Kounadis A.N., Gdoutos E.E. (Eds.) Recent Advances in Mechanics

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XXX Contributors

O.A. Stapountzi,

Mechanical Engineering

Department, Imperial College,

London, UK

D. Šumarac,

Faculty of Civil Engineering,

University of Belgrade,

Serbia

W.-C. Wang,

Department of Power Mechanical

Engineering, National Tsing Hua

University, Taiwan, Republic of China

J.G. Williams,

School of Aerospace, Mechanical &

Mechatronic Engineering,

University of Sydney, Australia

Part I

Mathematical Methods in Applied

Mechanics

Application of Reciprocity Relations to

Laser-Based Ultrasonics

Jan D. Achenbach

Department of Mechanical Engineering, Northwestern University,

Evanston, IL 60208, USA

achenbach@northwestern.edu

Abstract. An application of the global reciprocity theorem for thermo-anisotropic

elastodynamics of inhomogeneous solids is discussed. The theorem relates body

forces, surface tractions, displacements, temperatures and strains of two solutions

of the field equations, states A and B, by integrals over a volume V and its bound-

ing surface S. The difference between various anisotropies enters via the stress

temperature tensor, which is different for different cases of anisotropy. The recip-

rocity theorem is used to analyze the dynamic response to high-intensity heating

of a small surface region of a half-space that is transversely isotropic and whose

elastic moduli and mass density depend on the depth coordinate.

1 Introduction

The reciprocity theorem for classical linearized elasticity has been known for

more than a century. As stated in the book by Kovalenko [1], a first statement of

the reciprocity theorem for isotropic thermoelasticity was presented by Maizel’

[2]. Despite its long existence, the theorem was not used extensively to actually

solve problems. A recent book by Achenbach [3] presents, however, novel uses of

reciprocity relations for the actual determination of elastodynamic fields.

As an example, we consider the surface-wave motion generated by line-focus

laser irradiation of a transversely isotropic half-space whose mechanical properties

depend on the depth component z.

Due to length limitations of this paper, we are not able to elaborate as much as

would be desirable, but details can be found in Refs. [4] and [5].

2 Thermo-anisotropic Elasticity

When a free cube of an anisotropic material is subjected to a uniform temperature

T, the thermal strains,

, may be written in the general form

4 J.D. Achenbach

ks ks ks sk

εα αα

(1)

where

are the components of the thermal expansion tensor. For a material

with axes of symmetry that are aligned with the Cartesian coordinate axes, we

may write

12 23 13

ααα

==≡

(2)

On the basis of a simple intuitive argument the thermo-anisotropic stress-strain

relations may then be written as

(),

ij ijks ks ks

τεα

=−

ij ijks ks ij

CmT

τε

=−

(3a,b)

Where

ij ijks ks

(4)

are the components of the stress-temperature tensor. For the case that Eq.(2) holds

the general stress-strain relation (6) reduces to

ij ijks ks ij

CmT

τε

=−

(5)

Where

11 11 22 22 33 33ij ij ij ij

mC C C

ααα

=+ +

(6)

3 Reciprocity Theorem

For convenience of presentation we use indicial notation in this section. For the

steady-state time-harmonic case, body forces, displacements, surface tractions and

temperatures are defined by

(,) ()

ftfe

−ω

=xx, ( , ) ( )

utu e

−

=xx, (,) ()

ttte

−

=xx,

(,) ()

TtTe

−

=xx

(7)

Let us now consider a solid elastic body of volume V and bounding surface S. The

two elasticodynamic states, are indicated by superscripts A and B. As shown in

Ref [5] the reciprocity theorem for the two states may then by written as

()()

AB BA AB BA

jj jj ii ii

fu fu dV tu tu dS−+−

∫∫

()

AB BA

ij ij ij

mT T dV

εε

=− −

∫

(8)

The terms in the integrals, depend on

x and z. For an isotropic material the recip-

rocity theorem in the form given here was apparently first derived by Maizel’ [2].

The quantities indicated by superscripts

A can be thought of as the solution of

an actual problem, while

B indicates an auxiliary solution, or in our terminology, a

virtual wave. Let us consider the case that the disturbance for the actual problem is

Application of Reciprocity Relations to Laser-Based Ultrasonics 5

generated strictly by a distribution of temperature

()

T x , i.e., 0≡

f . The

virtual wave can be taken as isothermal (

T ≡ ) and with zero body forces, i.e.,

f ≡ . Equation (8) then reduces to

()

AB B

BA A

ii ij

tu tu dS m T dV

−=−

∫∫

(9)

4 Transverse Isotropy with Depth Dependent Properties

The stress-strain relations are referred to Cartesian coordinates, x and z. The z –

axis is the axis of symmetry of the transverse isotropy. Relative to the x, z system

the stress-strain relations for plane strain of the inhomogeneous transversely iso-

tropic solid may be written as:

() ()

11 13xx

Cz Cz mT

∂∂

=+−

∂∂

, and

() ()

13 33zz

Cz Cz mT

∂∂

=+−

∂∂

(10a,b)

()

44xz

∂∂

⎛⎞

⎜⎟

∂∂

⎝⎠

(11)

It is noted that the elastic behavior depends on the depth

z. Thus the material is

inhomogeneous and the formulation includes functionally graded materials. In

addition the mass density,

()

is also taken to depend on z. Here m

and m

are

components of the stress-temperature tensor:

() () ()

11 12 13xx z

mCzCzCz

αα

⎡⎤

=++

⎣⎦

, and

() ()

13 33

zx z

mCzCz

αα

(12a,b)

Where

and

are components of the thermal expansion tensor.

5 Heating of the Boundary

Let us consider the case that thermal conductivity may be ignored, and that

(,)

Txt is applied only to a boundary, which in this case is taken as the plane

z = . The assumption that the thermal conductivity may be ignored, would gen-

erally be a dubious one. However, if the heat is applied with a high-intensity

pulse, as will be considered here, the resulting wave motion is also an intensive

pulse which propagates away from the source of heat with a high velocity. The

6 J.D. Achenbach

principal pulse then comes directly from the source region. Additional wave mo-

tion due to diffusion of the heat will be of smaller magnitude and arrive later.

When

T is applied on the boundary, the volume integral on the right hand side

of Eq. (9) is converted into a surface integral. The vanishing of surface traction of

state

B, the virtual wave, has, however, implications for the form of

. For

example, for a transversely isotropic material we have at 0

z = :

13 33

zxz

τεε

=+=

(13)

Hence

εε

=−

(14)

From Eqs. (5)-(6) and (11) it then follows that

BBB

ij x x z x x

mDD D

εε ε ε

=− =−, where

13 13 11 12 33

2()

CC C C C

−+

(15a,b)

Equation (15) is identical to the one obtained earlier by using an intuitive argu-

ment of thermal expansion of a surface element in Ref. [4].

When heat conduction is ignored, the temperature is governed by

∂

(16)

where

and

c are the mass density and the specific heat at constant deforma-

tion, and

q represents the heat deposition per unit volume and per unit time. The

rate of heat production by mechanical deformation has been neglected. For the

two-dimensional case of line illumination at 0

x = ,

0z =

, a suitable expression

for

q is often taken as

(1 ) ( ) ( )

qE R xgt

=− ,

(17)

where

E is the energy of the laser pulse per unit length in the direction normal to

the

x,z plane, and

is the surface reflectivity. Integration of Eq.(15) then yields

() (1 ) ( ) ()

Tt R x gsds

=−

∫

(18)

The function g(s) can be represented by a Fourier integral

() ()

ggtedt

∞

−∞

∫

(19)

Application of Reciprocity Relations to Laser-Based Ultrasonics 7

In the frequency domain we write

() (1 ) ()

TRg

ωω

ρω

=− − ,

(20)

For a line temperature application of the type given by Eq.(19), Eq.(9) becomes

() ( )

()(),0

BA B

ii x

tu tu dS DT x xz dx

ωδε

∞

−∞

−= =

∫∫

(21)

6 Surface Waves due to Laser Irradiation

The line focus irradiation generates a two-dimensional, plane strain elastodynamic

state. The corresponding radiated surface waves, which we call state A, may be writ-

ten as

() ()

Aikx

uxz iAVze

=±

, and

() ()

Aikx

wxz AWze

(22a,b)

Here the plus and minus signs apply for propagation in the positive and negative

x- directions, i.e., for x > 0 and x < 0, respectively. The corresponding expressions

for the stresses are

() ()

11 11

ik x

xz AT ze

, and

() ()

ik x

xz iAT ze

=±

(23a,b)

For the special case that all elastic constants and the mass density have the same

exponential dependence on z, details for these expressions are given in Ref. [6].

Following a method developed in Ref. [3], the surface waves generated by line-

focus irradiation will be obtained by an application of the reciprocity theorem for

time-harmonic wave motion given by Eq. (20). In the present application, State A

is the system of surface waves generated by the laser illumination, as given by

Eqs. (21)-(22). For State B, the virtual wave, we select a surface wave propagat-

ing in the positive x- direction:

() ()

Bikx

uxz iBVze=

, and

() ()

Bikx

wxz BWze=

(24)

() ()

11 11

Bikx

xz BT ze

, and

() ()

Bikx

xz iBT ze

(25)

For the problem at hand, and considering the contour shown in Fig. 1 by dashed lines,

the reciprocity relation, Eq. (20), includes contributions from the lines at

,0 , ,0xa z xb z=≤<∞=≤<∞ and from the temperature term at

0, 0xz==

. There are no other contributions from 0z = and from the line

constantz = as z

→∞. We obtain from Eq. (20)

()()

0| |,

AB x b AB x a

kDT V F dz F dz

∞∞

−=−

∫∫

(26)

Where

11 1 11 1

(,)

BABBABA

Bzz

Fxzu w u w

ττττ

=+−−

(27)

8 J.D. Achenbach

Fig. 1. Focused laser-beam irradiation of a solid body with contour for reciprocity relation

As has been noted in earlier calculations, see Ref. [3], the integration along

,0xa z=≤<∞

and , 0xb z=≤<∞, only yield contributions from counter-

propagating waves. By the present choice of the virtual wave, State A and State B

are counter-propagating at

,0xa z=≤<∞

. Substituting of the expressions

for States A and B into Eq. (19) then yields

()()

02,kDT V iAI

−=

where

() () () ()

11 1

ITzVzTzWzdz

∞

=−

⎡

⎤

⎣

⎦

∫

(28a,b)

The integral involves exponential terms and can be easily evaluated, see Ref. [6].

Fourier analysis can be used to determine the surface wave displacement gen-

erated by a pulsed laser beam.

References

[1] Kovalenko, A.D.: Thermoelasticity, Wolters-Noordoff Publishing, Groningen, the

Netherlands (1969)

[2] Maizel, V.M.: Generalization of the Betti-Maxwell theorem to the case of thermal

stresses and some applications. In: DAN SSSR, vol. 30 (1941)

[3] Achenbach, J.D.: Reciprocity in Elastodynamics. Cambridge University Press, Cam-

bridge (2003)

[4] Achenbach, J.D.: The Thermoelasticity of Laser-Based Ultrasonics. Journal of Ther-

mal Stresses 28, 713–728 (2005)

[5] Achenbach, J.D.: Application of the Reciprocity Theorem to Analyze Ultrasound Gen-

erated by High-Intensity Surface Heating of Elastic Bodies. Journal of Thermal

Stresses 30, 841–853 (2007)

[6] Kulkarni, S.S., Achenbach, J.D.: Application of the Reciprocity Theorem to Determine

Line-Load Generated Surface Waves on an Inhomogeneous Transversely Isotropic

Half-Space. Wave Motion 45, 350–360 (2008)