutet_λρ(∞)n_ρ(M_◦)for t_λ(M_◦)andc_λ +ε3ρλx_ρω+ e_κβ(∞)x_β foru_λ(M_◦). This implies

I_κ=⁽

1)

I_κ+⁽

2)

I_κ+⁽

3)

I_κ+⁽

4)

I_κ =

− lim

eR→∞

I

LR

c_λT_λκ(M_◦,Q)ds_M◦− lim

eR→∞ε3ρλeRωI

L_R

n_ρ(M^o)T_λκ(M_◦,Q)dsM_◦+ lim

eR→∞t_λρ(∞)I

L_R

n_ρ(M_◦)U_λκ(M_◦,Q)ds_M◦ − lim

eR→∞e_λβ(∞)eR I

L_R

n_β(M^o)T_λκ(M_◦,Q)dsM_◦. (21)

R1.: In accordance with Eq. (16), the stresses and strains are taken as constant quantities which are therefore independent of the arc coordinates.

Since I

L_R

T_λκ(M_◦,Q)ds_M◦ = −δκλ

and

ε3ρλ

I

L_R

r_ρT_λκ(M_◦,Q)ds_M◦ =0

which is the moment about the origin of the stresses due to a unit force applied atQ, one can write

(1)

I_κ+⁽

2)

I_κ =c_κ− lim

eR→∞ε3ρλωI

LR

(ξ_ρ+r_ρ)T_λκ(M_◦,Q)ds_M◦ = c_κ− lim

eR→∞

ε3ρλωξ_ρI

LR

T_λκ(M_◦,Q)ds_M◦+ ε3ρλωI

L_R

r_ρT_λκ(M_◦,Q)ds_M◦

=c_κ+ε3ρκωξρ (22) which clearly shows that (22) reproduces the rigid body motion.

Determining the limits⁽

3)

I_κand⁽

4)

I_κrequires long formal trans- formations. For this reason we confine ourselves to basic argu- ment and the results of the most important steps.

First, we have to expandU_λκ andT_λκ into series in terms of

eR to the power1,0,−1,−2 etc. These transformations are based on the relations:

n_α(M^o)=n_α (23a)

r_α=x_α(M^o)−ξα(Q)=x_α−ξα=eR

n_α− ξα eR

, (23b)

ρα=

qλαr₁²+r₂² '

'eR

qλαn²₁+n²₂ (

1− 1 eR

λαn₁ξ1+n₂ξ2 λαn²₁+n²₂ + 1

2eR²

λαξ₁²+ξ₂² λαn²₁+n²₂ )

, (23c) lnρα'lneR+1

2ln

λαn²₁+n²₂

− 1

eR

λαn₁ξ1+n₂ξ2 λαn²₁+n²₂ + 1

2eR²

λαξ₁²+ξ₂²

λαn²₁+n²₂, (23d) 1

ρ_α² ' 1

eR²

λαn²₁+n²₂

1+ 2 eR

λαn₁ξ1+n₂ξ2 λαn²₁+n²₂ − 1

eR²

λαξ₁²−ξ₂² λαn²₁+n²₂

!

(23e)

and

arctan(x+εa)=arctanx+ εa 1+x²,

whereais constant andεis a very small quantity.

Consequently

arctan

√λ1−√ λ2

r₁r₂

√λ1

√λ2r₁²+r₂² 'arctan

√λ1−√ λ2

n₁n₂

√λ1λ2n²₁+n²₂ +

√λ1−√ λ2

eR

√

λ1λ2n²₁n₂−n³₂ ξ1−

√

λ1λ2n³₁−n₁n²₂ ξ2 n⁴₁λ1λ2+n⁴₂+(λ1+λ2)n²₁n²₂ . (23f)

Using relations (23a),. . ., (23f) for a sufficiently large _eR we have

U₁₁(M_◦,Q)' D

"

pλ1A²₂ lneR+1 2ln

λ1n²₁+n²₂

− 1 eR

λ1n₁ξ1+n₂ξ2 λ1n²₁+n²₂

!

−

pλ2A²₁ lneR+1 2ln

λ2n²₁+n²₂

− 1 eR

λ2n₁ξ1+n₂ξ2 λ2n²₁+n²₂

!#

, (24)

U₂₂(M_◦,Q)'

−D

"

A²₁

√λ1

lneR+1 2ln

λ1n²₁+n²₂

− 1 eR

λ1n₁ξ1+n₂ξ2 λ1n²₁+n²₂

!

− A²₂

√λ2

lneR+1 2ln

λαn²₁+n²₂

− 1 eR

λαn₁ξ1+n₂ξ2 λαn²₁+n²₂

!#

, (25)

U₁₂(M_◦,Q)'D A₁A₂

"

arctan

√λ1−√ λ2

n₁n₂

√λ1λ2n²₁+n²₂ +

√λ1−√ λ2

eR ×

√

λ1λ2n²₁n₂−n³₂ ξ1−

√

λ1λ2n³₁−n₁n²₂ ξ2 n⁴₁λ1λ2+n⁴₂+(λ1+λ2)n²₁n²₂



 (26)

and

T₁₁(M_◦,Q)= D eR

√λ2A₁ λ2n²₁+n²₂

"

1+ 2 eR

λ2n₁ξ1+n₂ξ2 λ2n²₁+n²₂ − 1

eR(ξ1n₁+ξ2n₂)

#

−

− D eR

√λ1A₂ λ1n²₁+n²₂

"

1+ 2 eR

λ1n₁ξ1+n₂ξ2 λ1n²₁+n²₂ − 1

eR(ξ1n₁+ξ2n₂)

# , (27)

T₂₁(M_◦,Q) = D eR

√λ1A₂ λ1n²₁+n²₂×

"

(λ1−1)n₁n₂ 1+ 2 eR

λ1n₁ξ1+n₂ξ2 λ1n²₁+n²₂

!

− 1

eR(λ1ξ1n₂−ξ2n₁)

#

− D

eR

√λ2A₁ λ2n²₁+n²₂ ×

"

(λ2−1)n₁n₂ 1+ 2 eR

λ2n₁ξ1+n₂ξ2 λ2n²₁+n²₂

!

− 1

eR(λ2ξ1n₂−ξ2n₁)

# , (28)

T₁₂(M_◦,Q)= D eR

A₁√ λ1 λ1n²₁+n²₂×

"

1− 1

λ1

n₁n₂ 1+ 2 eR

λ1n₁ξ1+n₂ξ2 λ1n²₁+n²₂

!

− 1 eR

ξ1n₂− 1

λ1ξ2n₁ #

×

−D A₂√ λ2 λ2n²₁+n²₂×

"

1− 1

λ2

n₁n₂ 1+ 2 eR

λ1n₁ξ1+n₂ξ2 λ1n²₁+n²₂

!

− 1 eR

ξ1n₂− 1

λ2ξ2n₁ #

, (29)