User:Egm6936.s09/Dynamic stability

=Dynamic stability analysis of an elastic composite material having a negative-stiffness phase=

The general solutions for the displacement field
In Kochman 2009 analyze a composite material consisting of two homogeneous, isotropic, linear elastic phases, i.e, an infinitely long circular cylindrical inclusion of radius $$\displaystyle a $$ and Lame's elastic moduli $$\displaystyle \lambda^I, \mu^I$$ with a concentric coating of thickness $$\displaystyle d$$ and elastic moduli $$\displaystyle \lambda^{II}, \mu^{II}$$. The composite is assumed to be in a state of plane strain. Its dynamic perturbing displacements are ruled by Cauchy equation of motion without body force:


 * {| style="width:100%" border="0"

$$   \displaystyle \rho\ddot{\mathbf u}=\nabla\cdot \boldsymbol \sigma $$     (1)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

where $$\displaystyle \mathbf u$$ is the perturbing displacment vector field, restricted to being infinitesimal and having infinitesimal gradients. The dots denote time derivatives, $$\displaystyle \boldsymbol \sigma$$ is the stress tensor, $$\displaystyle \rho$$ is the density. For simplicity, the same density is assumed for both materials since different density will not affect the stability requirements.

The Hooke's law gives:


 * {| style="width:100%" border="0"

$$   \displaystyle \boldsymbol \sigma=2\mu \boldsymbol \epsilon + \lambda tr(\boldsymbol \epsilon) \mathbf 1 $$     (2)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

where $$\displaystyle \boldsymbol \epsilon = \text{sym} (\nabla \mathbf u)$$ is the strain tensor and $$\displaystyle \mathbf 1$$ is the identity tensor.

After putting Eq.(2) into Eq.(1), the resulting equation is usually named after Navier.

Employing the polar coordinates with the origin at the center of the inclusion, Kochmann got the following PDEs equations for displacements $$\displaystyle \mathbf u = (u_r,u_\theta)$$ w.r.t to polar coordinates $$\displaystyle r,\theta$$. These equations are valid for each material phase with different Lame's values $$\displaystyle \lambda, \mu$$.


 * {| style="width:100%" border="0"

$$   \displaystyle \rho(\ddot{\mathbf u})_r = \frac {\lambda + \mu}{r^2}[r^2 u_{r,rr} + r u_{r,r} - u_r + r u_{\theta,r} - u_\theta] + \frac {\mu}{r^2}[r^2 u_{r,rr} + r u_{r,\theta\theta} - u_r + r u_{r,r} - 2 u_{\theta,\theta}] $$     (3a)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }


 * {| style="width:100%" border="0"

$$   \displaystyle \rho(\ddot{\mathbf u})_\theta = \frac {\lambda + \mu}{r^2}[r u_{r,r\theta} + u_{r,\theta} + u_{\theta,\theta\theta}] + \frac {\mu}{r^2}[r^2 u_{\theta,rr} + u_{\theta,\theta\theta} - u_\theta + r u_{\theta,r} + 2 u_{r,\theta}] $$     (3b)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

The general solutions are sought in the following forms:


 * {| style="width:100%" border="0"

$$   \displaystyle u_r = f(r)e^{im\theta}e^{i\omega t}, \qquad u_\theta = g(r)e^{im\theta}e^{i\omega t} $$ (4)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

Substituting (4) to (3) to get as Kochmann :


 * {| style="width:100%" border="0"

$$   \displaystyle f''+\frac{f'}{r}-[\frac{\lambda+2\mu + m^2\mu}{(\lambda+2\mu)r^2}-\frac{\rho\omega^2}{\lambda+2\mu}]f = \frac{im}{\lambda+2\mu}[-(\lambda+\mu)\frac{g'}{r}+(\lambda+3\mu)\frac{g}{r^2}] $$     (5a)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }


 * {| style="width:100%" border="0"

$$   \displaystyle g''+\frac{g'}{r}-[\frac{\mu + m^2(\lambda+2\mu)}{\mu r^2}-\frac{\rho\omega^2}{\mu}]g = -\frac{im}{\mu}[(\lambda+\mu)\frac{f'}{r}+(\lambda+3\mu)\frac{f}{r^2}] $$     (5b)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

Then Kochmann in used the infinite power series to find the solution for the system of ODEs in (7):


 * {| style="width:100%" border="0"

$$   \displaystyle f = \sum_{n=0}^{\infty}a_n r^{k+n}, \qquad g = \sum_{n=0}^{\infty}b_n r^{k+n} $$     (6)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

To satisfied these equation (5), the values of $$\displaystyle k$$ must be:


 * {| style="width:100%" border="0"

$$   \displaystyle k = \pm \left | m \right | \pm 1 $$     (7)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

and the coefficients of the series also satisfied:


 * {| style="width:100%" border="0"

$$   \displaystyle b_0 = a_0 im \frac{\lambda(1+k)+\mu(3+k)}{m^2(\lambda+2\mu)+\mu(1-k^2)}, \quad b_1 = a_1 = 0 $$     (8)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

and for $$\displaystyle n \geqslant 2$$


 * {| style="width:100%" border="0"

$$   \displaystyle a_{n+2} = \frac{\rho \omega^2}{\mu (\lambda + 2 \mu)} \frac{(-(1+k+n)(3+k+n)\mu + m^2(\lambda + 2\mu))a_n + im\left ( (1+k+n)\lambda + (-1+k+n)\mu \right )b_n}{(1+k-m+n)(3+k-m+n)(1+k+m+n)(3+k+m+n)} $$     (9a)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }


 * {| style="width:100%" border="0"

$$   \displaystyle b_{n+2} = \frac{\rho \omega^2}{\mu (\lambda + 2 \mu)} \frac{im\left ( (3+k+n)\lambda + (5+k+n)\mu \right )a_n + (m^2 \mu-(1+k+n)(3+k+n)(\lambda + 2\mu)b_n) }{(1+k-m+n)(3+k-m+n)(1+k+m+n)(3+k+m+n)} $$     (9b)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

From (8) and (9), it is easy to see that all "odd" coefficients are zero.

For each value of $$\displaystyle k$$, writing the coefficients $$\displaystyle a_n, b_n$$ in a non-recursive form and making use of the definition of the Bessel's functions:


 * {| style="width:100%" border="0"

$$   \displaystyle J_m(z)= \left( \frac{z}{2} \right)^m \sum_{n=0}^{\infty}\frac{(-1)^n z^{2n}}{2^{2n}n!(m+n)!} $$     (10)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

Kochmann ended up with the following genenal forms:


 * {| style="width:100%" border="0"

$$   \displaystyle f_{mn}(r)= A_{mn}\frac{J_{\left | m \right |}(\xi_{\omega_{mn}}r)}{r} - B_{mn} J'_{\left | m \right |}(\chi_{\omega_{mn}}r) + C_{mn}\frac{Y_{\left | m \right |}(\xi_{\omega_{mn}}r)}{r} - D_{mn} Y'_{\left | m \right |}(\chi_{\omega_{mn}}r) $$     (11a)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }


 * {| style="width:100%" border="0"

$$   \displaystyle g_{mn}(r)= \frac{i}{m} \left [ -m^2 B_{mn}\frac{J_{\left | m \right |}(\chi_{\omega_{mn}}r)}{r} + A_{mn} J'_{\left | m \right |}(\xi_{\omega_{mn}}r) - m^2 D_{mn}\frac{Y_{\left | m \right |}(\chi_{\omega_{mn}}r)}{r} + C_{mn} Y'_{\left | m \right |}(\xi_{\omega_{mn}}r) \right ] $$     (11b)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

where


 * {| style="width:100%" border="0"

$$   \displaystyle \xi_\omega := \sqrt{\frac{\rho \omega^2}{\mu}}, \qquad \chi_\omega := \sqrt{\frac{\rho \omega^2}{\lambda + 2\mu}} $$     (12)
 * style="width:95%  |
 * style="width:95%  |
 * 
 * }

and the prime denotes differentiation with respect to $$\displaystyle r$$ and $$\displaystyle A_{mn}-D{mn}$$ are undetermined complex constants, one independent set for each $$\displaystyle m,n$$ pair. $$\displaystyle J_m,Y_m$$ are the Bessel functions of the first and second kind respectivley.

Since the displacement field must be single-valued, i.e $$\displaystyle \mathbf u(r,0,t)=\mathbf u(r,2\pi,t)$$, $$\displaystyle m$$ must be integer-valued. Also, Kochmann used the complex representation for the solution, he required that all displacements are real and hence the following conditions must be satisfied:


 * {| style="width:100%" border="0"

$$   \displaystyle f_{-m-n} = \overline{f_{mn}}, \qquad f_{m-n} = \overline{f_{-mn}}, \qquad g_{-m-n} = \overline{g_{mn}}, \qquad g_{m-n} = \overline{g_{-mn}}, \qquad $$     (13)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

Then Kochmann said that (13) are satisfied when all constants $$\displaystyle A_{mn}-D{mn}$$ are real. ?????

Finally, summation of all admissible values of $$\displaystyle \omega $$ gives the general solutions as :


 * {| style="width:100%" border="0"

$$  \displaystyle u_r(r,\theta,t) =\frac{1}{r} \sum_{m = -\infty}^{\infty} \sum_{n = 0}^{\infty} \left \{ A_{mn}J_{\left | m \right |}(\xi_{\omega_{mn}}r) - B_{mn} r J'_{\left | m \right |}(\chi_{\omega_{mn}}r) + C_{mn}Y_{\left | m \right |}(\xi_{\omega_{mn}}r) - D_{mn} rY'_{\left | m \right |}(\chi_{\omega_{mn}}r) \right \}e^{im\theta}e^{i\omega_{mn}t} $$     (14a)
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle u_\theta(r,\theta,t)= \frac{i}{r} \sum_{m = -\infty}^{\infty} \sum_{n = 0}^{\infty} \left \{ A_{mn}\frac{r}{m} J'_{\left | m \right |}(\xi_{\omega_{mn}}r) - m B_{mn}J_{\left | m \right |}(\chi_{\omega_{mn}}r) + C_{mn} \frac{r}{m} Y'_{\left | m \right |}(\xi_{\omega_{mn}}r) - m D_{mn}Y_{\left | m \right |}(\chi_{\omega_{mn}}r)  \right \}e^{im\theta}e^{i\omega_{mn}t} $$     (14b)
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * <p style="text-align:right">
 * }

Using Hook's law, the stress components entering the traction vector on surfaces $$\displaystyle r = const $$:


 * {| style="width:100%" border="0"

$$  \displaystyle \sigma_{rr} = (\lambda+2\mu)u_{r,r} + \lambda \frac{u_r+u_{\theta,\theta}}{r}, \qquad \sigma_{r\theta} = \mu \left ( u_{\theta,r} + \frac{u_{r,\theta} - u_{\theta}}{r}, \right ) $$     (15)
 * style="width:95% |
 * style="width:95% |
 * <p style="text-align:right">
 * }

Kochmann got:


 * {| style="width:100%" border="0"

$$  \displaystyle \sigma_{rr} = \frac{\mu}{2}\sum_{m = -\infty}^{\infty} \sum_{n = 0}^{\infty} \begin{Bmatrix} A_{mn}\xi_{\omega_{mn}}^2 [J_{\left | m \right | - 2}(\xi_{\omega_{mn}}r)- J_{\left | m \right | + 2}(\xi_{\omega_{mn}}r)] - B_{mn} \chi_{\omega_{mn}}^2 [J_{\left | m - 2 \right |}(\chi_{\omega_{mn}}r) + J_{\left | m +2 \right |}(\chi_{\omega_{mn}}r) ]\\ + C_{mn} \xi_{\omega_{mn}}^2 [Y_{\left | m \right | - 2}(\xi_{\omega_{mn}}r)- Y_{\left | m \right | + 2}(\xi_{\omega_{mn}}r)] - D_{mn} \chi_{\omega_{mn}}^2 [Y_{\left | m - 2 \right |}(\chi_{\omega_{mn}}r) + Y_{\left | m +2 \right |}(\chi_{\omega_{mn}}r) ] \\ + 2(\frac{\lambda}{\mu}+1)\chi_{\omega_{mn}}^2 [B_{mn}J_{\left | m \right |}(\chi_{\omega_{mn}}r) + D_{mn}Y_{\left | m \right |}(\chi_{\omega_{mn}}r) ] \end{Bmatrix}e^{im\theta}e^{i\omega_{mn}t} $$     (16a)
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle \sigma_{r\theta} = \frac{i\mu}{2}\sum_{m = -\infty}^{\infty} \sum_{n = 0}^{\infty} sign(m)\begin{Bmatrix} A_{mn}\xi_{\omega_{mn}}^2 [J_{\left | m \right | - 2}(\xi_{\omega_{mn}}r) + J_{\left | m \right | + 2}(\xi_{\omega_{mn}}r)] - B_{mn} \chi_{\omega_{mn}}^2 [J_{\left | m - 2 \right |}(\chi_{\omega_{mn}}r) - J_{\left | m +2 \right |}(\chi_{\omega_{mn}}r) ]\\ + C_{mn} \xi_{\omega_{mn}}^2 [Y_{\left | m \right | - 2}(\xi_{\omega_{mn}}r)+ Y_{\left | m \right | + 2}(\xi_{\omega_{mn}}r)] - D_{mn} \chi_{\omega_{mn}}^2 [Y_{\left | m - 2 \right |}(\chi_{\omega_{mn}}r) - Y_{\left | m +2 \right |}(\chi_{\omega_{mn}}r) ] \end{Bmatrix}e^{im\theta}e^{i\omega_{mn}t} $$     (16b)
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * <p style="text-align:right">
 * }

Dynamic stability analysis of a homogeneous circular cylinder fixed at the boundary
Now consider, first the case of a homogeneous, isotropic, linear elastic circular cylindrical body with radius $$\displaystyle a$$ and elastic moduli$$\displaystyle \lambda,\mu$$ restricted to plane strain deformations. The general form of the displacement field in this case is obtained from (14) by removing terms which lead to singularity at the center $$\displaystyle r=0$$(Terms of Bessel-second function), after normalizing $$\displaystyle r $$ by $$\displaystyle a $$are:


 * {| style="width:100%" border="0"

$$   \displaystyle u_r(r,\theta,t) =\frac{1}{2} \sum_{m = -\infty}^{\infty} \sum_{n = 0}^{\infty} \left \{ A_{mn} \frac{\xi_{\omega_{mn}}r}{\left | m \right |} [J_{\left | m \right | +1}(\xi_{\omega_{mn}}ar) + J_{\left | m \right |-1}(\xi_{\omega_{mn}}ar)] + B_{mn} \chi_{\omega_{mn}} [J_{\left | m  \right | +1}(\chi_{\omega_{mn}}ar) - J_{\left | m \right |-1}(\chi_{\omega_{mn}}ar)] \right \}e^{im\theta}e^{i\omega_{mn}t} $$     (17a)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$   \displaystyle u_\theta(r,\theta,t) = -\frac{i}{2} \sum_{m = -\infty}^{\infty} \sum_{n = 0}^{\infty} \left \{ A_{mn} \frac{\xi_{\omega_{mn}}r}{\left | m \right |} [J_{\left | m \right | +1}(\xi_{\omega_{mn}}ar) - J_{\left | m \right |-1}(\xi_{\omega_{mn}}ar)] + \frac{m}{\left |m \right|}B_{mn} \chi_{\omega_{mn}} [J_{\left | m  \right | +1}(\chi_{\omega_{mn}}ar) + J_{\left | m \right |-1}(\chi_{\omega_{mn}}ar)] \right \}e^{im\theta}e^{i\omega_{mn}t} $$     (17b)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

The above equations (17) used the following useful identities of the Bessel-first order:


 * {| style="width:100%" border="0"

$$   \displaystyle \frac{J_m(\kappa r)}{r} = \frac{\kappa}{2m} [J_{m+1}(\kappa r)+J_{m-1}(\kappa r)], \qquad J'_m(\kappa r) = \frac{\kappa}{2} [J_{m-1}(\kappa r)-J_{m+1}(\kappa r)] $$     (18)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

Now applying the boundary conditions:


 * {| style="width:100%" border="0"

$$   \displaystyle u_r(r=1,\theta,t) = 0, \qquad u_\theta(r=1,\theta,t) = 0 $$     (19)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

Defining


 * {| style="width:100%" border="0"

$$   \displaystyle k= \sqrt{\frac{\mu}{\lambda+2\mu}} $$     (20)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

Kochmann rewrote the B.Cs in the matrix form:


 * {| style="width:100%" border="0"

$$   \displaystyle \xi_{\omega_{mn}}\begin{pmatrix} J_{\left | m \right | +1}(\xi_{\omega_{mn}}a) & \left | m \right | k J_{\left | m  \right | +1}(k\xi_{\omega_{mn}}a) \\ -J_{\left | m  \right | -1}(\xi_{\omega_{mn}}a) & \left | m \right | k J_{\left | m \right | -1}(k\xi_{\omega_{mn}}a) \end{pmatrix} \begin{pmatrix} A_{mn}\\ B_{mn} \end{pmatrix} = \begin{pmatrix} 0\\0 \end{pmatrix} $$     (21)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

To have non-trivial solutions, the determinant of the matrix of the coefficients $$\displaystyle A_{mn},B_{mn}$$ must be zero or:


 * {| style="width:100%" border="0"

$$   \displaystyle J_{\left | m \right | +1}(j_{mn}) J_{\left | m  \right | - 1}(kj_{mn}) + J_{\left | m  \right | -1}(j_{mn}) J_{\left | m  \right | + 1}(kj_{mn}) = 0 $$     (22)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

where $$\displaystyle j_{mn}=\xi_{\omega_{mn}}a$$.

Kochmann argued that equation (22) must have real solution for $$\displaystyle \omega_{mn}$$ since imaginary solutions will lead to unstable solution (see the form of the solution (5). This condition is satisfied if $$\displaystyle k$$ is real, i.e all roots $$\displaystyle j_{mn}$$ are real for all admissible integer values of $$\displaystyle m$$. This is easily shown using the sum representation of the Bessel function of the first kind:



\displaystyle J_{\left | m \right |+1}(ix) J_{\left | m  \right |-1}(kix) + J_{\left | m  \right |-1}(ix) J_{\left | m  \right |+1}(kix)=(-1)^m k^{\left | m \right |-1} x^{2\left | m \right |}2^{-2\left | m \right |} \sum_{n = 0}^{\infty} \sum_{l = 0}^{\infty} \frac{x^{2(n+l)}(k^2l+k^{2n+2})}{2^{2n+2l}n!l!(n+\left | m \right | +1)!(l+\left | m \right | -1)!} $$


 * {| style="width:100%" border="0"

$$   \displaystyle =\left\{\begin{matrix} >0, \text{if m is even}\\ <0, \text{if m is odd}\\ =0, \text{if x=0} \end{matrix}\right. $$     (23)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

Kochmann as well definites $$\displaystyle \bar j_{mn}=\chi_{\omega_{mn}}a$$ where the bar here and in the sequel does not indicate complex conjugate. So (22) rewrites as:


 * {| style="width:100%" border="0"

$$   \displaystyle J_{\left | m \right | +1}(\bar j_{mn}/k) J_{\left | m  \right | - 1}(\bar j_{mn}) + J_{\left | m  \right | -1}(\bar j_{mn}/k) J_{\left | m  \right | + 1}(\bar j_{mn}) = 0 $$     (24)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

Due to Kochmann's definition ,


 * {| style="width:100%" border="0"

$$   \displaystyle \omega_{mn}=\pm \sqrt{\frac{\mu}{a^2\rho}j_{mn}^2}, \qquad \omega_{mn}=\pm \sqrt{\frac{\lambda+2\mu}{a^2\rho}j_{mn}^2} $$     (25)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

Solving (21) to get the general displacment field:


 * {| style="width:100%" border="0"

$$   \displaystyle u_r(r,\theta,t) =\frac{1}{ar} \sum_{m = -\infty}^{\infty} \sum_{n = 0}^{\infty} A_{mn} \begin{Bmatrix} \left[J_{\left | m \right |}(j_{mn}r) + \frac{J_{\left | m  \right | +1}(j_{mn})}{J_{\left | m  \right | +1}(kj_{mn})\left |m \right |k} rJ'_{\left | m  \right |}(kj_{mn}r)\right]e^{\pm i \sqrt{\frac{\mu}{a^2\rho}j_{mn}^2}t}\\ + \left[J_{\left | m \right |+1}(\bar j_{mn}r/k) + \frac{J_{\left | m  \right | +1}(\bar j_{mn}/k)}{J_{\left | m  \right | +1}(\bar j_{mn})\left |m \right |k} rJ'_{\left | m  \right |}(\bar j_{mn}r)\right ]e^{\pm i \sqrt{\frac{\lambda+2\mu}{a^2\rho}\bar j_{mn}^2}t} \end{Bmatrix} e^{im\theta} $$     (26a)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }


 * {| style="width:100%" border="0"

$$   \displaystyle u_\theta(r,\theta,t) =\frac{i}{ar} \sum_{m = -\infty}^{\infty} \sum_{n = 0}^{\infty} sign(m)A_{mn} \begin{Bmatrix} \left[ \frac{1}{\left | m \right |}rJ'_{\left | m \right |}(j_{mn}r) + \frac{J_{\left | m  \right | +1}(j_{mn})}{J_{\left | m  \right | +1}(kj_{mn})k} J_{\left | m  \right |}(kj_{mn}r)\right]e^{\pm i \sqrt{\frac{\mu}{a^2\rho}j_{mn}^2}t}\\ + \left [\frac{1}{\left | m \right |}rJ'_{\left | m \right |}(\bar j_{mn}r/k) + \frac{J_{\left | m \right | +1}(\bar j_{mn}/k)}{J_{\left | m  \right | +1}(\bar j_{mn})k} J_{\left | m  \right |}(\bar j_{mn}r)\right]e^{\pm i \sqrt{\frac{\lambda+2\mu}{a^2\rho}\bar j_{mn}^2}t} \end{Bmatrix} e^{im\theta} $$     (26b)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

Avoiding exponential time growth of the displacement field in (26) hence requires:


 * {| style="width:100%" border="0"

$$   \displaystyle \omega_{mn}=\pm \sqrt{\frac{\mu}{a^2\rho}j_{mn}^2}\in R, \qquad \omega_{mn}=\pm \sqrt{\frac{\lambda+2\mu}{a^2\rho}j_{mn}^2} \in R	$$ (27)
 * style="width:95%  |
 * style="width:95%  |
 * <p style="text-align:right">
 * }

or


 * {| style="width:100%" border="0"

$$  \displaystyle \mu >0, \qquad \lambda+2\mu>0 $$     (28)
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * <p style="text-align:right">
 * }

=References =