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Linear Theory of Shells Taking into Account Surface Stresses

Holm Altenbach, Victor Eremeyev, Nikita Morozov

To cite this version:

Holm Altenbach, Victor Eremeyev, Nikita Morozov. Linear Theory of Shells Taking into Account

Surface Stresses. Doklady Physics, 2009, 54 (12), pp.531-535. �hal-00824864�

(2)
(3)

532

DOKLADY PHYSICS Vol. 54 No. 12 2009 ALTENBACH et al.

Here,

is the surface strain tensor, u

S

= is the dis placement vector on Ω

2

, ∇

S

is the surface gradient operator [13], A = I – n ⊗ n, I is the threedimensional unit tensor, n is the externalnormal vector to Ω

2

, λ

S

, and μ

S

are the surface elastic moduli (the analogues of the Lamé surface constants), and τ

0

is the residual sur face stress. The function U in Eq. (2) coincides with the strain energy of a membraneshell, and

is equal to the linear stretch tensor [14].

The totalenergy functional for a body with the sur face stresses has the form

Here, f is the vector of external volume forces.

The stationarity condition δ

J(u) = 0,

∀δ u: δ = 0, leads to the equilibrium equations and the static boundary conditions:

(3) The stress and surfacestress tensors in Eq. (3) are given by formulas

It is also possible to show that an arbitrary solution of boundaryvalue problem (1) and (3) is the stationary point of the functional J(u) on the kinematically admissible displacement fields u.

u

Ω

2

J ( ) u = E ( ) uA ( ) u , E ( ) u W ( )

ε

d V

V

U ( ) Ω

d ,

Ω2

+

=

A ( ) u f

V

udV

ϕ

· u d Ω .

Ω2

+

=

u

Ω

1

∇ ·

σ

+ f =

0,

( n ·

σ

– ∇

S

·

τ

)

Ω2

=

ϕ

.

σ

W

ε

λ Itr

ε

+ 2 με ,

= =

τ

U

∂⑀ τ

0

A + λ

S

Atr

+ 2 μ

S

.

= =

Further, we require fulfillment of the conditions of positive definiteness of the functions W(

ε

) and U( ):

(4) The fulfillment of inequalities (4) results in the follow ing restrictions on the elastic moduli:

(5) It is necessary to note that assumptions (4) should be fulfilled independently from each other. If, in par ticular, we assume that U < 0 for certain deformations, or inequalities (5)

3, 4

are violated, it is possible to show that J(u) proves to be unbounded from below.

2. For the transition to the shelltheory equations,

we consider a threedimensional body, one of the characteristic sizes of which is much less than oth ers—the socalled shelllike body (Fig. 1). The shell like body volume V is bounded by two faces Ω

±

and the lateral surface Ω

ν

. We introduce also a middle (base) surface ω , which is equidistant from Ω

±

. The lateral surface Ω

ν

represents the ruled surface formed by the motion of the normal n to ω along its contour γ ≡ ∂ω . It is convenient to present the radius vector r of the shellbody points as [13, 14]

where

ρ

is the radius vector of points of the base sur face ω , n is the vector of the normal to ω , z is the coordi nate counted from the normal to ω , z [–h/2, h/2], h is the shell thickness, and q

1

and q

2

are the Gaussian coor dinates at ω . The radius vectors of Ω

±

are r

±

=

ρ ±

nh/2, respectively.

We display certain auxiliary formulas related to the description of tensor fields near the surface ω [13, 14].

The basic and dual bases and the surface nabla opera tor at ω are given by the formulas

where is the Kronecker delta. We use the quantities

q1

, q

2

, and z as the curvilinear coordinates in the vicin ity of ω . Then the following formulas hold:

W ( )

ε

> 0 ∀

ε

0,

U ( )

> 0 ∀

0.

μ > 0, 3 λ + 2 μ > 0, μ

S

> 0, μ

S

+ λ

S

> 0.

r =

ρ

( q

1

, q

2

) + zn,

ρα

∂ρ

q

α

,

ρα

·

ρβ

δ

αβ

, α β , 1 2, ,

= = =

S ρα

q

α

= , δ

αβ

r

α

r

q

α

ρα

zn

q

α

+ ( AzB ) ·

ρα

,

= = =

r

3

= r

3

= n,

r

α

= ( AzB )

1

·

ρα

, r

α

· r

β

= δ

αβ

, B = – ∇

S

n,

r

α

q

α

n

z

+ ( AzB )

1

· ∇

S

n

z + ,

= =

S

±

A h

2 B

⎝ + ⎠

⎛ ⎞

1

·

ρα

q

α

A h

2 B

⎝ + ⎠

⎛ ⎞

1

· ∇

S

,

= =

γ w

ρ r n

Ω+

q1 q2 n+

h

z n

Ω

i1 i2 i3

Fig. 1. Shelllike body.

(4)

where B is the curvature tensor of the surface ω . We assume that the following surface stresses act on Ω

2

= Ω

+

∪ Ω

:

(6) In Eq. (6), n

±

are the vectors of the normal to Ω

±

(see Fig. 1), and

ϕ±

are the surface stresses and surface loads at Ω

±

, respectively,

Here, and are the surface elastic moduli, and are the residual stresses at Ω

±

. We note that the sur face gradient operators at Ω

±

, differ from each other, in general.

For passing to the twodimensional equations of shell theory, we use a procedure known in shell the ory—the integration of equilibrium equations (3) over thickness with taking into account boundary condi tions (6) used also in [12]. Integrating Eqs. (3) over z with taking into account Eq. (6), we obtain

(7) where T = 〈 (A – zB)

–1

·

σ〉

is the stress resultants tensor, q =

G+ϕ+

– G

ϕ

+ 〈 f 〉 is the surface density of the external forces acting to the shell,

Taking the vector product of equilibrium equa tion (3) and zn at the left and integrating over the thickness, we obtain the second equilibrium equation

(8)

The tensor M = – 〈 (A – zB)

–1

· z

σ ×

n 〉 is the stress cou ples tensor, while

is the surface moment distributed over ω , the sub script × designates the vector invariant of the second order tensor, in particular, the vector invariant is cal culated for the dyad formed by the vectors a and b from the formula (a ⊗ b)

×

= a × b.

The presence of terms related to the fact that

τ±

dis tinguishes equilibrium Eqs. (7) and (8) from the equi librium equations of the linear theory of shells. Trans

Az B

( )

1

· A ( – zB ) = ( AzB ) · A ( – zB )

1

= A,

n

±

·

σ

S

±

·

τS

+

±

( )

Ω±

=

ϕ±

.

τS

±

τ±

τ

0

±

A λ

S

±

Atr

±

2 μ

S

±±

,

+ +

= 2

±

S

±

u

S±

( ) · A A ·

S

±

u

S±

( )

T

. +

= μ

S

±

λ

S

±

τ

0

±

S

±

S

· T G

+

S

+

·

τ+

G

S

·

τ

q

+ + + =

0,

… ( )

〈 〉 ( ) … G z, d G

h/2 h/2

G z ( ) det A ( zB ) ,

= =

G

±

G h 2

⎝ ⎠ ±

⎛ ⎞ .

=

S

M

T×

m h G 2

+

n

+ + + ∇

S

+

τ+

×

h

G 2

n

S

τ

× =

0,

m h

2 G

+

n

ϕ+

h

2 G

n ×

ϕ

+ 〈 zn × f

× +

=

forming them while taking into account the assump tion that h || B ||

1, we obtain

(9) where the effective stress resultants and couples ten sors T* and M* are introduced

(10) For description the shell deformations, we assume the displacementfield approximation linear in thick ness used in the theory of plates and shells with taking into account the transverse shear (see, for example, [15]):

(11) Here it is assumed that the rotation vector

ϑ

is kine matically independent of the displacement vector of the shell midsurface w. Equation (11) leads to the for mulas

(12) where

are the twodimensional tensors of extension–shear and bending–torsion deformations. Using Eq. (12), we obtain for the surface stresses

τ±

the expressions

In the case of the shell with identical surface properties, i.e., when = = τ

0

, = = μ

S

, = = λ

S

, we obtain the stress resultants and couples tensors gen erated by the surfacestress action:

(13)

With taking into account Eqs. (10), it follows from Eq. (13) that the surface stresses render no effect on the transverse shear forces because T

S · n = 0 and have

little or no effect on the stiffness and the transverse shear of the shell. From Eq. (13), it can be seen also that the residual surface stresses τ

0

do not affect the shell stiffness, although, naturally they affect its stress state.

S

T* + q =

0,

S

M* + T

×

* + m =

0,

T* = T + T

S

, M* = M + M

S

, T

S τ+

+

τ

, M

S

h

2 (

τ+

τ

)

– × n.

= =

u ( q

1

, , q

2

z ) = w ( q

1

, q

2

) – z

ϑ

( q

1

, q

2

) , n

ϑ

= 0.

u

S± w

h 2

ϑ

,

±

+

h

2

κ

, +

= =

1

2 ( ∇

S

w A ⋅ + A ⋅ ( ∇

S

w )

T

) ,

=

κ

1

2 ( ∇

Sϑ

A + A ⋅ ( ∇

Sϑ

)

T

)

=

τ±

τ

0

±

A λ

±

S

Atr

2 μ

± S

h

2 λ

±

S

Atr

κ

2 μ

± Sκ

( + ) .

+

+ +

=

τ

0 +

τ

0

μ

+

S

μ

S

λ

+

S

λ

S

T

S

= 2 τ

0

A + C

1S

+ C

2S

Atr

, M

S

= – [ D

1Sκ

+ D

2S

Atr

κ

] × n, C

1S

= 4 μ

S

, C

2S

= 2 λ

S

, D

1S

= h

2

μ

2

,

D

2S

h

2

λ

S

. 2

=

(5)

534

DOKLADY PHYSICS Vol. 54 No. 12 2009 ALTENBACH et al.

For T and M, we accept the constitutive equations in one of the simplest forms given, for example, in [15]

(14)

(15) Here, W

S

is the surface strain energy density, C and D are the fourthorder tensors determining the tangen tial and bending stiffness of the shell,

γ

is the trans verseshear vector:

γ

= ∇

S

(w · n) –

ϑ

, while Γ is the transverseshear stiffness. For an isotropic shell [15]

where

e

1

and e

2

are the unit vectors lying in the plane tangent to ω (e

1

· e

2

= e

1

· n = e

2

· n = 0). The components C

11

,

C22

, D

22

, D

33

, and Γ are given by the formulas [15]

T A ⋅ 1

2 ( M··B ) A

– × nW

S

, ∂⑀

=

T n ⋅ ∂ W

S

, ∂γ MW

S

, ∂κ

= =

2W

S

=

··C··

+

κ

··D··

κ

+ Γγ γ ⋅ .

C = C

11

a

1

a

1

+ C

22

( a

2

a

2

+ a

4

a

4

) , D = D

22

( a

2

a

2

+ a

4

a

4

) + C

33

a

3

a

3

,

a

1

= Ae

1

e

1

+ e

2

e

2

, a

2

= e

1

e

1

e

2

e

2

, a

3

= – A × n = e

1

e

2

e

2

e

1

,

a

4

= e

1

e

2

+ e

2

e

1

,

C

11

Eh 2 1 ( – ν )

, C

22

Eh

2 1 ( + ν )

= = ,

D

22

Eh

3

24 1 ( + ν )

, D

33

Eh

3

24 1 ( – ν )

, Γ k μ h,

= = =

E 2 μ ( 1 + ν ) , ν λ 2 ( λ μ + )

= = ,

CC

11

+ C

22

Eh 1 – ν

2

= ,

where C and D are the tangential and bending stiffness parameters,

E and

ν are the Young’s modulus and Poisson ratio of the shell material, respectively; and k is the analogue of the transverseshear factor [15]. The effective tangential and bending stiffness parameters are equal to

Ceff C1

+ C

2

= C + 4 μ

S

+ 2 λ

S

,

Deff D1

+ D

2

= D + h

2

μ

S

+ .

Constitutive equations (14) and (15) make it possi ble to write the equilibrium equations for the shell and plate with taking into account surface stresses (9) in terms of displacements w and rotations

ϑ

. In particular, the equation for the deflection w = w · i

3

in the case of the plate (n = i

3

) can be reduced to the form

For the quantitative estimate of the results of the surfacestress effect, we use the data for aluminum [1]:

μ = 34.7 GPa, ν = 0.3, λ

S

= –3.48912 N/m, and μ

S

= 6.2178 N/m. We consider the dependences of D

eff

, C

1

,

C2

, D

1

, and D

2

on the thickness h. The plot for the bending stiffness D

eff

is shown in Fig. 2, and the plots for the dimensionless values = , = ,

= , and = are shown in Fig. 3.

As it follows from Figs. 2 and 3, the surfacestress effect is almost negligible for h > 50 nm. They render the greatest effect for h < 20 nm. In addition, it can be seen that the surface stresses differently affect the stiff ness parameters —some of them increase (C

1

, D

1

),

DD

11

+ D

22

Eh

3

12 1 ( – ν

2

)

,

=

h

2

λ

S

2

D

eff

ΔΔ w

S

m D

eff

Γ Δ q

n

– + q

n

,

=

q

n

= q i

3

, Δ = ∇

S

⋅ ∇

S

.

C

1

C

1

C 1 ( – ν )

C

2

C

2

C ν D

1

D

1

D 1 ( – ν )

D

2

D

2

D ν

3

0 2

1

10 20 30 40 50

h, nm D

Deff

Fig. 2. Dependence of bending stiffness on thickness.

3

0 2

1

10 20 30 40 50

h, nm

C1

C2

D2

D1

D Deff

Fig. 3. Dependences of stiffness parameters on thickness.

(6)

while others (C

2

, D

2

) decrease, the thicknesses being zero for certain values. Condition (5) guarantees a positive sign for C

eff

and D

eff

for arbitrary values of h.

Thus, we obtained the twodimensional equilib rium equations for plates and shells with taking into account the transverse shear and the presence of sur face stresses. We presented the relations for the stress resultants and couples tensors and found the expres sions for effective stiffness parameters of shells. In par ticular, it was shown that the plate stiffness substan tially changes with taking into account the surface stresses, which agrees with the results of the theoretical analysis and the experimental data known in the liter ature (see, for example, [1]). In particular, it is shown that the shell bending stiffness substantially grows for the nanometer thicknesses.

ACKNOWLEDGMENTS

This work was supported by the Russian Founda tion for Basic Research, project no. 090100459, and DFG (AL 341/311).

REFERENCES

1. H. L. Duan, J. Wang, and B. L. Karihaloo, Adv. Appl.

Mech. 42, 1 (2008).

2. E. Orowan, Proc. Roy. Soc. A (London) 316 (1527), 473 (1970).

3. M. E. Gurtin and A. I. Murdoch, Arch. Rat. Mech.

Anal. 57 (4), 291 (1975).

4. Ya. S. Podstrigach and Yu. Z. Povstenko, Introduction in Mechanics of Surface Phenomena in Deformable Solids (Naukova Dumka, Kiev, 1985) [in Russian].

5. D. J. Steigmann and R. W. Ogden, Proc. Roy. Soc. A (London) 455 (1982), 437 (1999).

6. K. Dahmen, S. Lehwald, and H. Ibach, Surf. Sci. 446 (1–2), 161 (2000).

7. R. E. Miller and V. B. Shenoy, Nanotecnology 11 (3), 139 (2000).

8. J. G. Guo and Y. P. Zhao, J. Appl. Physics 98 (7), 074306 (2005).

9. P. Lu, L. H. He, H. P. Lee, and C. Lu, Int. J. Solids Struct. 43 (16), 4631 (2006).

10. D. W. Huang, Int. J. Solids Struct. 45 (2), 568 (2008).

11. C. F. Lu, C. W. Lim, and W. Q. Chen, Int. J. Solids Struct. 46 (5), 1176 (2009).

12. V. A. Eremeyev, H. Altenbach, and N. F. Morozov, Dokl. Phys. 54 (2), 98 (2009) [Dokl. Akad. Nauk 424 (5), 618 (2009)].

13. V. A. Eremeyev and L. M. Zubov, Mechanics of Elastic Shells (Nauka, Moscow, 2008) [in Russian].

14. V. V. Novozhilov, K. F. Chernykh, and E. I. Mikhaіlovskiі, Linear Theory of Shells (Politekhnika, Leningrad, 1991) [in Russian].

15. P. A. Zhilin, Applied Mechanics: Fundamentals of The ory of Shells (Izd. Politekhn. Univ., St. Petersburg, 2006) [in Russian].

Translated by V. Bukhanov

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