Dynamic Green's Functions of an Axisymmetric

Dynamic Green’s Functions of an Axisymmetric Thermoelastic Half-Space by a Method of Potentials

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

Yazdan Hayati1; Morteza Eskandari-Ghadi2; Mehdi Raoofian3; Mohammad Rahimian4; and Alireza Azmoudeh Ardalan5 Abstract: With the aid of a new complete scalar potential function, an analytical formulation for thermoelastic Green’s functions of an axisymmetric linear elastic isotropic half-space is presented within the theory of Biot’s coupled thermoelasticity. By using the potential function, the governing equations of coupled thermoelasticity are uncoupled into a sixth-order partial differential equation in a cylindrical coordinate system. Then, by using Hankel integral transforms to suppress the radial variable, a sixth-order ordinary differential equation is received. By solving this equation and considering boundary conditions, displacements, stresses, and temperature are derived in the Hankel integral transformed domain. By applying the theorem of inverse Hankel transforms, the solution is obtained generally for arbitrary surface timeharmonic traction and heat distribution. Subsequently, point-load Green’s functions for the displacements, temperature, and stresses are given in the form of some improper line integrals. For more investigations, the solutions are also determined analytically for uniform patch-load and patch-heat distributed on the surface. For validation, it is shown that the derived solutions could be degenerated to elastodynamic and quasistatic thermoelastic cases reported in the literature. Numerical evaluations of improper integrals, which have some branch points and pole, are carried out using a suitable quadrature scheme by Mathematica software. To show the accuracy and efficiency of numerical algorithm, a numerical evaluation from this study is compared with the results of an existing elastodynamic case, where excellent agreement is achieved. DOI: 10.1061/(ASCE)EM.1943-7889.0000540. © 2013 American Society of Civil Engineers. CE Database subject headings: Greens function; Wave propagation; Thermodynamics; Axisymmetry; Half space. Author keywords: Green’s function; Wave propagation; Axisymmetric; Thermoelastodynamic; Potential method.

Introduction Coupled thermoelasticity theory deals with the mutual interactions between the deformation and temperature fields of a body. The first insight into this theory is attributed to Duhamel (Nowacki 1962), who introduced the dilatation term into the heat conduction equation. However, the general treatment in this theory was done in the pioneering paper of Biot (1956), which was based on the irreversible thermodynamics. Early investigation into thermoelasticity problems was based on simplifying assumptions by neglecting the influence of deformation 1

Graduate Student (M.S.), School of Civil Engineering, College of Engineering, Univ. of Tehran, 11155-4563 Tehran, Iran (corresponding author). E-mail: [email protected] 2 Associate Professor, School of Civil Engineering, College of Engineering, Univ. of Tehran, 11155-4563 Tehran, Iran. E-mail: [email protected] 3 Ph.D. Candidate, Dept. of Surveying Engineering, College of Engineering, Univ. of Tehran, 11155-4563 Tehran, Iran. E-mail: mraoofian@ ut.ac.ir 4 Professor, School of Civil Engineering, College of Engineering, Univ. of Tehran, 11155-4563 Tehran, Iran. E-mail: [email protected] 5 Professor, Dept. of Surveying and Geomatics Engineering, Center of Excellence in Geomatics, Engineering and Disaster Prevention, College of Engineering, Univ. of Tehran, 11155-4563 Tehran, Iran. E-mail: Ardalan@ ut.ac.ir Note. This manuscript was submitted on February 5, 2012; approved on September 5, 2012; published online on September 6, 2012. Discussion period open until February 1, 2014; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Mechanics, Vol. 139, No. 9, September 1, 2013. ©ASCE, ISSN 0733-9399/ 2013/9-1166–1177/$25.00.

on temperature field, in which the temperature field is obtained independent of displacement components. In this case, the temperature field is obtained by solving the heat conduction equation and is directly used in equations of motion to determine displacements. This class of thermoelastic problems is called uncoupled thermoelasticity, whereas the general case of thermoelastodynamics is denoted as coupled thermoelastics. For thermoelastodynamic studies in isotropic media, one can mention the works done by Biot (1956), Deresiewicz (1958), Zoreski (1958), Verruijt (1967), Georgiadis et al. (1999), Lykotrafitis et al. (2001), Carlson (1972), and Chandrasekharaiah (1980). Biot (1956) derived the basic equations of coupled thermoelasticity and was the first to show that the governing equations of the thermoelasticity theory are the same as the theory of elasticity of porous materials. The latter is often applicable in seepage and consolidation in geomechanics. Biot has shown that the temperature increment in thermoelasticity plays the same role as the pore fluid pressure in poroelasticity. Biot also, in terms of potential functions, presented a complete general solution for coupled thermoelasticity in the absence of a heat source and the inertia effect, where the completeness of his solution was shown by Verruijt (1967). Deresiewicz (1958) and Zoreski (1958) separately presented a complete solution for thermoelasticity problems in isotropic media. Georgiadis et al. (1999) studied thermoelastodynamic disturbances in a half-space under the action of a buried thermal/mechanical line source. Lykotrafitis et al. (2001) derived three-dimensional (3D) thermoelastic Green’s functions under the action of both thermal and mechanical buried point sources within Biot’s thermoelastic theory by using the Laplace transform. This study is based on the Biot’s coupled thermoelasticity theory and could be applicable as integral kernels in boundary integral equations, which may be useful in the numerical treatment of more

1166 / JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013

J. Eng. Mech. 2013.139:1166-1177.

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

complicated thermoelastodynamic problems involving half-space geometries. Thermoelasticity has many applications, some of which are in the fields of engineering and physical science, including earthquake engineering, geophysics, soil dynamics, and astronautics (Singh 2010). For example, in the field of geophysics, the observations of thermoelastic deformations derived from satellite radar interfrometry (SAR) or a global positioning system (GPS) in geothermal areas such as active volcanoes could provide worthwhile information about the characteristics of source mechanism beneath the Earth (Fialko and Simons 2000; Furuya 2005). Also, some evidence has revealed that there is connection between the earthquake and the process of change of temperature within the Earth (Lubimova and Magnitzky 1964). Moreover, thermoelastic stresses are some of the major sources of uplift and flexure in the continental Lithosphere (Bill 1983; Hinojosa and Mickus 2002), whereas in the oceanic lithosphere, the temperature profile may also be used to predict ocean floor topography (Ranalli 1995). In all the planetary masses, the disintegration of heavy radioactive elements provide a vast amount of thermal energy that radiated into the space. The radiative cooling and radiogenic heating of planets causes stresses that may be large enough to alter even their sizes (Lanzano 1985). The high velocities of modern aircraft give rise to aerodynamic heating, which produces intense thermal stresses, reducing the strength of the aircraft structure. Moreover, in the nuclear field, the extremely high temperatures and temperature gradients originating inside nuclear reactors influence their design and operations (Nowinski 1978). A potential method is a common and easy way to uncouple a set of coupled linear partial differential equations. Goodier (1937) proposed a solution for the stationary and quasi-static thermoelastic problems in isotropic materials which results in Poisson’s equations. Another potential function for this class of problems was suggested by McDowell and Sternberg (1957), which is based on the Boussinesq-Papkovich stress function. Eskandari-Ghadi et al. (2013) proposed, in the absent of heat source, a complete representation including two scalar potential functions for thermoelastodynamic problems; in this study, the special form of this solution for the axisymmetric problem is used. Using this potential function, equations of motion accompanied with the heat conduction equation are reduced to a sixth-order partial differential equation, governing the potential function. By using the Hankel integral transform to suppress the radial variable in cylindrical coordinates, this partial differential equation is reduced to a sixth-order ordinary differential equation in terms of depth. After solving this ordinary differential equation, displacements, temperature, and stresses are derived in the Hankel domain. Applying the inverse theorem for Hankel transform, the responses are derived in the physical domain as improper integrals. To prove the validity of the analytical solution presented in this paper, the solutions are simplified for two specific cases reported in the literature. For the elastodynamic case, the Green’s functions are degenerated to results of Lamb (1904) for a vertical point-load applied at the origin. Also, for the case of an axisymmetric quasi-static thermoelastic problem, the results are reduced to Haojiang et al. (2000). Because of the complexity in the involving integrands as a result of thermomechanical coupling, the inversion of Hankel transforms are done numerically via Mathematica software. Some results about wave propagation in thermoelastic media are presented and compared with elastodynamics. Numerical results are depicted graphically for different cases of load and heat distributions. To show the efficiency and validity of the numerical integration algorithm, the vertical displacement is numerically evaluated because of a uniform vertical patch load for special elastodynamic case and compared with Rahimian et al. (2007), where very good agreement is achieved.

Problem Description An axisymmetric thermoelastic isotropic half-space defined as z $ 0 in the cylindrical coordinate system (r, u, z) is assumed to be under the effects of normal traction and temperature at z 5 0 in such a manner that the problem is treated in the axisymmetric case (Fig. 1). According to the linear theory of coupled thermoelasticity, the basic equations ignoring body forces and heat supply are as follows (Biot 1956): m=2 u þ ðl þ mÞ=e 2 b=T ¼ r ∂2 u=∂t 2

(1)

k=2 T ¼ c ∂T=∂t þ T0 b ∂e=∂t

(2)

where u 5 displacement vector consist of ur and uz , the radial and axial components, respectively; r 5 mass density; = 5 gradient operator; =2 5 ∂2 =∂r2 1 ∂=ðr∂rÞ 1 ∂2 =∂z2 5 Laplacian; b 5 ð2m 1 3lÞa 5 thermal-stress coefficient; a 5 thermal expansion coefficient; e 5 err 1 euu 1 ezz 5 dilatation; c 5 specific heat; T 5 T1 2 T0 5 change of temperature; T1 5 absolute temperature; T0 5 reference temperature; k 5 thermal conductivity, and l and m 5 Lame’s constants. Also, the strain-displacement relationships in the cylindrical coordinates for axisymmetric case are err ¼

∂ur , ∂r

euu ¼

eru ¼ euz ¼ 0

ur , r

ezz ¼

∂uz , ∂z

2erz ¼

∂ur ∂uz þ , ∂z ∂r (3)

Substituting Eq. (3) into Eqs. (1) and (2), the governing equations in terms of displacements and temperature are obtained as follows: 
2 ∂ ur 1 ∂ur ur ∂2 u 2 þ m 2r þ ð2m þ lÞ 2 2 r ∂r r ∂r ∂z ∂2 uz ∂2 u ∂T ¼ r 2r , 2b þ ðm þ lÞ ∂r ∂r∂z ∂t  
2
2 2 ∂ uz 1 ∂uz ∂ uz ∂ ur 1 ∂ur þ ð2m þ lÞ þ þ þ ðm þ lÞ m r ∂r ∂r2 ∂z2 ∂r∂z r ∂z   2 2 2 ∂ u ∂ u ∂ u 2 b ∂T ¼ r 2z , 2bT0 r 2 bT0 z þ k=2 2c ∂ T ¼ 0 ∂z ∂t ∂t ∂r∂t ∂z∂t (4)

Fig. 1. Isotropic half-space subjected to vertical traction and temperature at the surface

JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013 / 1167

J. Eng. Mech. 2013.139:1166-1177.

Potential Function

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

Because of the difficulty in dealing with the coupled partial differential equations in Eq. (4), the potential method plays a major role in the solution of related coupled boundary value problems. Because of simplicity and completeness, the scalar potential function proposed by Eskandari-Ghadi et al. (2013) is used in this treatment to uncouple the system of Eq. (4). In terms of potential function Fðr, z, tÞ, the displacements and the temperature could be expressed as follows:

where j 5 Hankel transform parameter, and Jm 5 Bessel function of the first kind and mth order. Applying the zeroth-order Hankel transform to Eq. (9) and after some algebraic manipulation, the following sixth-order ordinary differential equation with respect to depth is found: 

∂6 þ I2 ðjÞ∂4 þ I1 ðjÞ∂2 þ I0 ðjÞ F 0 ðj, zÞ ¼ 0

where ∂n 5 d n =dzn and I2 ðjÞ ¼ a1 2 3j2 2 b2 T0 p=kð2m þ lÞ   I1 ðjÞ ¼ a2 2 2a1 j2 þ 3j4 2 b2 T0 p a3 2 2j2 kð2m þ lÞ   I0 ðjÞ ¼ a4 2 a2 j2 þ a1 j4 2 j6 2 b2 T0 p j4 2 a3 j2 kð2m þ lÞ

 ∂2 Fðr, z, tÞ  , ur ðr, z, tÞ ¼ 2 1=m2 kðm þ lÞ□2T 2 b2 T0 ð∂=∂tÞ ∂r∂z


 
 k □2 l þ 2m =2 þ ∂2 2 r ∂2 uz ðr, z, tÞ ¼ 0 2 T r m ∂z2 ∂t m


2   b ∂ 2 =r Fðr, z, tÞ, 2 T0 ∂t m 2
  b ∂2 ∂ Fðr, z, tÞ Tðr, z, tÞ ¼ T0 =2 2 r0 2 ∂t ∂z∂t m

(12) also 
pr pr a1 ¼ 2p c þ þ k m 2m þ l (5) a2 ¼

where r0 ¼ r=m,

cT ¼ k=c,

□2T ¼ =2 2 ∂ , cT ∂t

= ¼ 2

=2r

(6)

2 =2r ¼ ∂ 2 þ ∂ r∂r ∂r

Substituting Eq. (5) into Eq. (4), a sixth-order partial differential equation that governs the potential function F is obtained as (Eskandari-Ghadi et al. 2013) 

p3 r½ð3m þ lÞc þ prk mkð2m þ lÞ

a3 ¼ 2r0 p2 ,

2 þ ∂ 2, ∂z

  kð2m þ lÞ□2T □21 □22 2 b2 T0 =2 ð∂=∂tÞ =2 2 r0 ∂2 =∂t 2  Fðr, z, tÞ ¼ 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where □p =2ffi 2 ∂2 =ðc2i ∂t2 Þ, ði 5 1, 2Þ; and c1 5 ð2m 1 lÞ=r i 5 ffiffiffiffiffiffiffiffi and c2 5 m=r 5 dilatational and shear wave speeds, respectively. In the case of time-harmonic motion with a time factor ept , one can express the displacements, temperature, stresses, and potential function in the form of

a4 ¼

(13)

2p5 r2 c mkð2m þ lÞ

Considering the regularity condition at infinity, the solution for Eq. (11) is given by F 0 ðj, zÞ ¼ a1 ðjÞe2l1 z þ a2 ðjÞe2l2 z þ a3 ðjÞe2l3 z

(14)

where li , ði 5 1, 2, 3Þ 5 roots of the following polynomial equation: l6 þ I2 ðjÞl4 þ I1 ðjÞl2 þ I0 ðjÞ ¼ 0

(7)

½u, T, s, Fðr, z, tÞ ¼ ½u, T, s, Fðr, zÞe pt , etc:

(11)

(15)

which is determined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffi b1 b2 2 p b3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 ¼ j2 þ r0 p2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffi l3 ¼ b1 b2 þ p b3

l1 ¼

(16)

(8) with

where p5 pffiffiffiffiffiffi ffi iv, v 5 circular frequency of the harmonic motion, and i 5 21. With these definitions, one can rewrite Eq. (7) as follows: h

 i 2 2 2 kð2m þ lÞ□ T □ 1 □ 2 2 b2 T0 p=2 =2 2 r0 p2 Fðr, zÞ ¼ 0 2

(9)

b1 ¼ 1=½2kð2m þ lÞ     b2 ¼ ð2m þ lÞ pc þ 2kj2 þ p kpr þ b2 T0 b3 ¼ b4 T02 þ ½cð2m þ lÞ 2 kpr2 þ 2b2 T0 ½cð2m þ lÞ þ kpr (17)

2

where □ i 5 =2 2 p2 =c2i , ði 5 1, 2Þ and □ T 5 =2 2 p=cT . To solve Eq. (9), Hankel integral transforms are used to suppress the radial variable. The mth-order Hankel integral transform of Fðr, zÞ with respect to the radial variable is defined as (Sneddon 1972) ð‘ F ðj, zÞ ¼ rJm ðjrÞFðr, zÞ dr m

0

(10)

and ai ðjÞ, ði 5 1, 2, 3Þ 5 some unknown functions to be determined using the boundary conditions. From Eq. (16), it can be seen that l2i , ði 5 1, 2, 3Þ are single valued but li , ði 5 1, 2, 3Þ are multivalued functions. These functions have some branch points that may be located in the common path of integration, and thus the branch cuts must be introduced to make the functions be single valued at those points (Fig. 2). In this paper, the branch points are determined from li ðjÞ 5 0, ði 5 1, 2, 3Þ, which results in

1168 / JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013

J. Eng. Mech. 2013.139:1166-1177.

jl1 ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii b1 v krv þ i 2b4 þ ib5 þ b6 ,

jl3 ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii b1 v krv þ i 2b4 2 ib5 þ b6

pffiffiffiffiffiffiffiffiffi jl2 ¼ v r=m,

(18)

pffiffiffiffiffiffiffiffiffi jl2 ¼ v r=m,

jl3 ¼

rffiffiffiffiffiffiffiffiffiffiffi 2icv k

(20)

In this case, jl1 and jl2 are collapsed on those reported by Pak (1987) for dilatational and shear wave numbers, respectively.

Boundary Conditions and General Solution

where

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r jl1 ¼ v , 2m þ l

 b4 ¼ cð2m þ lÞ þ b2 T0  b5 ¼ 2krv b2 T0 2 cð2m þ lÞ  2 b6 ¼ b2 T0 þ cð2m þ lÞ 2 ðkrvÞ2

(19)

According to Eq. (18) and the numerical results, which are presented in the section Numerical Results and Discussion, the branch point jl1 corresponds to the coupled dilatational (P), and thermal wave numbers and the branch point jl2 corresponds to the shear (S) wave number. As can be seen from Eq. (18), the shear wave is not affected by thermal coupling; however, contrary to the elastodynamic case, here the dilatational wave number undergoes some damping and dispersion because of thermomechanical coupling. In addition, the branch point jl3 is a conjugate complex number and has a sufficiently large imaginary part, whereas the imaginary part of jl1 is an extremely small value and thus can be neglected (Table 1). For this reason, jl1 is located on the path of integration and causes a disturbance related to the dilatational (P) wave, whereas jl3 is not located on the path of integration (real j-axis) and thus does not cause any disturbance related to wave motions (Figs. 3–5). On the other hand, jl2 is a pure real number and also located on the path of integration and makes a disturbance for the shear ðSÞ wave. To be consistent with Eq. (14), one can define branch cuts for li , ði 5 1, 2Þ on the complex j-plane as Fig. 2 with branch points emanating from jli , ði 5 1, 2Þ such that the real parts of li , ði 5 1, 2Þ are always nonnegative (Pak 1987; Rahimian et al. 2007). Under this condition, eli z , ði 5 1, 2, 3Þ terms become inadmissible because of the radiation conditions and are thus omitted from Eq. (14). For the zero thermal stress coefficient (b 5 0), the branch points are reduced to

An arbitrary time-harmonic vertical traction and change of temperature with intensities gðrÞe pt and f ðrÞe pt , respectively, are assumed to be applied on a finite patch p0 located at the surface of the medium, where z 5 0 (Fig. 1). Thus, the boundary conditions are written as srz ðr, z ¼ 0Þ ¼ 0,

"r

szz ðr, z ¼ 0Þ ¼ 2gðrÞ r 2 p0 szz ðr, z ¼ 0Þ ¼ 0, r Ï p0 Tðr, z ¼ 0Þ ¼ f ðrÞ, Tðr, z ¼ 0Þ ¼ 0,

r 2 p0 r Ï p0

(21a) (21b)

(21c)

where ept is omitted for brevity. The stress-strain-temperature relations are srr ¼ 2merr þ le 2 Tb suu ¼ 2meuu þ le 2 Tb szz ¼ 2mezz þ le 2 Tb srz ¼ 2merz , sru ¼ suz ¼ 0

(22)

Using the displacement- and temperature-potential relations and stress-displacement-temperature relationships, one can derive the following stress-potential relations
  b2 ∂2 2 mþl 2 T0 p srz ðr, zÞ ¼ 2 2 k=T ∂z m m


 2m þ l 2 ∂2 b2 ∂F T0 p=2r =r þ 2 2 r0 p2 2 ∂r ∂z m m   

2m þ l 2 ∂2 k =2T =r þ 2 2 r0 p2 szz ðr, zÞ ¼ ð2m þ lÞ m ∂z m þ k=2T

2


2   b b2 T0 p=2r 2 T0 p =2 2 r0 p2 m m

2

∂F l 2 2 2 kðm þ lÞ= = 2 b T p 0 T m2 r ∂z (23)

To use the boundary conditions [Eq. (21)], the stresses and the temperature may need to be written in the Hankel-transformed

Fig. 2. Branch cuts for l1 and l2

Table 1. Comparison of the Branch Points and Poles for Thermoelastic Case with Those of Elastodynamics v0 5 0:5 Problem cases Thermoelastodynamic Material 2 Elastodynamic Material 2

v0 5 3

j l1

j l2

j l3

jp

j l1

j l2

j l3

jp

0.01201

0.5

1,159:3721,159:37i

0.523413

0.07207

3

2839:8622839:86i

3.14052

0.28867

0.5

—

0.543832

1.7320

3

—

3.26299

JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013 / 1169

J. Eng. Mech. 2013.139:1166-1177.

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

where u1r , s1rz 5 first-order Hankel transforms of ur , srz , and u0z , T 0 and s0zz 5 zeroth-order Hankel transforms of uz , T and szz , respectively. In addition ðm þ lÞk , g2 ¼ k=m, g3 ¼ bT0 p=m, g1 ¼ j m2

Fig. 3. Real (solid line) and imaginary (dashed line) parts of integrand function of point-load Green’s function for displacement uz and Material 2 with v0 5 0:5

g4 ¼ mðg1 2 jg2 Þ, g5 ¼ ð2m þ lÞg2   2j  h1 ¼ 2 kðm þ lÞ j2 þ pc=k þ b2 T0 p , m   2  2k 2 2 2m þ l r0 p þ j þ pc=k þ j h2 ¼ m m   2 h3 ¼ 2 bT0 p j þ r0 p2 m, h4 ¼ mðh1 2 jh2 Þ,

(25)

h5 ¼ ljg1 þ ð2m þ lÞh2 2 bg3 h i pc 2 2m þ l b2 T0 pj2 l2 ¼ k j2 þ j þ r0 p2 þ m m2 k m l4 ¼ 2 jml2 ,

Fig. 4. Real (solid line) and imaginary (dashed line) parts of integrand function of point-load Green’s function for displacement uz and Material 2 with v0 5 3

l5 ¼ ljh1 þ ð2m þ lÞl2 2 bh3

Using Eq. (14) and substituting Eqs. (24c)–(24e) into the transformed form of the boundary conditions [Eq. (21)] results in a 3 3 3 system of linear algebraic equations for ai ðjÞ, i 5 1, 2, 3 as a1 ðjÞx1 þ a2 ðjÞx2 þ a3 ðjÞx3 ¼ 0 (26) a1 ðjÞy1 þ a2 ðjÞy2 þ a3 ðjÞy3 ¼ 2g0 ðjÞ a1 ðjÞz1 þ a2 ðjÞz2 þ a3 ðjÞz3 ¼ f 0 ðjÞ where g0 ðjÞ and f 0 ðjÞ 5 zeroth-order Hankel transforms of gðrÞ and f ðrÞ, respectively, and   xi ¼ g4 l4i þ h4 l2i þ l4 , yi ¼ 2 g5 l5i þ h5 l3i þ l5 li , (27)   zi ¼ 2 g3 l3i þ h3 li , ði ¼ 1, 2, 3Þ Solving Eq. (26) for a1 ðjÞ, a2 ðjÞ and a3 ðjÞ results in   a1 ðjÞ ¼ g0 ðjÞðx2 z3 2 x3 z2 Þ þ f 0 ðjÞðx2 y3 2 x3 y2 Þ IðjÞ   a2 ðjÞ ¼ g0 ðjÞðx3 z1 2 x1 z3 Þ þ f 0 ðjÞðx3 y1 2 x1 y3 Þ IðjÞ  0  a3 ðjÞ ¼ g ðjÞðx1 z2 2 x2 z1 Þ þ f 0 ðjÞðx1 y2 2 x2 y1 Þ IðjÞ

Fig. 5. Real (solid line) and imaginary (dashed line) parts of integrand function of point-load Green’s function for displacement uz and Material 2 with v0 5 0:5

domain. Therefore, by applying Hankel integral transforms into the displacement-, temperature-, and stress-potential relationships [Eq. (23) and Eq. (5) combined with Eq. (8)], the displacements, temperature, and stresses are derived in the Hankel domain as  (24a) u1r ðj, zÞ ¼ g1 ∂3 þ h1 ∂ F 0 ðj, zÞ  u0z ðj, zÞ ¼ g2 ∂4 þ h2 ∂2 þ l2 F 0 ðj, zÞ

 s1rz ðj, zÞ ¼ g4 ∂4 þ h4 ∂2 þ l4 F 0 ðj, zÞ

(24d)



¼ g5 ∂ þ h5 ∂ þ l5 ∂ F 0 ðj, zÞ 5

3

0

ð‘ uz ðr, zÞ ¼ jJ0 ðjrÞu0z ðj, zÞdj 0

ð‘

Tðr, zÞ ¼ jJ0 ðjrÞT 0 ðj, zÞdj 0

(24c)

s0zz ðj, zÞ

with IðjÞ 5 x3 ðy1 z2 2 y2 z1 Þ 1 x2 ðy3 z1 2 y1 z3 Þ 1 x1 ðy2 z3 2 y3 z2 Þ. Substituting a1 ðjÞ, a2 ðjÞ, and a3 ðjÞ from Eq. (28) into Eq. (14), the function F 0 ðj, zÞ is determined, from which the stresses, displacements, and temperature are known in the Hankel-transformed domain. Applying the theorem of inverse Hankel transforms (Sneddon 1972), the same functions are obtained in real space in the form of some definite integrals as ð‘ ur ðr, zÞ ¼ jJ1 ðjrÞu1r ðj, zÞdj

(24b)

 T 0 ðj, zÞ ¼ g3 ∂3 þ h3 ∂ F 0 ðj, zÞ

(28)

ð‘

srz ðr, zÞ ¼ jJ1 ðjrÞs1rz ðj, zÞdj 0

ð‘ szz ðr, zÞ ¼ jJ0 ðjrÞs0zz ðj, zÞdj

(24e)

1170 / JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013

J. Eng. Mech. 2013.139:1166-1177.

0

(29)

Green’s Functions

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

Having determined the responses of the half-space caused by the general case of normal traction and temperature applied at the surface of the half-space, in this section, solutions are derived for three special cases: (1) point-load Green’s functions caused by a vertical point load applied at the origin; (2) patch-load solutions caused by a uniform vertical load of unit resultant applied on a circular disc of radius a, and (3) patch-heat solutions caused by a temperature distributed on a circular disc of radius a.

For a vertical point load R acting at the origin, one can show that RdðrÞ 2pr

f ðrÞ ¼ 0

For the case of a uniform vertical patch load of unit resultant acting in the z-direction 8 < 1 r,a 2 (33a) gðrÞ ¼ pa : 0 r$a f ðrÞ ¼ 0 "r

Case 1: Point-Load Green’s Functions Caused by a Vertical Point Load with Magnitude R Acting at the Origin

gðrÞ ¼

Case 2: Patch-Load Solutions Caused by a Uniform Vertical Load of Unit Resultant Applied on a Circular Disc of Radius a

Therefore, the zeroth-order Hankel transforms of gðrÞ and f ðrÞ are g0 ðjÞ 5 J1 ðjaÞ=ðpajÞ and f 0 ðjÞ 5 0. Substituting these functions into Eq. (28)

(30a)

a1ul ðjÞ ¼

J1 ðjaÞ ðx2 z3 2 x3 z2 Þ pajIðjÞ

(30b)

a2ul ðjÞ ¼

J1 ðjaÞ ðx3 z1 2 x1 z3 Þ pajIðjÞ

a3ul ðjÞ ¼

J1 ðjaÞ ðx1 z2 2 x2 z1 Þ pajIðjÞ

where dðrÞ 5 Dirac-delta function. Therefore, the zeroth-order Hankel transforms of gðrÞ and f ðrÞ are g0 ðjÞ 5 R=2p and f 0 ðjÞ 5 0. Substituting these functions into Eq. (28) results in a1pl ðjÞ ¼

R ðx z 2 x z Þ 2 3 3 2 2pIðjÞ

a2pl ðjÞ ¼

R ðx 3 z1 2 x1 z3 Þ 2pIðjÞ

a3pl ðjÞ ¼

R ðx z 2 x z Þ 1 2 2 1 2pIðjÞ

(31)

where in Eq. (31), the subscript pl denotes the case of applying a point load. Combining Eq. (31) with Eqs. (14), (24), and (29), the responses could be derived as

(34)

where in Eq. (34), the subscript ul 5 case of applying a uniform load. Combining Eq. (34) with Eqs. (14), (24), and (29), the displacements, temperature, and stresses are derived as those functions reported in Eq. (32), with a1pl ðjÞ, a2pl ðjÞ, and a3pl ðjÞ replaced by a1ul ðjÞ, a2ul ðjÞ, and a3ul ðjÞ, respectively. Case 3: Patch-Heat Solutions Caused by a Temperature Distributed on a Circular Disc of Radius a For the case of a uniform temperature ðu0 Þ distributed on a circular disc of radius a

ð‘

gðrÞ ¼ 0 "r

u0 r , a f ðrÞ ¼ 0 r$a

 ur ðr, zÞ ¼ jJ1 ðjrÞ g1 ∂3 þ h1 ∂ 0

  a1pl ðjÞe2l1 z þ a2pl ðjÞe2l2 z þ a3pl ðjÞe2l3 z dj

(33b)

(35)

Therefore, the zeroth-order Hankel transforms of gðrÞ and f ðrÞ are g0 ðjÞ 5 0 and f 0 ðjÞ 5 au0 J1 ðjaÞ=j. Substituting these functions into Eq. (28) results in

ð‘

 uz ðr, zÞ ¼ jJ0 ðjrÞ g2 ∂4 þ h2 ∂2 þ l2 0

  a1pl ðjÞe2l1 z þ a2pl ðjÞe2l2 z þ a3pl ðjÞe2l3 z dj ð‘

 Tðr, zÞ ¼ jJ0 ðjrÞ g3 ∂3 þ h3 ∂

a1ut ðjÞ ¼

au0 J1 ðjaÞ ðx2 y3 2 x3 y2 Þ jIðjÞ

a2ut ðjÞ ¼

au0 J1 ðjaÞ ðx3 y1 2 x1 y3 Þ jIðjÞ

a3ut ðjÞ ¼

au0 J1 ðjaÞ ðx1 y2 2 x2 y1 Þ jIðjÞ

0

  a1pl ðjÞe2l1 z þ a2pl ðjÞe2l2 z þ a3pl ðjÞe2l3 z dj ð‘  srz ðr, zÞ ¼ jJ1 ðjrÞ g4 ∂4 þ h4 ∂2 þ l4

(36)

ð‘

where in Eq. (36), the subscript ut 5 case of applying uniform temperature on the surface. Therefore, with the same procedure as before, the responses are derived as those functions reported in Eq. (32), with a1pl ðjÞ, a2pl ðjÞ, and a3pl ðjÞ replaced by a1ut ðjÞ, a2ut ðjÞ, and a3ut ðjÞ, respectively.

0

Degeneration of Solutions

0

  a1pl ðjÞe2l1 z þ a2pl ðjÞe2l2 z þ a3pl ðjÞe2l3 z dj  szz ðr, zÞ ¼ jJ0 ðjrÞ g5 ∂5 þ h5 ∂3 þ l5 ∂   a1pl ðjÞe2l1 z þ a2pl ðjÞe2l2 z þ a3pl ðjÞe2l3 z dj (32)

In the previous sections, the analytical results were obtained for the general time-harmonic thermoelastic case. In this section, the results

JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013 / 1171

J. Eng. Mech. 2013.139:1166-1177.

are degenerated to two specific cases and compared with the existing results reported in the literature. Those degeneration cases are (1) an elastodynamic case with a vertical point load R applied at the origin and (2) an axisymmetric quasi-static thermoelastic case, traction free with a temperature distributed on the surface.

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

Case 1: Elastodynamic Case with a Vertical Point Load R Applied at the Origin

where 2ð3m þ lÞ r p2 2 2j2 , ð2m þ lÞ 0    m cðjÞ ¼ r0 p2 þ j2 r0 p2 þ j2 ð2m þ lÞ pc , k

bðjÞ ¼

(39a)

 u0ze ðj, zÞ ¼ g2 ∂2 þ h2e Fe0 ðj, zÞ

(39b)

s0zze ðj, zÞ ¼ g5 ∂3 þ h5e ∂ Fe0 ðj, zÞ

a2e ðjÞ ¼

g1 ðjÞR 2pIe ðjÞ

 2ðm þ lÞ 2 m þ l 2 2j 2 ks2 , a , g2 ðjÞ ¼ m m 2
mþl Ie ðjÞ ¼ k aFðjÞ m

(41)

(42)

and FðjÞ 5 ð2j2 2 ks2 Þ2 2 4j2 ab. Using Eq. (41), Fe0 ðj, zÞ is completely obtained, which may be replaced in Eqs. (39a) and (39b) for determining the transformed displacements that via the use of the Hankel inverse theorem results in the displacements as ur ðr, zÞ ¼ 2R 2pm

ð‘ jg3 ðz, jÞJ1 ðjrÞdj 0

u1re ðj, zÞ ¼ ½g1 ∂Fe0 ðj, zÞ



2g2 ðjÞR , 2pIe ðjÞ

g1 ðjÞ ¼

(38)

As can be seen in Eq. (37), the whole operator for the potential function F 0 ðj, zÞ in the case of b 5 0 can be written in the form of two separated operators, one of which is the operator for elastodynamics and the other is for thermal effect. Denoting ½∂2 1 aðjÞF 0 ðj, zÞ by Fe0 ðj, zÞ, the thermal effect can be eliminated, and therefore, ½∂4 1 bðjÞ∂2 1 cðjÞFe0 ðj, zÞ 5 0, where the subscript e denotes the case of elastodynamics. Considering the radiation conditions, the latter equation results in Fe0 ðj, zÞ 5 a1e ðjÞe2l1e z 1 a2e ðjÞe2l2e z , where l2ie , ði 5 1, 2Þ are the roots for the characteristic solving equation l2e 1 bðjÞl2e 1 cðjÞ 5 0. By p e and ffiffiffiffiffiffiffiffiffiffi this equation forplffiffiffiffiffiffiffiffiffiffi j2 2 ks2 remembering that p 5 iv, l1e 5 a 5 j2 2 kd2 and l2e 5 bp5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi can be obtained, with kd 5 v=Cd , ks 5 v=Cs , Cd 5 ð2m 1 lÞ=r, pffiffiffiffiffiffi and Cs 5 m=r. As mentioned previously, Cd and Cs are dilatational and equivoluminal wave speeds, respectively, which were also presented in Pak (1987). Moreover, the displacements and stresses in the Hankel-transformed domain can be written in terms of Fe0 ðj, zÞ as follows:

 s1rze ðj, zÞ ¼ g4 ∂2 2 jmh2e Fe0 ðj, zÞ

a1e ðjÞ ¼ where

For degeneration to this case, it can be shown that if the thermalstress coefficient b is set to be zero (b 5 0), then Eq. (11) changes to  6 ∂ þ I2 ðjÞ∂4 þ I1 ðjÞ∂2 þ I0 ðjÞ F 0 ðj, zÞ   (37) ¼ ∂4 þ bðjÞ∂2 þ cðjÞ ∂2 þ aðjÞ F 0 ðj, zÞ ¼ 0

aðjÞ ¼ 2 j2 2

zeroth-order Hankel transform of Eq. (30a), one may obtain a 2 3 2 system of linear algebraic equations to be solved for a1e ðjÞ and a2e ðjÞ, whose solution is

uz ðr, zÞ ¼

R 2pm

(43)

ð‘ jg4 ðz, jÞJ0 ðjrÞdj 0

where   2 2 2jab 2bz j 2j 2 ks 2az e e 2 g3 ðz, jÞ ¼ FðjÞ FðjÞ   2 2 22j2 a 2bz a 2j 2 ks 2az e e g4 ðz, jÞ ¼ þ FðjÞ FðjÞ For z 5 0, Eq. (44) is reduced to   2j 2j2 2 ks2 2 2ab , g3 ð0, jÞ ¼ FðjÞ

g4 ð0, jÞ ¼

2aks2 FðjÞ

(44)

(45)

Eventually, by substituting Eq. (45) into Eq. (43), the displacements are derived for z 5 0 as follows: ur ðr, 0Þ ¼ R 2pm

(39c)

ð‘

  j2 2j2 2 ks2 2 2ab J1 ðjrÞdj FðjÞ

0

(39d)

uz ðr, 0Þ ¼

2R 2pm

ð‘

(46) ajks2 FðjÞ

J0 ðjrÞdj

0

where

h5e



2m þ l 2 j 2 ks2 , m  k ¼ ð2m þ lÞks2 2 ð4m þ 3lÞj2 m

h2e ¼ 2k m

(40)

Also if b 5 0, according to Eq. (25), g3e 5 h3e 5 0 and considering Eq. (24c), the change of temperature that is coupled with the displacements is determined to be zero. By using Eqs. (39c) and (39d), respectively, in the first- and zeroth-order Hankel transforms of Eqs. (21a) and (21b) and the

The displacements given in Eq. (46) are exactly the same as the results presented in Lamb (1904). It should be noted that ur , uz , ks , r, a, and b in this paper are equivalent, respectively, to q0 , w0 , k, v, a, and b in the notation of Lamb. Case 2: Axisymmetric Quasi-Static Thermoelastic Case, Traction Free with a Temperature Distributed on the Surface In the quasi-static problem, the inertia effect is neglected; in other words, the density is set to be zero (r 5 0), so in this case with

1172 / JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013

J. Eng. Mech. 2013.139:1166-1177.

pt a temperature f ðrÞe distributed on the surface, l1qs 5 l2qs 5 j and ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l3qs 5 j 1 p=k0 , where k0 5 kð2m 1 lÞ=½cð2m 1 lÞ 1 b2 T0  and the subscript qs denotes the case of quasi-static. Because of the existence of repeated roots l1qs 5 l2qs 5 j, the solution for Eq. (14) is changed to

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

0 ðj, zÞ ¼ a1qs ðjÞe2jz þ za2qs ðjÞe2jz þ a3qs ðjÞe2l3qs z Fqs

given by Haojiang et al. where the first two equations of Eq. (31) in Haojiang et al. must be changed as follows: 2rc1 þ 2rc2 þ Bc3 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ p=k0 c2 þ ðB 2 2Þc3 ¼ 0 2rc1 þ 2

(51)

(47)

By assuming the traction-free surface, the external traction is set to be zero, which means gðrÞ 5 0. Therefore, the boundary conditions 0 ðj, z 5 0Þ 5 f 0 ðjÞ. Substituting are ½s1rzqs , s0zzqs ðj, z 5 0Þ 5 0 and Tqs Eq. (47) into Eqs. (24c)–(24e) and satisfying the latter boundary conditions, the functions a1qs , a2qs , and a3qs are determined as  f 0 ðjÞ  a1qs ¼ x2qs y3qs 2 x3qs y2qs Iqs ðjÞ

j, uz , and T in this paper are, respectively, equivalent to r, w, and u in the notation of Haojiang et al.

Numerical Results and Discussion

(50)

As seen in Eq. (32), the solutions for displacements, stresses, and temperature are expressed in the form of some improper line integrals. Because of the existence of complex integrands, the integrals cannot be analytically given in the closed form, so a suitable numerical quadrature scheme should be used for evaluation of the integrals (Rahimian et al. 2007; Eskandari-Ghadi et al. 2010). On the other hand, the numerical evaluation of integrals needs careful consideration because of the presence of singularities within the range of integration and the oscillatory behavior of the integrands involving the product of the Bessel functions. The singularity happens because of the existence of the branch points and poles lying on the path of integration. In this paper, for numerical evaluation of integrals, the method introduced in Pak (1987) is used successfully. As indicated in Pak (1987), one can evaluate the improper integrals by means of residual method and contour integration. Therefore, for numerical integration, the following stages are considered: (1) finding the poles and branch points located in the path of integration; (2) integrating from zero to a point behind the pole and continuing the integration from a point after the pole to a sufficiently large value; and (3) adding the contribution from residue at the pole to the final sum. This means that the integral is decomposed into three integrals, two of which are line integrals and the last one is the integral over a small semicircle of radius ɛ above the pole, where ɛ is a sufficiently small value. Here, the two line integrals are evaluated directly using Mathematica software, and the integral over a small semicircle is evaluated using the residual method. For evaluating the two line integrals, sufficiently large and sufficiently small values for upper limit of the last integral and ɛ, respectively, must be defined, and then the PrincipalValue method in the NIntegrate option of Mathematica as {PrincipalValue,SingularPointIntegrationRadius → ɛ} is used. In general, there exist three branch points located at jli , ði 5 1, 2, 3Þ where jl1 and jl2 are located on the path of integration. For the special case of bp5 0, the branch points are reduced ffiffiffiffiffiffiffiffiffiffiffi to jl1 5 kd , jl2 5 ks , and jl3 5 2ivc=k, where jl1 and jl2 are those for elastodynamics (Pak 1987) and jl3 is for the thermal effect. In addition, integrals have a simple pole at jp , which may be obtained as the solution of the equation IðjÞ 5 0 for j. In the elastodynamic corresponds to the Rayleigh wave (Pak 1987), which is case, p jpffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffiffiffi jp 5 3 1 3 r=mv=2. As stated before, the path of integration by semicircle of radius ɛ around the pole should be changed. The integrands are strongly singular at the poles, but they are weakly singular at the branch points (Figs. 3 and 4). Therefore, just the small semicircles over the poles should be evaluated by using the residual method. Because the pole is of the first order, the integrand can be written as qðjÞ=IðjÞ, where qðjÞ is an analytical function at jp , and IðjÞ is defined earlier. According to the residue theorem, the integral over the small semicircle of radius ɛ at the pole is equal to 2piResðjp Þ, where Resðjp Þ 5 lim ½qðjÞ=ðdIðjÞ=djÞ. For weakly

Eq. (50) should be the same as the results presented by Haojiang et al. (2000). However, there are some typographical errors in the expressions

singular behavior of branch points, the MaxRecursion option of the Mathematica software is used in the numerical integration. To do so, the default value of the MaxRecursion is increased to increase the

a2qs ¼

 f 0 ðjÞ  x3qs y1qs 2 x1qs y3qs Iqs ðjÞ

 f 0 ðjÞ  x1qs y2qs 2 x2qs y1qs Iqs ðjÞ      Iqs ðjÞ ¼ z2qs x3qs y1qs 2 x1qs y3qs þ z3qs x1qs y2qs 2 x2qs y1qs a3qs ¼

(48) where in Eq. (48), the functions xiqs , yiqs , ziqs , ði 5 1, 2, 3Þ are as follows: h i xiqs ¼ g4qs l4iqs þ h4qs l2iqs þ l4qs , ði ¼ 1, 3Þ h i x2qs ¼ 24g4qs j3 2 2h4qs j i h yiqs ¼ 2 g5qs l5iqs þ h5qs l3iqs þ l5qs liqs , ði ¼ 1, 3Þ (49) h i y2qs ¼ 5g5qs j4 þ 3h5qs j2 þ l5qs i h z1qs ¼ 0, z2qs ¼ 3g3qs j2 þ h3qs , i h z3qs ¼ 2 g3qs l33qs þ h3qs l3qs By setting r 5 0 in Eq. (25), the functions giqs , hiqs , ði 5 1, 2, 3, 4, 5Þ and liqs , ði 5 2, 4, 5Þ are determined, from which the functions xiqs , yiqs , ziqs , ði 5 1, 2, 3Þ are known. Then with the same procedure as before, the displacements and the temperature are obtained as follows: ð‘ h i urqs ðr, zÞ ¼ jJ1 ðjrÞ g1qs ∂3 þ h1qs ∂ 0

  a1qs e2jz þ za2qs e2jz þ a3qs e2l3qs z dj

ð‘

h i uzqs ðr, zÞ ¼ jJ0 ðjrÞ g2qs ∂4 þ h2qs ∂2 þ l2qs 0

  a1qs e2jz þ za2qs e2jz þ a3qs e2l3qs z dj ð‘ h i Tqs ðr, zÞ ¼ jJ0 ðjrÞ g3qs ∂3 þ h3qs ∂ 0

  a1qs e2jz þ za2qs e2jz þ a3qs e2l3qs z dj

j → jp

JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013 / 1173

J. Eng. Mech. 2013.139:1166-1177.

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

subdivisions of the integral; therefore, the numerical integration could be converged. For viscoelastic problems, the elasticity moduli are complexvalued quantities. Therefore, the branch points and poles are also complex-valued. Under these conditions, the real j-axis is free from any singularities, and the numerical integration is performed easily. One may also use a small imaginary part in the argument of integration to remedy the singular behavior of the integrand function. For example, Rajapakse and Wang (1993) used 1% complex part in their nondimensional material parameters to eliminate the singularities from the path of integration. This may resolve the problem encountered in treating with singular behavior; however, it may result in some uncertainties in the numerical evaluations. To illustrate some results, several isotropic materials with different elastic and thermal properties are considered, as given in Table 2. The thermal properties of these materials can be found in Das and Lahiri (2009). k 5 69 Wm21 degrees21 , r 5 8,836 kgm23 , c 5 427 Jkg21 degrees21 , and T0 5 298 k are used for all materials listed in Table 2. To show the efficiency and validity of the numerical evaluations, the vertical displacement uz for the case of elastodynamics, b 5 0, is evaluated and compared with the solution presented in Rahimian et al. (2007). In the numerical evaluation to verify the elastodynamic case, if b 5 0, infinity was encountered; therefore, a sufficiently small value for b is required and 1025 has been selected. Fig. 6 compares the displacement pffiffiffiffiffiffiffiffiffi uz along the r-axis with dimensionless frequency v0 5 av r=m 5 3 and y 5 0:25 (Material 1 in Rahimian et al. 2007), because of a vertical patch load of unit resultant applied on a circular disc of radius a. Displacement is normalized as pmauz =R. As shown in the figure, the agreement between the two results is excellent. Table 1 compares the branch points and poles for the thermoelastic case (Material 2) with those of elastodynamics for frequencies v0 5 0:5 and 3:0. As stated previously, jl1 , jl2 , and jp correspond to dilatational (P), shear (S), and Rayleigh (R) wave numbers, respectively. As can be inferred from the table, the dilatational and Rayleigh wave numbers are affected by thermomechanical coupling and undergo some damping and dispersion. However, this damping

for P wave number is significant and for the Rayleigh wave number is negligible. Based on these points, P and Rayleigh wave speeds for the thermoelastic case are more than those of elastodynamics, and these two waves arrive sooner in the thermoelastic case than in elastodynamics. It can be seen that the shear wave (S) is not affected by thermomechanical coupling. Figs. 3–5 illustrate the variation of the integrand function of point-load Green’s function for displacement uz with respect to j for frequencies v0 5 0:5 and 3:0 and for Material 2. As can be seen from these figures, jl1 , jl2 , and jp given in Table 1 for the thermoelastic case, correspond to P, S, and Rayleigh waves. Fig. 5 indicates that the behavior of the integrand function at branch point jl3 5 1159:3721159:37i is not singular, and both real and imaginary parts of the integrand function are smooth at this point, whereas for jl1 , jl2 , and jp , the figure is not smooth, and the integrand function is strongly singular at jp and is weakly singular at jl1 and jl2 . Also, it can be inferred from these figures that the wave numbers are completely dependent on the frequency of excitation. To illustrate the numerical results graphically, some dimensionless parameters are defined. For point-load Green’s functions, the dimensionless displacement, stress, and temperature are defined as pmLuz =R, pL2 szz =R, and pL2 bT=R, respectively, where R is the resultant force and L represents a parameter of length. In the case of a uniform temperature u0 5 1 K applied on a circular disc of dimensionradius a; muz =ðbau0 Þ, bT=m, and szz =ðbu0 Þ are p ffiffiffiffiffiffiffiffiffi less parameters. pffiffiffiffiffiffiffiffiffi Dimensionless frequencies v0 5 Lv r=m and v0 5 av r=m are used for applying point-load and patch-heat, respectively, where L and a represents the unit of length and radius of

Table 2. Synthetic Material Elastic and Thermal Coefficients Material number 1 2 3 4

E ðNm22 Þ

y

b ðNm22 degrees21 Þ

5 3 1010 5 3 1010 5 3 1010 5 3 1010

0 0.25 0.499 0.25

7:04 3 106 7:04 3 106 7:04 3 106 7:04 3 106

Fig. 6. Comparison of displacement uz along r-axis subjected to uniform vertical load of resultant R applied on disc of radius a with v0 5 3

Fig. 7. Real and imaginary parts of point-load Green’s function for displacement uz along z-axis subjected to vertical point load R with v0 5 0:5

Fig. 8. Real and imaginary parts of point-load Green’s function for stress szz along z-axis subjected to vertical point load R with v0 5 0:5

1174 / JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013

J. Eng. Mech. 2013.139:1166-1177.

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

circular patch, respectively. Numerical results given in Figs. 7–19 illustrate the effects of Poison’s ratio, frequency, and thermal-stress coefficient on the responses. Figs. 7–12 depicted the point-load Green’s functions for normalized displacement, stress, and temperature, and Figs. 13–19 illustrate normalized displacement, stress, and temperature caused by a uniform temperature distributed on a disc of radius a. Both real and imaginary parts of displacements, stresses, and temperature tend to zero with increasing depth or radial distance.

Also, as the frequency of excitation increases, both the real and imaginary parts of these functions show more oscillatory behavior. As can be seen from Figs. 7–9, these figures are singular at the origin, where the point load is applied. Figs. 13–15 indicate that in the case of applying a uniform temperature on a disc of radius a, normalized displacement and temperature for z 5 0 are approximately constant and nonzero within the circular disc, and they are zero when out of it, whereas for z 5 a, another behavior is seen.

Fig. 9. Real and imaginary parts of point-load Green’s function for temperature T along z-axis subjected to vertical point load R with v0 5 0:5

Fig. 12. Real and imaginary parts of point-load Green’s function for stress szz and z 5 L subjected to vertical point load R with v0 5 0:5

Fig. 10. Real and imaginary parts of point-load Green’s function for displacement uz and r 5 L subjected to vertical point load R with v0 5 0:5

Fig. 11. Real and imaginary parts of point-load Green’s function for displacement uz and r 5 L subjected to vertical point load R with v0 5 3

Fig. 13. Real part of temperature T along r-axis subjected to uniform temperature u0 applied on disk of radius a with v0 5 0:5

Fig. 14. Real and imaginary parts of displacement uz along r-axis subjected to uniform temperature u0 applied on disk of radius a with v0 5 0:5

JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013 / 1175

J. Eng. Mech. 2013.139:1166-1177.

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

Fig. 15. Real and imaginary parts of displacement uz for z 5 a and subjected to uniform temperature u0 applied on disk of radius a with v0 5 0:5

Fig. 18. Real and imaginary parts of displacement uz for z 5 a and subjected to uniform temperature u0 applied on disk of radius a with v0 5 3

Fig. 16. Real and imaginary parts of displacement uz along z axis subjected to uniform temperature u0 applied on disk of radius a with v0 5 0:5

Fig. 19. Real and imaginary parts of displacement uz along z-axis subjected to uniform temperature u0 applied on disk of radius a with v0 5 3

solution presented here, degenerations of the formulation have been made for elastodynamic and quasi-static thermoelastic cases, which are those previously reported in the literature. Also, to show the accuracy of the numerical results, the solution is compared with an existing numerical solution for the elastodynamic case, where very good agreement is achieved. Numerical results are also depicted graphically for different cases of point-load and patch-heat to show the effect of frequency and material properties on the responses. It is observed that thermoelastic wave propagation is different from elastodynamics.

Acknowledgments Fig. 17. Real and imaginary parts of stress szz along z-axis subjected to uniform temperature u0 applied on disk of radius a with v0 5 3

The partial support from the University of Tehran during this work is gratefully acknowledged.

Conclusion

Notation

Dynamic thermoelastic Green’s functions of an axisymmetric linear elastic isotropic half-space caused by point-load, patch-load, and patch-heat are derived using a method of potentials. All these Green’s functions are expressed in the form of some improper line integrals, which may applicable as integral kernels in the numerical methods such as the well-known boundary element method for numerical treatment of related and more complicated thermoelastodynamic problems. To confirm the validity of the analytical

The following symbols are used in this paper: a 5 radius of circular disc; Cd 5 dilatational wave speed; Cs 5 shear wave speed; c 5 specific heat; E 5 Young’s modulus; e 5 dilatation; eij 5 strain components;

1176 / JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013

J. Eng. Mech. 2013.139:1166-1177.

Downloaded from ascelibrary.org by University of California, San Diego on 10/15/13. Copyright ASCE. For personal use only; all rights reserved.

Jm 5 Bessel function of first kind and mth order; k 5 thermal conductivity; T 5 change of temperature; T0 5 reference temperature; T1 5 absolute temperature; t 5 time variable; ur , uz 5 displacement components; a 5 thermal expansion coefficient; b 5 thermal-stress coefficient; l, m 5 Lame’s constants; j 5 Hankel transform parameter; r 5 mass density; sij 5 stress components; y 5 Poisson’s ratio; v 5 circular frequency of harmonic motion; v0 5 dimensionless frequency; = 5 gradient operator; and =2 5 Laplacian.

References Bill, B. G. (1983). “Thermoelastic bending of the lithosphere: Implication for basin subsidence.” Geophys. J. R. Astr. Soc., 75(1), 169–200. Biot, M. A. (1956). “Thermoelasticity and irreversible thermodynamics.” J. Appl. Phys., 27, 240–253. Carlson, D. (1972). “Linear thermoelasticity.” Encyclopedia of physics: Mechanics of solids II, C. Truesdell, ed., Springer, Berlin, 297–345. Chandrasekharaiah, D. S. (1980). “Wave propagation in a thermoelastic half-space.” Indian J. Pure Appl. Math., 12(2), 226–241. Das, N. C., and Lahiri, A. (2009). “Eigen value approach to three dimensional coupled thermoelasticity in a rotating transversely isotropic medium.” J. Math. Sci., 25(3), 237–257. Deresiewicz, H. (1958). “Solution of the equations of thermoelasticity.” Proc., 3rd U.S. National Congress on Applied Mechanics, ASME, New York, 287–291. Eskandari-Ghadi, M., Fallahi, M., Ardeshir-Behrestaghi, A. (2010). “Forced vertical vibration of rigid circular disc on a transversely isotropic half-space.” J. Eng. Mech., 136(7), 913–922. Eskandari-Ghadi, M., Sture, S., Rahimian, M., and Forati, M. (2013). “Thermoelastodynamics with scalar potential functions.” J. Eng. Mech., (Apr. 3, 2013). Fialko, Y., and Simons, M. (2000). “Deformation and seismicity in the Coso geothermal area Inyo County, California: Observation and modeling using satellite eadar interferometry.” J. Geophys. Res., 105(B9), 21781– 21794.

Furuya, M. (2005). “Quasi-static thermoelastic deformation in an elastic half-spcae: Theory and application to InSAR observation at Izu-Oshima volcano, Japan.” Geophys. J. Int., 161(1), 230–242. Georgiadis, H. G., Rigatos, A. P., and Brock, L. M. (1999). “Thermoelastodynamic disturbances in a half-space under the action of buried thermal/mechanical line source.” Int. J. Solids Struct., 36(24), 3639–3660. Goodier, J. N. (1937). “On the integration of the thermoelastic equations.” Phil. Mag., 23(157), 1017–1032. Haojiang, D., Fenglin, G., and Pengfei, H. (2000). “General solutions of coupled thermoelastic problem.” J. Appl. Math. Mech., 21(6), 631–636. Hinojosa, J. H., and Mickus, K. L. (2002). “Thermoelastic modeling of lithospheric uplift: A finite-difference numerical solution.” J. Comput. Geosci., 28(2), 155–167. Lamb, H. (1904). “On the propagation of tremors over the surface of an elastic solid.” Phil. R. Soc. Lond. Ser. A, 203(354–371), 1–42. Lanzano, P. (1985). Thermo-elastic deformations of the Earth’s lithosphere, U.S. Naval Research Laboratory, Washington, DC. Lubimova, E. A., and Magnitzky, V. A. (1964). “Thermoelastic stresses and the energy of earthquakes.” J. Geophys. Res., 69(16), 3443–3447. Lykotrafitis, G., Georgiadis, H. G., and Brock, L. M. (2001). “Threedimensional thermoelastic wave motions in a half-space under the action of a buried source.” Int. J. Solids Struct., 38(28–29), 4857–4878. Mathematica 7 [Computer software]. Champaign, IL, Wolfram Research Inc. McDowell, E. L., and Sternberg, E. (1957). “On the steady state thermoelastic problem for halfspace.” Q. Appl. Math., 14(7822), 381–398. Nowacki, W. (1962). Thermoelasticity, Addison-Wesley, Reading, MA. Nowinski, J. (1978). Theory of thermoelasticity with applications, Sijthoff and Noordhoff International Publishers, Alphen aan den Rijn, Netherlands. Pak, R. Y. S. (1987). “Asymmetric wave propagation in an elastic half-space by a method of potentials.” J. Appl. Mech., 54(1), 121–126. Rahimian, M., Eskandari-Ghadi, M., Pak, R. Y. S., and Khojasteh, A. (2007). “An elastodynamic potential method for a transversely isotropic solid.” J. Eng. Mech., 133(10), 1134–1145. Rajapakse, R. K. N. D., and Wang, Y. (1993). “Green’s functions for transversely isotropic elastic half space.” J. Eng. Mech., 119(9), 1724– 1746. Ranalli, G. (1995), Rheology of the Earth, 2nd Ed., Chapman & Hall, London. Singh, B. (2010). “Wave propagation in an initially stressed transversely isotropic thermoelastic solid half-space.” Appl. Math. Comput., 217(2), 705–715. Sneddon, I. N. (1972). The use of integral transforms, McGraw Hill, New York. Verruijt, A. (1967). “The completeness of Biot’s solution of the coupled thermoelastic problem.” J. Quart. Appl. Math, 26(4), 485–490. Zoreski, H. (1958). “Singular solutions for thermoelastic media.” Bull. Acad. Polon. Sci. Ser. Sci. Tech., 6(6), 331–339.

JOURNAL OF ENGINEERING MECHANICS © ASCE / SEPTEMBER 2013 / 1177

J. Eng. Mech. 2013.139:1166-1177.