## Abstract

The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. *J. Mech. Phys. Solids* (doi:10.1016/j.jmps.2016.02.025)) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.

## 1. Introduction

One of the seminal solutions of twentieth century solid mechanics is the Eshelby ellipsoidal inclusion [1] with transformation strain, which has the constant constrained strain property in the interior domain (Eshelby property); it allows for the solution of expanding inhomogeneities with transformation strain through the Eshelby tensor [1,2] by the application of Eshelby's equivalent inclusion method [1,2]. The applications have been widespread throughout material science with extensive application to micro- and macro-mechanics including quantum dots [3,4].

The dynamic generalization of the Eshelby inclusion is the dynamically self-similarly expanding inclusion from the zero dimension (subsonically) with eigenstrain spatially and temporally constant in the interior domain of the expanding inclusion. It constitutes the evolution of the phase change singularity (discontinuity in the strain). In Ni & Markenscoff [5], the solution was obtained by the Radon transform, in which it was explicitly shown that it exhibits the Eshelby property of constant strain (and stress) in the interior domain of the expanding inclusion, and that the particle velocity is zero in the interior domain (lacuna), a property pointed out by Burridge & Willis [6]. This was also argued by Markenscoff [7] from dimensional analysis and analytic arguments alone. As shown by Ni & Markenscoff [8], for self-similar expanding motion, the governing system of partial differential equations (in the variables ** z**=

**/**

*x**t*) becomes elliptic in the region |

**|<**

*z**b*(where

*b*is the shear wave speed), and for subsonic expanding motion the inclusion lies within this region of ellipticity. The solution for the displacement gradient leads to the dynamic Eshelby tensor, which is exhibited here, and, which, in turn, allows for the solution of a dynamically expanding ellipsoidal inhomogeneity where the material undergoes chemical reaction and phase change (i.e. as in [9,10]).

The dynamic Eshelby tensors obtained by Mikata & Nemat-Nasser [11] and Michelitsch et al. [12] deal with time-dependent eigenstrain in an ellipsoidal inclusion fixed in space. This is a different physical problem and is unrelated to the one of the self-similarly expanding inclusion treated here. In our opinion, though, the name ‘Eshelby tensor’ may be naturally given when the constant stress Eshelby property holds, and in dynamics this is the case here for the self-similarly expanding ellipsoidal inclusion.

In this work, the system of partial differential equations governing the self-similarly expanding ellipsoidal inclusion is solved by adapting the method of Burridge & Willis [6] for an elliptic crack, which is based on the introduction of the operator

The applications of dynamically expanding inclusions with transformation strain may include phenomena such as inducement of martensitic transformations and chemical phase transition under dynamically applied loading [13]. Yang et al. [13] used the static Eshelby tensor in their modelling, but the above-mentioned fields with inertia are more appropriate. Expanding regions of transformation strain have application to geophysics and seismology, where the transformation strain concept is the fundamental basis for the concept of seismic moment, in particular for the modelling of deep earthquakes [14,15]. The driving force on the expanding phase ellipsoidal boundary can be obtained from the solution for the expanding ellipsoid, as it was obtained by Markenscoff & Ni [16,17] for a spherical expanding inclusion. It constitutes the mechanical rate of energy (per unit area) required for this motion, and would be provided by some other source of energy (i.e. thermal and chemical). The importance of the self-similar motion lies in the fact that it grasps the early response of the system [18].

## 2. The field solutions

We consider an ellipsoidal Eshelby inclusion (with transformation strain spatially and temporally uniform) centred at the origin of a Cartesian coordinate system (*x*_{1},*x*_{2},*x*_{3}), with zero dimension at time *t*=0 and expanding in a self-similar manner in an infinite linear elastic medium. At time *t*, the inclusion occupies the ellipsoid *t*/*s*_{1}, *t*/*s*_{2} and *t*/*s*_{3}, i.e. with axis expansion velocities 1/*s*_{1}, 1/*s*_{2} and 1/*s*_{3}, assumed to be subsonic. The uniform transformation strain is non-zero in the interior of the expanding ellipsoid, which is expressed as
*H* denotes the Heaviside step function, and, unless stated otherwise, the summation convention is used throughout with the Latin indices in the range 1,2,3. The governing equations for this problem are [5]

By using the Radon transform, Ni & Markenscoff [5] obtained the solution of the governing equation (2.2). Here we will give an alternative way to obtain the solution. As discussed by Burridge & Willis [6], the self-similarly expanding ellipsoidal phase change is analogous to the self-similarly expanding elliptic crack. We follow and adapt the method of Burridge & Willis [6] to obtain the field solution for the three-dimensional problem of the self-similarly expanding ellipsoidal inclusion. Define the operator *M* as
*M* has the following property:
** y**=(

*s*

_{1}

*x*

_{1},

*s*

_{2}

*x*

_{2},

*s*

_{3}

*x*

_{3}) and

*y*=|

**|, and we have for**

*y**t*>0

*R*

^{3}

*E*(

**,**

*y**t*) for

**∈**

*y**R*

^{3}given as (e.g. [19,20])

By applying the operator *M*^{2} to both sides of equation (2.2) and using equation (2.7), we have

The right-hand side of the above equation is expanded in plane wave functions [21] according to
** ξ** is a dimensionless vector and

*Ω*is the unit sphere, so that

*K*

_{j}(

**)=**

*ξ**C*

_{jklm}

*ϵ*

^{*}

_{lm}

*ξ*

_{k}. The problem (2.17) with (2.18) and (2.21) then reduces to solving the matrix

**for each plane wave component separately, which satisfies**

*A***, the solution**

*A***of (2.17), (2.18) and (2.21) is expressed as**

*u*As in Burridge & Willis [6], analogously to [22], ** A** is obtained as the contour integral

*f*is an analytic function of the complex variable

*γ*in a neighbourhood of the real axis and

*Γ*is a contour lying in the domain of regularity of

*f*(

*γt*+

**⋅**

*ξ***) and encircling all the poles of**

*x*

*L*^{−1}and

*M*

^{−2}for subsonic motion. We note that

*γ*has the dimension of velocity and the function

*f*(

*γt*+

**⋅**

*ξ***) has the dimension of length.**

*x*By Cauchy's theorem, the contour integral in equation (2.26) is evaluated as the sum of the residues at the poles of *L*^{−1} and *M*^{−2}, which are all real. In fact, in an isotropic medium, *L*^{−1}(*γ*,** ξ**) is expressed as

**|=1, we have**

*ξ**L*

^{−1}are

*γ*=±

*a*,±

*b*.

For the operator *M*, defined in equation (2.6), we have
*γ*=±*p*_{ξ} with *f*(*γt*+** x**⋅

**) on the real axis contributes to the contour integral in equation (2.26). As a boundary value on the real axis of an analytic function, we can choose the homogeneous function of order 1 [6],**

*ξ*From equations (2.26) and (2.32), we have the contour integral
*γ*=±*a*,±*b* of *L*^{−1}, and *A*^{M}_{γ=±pξ}, the contributions of the residues at the double poles *γ*=±*p*_{ξ} of *M*^{−2}. With the aid of equations (2.27)–(2.29), we have the evaluation of ** ξ**|=1,

*γ*=

*p*

_{ξ}of

*M*

^{−2}, we have

**|=1, in the isotropic medium, we have**

*ξ*Similarly, we have

Hence, evaluating ** A** in equation (2.25), by means of equations (2.34)–(2.37) and equations (2.44) and (2.45), we obtain the solution of the displacement for the self-similar subsonically expanding ellipsoidal inclusion in an isotropic medium

*b*.

Furthermore, the expression of the displacement gradient for the self-similarly subsonically expanding ellipsoidal inclusion in an isotropic medium is given in the form

## 3. The dynamic Eshelby tensor

From (2.48), the displacement gradient in the interior domain ** x** and

*t*, and is written as

We extend the definition of the Eshelby tensor [1] to elasto-dynamics, and define the dynamic Eshelby tensor *S*_{mlns} by relating the total strain *ϵ*^{*}_{ns} according to
*S*_{mlns} is constant (independent of ** x** and

*t*) in the interior domain of the self-similarly expanding ellipsoidal inclusion, as proven by Ni & Markenscoff [5] and as is apparent from the solution in equation (3.2).

Then, from equations (3.2) and (3.3), the expression of the dynamic Eshelby tensor follows for the self-similarly expanding ellipsoidal inclusion

Because the integral domain in equation (3.4) is the whole-unit sphere |** ξ**|=1, due to symmetry, there are only 12 non-zero components of the interior dynamic Eshelby tensor

## 4. The static Eshelby inclusion as the limit of the dynamically self-similarly expanding ellipsoidal inclusion

We show that, by applying a limiting procedure, the classic solution of the displacement gradient of the static ellipsoidal Eshelby inclusion [1] is obtained from the solution of the displacement gradient for the self-similarly subsonically expanding ellipsoidal inclusion given above. Here, we only focus on the case when the field point is in the interior domain.

We define the ‘static limit’ as the limit obtained by the procedure
*a*_{k} are set to be finite constants, and are equal to the semi-axes of the corresponding static ellipsoidal inclusion defined by

Noting (2.49), we have
*α*,*β*=1,2,3, and the repeated Greek indices *α* and *β* are not summed.

To evaluate the integrals (4.10)–(4.12), it is convenient to apply the transformation of the variables indicated in Mura [23], ch. 3:
** η**=(

*η*

_{1},

*η*

_{2},

*η*

_{3}) is a unit normal on the unit sphere, and the corresponding change of the surface element is given as

By using the transformation of variables (4.13)–(4.15) in the calculation of the integrals (4.10)–(4.12), we will prove the following identities:
*I*_{α},*I*_{αα} and *I*_{αβ}=*I*_{βα} are the Eshelby elliptic integrals [1,2]

To prove the identities (4.16) and (4.17), we use the transformation of the variables (4.13), and equations (4.14), and (4.15) in the integral *M*_{α}, and obtain
*η*_{0} in terms of ** η**, from (4.13), it follows that

**: |**

*ξ***|=1 gives**

*ξ*To derive (4.17) for *α*=*β*, again using (4.13)–(4.15), we transform the variables in integral (4.11) and obtain
*a*_{α}, i.e.
*I*_{α} has the evaluation (4.18), we calculate the derivative in equation (4.27) as follows:
*α*=*β*. In a similar way, we can obtain (4.17) for *α*≠*β*.

By using the identities (4.16) and (4.17), from equations (4.6)–(4.9), we obtain the static limits of the components of the dynamic Eshelby tensor in the interior domain
*ν*=λ/[2(λ+*μ*)], and where the relation
*S*_{1122}≠*S*_{2211}.

From equation (3.2) and the limiting procedure (4.1), we have the static limit of the displacement gradient for the self-similarly expanding ellipsoidal inclusion

Recall that the displacement gradient for the static ellipsoidal inclusion given in [1] (also see [23]) is written as
*g*_{βjk} is defined by
*Static* representing the static solution.

We will prove that the static limit of the displacement gradient (4.35) for the self-similarly expanding ellipsoidal inclusion is identical to the displacement gradient for the static ellipsoidal inclusion (4.36), i.e. we will prove that

In the proof, we use the relations
*k* are not summed. Relation (4.39) for *α*=*β* follows from definition (4.16); when *α*≠*β*, the integral on the right-hand side of (4.39) is zero by symmetry. We next need to prove (4.40). To this effect, we will use definitions (4.11) and (4.17) for *M*_{αα} and *M*_{αβ}, respectively. In fact, first, if *α*=*β*=*j*=*k*, then every term inside the brackets on the right-hand side of equation (4.40) equals that on the left-hand side of equation (4.40), so (4.40) holds. In the case when, within *α*,*β*,*j*,*k*, there are two different pairs of equal numbers, such as *α*=*β* and *j*=*k*, but *α*≠*j*, then there will be only one non-zero term in the brackets on the right-hand side of equation (4.40), which must be equal to *M*_{ij} for some *i* and *j* with *i*≠*j*. Then, according to definition (4.17), *M*_{ij}/3 equals the integral on the left-hand side of (4.40). Finally, otherwise, when *α*,*β*,*j*,*k* neither are all equal to each other nor contain two different pairs of equal numbers, then both sides of (4.40) are zero.

By the same arguments, we have
*k* are not summed.

On the right-hand side of (4.40), using the identities (4.16) and (4.17) to replace *M*_{αk} and *M*_{αl} by *I*_{α}, *I*_{αk} and *I*_{αl}, and applying (4.41), we obtain

Using equations (4.25), (4.39) and (4.42), we calculate the static limit (4.35) and obtain

## 5. Conclusion

The Eshelby inclusion problem is shown to be the static limit of the self-similarly dynamically expanding ellipsoidal inclusion (subsonically), which is its dynamic generalization, exhibiting the same property of constant constrained strain (and stress) in the interior domain of the expanding inclusion with constant transformation strain. The field solution was obtained by adapting the method of Burridge & Willis [6] for an expanding elliptic crack to a three-dimensional expanding ellipsoidal inclusion. The dynamic Eshelby tensor for the ellipsoid, obtained from the field solution, is shown here in integral form. It allows for the analytic solution (by the Eshelby equivalent inclusion method) of expanding inhomogeneities (i.e. regions undergoing chemical reactions with phase changes). The self-similar solution grasps the early response of the system [18], and applications include dynamic phase transformation under dynamic loading and the modelling of deep earthquake sources [9,10,14,15].

## Authors' contributions

The authors consider that this paper is a unity and do not wish to split the contributions. Both authors agree on this and the material to be published.

## Competing interests

We have no competing interests.

## Funding

This work was supported by National Science Foundation grant no. CMS 1129888.

- Received April 8, 2016.
- Accepted June 10, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.