Elastic fields due to a suddenly expanding spherical inclusion within Mindlin’s first strain-gradient theory

Abstract

The notion of dynamic inclusion can be utilized to represent phenomena in micromechanics of solids such as phase transformations, in particular when subjected to dynamic loading. If the size of a dynamic inclusion is comparable to the inherent length parameters of its constituent material, then the size effect manifests its significance. The present paper is, hence, dedicated to the employment of the complete form of Mindlin’s first strain gradient theory of elasticity to address a special class of dynamic inclusion problems. Specifically, in this paper, an analytical solution is obtained for the elastic displacement field developed in an infinitely extended isotropic medium due to a suddenly expanding spherical inclusion that undergoes a constant and uniform distribution of classical dilatational eigenstrains. The employment of such a theoretical framework results in eliminating the classical singularities of the strain and stress fields of the problem. The analysis presented in this paper can, moreover, account for the effect of microinertia on the response of the medium to the sudden expansion of the inclusion, by incorporating an additional characteristic length into the equation of motion of the medium. An extended version for Hadamard conditions associated with the adopted theory is also derived, and subsequently, it is shown that the obtained solution of the considered problem satisfies these conditions.

Appendix

This appendix is dedicated to the determination of the unknown functions $\Theta _0(r)$ and $\Theta _1(r;t)$ by solving differential equations (34a) and (34b). It is desired here to use the method of Green’s function for such a purpose. To this end, let us first consider Eq. (34a), and suppose that ${\mathscr {G}}_0(r,r')$ denotes its associated Green’s functions. Then, it can be shown that multiplying both sides of that equation by $r^2\,{\mathscr {G}}_0(r,r')$ followed by integrating them from 0 to $\infty$ with respect to r gives rise to

$$\begin{aligned} \int _0^\infty r^2\,\Theta _0(r)\,{\mathscr {G}}_0(r,r')\,\mathrm{d}r-\ell _4^2\int _0^\infty \frac{\mathrm{d}}{\mathrm{d}r}\bigg [\,r^2\frac{\mathrm{d}}{\mathrm{d}r}\Theta _0(r)\bigg ]\,{\mathscr {G}}_0(r,r')\,\mathrm{d}r=\epsilon _0\int _0^{R_0}r^2\,{\mathscr {G}}_0(r,r')\,\mathrm{d}r\,. \end{aligned}$$

(A-1)

If the second integral on the left-hand side of the above equation is expanded by integration by parts twice, this equation can be rewritten as

(A-2)

Now, let us suppose that the Green’s function ${\mathscr {G}}_0(r,r')$ is the solution of the following equation,

$$\begin{aligned} \Bigg (1-\ell _4^2\,\bigg [\,\frac{\partial ^2}{\partial r^2}+\frac{2}{r}\,\frac{\partial }{\partial r}\,\bigg ]\Bigg )\,{\mathscr {G}}_0(r,r')=\delta (r-r')\quad \text{ for }\quad 0\le r\quad \text{ and }\quad 0\le r'\,, \end{aligned}$$

(A-3)

and satisfies the conditions,

$$\begin{aligned}&\underset{r\rightarrow \,\infty }{\lim }\ {\mathscr {G}}_0(r,r')=0\,, \end{aligned}$$

(A-4a)

$$\begin{aligned}&\underset{r\rightarrow \,\infty }{\lim }\ \dfrac{\partial }{\partial r}{\mathscr {G}}_0(r,r')=0\,. \end{aligned}$$

(A-4b)

Accordingly, one can readily conclude from Eq. (A-2) that

$$\begin{aligned} \Theta _0(r)=\frac{\epsilon _0}{r^2}\int _0^{R_0}\!{\mathscr {G}}_0(r',r)\,r'^2\,\mathrm{d}r'\,. \end{aligned}$$

(A-5)

Hence, the determination of the expression for $\Theta _0(r)$ requires to obtain the solution of the Green’s function. To this end, let us re-express Eq. (A-3) as

$$\begin{aligned} \Bigg (1-\ell _4^2\,\bigg [\,\frac{\partial ^2}{\partial r^2}+\frac{2}{r}\,\frac{\partial }{\partial r}\,\bigg ]\Bigg )\,{\mathscr {G}}_0(r,r')=\left\{ \begin{array}{ll} 0&{}\quad 0\le r<r'\,,\\[7pt] 0&{}\quad 0\le r'<r\,.\end{array}\right. \end{aligned}$$

(A-6)

Then, it is appropriate to consider the following general expressions for the solution of the above equation.

$$\begin{aligned} {\mathscr {G}}_0(r,r')=\left\{ \begin{array}{ll} \dfrac{A_1}{r}\,\exp \!\left( \dfrac{r}{\ell _4}\right) +\dfrac{A_2}{r}\,\exp \!\left( \!-\dfrac{r}{\ell _4}\right) &{}0\le r<r'\,,\\ \dfrac{A_3}{r}\,\exp \!\left( \dfrac{r}{\ell _4}\right) +\dfrac{A_4}{r}\,\exp \!\left( \!-\dfrac{r}{\ell _4}\right) &{}0\le r'<r\,,\end{array}\right. \end{aligned}$$

(A-7)

where $A_1$, $A_2$, $A_3$, and $A_4$ are unknown constants that must be determined by applying a set of proper conditions. First of all, note that the Green’s function ${\mathscr {G}}_0(r,r')$ must have bounded values over its entire domain, particularly at $r=0$ and $\infty$. Thus,

$$\begin{aligned} A_1+A_2&=0\,, \end{aligned}$$

(A-8a)

$$\begin{aligned} A_3&=0\,. \end{aligned}$$

(A-8b)

Let us mention here that setting $A_3=0$ satisfies both conditions given by Eqs. (A-4a) and (A-4b). Moreover, note that the Green’s function ${\mathscr {G}}_0(r,r')$ must be continuous at $r=r'$. The latter condition in view of the expressions given by Eq. (A-7) and using Eqs. (A-8) results in

$$\begin{aligned} 2\,A_1\,\sinh \!\left( \frac{r'}{\ell _4}\right) -A_4\,\exp \!\left( \!-\dfrac{r'}{\ell _4}\right) =0\,. \end{aligned}$$

(A-9)

Furthermore, by integrating both sides of Eq. (A-3) from $r'-\delta r'$ to $r'+\delta r'$ with respect to r and, then, taking their limits as $\delta r'\rightarrow 0^+$, it is obtained that

$$\begin{aligned} \lim _{\delta r'\rightarrow \,0^+}\Bigg \{\,\int _{r'-\delta r'}^{\,r'+\delta r'}{\mathscr {G}}_0(r,r')\,\mathrm{d}r&-\ell _4^2\int _{r'-\delta r'}^{\,r'+\delta r'}\frac{\partial ^2}{\partial r^2}{\mathscr {G}}_0(r,r')\,\mathrm{d}r -2\,\ell _4^2\int _{r'-\delta r'}^{\,r'+\delta r'}\frac{1}{r}\,\frac{\partial }{\partial r}{\mathscr {G}}_0(r,r')\,\mathrm{d}r\Bigg \}=1\,. \end{aligned}$$

(A-10)

After evaluating the second integral on the left-hand side of the above equation in conjunction with expanding the last integral by integration by parts, one can write

$$\begin{aligned} \lim _{\delta r'\rightarrow \,0^+}\left\{ \,\int _{r'-\delta r'}^{\,r'+\delta r'}\bigg (1-2\,\frac{\ell _4^2}{r^2}\bigg )\,{\mathscr {G}}_0(r,r')\,\mathrm{d}r-\ell _4^2\,\left[ \,\frac{\partial }{\partial r}{\mathscr {G}}_0(r,r')+\frac{2}{r}\,{\mathscr {G}}_0(r,r')\right] \!\bigg |_{\,r'-\delta r'}^{\,r'+\delta r'}\right\} =1\,. \end{aligned}$$

(A-11)

With due attention to the continuity of the Green’s function ${\mathscr {G}}_0(r,r')$ at $r=r'$, the above equation reduces to

$$\begin{aligned} \lim _{\delta r'\rightarrow \,0^+}\left[ \,\frac{\partial }{\partial r}{\mathscr {G}}_0(r,r')\right] \!\bigg |_{\,r'-\delta r'}^{\,r'+\delta r'}=-\frac{1}{\ell _4^2}\,. \end{aligned}$$

(A-12)

Thereafter, using Eqs. (A-7) and (A-8), it can be shown that

$$\begin{aligned} 2\,A_1\,\frac{\ell _4}{r'}\left[ \,\cosh \!\left( \frac{r'}{\ell _4}\right) -\frac{\ell _4}{r'}\,\sinh \!\left( \frac{r'}{\ell _4}\right) \right] +A_4\,\frac{\ell _4}{r'}\left[ \,1+\frac{\ell _4}{r'}\right] \exp \!\left( \!-\dfrac{r'}{\ell _4}\right) =1\,. \end{aligned}$$

(A-13)

Then, solving Eqs. (A-9) and (A-13) for $A_1$ and $A_4$ leads to

$$\begin{aligned} A_1&=\frac{r'}{2\,\ell _4}\,\exp \!\left( \!-\dfrac{r'}{\ell _4}\right) , \end{aligned}$$

(A-14a)

$$\begin{aligned} A_4&=\frac{r'}{\ell _4}\,\sinh \!\left( \dfrac{r'}{\ell _4}\right) . \end{aligned}$$

(A-14b)

Using the above results together with Eqs. (A-8), the Green’s function ${\mathscr {G}}_0(r,r')$ takes the following final form.

$$\begin{aligned} {\mathscr {G}}_0(r,r')=\frac{r'}{r\,\ell _4}\,\Bigg [\,H\big (r'-r\big )\,\exp \!\left( \!-\frac{r'}{\ell _4}\right) \,\sinh \!\left( \frac{r}{\ell _4}\right) +H\big (r-r'\big )\,\exp \!\left( \!-\frac{r}{\ell _4}\right) \,\sinh \!\left( \frac{r'}{\ell _4}\right) \!\Bigg ]\,. \end{aligned}$$

(A-15)

Subsequently, by the substitution of the expression for ${\mathscr {G}}_0(r,r')$ from the above equation into Eq. (A-5), it is obtained that

$$\begin{aligned} \Theta _0(r)=&\,\epsilon _0\,H\big (R_0-r\big ) +\frac{\epsilon _0}{2}\left( \frac{\ell _4}{r}+\frac{R_0}{r}\right) \Bigg [\exp \!\left( \!-\frac{r+R_0}{\ell _4}\right) -H\big (R_0-r\big )\,\exp \!\left( \frac{r-R_0}{\ell _4}\right) \!\Bigg ]\nonumber \\&-\frac{\epsilon _0}{2}\,H\big (r-R_0\big )\left( \frac{\ell _4}{r}-\frac{R_0}{r}\right) \exp \!\left( \!-\frac{r-R_0}{\ell _4}\right) . \end{aligned}$$

(A-16)

Likewise, it can be shown that the Green’s function problem corresponding to Eq. (34b) is expressible as

$$\begin{aligned} \Bigg (1-\ell _4^2\bigg [\,\frac{\partial ^2}{\partial r^2}+\frac{2}{r}\,\frac{\partial }{\partial r}\,\bigg ]\Bigg )\,{\mathscr {G}}_1(r,r';t)=\delta (r-r')\quad \text{ for }\quad 0\le r\le R_{\mathrm{m}}(t)\quad \text{ and }\quad 0\le r'\le R_{\mathrm{m}}(t)\,, \end{aligned}$$

(A-17)

where ${\mathscr {G}}_1(r,r';t)$ is the Green’s function associated with the function $\Theta _1(r;t)$. The Green’s function ${\mathscr {G}}_1(r,r';t)$ must, similarly, be continuous at $r=r'$, and it must also have a bounded value at $r=0$. Furthermore, it must vanish at $r=R_{\mathrm{m}}(t)$, where $R_{\mathrm{m}}(t)$ has formerly been defined by Eq. (35). By applying these conditions to Eq. (A-17), the below-given solution is, thus, obtained for ${\mathscr {G}}_1(r,r';t)$.

$$\begin{aligned} {\mathscr {G}}_1(r,r';t)=&\,\frac{r'}{2\,r\,\ell _4}\,\mathrm{csch}\!\left( \frac{R_{\mathrm{m}}(t)}{\ell _4}\right) \Bigg [\,H\big (r'-r\big )\,\Bigg \{\cosh \!\left( \frac{r-r'+R_{\mathrm{m}}(t)}{\ell _4}\right) -\cosh \!\left( \frac{r+r'-R_{\mathrm{m}}(t)}{\ell _4}\right) \!\Bigg \}\nonumber \\&+H\big (r-r'\big )\,\Bigg \{\cosh \!\left( \frac{r-r'-R_{\mathrm{m}}(t)}{\ell _4}\right) -\cosh \!\left( \frac{r+r'-R_{\mathrm{m}}(t)}{\ell _4}\right) \!\Bigg \}\Bigg ]\,. \end{aligned}$$

(A-18)

Now, the function $\Theta _1(r;t)$ can similarly be obtained in terms of its associated Green’s function via

$$\begin{aligned} \Theta _1(r;t)=\frac{\epsilon _0}{r^2}\,\int _{R_0}^{R(t)}\!{\mathscr {G}}_1(r',r;t)\,r'^2\,\mathrm{d}r'\,. \end{aligned}$$

(A-19)

Subsequently, by substituting Eq. (A-18) into the above integral, it can be shown that

$$\begin{aligned} \Theta _1(r;t)=&\,\epsilon _0\,H\big (r-R_0\big )\,H\big (R(t)-r\big )\nonumber \\&+\frac{\epsilon _0}{2}\,\mathrm{csch}\!\left( \frac{R_{\mathrm{m}}(t)}{\ell _4}\right) \frac{\ell _4}{r}\,\Bigg \{H\big (R_0-r\big )\Bigg [\cosh \!\left( \frac{r-R_0+R_{\mathrm{m}}(t)}{\ell _4}\right) +\frac{R_0}{\ell _4}\,\sinh \!\left( \frac{r-R_0+R_{\mathrm{m}}(t)}{\ell _4}\right) \!\Bigg ]\nonumber \\&-H\big (R(t)-r\big )\Bigg [\cosh \!\left( \frac{r-R(t)+R_{\mathrm{m}}(t)}{\ell _4}\right) +\frac{R(t)}{\ell _4}\,\sinh \!\left( \frac{r-R(t)+R_{\mathrm{m}}(t)}{\ell _4}\right) \!\Bigg ]\nonumber \\&+H\big (r-R_0\big )\,H\big (R_{\mathrm{m}}(t)-r\big )\Bigg [\cosh \!\left( \frac{r-R_0-R_{\mathrm{m}}(t)}{\ell _4}\right) +\frac{R_0}{\ell _4}\,\sinh \!\left( \frac{r-R_0-R_{\mathrm{m}}(t)}{\ell _4}\right) \!\Bigg ]\nonumber \\&-H\big (r-R(t)\big )\,H\big (R_{\mathrm{m}}(t)-r\big )\Bigg [\cosh \!\left( \frac{r-R(t)-R_{\mathrm{m}}(t)}{\ell _4}\right) +\frac{R(t)}{\ell _4}\,\sinh \!\left( \frac{r-R(t)-R_{\mathrm{m}}(t)}{\ell _4}\right) \!\Bigg ]\nonumber \\&-H\big (R_{\mathrm{m}}(t)-r\big )\Bigg [\cosh \!\left( \frac{r+R_0-R_{\mathrm{m}}(t)}{\ell _4}\right) -\frac{R_0}{\ell _4}\,\sinh \!\left( \frac{r+R_0-R_{\mathrm{m}}(t)}{\ell _4}\right) \nonumber \\&-\cosh \!\left( \frac{r+R(t)-R_{\mathrm{m}}(t)}{\ell _4}\right) +\frac{R(t)}{\ell _4}\,\sinh \!\left( \frac{r+R(t)-R_{\mathrm{m}}(t)}{\ell _4}\right) \!\Bigg ]\,. \end{aligned}$$

(A-20)