TRANSIENT ELASTIC FIELD DUE TO ASPHERICAL DYNAMIC INCLUSION

WITH AN ARBITRARY TIME PROFILEByY. MIKATA

(Department of Mechanical Engineering and Mechanics, Old DominionUniversity, Norfolk, Virginia 23529-0247, USA)

[Received 24 September 1991. Revise 2 January 1992]

SUMMARYThe elastodynamic field of an infinite isotropic elastic medium is investigated when a

spherical portion of the medium experiences a dynamic phase transformation. The phasetransformation is modelled as a spatially uniform eigenstrain over a sphere multipliedby an arbitrary function of time. The objective is to determine the elastic fields insideand outside the sphere when the dynamic eigenstrain is given. Using a Green's-functionapproach, an exact closed-form solution is obtained for this problem. As special cases,a time-harmonic solution as well as an elastodynamic response of the medium due toa sudden phase transformation are discussed.

1. Introduction

ELASTODYNAMIC response of a material is investigated when a spherical portionof the material undergoes a dynamic phase transformation. The time response ofthe phase transformation is arbitrary. The present problem was motivated bythe desire to understand the dynamic response of transformation-toughenedceramics. These have attracted considerable attention in recent years (1 to 4).Toughening in ceramics occurs when a zirconia particle undergoes a stress-induced phase transformation which accompanies a volumetric expansion. Mostof the studies on this phenomenon have been for quasi-static loadings (forexample, (5 to 7). In a previous paper (8), in an attempt to analyse the dynamicbehaviour of transformation-toughened ceramics, a time-harmonic elastodynamicfield due to a dynamically transforming spherical inclusion was investigated.In a related paper (9), interaction between a harmonic stress wave and adynamically transforming spherical inhomogeneity has been treated. The presentpaper is an extension of (8) in the sense that the elastodynamic response due to anarbitrary temporal progression of the phase transformation is investigatedinstead of the time-harmonic behaviour of the material.

The dynamic phase transformation is modelled as a spatially uniformeigenstrain (or transformation strain) over a sphere multiplied by an arbitraryfunction of time. Our objective is to determine the elastic fields inside andoutside the sphere within an infinite isotropic elastic medium when the dynamiceigenstrain is given. An exact closed-form solution is obtained for this problem.

[Q. Jl Mtch. appL M»tlt, Vol 46, PL 2, 1993] © Oxford Unhersity Press 1993

276 Y. MIKATA

Our solution is a generalization of Willis's solution (10), where only thespherically symmetric deformation was considered.

2. Statement of the problem

Consider a dynamic inclusion Q with an eigenstrain e* which is embeddedin an infinite isotropic elastic medium. Here we use the term 'inclusion' todenote a subset of a matrix, which has an eigenstrain (or transformation strain),but has the same elastic properties as the matrix. Our objective is to determinethe elastic fields inside and outside the inclusion Q. The following eigenstrainis considered:

0, x e R 3 - Q ,

where ef, are constant eigenstrain components over Q, and f(t) is an arbitraryfunction of time t. By using the Fourier-transform method (see (11, (8.11)),the displacement field is given by

Too Too

*, t) = - CmMet,(x', t')Gim,n(x -x',t- t')dx'dt',J — oo •* — oo

(2.2)

where Einstein's summation convention is used, a comma implies differentiation,and

Anp \_ dxfix^ x P dx,dxm ax

r X < <X (2.3)

tx P

0 otherwise,

C i C C \ I 1 C C / ^ A \

a= /ZCLJ:, p= £, x = \x\ = (xkxk)*. (2.5)VP

Here Cmtkl are the elasticity tensor components of the medium, \i and X areLame constants, and Gim(x,t) is the three-dimensional elastodynamic Green'sfunction. The above form (2.3) can be derived from (11, equation (9.34)). It isnoted here that (2.3) is similar to the equation (34) given by Willis (10). Thereare, however, a couple of minor misprints in the equation (34). It may be alsonoted that there are several minor misprints in equation (38) in (10).

TRANSIENT ELASTIC FIELD 277

3. Spherical inclusion

Substituting (2.1) into (2.2), and applying Gauss's theorem to the resultingexpression, we have

Ui(x,t) = CnmUetl f °° f{f)df I nn(x')Gim(x - x',t - t')dS, (3.1)J -oo Jan

where nn is the nth component of the outward normal to the inclusion surfacedQ. When the inclusion Q is a sphere of radius a, we have

By setting v

where

Substituting

where

p1

p1

p1

n (x') - —a

= t — t', (3.1) is rewritten as

* f°°J - 0 0

Eimn(*,v)=\ « . ( x ' ) G J xJan

(2.3) into (3.4), we have

47rpL a

,! _ f 32 1

Jr•L(x,v)=! nn(x')8(\x-x'\-Pv

)Elmn(x,v)dv,

-\',v)dS.

(x, - x',)(xM - x^;1 | x - x ' | 3

d2

(3.2)

(3.3)

(3.4)

(3.5)

(3.6.1)

]-dS, (3.6.2)

(3.6.3)

and

r = /4nB, y4 = {x'Mx'l = a}, B = {x'\pV < |x - x'| < at;}. (3.7)

The evaluation of the integrals Flimn, F ^ , and Ff^, is the major result in this

paper, and is carried out in the following.Let us first note that the domain of integration r ( c dil) is, in general, a strip

on the surface of the sphere, which has axial symmetry with respect to the lineL (see Fig. 1), given by

L.A=*i = A_ ( 3 8 )

X, X2 X3

278 Y. MIKATA

FIG. 1. Domain of the integration r, and the line of symmetry L

From the above observation, it is natural to select a new coordinate systemx'i[i = 1,2,3) such that the i'3-axis coincides with the line of symmetry L (seeFig. 2). The relation between the two coordinate systems is

A = (3.9)

where

9 = (9tj) =

xy

x2x3

xy

_yX

y

y

0

X

*2

X

X

(3.10)

and x = *Jx\ + x\ + x\, y = yjx\ + x\. The matrix g is orthogonal. Thuswe have

| x - x ' | = |0(x-* ' ) l = | x - x ' | . (3.11)

Let us first concentrate on F,1,̂ ,. Using (3.6.1), we have

FLK v) = 3LxiXmGsm - axfil, - axmG5

mi + a2G^ - 5imG3n, (3.12)

TRANSIENT ELASTIC FIELD 279

FIG. 2. Original coordinates x\ and new coordinates )t\

where

r | x - x'|;dS, *m* Jrlx-x'| (3.13)

-5 _ f ninmnn>imn — ' 77

Jr lx-x ' |:dS, Gi =

Let us introduce the spherical coordinates (r, 6, <p) in the new coordinate systemj?J. The relationship between the coordinates $.\ and the spherical coordinates(r,6,4>) is given by

£\ = rs\n6cos<p, X'2 = r sin 0 sin 4>, i'3 = rcos0. (3.14)

By using the spherical coordinates, (3.13) can be rewritten as

f .y^.5dS = aJ r | X — X | |X — X

(3.15)

280 Y. MIKATA

FIG. 3. Polar angles 0, and #2. a nd t n e domain of the integration F

where ^i and 92 are the polar angles corresponding to the boundaries of thestrip F (see Fig. 3). The normal vector A is given by

n = (sin 9 cos <f>, sin 6 sin (j>, cos 6).

Using (3.16), we obtain, after some algebraic manipulations,

(3.16)

o2u

cos 0,

= 7i[sin20<5M + (2cos2 0 - sin2 9)8p

Sq3Srp

(3.17)

+ (5 cos3 9 - 3 cos 6)Sp35q35r3].

Carrying out the integrals (3.15) using (3.17), and substituting the resultingexpressions into (3.12), and setting

z = |x - x'| = yjx2 + a2 - laxcosd, z} = Jx2 + a2 - 2axcos9j, (3.18)

we finally obtain

FL(M) = [£!«W];i, (3-19)where

l ^ ^ ( ) ^ \ (3.20a)

TRANSIENT ELASTIC FIELD 281

H^z) = -±-\--{x2 - a2)2(x2 + 5a2) - -(x4 + 6(ax)2 - 15a4)8x*|_ z z

- 5z3 ,

3]J

9z(x2 + 5a2)

)3 +

8x

-z(x2 +9a2) z3 .

(3.20b)

It should be noted here that z{ and z2 depend on y through 0, and 02. Similarly,we obtain

{ f W ^ ^ - " a (3,,)0

where

r . : | x -x ' | = «o, r , : | x - x ' | = /?», (3.23)

and

f

H A a) = — [/(ay){x4 - 2x2(7a2 - (ay)2) + 5(a2 - (av)2)2} + 16a2x4],8x4 '

H5(a) = -^[/(ay){x4 + 4(ax)2 - (a2 - (au)2)2} - 8a2x4],

8x4

/(z) = x2 + a2 - z2.

(3.24)

282 Y. MIKATA

We now turn to the determination of the boundary of the strip F. We havetwo cases: (I) the observation point x is inside the inclusion ft, and (II) x isoutside ft. Six wave-front diagrams are shown for each of these two cases, inFigs 4 and 5, respectively, to determine the strip F. In these figures, theobservation point x is located on the positive i'3-axis without loss of generality.

(i)

(ii) i | a - J T | < i> <min{^(a +x).jj\a - x

FIG. 4. Wave-front diagrams for the observation point x inside the inclusion

TRANSIENT ELASTIC FIELD 283

(iii) j | \a - x | < v 4. - (a + x)

(iii)' ^

FIG. 4. Continued

284 Y. MIKATA

- *3

fiv

\\ ~a

7

\

i

a v ^ \

(iv) m a x { i ( a + j r ) , ^\a -x\}< v < I (

• • Jti

(v) v > Ua + JC)

FIG. 4. Continued

TRANSIENT ELASTIC FIELD 285

(ii) I \a - x \ 4. v < m i n { i (a + x ) , ± \a - x \ )

FIG. 5. Wave-front diagrams for the observation point x outside the inclusion

286 Y. MIKATA

FIG. 5. Continued

TRANSIENT ELASTIC FIELD 287

, i | a -x\}< V < ± ( a + x)

*• x;

FIG. 5. Continued

288 Y. MIKATA

Using(3.19), (3.21) and (3.22) with the help of Figs 4and 5, we finally arrive at

(i) B ^ I | a - x | , Ffnn(x,v) = Fln(x,v) = Ffmn(x,v) = O; (3.25.1)a

( i i ) ! | f l - x | ^ » ^ m i n j ( a + x ) |

(iii) I | f l -

= Efm»(a), Ffmn(x,v) = 0; (3.25.2)

(3.25.3)

Ffmn(x,v) = Ffmn(x,v) = O; (3.25.4)

(iv) Ak U ] Uk + ),ax\]^v^Ua + x\ FL(x,») [ £ L , ( Z ) ] J B ,

FL(x,«) = 0, /rL,(x,o) = £L,(/J); (3.25.5)

(v) y ^ (a + x ) , F/mn(x,t;) = FL,(x )y) = FL I(x,U) = O. (3.25.6)

P

Substituting (3.25) into (3.5), we obtain

(i) o < I | a - x | , Elnu,(x,v) = 0; (3.26.1)a

(ii) -\a - x\ ̂ v < min -^-(a + x),-|a - x|a [a p J

7^£L(^)]r:-^ + -EL(«)1; (3-26.2)

(iii) - | a - x | ^ D < - ( a + x),P a

£<™(x,y) = -t-lvlEU*)!?. + -ELXOL) +l-EUp)\; (3.26.3)

(iii)' i(fl + x)<o^i|fl-x|, UM)=-^[£!jz)]f t ; (3.26.4)a p 4

TRANSIENT ELASTIC FIELD 289

(iv) max \l(a + x),-\a-(.« P

1

~P

(v) v>-(P

4np\

c), Eimn(x,v) = <

(3.26.5)

(3.26.6)

Equations (3.3) and (3.26) represent the displacement field due to the sphericaldynamic inclusion Q with a spatially uniform but arbitrary time-profileeigenstrain e*,(x, t) given by (2.1).

By substituting (3.20) and (3.24) into (3.26), we have, as a simplified form,

(i) !><I| f l-x| , Elmn(x,v) = 0;a

(ii) -\a-x\

(iii) - | a -P

16px |_a

1

a

1

\ - Yimn(av) -\a-x\

Jlmn >

(iii)' -(p

(iv) maxi-(a + x),-\a - x\ \ < v < -(a + x),

-Y, (/P

Eimn{x,v) =

1

4

a3.;

2px4 | a - x

1

(v) v^-(aP

where

16px4|

0, Elmn(x,v) = 0,

imnX X X X

g(z) = z4 - 2z2(x2 + 3a2) + (x2 - a2)(x2 + 3a2),

h(z) = -5z 4 + 6z2(x2 + 5a2) - {x4 + 6(ax)2 - 15a4}.

(3.27.1)

(3.27.2)

(3.27.3)

(3.27.5)

(3.27.6)

(3.28)

290 Y. MIKATA

4. Dilatational eigenstrain

Let us consider the following dilatational eigenstrain:

et, = e*5kl, (4.1)

where e* signifies the magnitude of dilatation. Substituting (2.4) and (4.1)into (3.3), we have

J — cuf(x,t) = (2fi + U)e* | f{t - v)Eimm(x, v)dv. (4.2)

Using (3.28), we have

Ytmm(z) = 4xx,/(z), Clmm = 0. (4.3)

Substituting (3.27) along with (4.3) into (4.2), we obtain

u,(x, t) = ^ - '- '- f(t - v)l(otv)dv. (4.4)

Equation (4.4) represents the displacement field due to the spherical inclusionQ with a dilatational eigenstrain e?,(x, t) given by (2.1) and (4.1). It should bementioned here that the radial component of the displacement (4.4) if rewrittenin spherical coordinates coincides with the result given by Willis (see (10, (41)).

5. Time-harmonic solution

As a special case of the solution (4.4), let us consider a time-harmoniceigenstrain. Thus we set

= «-"», (5.1)

where co denotes an angular frequency. Substituting (5.1) and (3.24.6) into (4.4),and performing the integration, we obtain

ui(x,t) = ui(x)e-ia", (5.2)

where

ui(x) = R^j1(kx)hi{ka), xeQ,

(5.3)

u,(x) = R—ji (kafh^kx), xeR — £2,x

with

1 + v , coR = i- fcaV, k = - . (5.4)

1 — v aHere v is the Poisson ratio, and jx and /ij are spherical Bessel functions. Notethat (5.2) coincides with the result obtained by Mikata and Nemat-Nasser (see(9,(31)).

TRANSIENT ELASTIC FIELD 291

6. Suddenly transformed inclusion

Let us now consider a suddenly transformed spherical inclusion with anarbitrary eigenstrain. Thus we set

= H(t), (6.1)

where H{t) is the Heaviside step function. Substituting (3.26) and (6.1) into(3.3), and after fairly lengthy but straightforward calculations, we finally arrive at

" ( (x ,0 = Cm i u AV i m , (x,r) , (6.2)where

(i) t^-\a-x\, J imn(x,t) = 0; (6.3.1)a

(ii) -\a - x\ < t < min\-(a + x),-\a - x\ \,a (a P J

(iii) -\a-x\^tK-(P a

1 - 2v

(iii)' -(a + x) ^ t ^-\a - x\,a p

(iv) max \-(a + x),- |a - x\ \ ^ t ^ -{a + x),L P ) P

(6.3.2)

(6.3.3)

16(2/1

] ^ } C ( B U I ] ; (6.3.4)

(6.3.5)

(v) t>-(a + x), Jinu,(x,t) = JlmB(x,^(a + x)j, (6.3.6)

292 Y. MIKATA

and

Mlmn{z) = Pln,(z) + ItfC^, P^iz) = S(z)Aba + T(z)Blmm \

S(z) = z3(x2 + 3a2) + z(x2 - a2)(x2 + 3a2),

T(z) = - z 5 + 2z3(x2 + 5a2) - z{x4 + 6(ax)2 - 15a4},

Uz)= - - + z(x2+a2), R(z) = 4a3z2,

Q(x)= -(l-2v)[/(z)]»||fl_-^l = 4a3(a-

7. Willis's solution

As a special case of the solution (6.2), let us consider the dilatational eigenstrainef,, given by (4.1). Substituting (2.4) and (4.1) into (6.2), we have

U((x,t) = (2/i + 3A)e*JiBM(x)t). (7.1)

Using (3.28) and (6.4), we have

PimmW = 4xx,L(z), C ^ = 0. (7.2)

Substituting (6.3) along with (7.2) into (7.1), we obtain

(i) £ < - | a - x | , u,(x,t) = 0, (7.3.1)a

(ii) - | a - x | s£ts : - (a + x),a a

4(2,

whose radial component coincides with Willis's solution (see (10, (44)). Itshould be pointed out here that the solution (7.3) can also be readily obtainedfrom (4.4) by setting f(t) = H(t).

8. Eshelby tensor

Equation (6.2) represents the transient displacement field due to a suddenlytransformed spherical inclusion. Thus for a given point in space, after a

TRANSIENT ELASTIC FIELD 293

sufficiently long time, the elastic field at that point is expected to become astatic elastic field due to the transformed spherical inclusion. This static elasticfield is given by (6.2) and (6.3.6). In this section, let us consider only the elasticfield inside the inclusion. Substituting (6.4) into (6.3.6), we have

,t) = ^ - l(v) t>Ua + x), Jbm(x,t) = ^- lOxAm-T-^-^- l , *<«• (8-1)P 30L 1-v J

By using (6.2) and (8.1), the strain field is obtained as

(v) t>-(a + x), e,j(x, t) = StJkletb x < a, (8.2)P

where

•Y-J +-J \c2 \ 5 X J 3x, / ( 8 3 )

5v - 1 „ „ 4 - 5v

1 5 ( 1 - v ) " " 1 5 ( 1 - v ) v " J' " J"

Thus we have recovered the static Eshelby tensor SIJkl (Eshelby (12), Mura (11)).

9. Numerical results and discussion

Numerical results are given for a suddenly transformed spherical inclusion,which was treated in section 6. Let us consider three cases with the followingeigenstrains

easel : e*, = e*2 = e*2 = e*, e j = 0 ( < / ; ) ,

case 2: e*x = l-5e*, e*2 = e*, £33 = 0-5e*, e,* = 0 (1 ^ _/),

case 3: ef3 = e*, efj = 0 (otherwise),

where e* is a reference strain magnitude, which is arbitrary. In all of the abovecases, Poisson's ratio v is taken as ^. In the subsequent figures, ux and uz standfor Uj and u3, respectively. In Fig. 6, the non-dimensional displacement ujae*is shown for case 1 as a function of non-dimensional time at/a at (a)x = (0-5a,0,0), (b) x = (a,0,0) and (c) x = (15a,0,0). It is seen from Fig. 6that the largest displacement is experienced at the boundary of the sphericalinclusion. In Figs 7 and 8, the non-dimensional displacements ujae* and ujae*are shown for cases 2 and 3, respectively, as a function of at/a at (a)x = (005a,0,0), (b) x = (0-5a,0,0), (c) x = (a,0,0) and (d) x = (1-5a,0,0). Itis seen from Figs 7 and 8 that the largest displacement is also experienced atthe boundary of the spherical inclusion. It is also noticed from Figs 7(a) and8(a) that there exist two peaks for the respective non-dimensional displacementat at/a = 1 and 2. The first peak at at/a = 1 corresponds to the arrival of the

294

i-u

0-8

% °"6

a" 0-4

0-2

0

/ ^ N x = (0-5a,

- / \

- /

/ ,

0,0) ;

-

-

i . . . .

Y. MIKATA

10

0 0-5 10 1-5 20 2-5 30

Time at/a

OL.0 0-5 10 1-5 20 2-5 3 0

Time at/a

10

0-8

, 0-6VI

o3 ' 0-4

0-2

• • • • | • • • •

eti ='& = eh = e*

x = (1-5a, 0,0)

01 , . . .0 0-5 10 1-5 2 0 2-5 30

Time at/a

FIG. 6. Non-dimensional displacement ujae* for case 1 at (a) x = (0-5a,0,0),(b) x = (a,0,0) and (c) x = (15a,0,0)

longitudinal wave, which is generated at the interface between the sphericalinclusion and the surrounding medium. Similarly, the second peak at at/a = 2corresponds to the arrival of the shear wave, which is also generated at theinterface. This type of behaviour is typical for the region sufficiently close tothe origin. Amplitudes of these pulses stay finite even at a point very close tothe origin. But the width of the pulse becomes thinner and thinner as theobservation point x approaches the origin. Therefore, they behave almost likea Dirac delta function in time at that point. At the origin, however, the

10

0-8

0-6

0-4

0-2

0

1-4

1-2

10

0-8

0-6

0-4

0-2

00

I I I I I I • • ' ' I • • • ' I • • • ' I

gf, = 15g*, g£ = e*.

TRANSIENT ELASTIC FIELD

1-4

1-2

10

295

ef3 = 0-5g*

x = (005a, 0, 0)

A0 0-5 10 1-5 20 2-5 30

Time atla

= l= 0

(a.

•5g*, gf; = e*. :•5g* :

0,0) -

2 3 4

Time at la

1-4

1-2

10

0-8

0-6

0-4

0-2

00

= l-5g*, gf, = e*= 0-5g*

x = (0-5a, 0, 0)

gf3 = 0-5g*

x = (1-5a, 0,0)

FIG. 7. Non-dimensional displacement ujae* for case 2 at (a) x = (0O5a,0,0),(b) x = (0-5a,0,0), (c) x = (a,0,0) and (d) x = (15a,0,0)

displacement stays always zero. This singular behaviour of the displacement nearthe origin produces large stresses however small the eigenstrain, which mightlead to a possible rupture of the material near the origin. This fact was pointedout first by Willis (10) for a spherically symmetric eigenstrain. Because of thespherical symmetry, however, Willis found only one such peak of the displace-ment corresponding to the arrival of the longitudinal wave. It is also interestingto note that for case 2, the first peak is larger than the second peak, and forcase 3, the second peak is larger than the first peak.

296 Y. MIKATA

0-2 -

0

u-y

0-7

# 0-5

3" 0-3

0 1

- 0 1

. . . . 1 . . . . 1 . . . . 1 .

eti = e*

x = (0-5a,

J

1 ' " • • • ' ' • .

-

0,0) --

-

-

. . . i . . . . i . . . .

0 0-5 10 1-5 2 0 2-5 30

Time at la Time alia

0-9

-0-11 i . . . . i0 1 2 3 4 5

Time alia

FIG. 8. Non-dimensional displacement ujae* for case 3 at (a) x = (0O5a,0,0),(b) x = (O5a,0,0), (c) x = (a,0,0) and (d) x = (15a, 0,0)

R E F E R E N C E S

1. I-WEI CHEN and P. E. REYES MOREL, J. Am. ceram. Soc. 69 (1986) 181-189.2. R. C. GARVTE, R. H. HANNINK and R. T. PASCOE, Nature 258 (1975) 703-704.3. F. F. LANGE, J. Mater. Sci. 17 (1982) 225-263.4. D. B. MARSHALL, J. Am. ceram. Soc. 65 (1986) 173-180.5. B. BUDIANSKY, J. W. HUTCHINSON and J. C. LAMBROPOULOS, Int. J. Sol. Sir. 19 (1983)

337-355.6. J. C. LAMBROPOULOS, J. Am. ceram. Soc. 69 (1986) 218-222.

T R A N S I E N T ELASTIC F I E L D 297

7. R. M. MCMEEKING and A. G. EVANS, ibid. 65 (1982) 242-246.8. Y. MIKATA and S. NEMAT-NASSER, J. appl. Mech. 57 (1990) 845-849.9. and , J. appl. Phys. 70 (1991) 2071-2078.

10. J. R. WILLIS, J. Mech. Phys. Solids, 13 (1965) 377-395.11. T. MURA, Micromechanics of Defects in Solids (Martinus Nijhoff, The Hague 1987).12. J. D. ESHELBY, Proc. R. Soc. A252 (1957) 561-569.