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\noindent
%M\'ecanique des sols et milieux poreux // {\it Soil Mechanics and Porous Media}


\bc
\large{\bf 
Dynamic growth of a spherical inclusion \\ 
in thermoelastic medium}
\ec

\normalsize
\bc
\vspace{3mm}
{\bf R.E. \ Wojnar} \\

Institute of Fundamental Technological  Research, Warsaw, Poland \\
e-mail: rwojnar@ippt.gov.pl

\ec


\noindent

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%{\small 
{\bf Abstract} \\ 
The paper deals with an initial value problem of dynamic uncoupled thermoelasticity
concerning a moving spherical thermal inclusion in an infinite solid. 
An extended Kosevich' theory 
of continuously distributed defects due to
prescribed plastic fields  is used.
Applying a generalization of the 
isothermal
elastodynamics with continuosly distributed defects, 
the displacement and stress fields due to 
a spherical thermal inclusion growing linearly with time are obtained.

\medskip
\vspace{0.3cm}
%\noindent
{\bf Keywords:} thermoelasticity / moving spherical inclusion \\ 




\renewcommand{\theequation}{1.\arabic{equation}}
\setcounter{equation}{0}
\vspace{.7cm}
\no

{\bf 1. Introduction}\\
\noindent
The dynamical thermal stresses produced by a spherical temperature inclusion in an infinite body were analyzed  by 
Piechocki and Ignaczak in 1960. These authors assumed that a temperature field is constant inside of a sphere and vanishes outside of the sphere; and it changes in time as the Dirac impulse function or as the Heaviside step function. For such a temperature they obtained a closed form of dynamical thermal stresses in the body.

The fact that a discontinuous temperature field can be treated as an inclusion, and the methods developed by
the theory of defects can be applied to it, were observed by Muskhelishvili in 1916. These ideas can be also found in the book by Maysel (1951) and in a paper by Willis (1964).

Willis analyzed an isothermal process of elastodynamics corresponding to a given deformation field on a finite region of $ \ {\bf E}^3 $. In particular, he investigated the stress waves produced by a spherical strain inclusion in  $ \ {\bf E}^3 $.

Wang, Sun and  Xiao (1999) analyzed a solution analogous to that of Willis in connection with discussion of the martensitic phase transitions.  

In the present paper we discuss the displacement and stress fields due to the expansion of a spherical temperature inclusion in an infinite body.  

The dynamical response of a body  to an isothermal mechanical inclusion or a temperature inclusion can be described by the extended theory of defects due to given plastic fields  developed by Kosevich (1962,1979,1981).

If the total displacement field of a solid with defects is
described by a 
vector  $u_{i}$, then the distortion 
tensor $w_{ij}$ is defined as, cf. Kosevich (1962,1979,1981), Landau and
Lifshits (1965), Ignaczak and Rao (1993) 
\bq
w_{ij} \equiv  u_{j,i} 
\eq 
In general  $ \, w_{ij} \neq  w_{ji}$
and the following equality holds true
\bq 
\epsilon _{ijm} w_{mk,j}= - \rho _{ik}  %\eqno (2)
\eq
where $\rho _{ik}$ denotes the dislocation density tensor, 
satisfying the condition
$ \ \rho _{ik,i} = 0 \ $ 
which for a single dislocation expresses the conservation of a Burgers
vector along the dislocation line.

A total distortion tensor  $w_{ij}$ may be 
postulated in the form
\bq  
w_{ij} = w^{E}_{ij} + w^{P}_{ij} + w^{\Theta}_{ij}
\eq 
where $w^{E}_{ij}$, $ w^{P}_{ij}$ and $ w^{\Theta}_{ij}$ represent the
elastic, plastic and thermal
parts of the distortion, respectively,
and neither $ \, w^{E}_{ij}$, $ w^{P}_{ij} \,$ or $ w^{\Theta}_{ij}$ are
given by the gradient of a vector field. Hence
the motion of a dislocation is related not only to a plastic
deformation but also to a thermal one.



\renewcommand{\theequation}{2.\arabic{equation}}
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\vspace{.7cm}
\no
{\bf 2. Elastic fields due to a thermal inclusion}
\vspace{.4cm}

Let a nonhomogeneous anisotropic linear thermoelastic body, 
occupying a three-dimen- sional region $ \, \Omega $, be subject to a dynamic
thermomechanical load. 
Let the displacement be described by a vector  
$ \, u_{i}= u_{i}({\bf x},t) $ and
the stress by 
$ \, s_{ij}= s_{ij}({\bf x},t). $ 
The temperature field is assumed to be known and is denoted by 
$ \, \Theta= \Theta({\bf x},t) $. 
If $ \, T \, $ denotes the absolute temperature, we have  
$ \ \Theta \equiv T - T_0 
$
where $ \, T_0 $ denotes a reference temperature. 

The equation of motion 
has the usual form
\renewcommand{\arraystretch}{1.5}
\bq 
\begin{array}{rll}
\rho \ddot u _{i} & = & s_{ik,k} + b_{i}   
\end{array}
\eq 
where $\rho = \rho({\bf x})$ and ${b_i}={b_i}({\bf x})$ denote the mass
density and body force vector fields. 

The following boundary condition is adjoined to (2.1)
\bq
\ba{rll }
s_{ij} n_j & = p_i \quad & {\rm on} \  \prt \Omega \\
%u_i & = U_i & {\rm on} \  \prt \Omega_u
\ea
\eq
where $ \, B_i$ is a prescribed function. 
The total strain is defined as
\be
e_{ij} \equiv {1 \over 2} \left(u_{i,j} + u_{j,i} \right)
\eq
and it is composed of three parts
\be
e_{ij} \equiv e^E_{ij} + e^P_{ij} + e^\Theta _{ij}
\eq
where $ \, e^E_{ij}$, $ \, e^P_{ij}$ and $ e^\Theta _{ij} \ $ denote the elastic and plastic
strain fields.
 
The stress $ \, s_{ij}$ is related to the strain $ \, e^E _{ij} $ 
and a temperature  $ \, \Theta $
by 
\be 
s_{ij} = C _{ijmn} e^E _{mn} - \gamma _{ij} \Theta  %\eqno (22)
\eq
where $ \, C_{ijmn} = C_{ijmn}({\bf x}) $ is the elastic tensor, and
\be
e^E_{ij} \equiv e_{ij} - e^P_{ij}
\eq
Moreover, $ \,\gamma _{ij}$ is related to
a stress-temperature tensor $ \, M_{ij}$ by 
$ \, \gamma _{ij} = - M_{ij}. 
$

>From (2.1), (2.4), (2.5) and (2.2) 
\be
\left( C_{ijmn} e_{mn} \right)_{,j} - \rho \ddot u _{i} = - f_i
\eq
where
\be
f_i = b_i - \left( C_{ijmn} e^P_{mn} + \gamma _{ij} \Theta \right)_{,j}
\eq
is a modified body force.

In further, we will assume that the medium is infinite 
$ \ \Omega = {\bf E}^3 $ and $ \ {\bf b} = {\bf 0} $. Moreover, the boundary condition is absent $ \ {\bf p} = {\bf 0} $ and the plastic deformation is absent $ \ {\bf e}^P = {\bf 0} $.


\renewcommand{\theequation}{3.\arabic{equation}}
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\vspace{.7cm}
\no
{\bf 3. Spherical thermal inclusion}
\vspace{.4cm}

We consider an infinite homogeneous thermoelastic body in which at the
instant $ \ t=0 \ $ certain spherical volume of radius $ \ a \ $ is
suddenly heated to a temperature $ \ \Theta^0 \ $ and the radius $ \ R \
$ of the heated volume expands linearly in time with a speed 
$ \ v > 0 \ $ 
\bq
R = a + v t
\eq
Thus the thermal field is given by the relation
\bq
\Theta \ = \ \Theta(r,t) \ = \ \Theta^0 \ H(a + vt - r) H(t) 
\eq
where$ \ \Theta^0 \ $ is a constant and $ \ H(t) \ $ denotes the Heaviside step function. 

The plastic deformations are assumed to be absent and in view of spherical 
symmetry
only one component of the displacement, the radial one in the spherical
coordinates  $ \ u_r \equiv u \ $ does not vanish 
and only one equation of motion is to be solved, cf. Nowacki (1975)
\bq
u_{,rr} + {2 \over r} \ u_{,r} + {2 \over r^2} \ u - \sigma^2 \ {\ddot u} = 
m \ \Theta_ {,r}  
\eq
where 
$$
\sigma^2 = {1 \over c_1^2} = {\rho \over \lambda + 2 \mu} \qquad
m = { \gamma \over  \lambda + 2 \mu} \qquad 
 \gamma = ( 3 \lambda + 2 \mu ) \alpha
$$
with $ \ \lambda $ and $ \mu \ $ denoting the Lam\'e constants and 
$ \ \alpha $ standing for the coefficient of linear thermal
expansion. The stress $ \ s \equiv s_{rr} \ $
is expressed by
\bq %3.4
s = 2 \mu \ u_{,r} + \lambda ( u_{,r} + {2 \over r} \ u ) \ - \ \gamma \ \Theta
\ \equiv \ 
2 \mu \ { 1 - \nu \over 1 - 2 \nu} \
 \left[  u_{,r} + { 2 \nu \over 1 -  \nu} \ { u \over r} \right] \ - \
  \gamma \ \Theta
\eq
where $ \ \nu \ $ denotes the Poisson ratio, and 
$ \lambda = \ds \ 2 \mu \ {\nu \over 1 - 2 \nu} $. \ 

If we choose the length and time units, $ \ \check x \ $ and 
$ \ \check t $, respectively, in such a way that 
$ \ {\check x}/{\check t} = c_1 $, then by 
letting
\bq
u = \Phi _{,r}
\eq
where $ \ \Phi \ $ stands for a displacement potential, and using (3.3) we obtain
$$
\Phi_{,rr} + {2 \over r} \ \Phi_{,r} - \ {\ddot \Phi} = \ \Theta 
$$
or
\bq
\left( r \Phi \right)_{,rr} - \ r {\ddot \Phi} = \ r \ \Theta 
\eq
To Eq. (3.6) the initial conditions are adjoined
\bq
\Phi_{t=0} \ = \ 0 \qquad {\dot \Phi}_{t=0} \ = \ 0 
\eq
The Laplace transform of Eq. (3.6) reads
\bq
\left( {\prt^2 \over \prt r^2} - p^2 \right) ( r  \ {\bar \Phi})  = 
\ r \ {\bar \Theta }
\eq
where 
$$
{\bar \Phi}(r,p) = \int_0 ^\infty {\rm e}^{- p t} \ \Phi(r,t) dt
$$
is the Laplace transform of $ \ \Phi(r,t)$.


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\vspace{.7cm}
\no
{\bf 4. Green function}
\vspace{.4cm}


We consider an auxiliary problem 
\bq
\left( {\prt^2 \over \prt r^2} - p^2 \right) ( r  \ {\bar \Phi}^*)  = 
\ r \ \delta (r - r_0) \equiv \ r_0 \ \delta (r - r_0)
\eq
The solution of this equation reads, cf. Nowacki (1975)
\bq
 r  \ {\bar \Phi}^* = \ - \ {2 r_0 \over \pi} \int_0 ^\infty
{\sin \alpha r_0 \ \sin \alpha r \over \alpha^2 + p^2 } \ d \alpha
\eq
or
$$
 r  \ {\bar \Phi}^* = { r_0 \over \pi} \ \left[ 
\int_0 ^\infty \
{\cos \alpha (r + r_0) \over \alpha^2 + p^2 } \ d \alpha \ - \
\int_0 ^\infty \
{\cos \alpha (r - r_0) \over \alpha^2 + p^2 } \ d \alpha \right]
$$
or 
\bq %4.3
{\bar \Phi}^* = {r_0 \over r} \ {1 \over 2 p} \left[
{\rm e}^{- p(r + r_0)} - {\rm e}^{ - p|r - r_0|} \right]
\eq
This is a unique solution of (4.1) for the infinite space.


%\newpage

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\setcounter{equation}{0}
\vspace{.7cm}
\no
{\bf 5. Solution of the problem}
\vspace{.4cm}

The appropriate displacement potential has the form
\bq
{\bar \Phi}(r,p) \ = \ \int_0 ^\infty \ 
    {\bar \Phi}^* (r,r_0,p) \ {\bar \Theta }(r_0,p) \ dr_0
\eq
where 
\bq
{\bar \Theta}(r,p) = \Theta^0 \ {1 \over p} \ 
\exp( \ - \ p \ \frac{r - a}{v}) 
\eq
is the Laplace transform of (3.2).
Hence
\bq
\ba{l}
{\bar \Phi}(r,p) \ = 
\ds
{\Theta^0 \over  r} \ 
{ v^2 \over 1 - v^2} \ 
\left\{ 
{2 v \over 1 - v^2} \left(
 - \ {1 \over p^4} \ \exp[- p (r - \frac{a}{v})] \ 
 + \ {1 \over p^4} \ \exp(- p \frac{r - a}{v}) 
                    \right) \  \right. \\ 
\hspace{9cm}  
  \left. \ds + \ r \ {1 \over p^3} \ \exp(- p \frac{r - a}{v})
\right\}
\ea
\eq
Applying the inverse Laplace transformation we get
\bq
\ba{l}
\Phi(r,t) \ = 
%\hspace{2cm}
\ds
{\Theta^0 \over  r} \ 
{ v^2 \over 1 - v^2} \ 
\left\{ 
{ v \over 1 - v^2} \ { 1 \over 3} \ 
\left[ 
- \ \left(t - \frac{r v - a}{v}\right)^3 \
   H\left(t - \frac{r v - a}{v} \right) \right. \right. \\
\hspace{7cm}
 \ +  \left. \left.
     \ds
 \  \left(t - \frac{r - a}{v}\right)^3 \ 
   H\left(t - \frac{r - a}{v}\right) \right]
\right.   \\
\hspace{7cm}
\left.
\ds 
 \ + \ r \ {1 \over 2} \ \left(t - {r - a \over v}\right)^2 \ 
H\left(t - { r - a \over v}\right)
\right\}
\ea
\eq
The displacements are determined by (3.5) and they are plotted in Figs. 1-4
for $ \ \Theta^0 = 1 \ $ and $ \ a = 0 $.
To obtain the radial stress $ \ s \ $,
the dimensionless Duhamel-Neumann relation (3.4) is used
 \bq 
s = \  u_{,r} \ + \ { 2 \nu \over 1 -  \nu} \ 
      { u \over r}  \ - \ \Theta
\eq
The temperature and stress units are selected as 
$ \ {\breve \Theta} = (\lambda + 2 \mu) \gamma^{-1} \ $ and
$ \ {\breve s} = \lambda + 2 \mu \ \equiv  \ 
2 \mu \ {1 - \nu \over 1 - 2 \nu} $, respectively.
The stress behaviour is shown in Figs. 5 and 6.



\centerbmp{8cm}{6cm}{c:/Andrzej/ryszard/peter/inkltem01.bmp}

Fig. 1. The radial displacement $ \ u \ $ at  $ \ r=1 \ $ and 
$ \ r=2 \ $ as a function of time  $ \ t \ $ for $ \ v={1 \over 2} \ $   \ (for $ \ \Theta^0 = 1 \ $, $ \ a = 0 \ $ and $ \ \nu = \frac{1}{3}$) 

\centerbmp{10cm}{6cm}{c:/Andrzej/ryszard/peter/inkltem02.bmp}


Fig. 2. The radial displacement $ \ u \ $ at $ \ r=1 \ $ and 
$ \ r=2 \ $ as a function of time $ \ t \ $ for $ \ v={3 \over 2} \ $ \ (for $ \ \Theta^0 = 1 \ $, $ \ a = 0 $ and $ \ \nu = \frac{1}{3}$) 



\newpage  

~

\vspace{-1cm}


\centerbmp{8cm}{5cm}{c:/Andrzej/ryszard/peter/inkltem03.bmp}


Fig. 3. The radial displacement $ \ u \ $ at $ \ t=1 \ $ and 
$ \ t=2 \ $ as a function of distance $ \ r \ $  for $ \ v={1 \over 2} \ $ \ (for $ \ \Theta^0 = 1 \ $, $ \ a = 0 $ and $ \ \nu = \frac{1}{3}$) 
 




\centerbmp{8cm}{5cm}{c:/Andrzej/ryszard/peter/inkltem04.bmp}

Fig. 4. The radial displacement $ \ u \ $ at  $ \ t=1 \ $ and 
$ \ t=2 \ $ as a function of distance $ \ r \ $  for $ \ v={3 \over 2} \ $ \ (for $ \ \Theta^0 = 1 \ $, $ \ a = 0 $ and $ \ \nu = \frac{1}{3}$) 


\centerbmp{7cm}{5cm}{c:/Andrzej/ryszard/peter/inkltem11.bmp}

Fig. 5. 
The radial stress $ \ s \ $ at $ \ t=1 \ $ and 
$ \ t=4 \ $ as a function of distance $ \ r \ $  for $ \ v={1 \over 2} \ $ \ (for $ \ \Theta^0 = 1 \ $, $ \ a = 0 $ and $ \ \nu = \frac{1}{3}$) 





\newpage

~

\vspace{-1cm}


\vspace{1.2cm}

\centerbmp{7cm}{5cm}{c:/Andrzej/ryszard/peter/inkltem2.bmp}

Fig. 6. 
The radial stress $ \ s \ $ at $ \ t=1 \ $ and
$ \ t=4 \ $ as a function  of distance $ \ r \ $  for $ \ v={3 \over 2} \ $ \ (for $ \ \Theta^0 = 1 \ $, $ \ a = 0 $ and $ \ \nu = \frac{1}{3}$) 




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\vspace{.7cm}
\no
{\bf 6. Discussion}
\vspace{.4cm}

\no
We observe  that the potential $\ \Phi $ from (5.4) generates the radial displacement  $ \ u = u(r,t) \ $ and the radial stress  $ \ s = s(r,t) \ $ which are finite for every 
$ \ r > a, t > 0 $. 

\no
The two wave fronts of the solution are clearly seen: one corresponding to
the temperature inclusion
boundary moving with the velocity $ \ v \ $ and the second moving with a
unit velocity and corresponding to the longitudinal elastic waves.


\no
The displacement $ \ u  \ $ is a continuous function for every
$ \ r > a, t > 0 $; 
while the stress $ \ s \ $ exhibits a finite jump on the $r$-axis corresponding to the spherical wave front 
 $ \ r = t v $.
\vspace{.4cm}

\no
{\bf Acknowledgement}: 
The author thanks Dr. Andrzej Ga{\char32l}ka for preparation of figures.


\vspace{.4cm}

\noindent
{\bf References}

\small{
\begin{enumerate}

%1
\item
\sc W. Piechocki {\rm and} J. Ignaczak, \rm Certain problems of dynamical distortion
in thermoelasticity, {\it Arch. Mech. Stos.} {\bf 12}, 259-278 (1960)

\item 
{\sc N.I. Muskhelishvili}, On thermal stresses in plane problem of theory of elasticity, Izv. Elektrotekhn. Inst., No 13, 23-37  (1916)
   
\item
{\sc V.M. Maysel}, The temperature problems of the theory of elasticity, Izd. UAN, Kiev 1951

\item %2
{\sc J.R. Willis}, \rm  Dislocations and inclusions, 
{\it J. Mech. Phys. Solids} {\bf 13}, 377-395 (1965)


%{3.} 
\item 
{\sc A.M. Kosevich}, {\it Equation of motion of a dislocation}, Zh. E. T. F. {\bf 43}, 637-
648 (1962)

%{4.} 
\item 
{\sc A.M. Kosevich}, {\it Crystal dislocations and the theory of
elasticity}, in: F.R.N.
Nabarro (ed.), Dislocation in solids, Vol. 1, North Holland, Amsterdam 1979


%{5.} 
\item 
{\sc A.M. Kosevich}, {\it Fizicheskaya mekhanika realnykh krystallov}, 
$ \ll$ Naukova Dumka $ \gg $, Kiyev 1981

%6
\item 
{\sc B. Wang, Q. Sun} and {\sc Z. Xiao}, On the dynamic growth of a
spherical inclusion with dilatational transformation strain in infinite
elastic domain, {\it J. Appl. Mech.} {\bf 66}, 879-884 (1999)


%{8.}
\item 
{\sc L. D. Landau} and {\sc E. M. Lifshitz}, {\it Teoriya uprugosti}, Izd. $ \ll $ Nauka $\gg $ Glavn.
Red. Fiz.-Mat. Literatury, 3rd edition, Moskva 1965; also English
translation: {\it Theory of elasticity}, Vol.7 of { Course of Theoretical Physics},
Pergamon Press, Oxford 1970

%{7} 
\item 
{\sc J. Ignaczak} and {\sc C.R.A. Rao}, {\it Stress characterization of elastodynamics
with continuosly distributed defects}, J. Elasticity {\bf 30},
219-250 (1993)


%9
\item 
{\sc W. Nowacki}, {\it Thermoelasticity}, ed. by P. Francis and R. B. Hetnarski,
Noordhoff International Publ., Leyden and PWN, Warszawa 1975


\end{enumerate}
\end{document}

We introduce the new system of units of length, time, temperature and stress
$$
{\breve x} = {\kappa \over c_1}                  \qquad
{\breve t} = {\kappa \over c^2_1}                \qquad
{\breve \Theta} = {\lambda + 2 \mu \over \gamma} \qquad
{\breve s} = 2 \mu \ {1 - \nu \over 1 - 2 \nu}
$$
and $ \ kappa \ $ denotes the diffusivity of the considered body. The unit
of velocity in this system is the velocity of the longititudinal wave 
$ \ c_1 $. 



\item R. Wojnar, {\it Distortion equation of motion in linear incompatible
elastodynamics}, Arch. Mech. {\bf 51}, 3-13 (1999)

%{10.} 
\item J. Ignaczak, {\it On the stress equation of motion in
the linear thermoelasticity}, Arch. Mech.Stos. ${\bf 15}, 691-695, (1963)$

Similarly, the boundary condition in these variables admits the form
\be
\ba{rll}
C_{ijmn} e_{mn} n_j & = F_i \quad & {\rm on} \ \prt \Omega \\ 
%u_i & = U_i & {\rm on} \  \prt \Omega_u
\ea
\eq
where
$ \ 
F_i = B_i + \left( C_{ijmn} e^P_{mn} + \gamma _{ij} \Theta \right) n_j
\ $
is a modified boundary condition.

