Construction of Modal Solution on a Heterogeneous Spherical Shell

A Chebfun script for creating a modal solution for linear elasticity on a radially heterogeneous spherical shell.

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Synopsis

The purpose of the MATLAB Chebfun script shell.m is to construct a high-order approximation of a modal solution to a radially heterogeneous, isotropic elastic shell. If $u_{r}$, $u_{\theta}$, $u_{\phi}$ are the displacements in a spherical coordinate system, then the solution is taken to be of the form $u_{r} = \cos(t) \phi(r)$ and $u_{\phi} = u_{\theta} = 0$.

The script specifies the material properties using the density $\rho$, Lamé's first parameter $\lambda$, and Lamé's second parameter $\mu$ (i.e., the shear modulus). The inner and outer radii of the shell are chosen so that the resulting boundary condition is traction free (free surface). Specifically, once $\phi(r)$ is determined the radii are chosen to be zeros of $$ \sigma_{rr} = \left(\lambda + 2\mu\right) \phi_{,r} + \frac{2}{r}\lambda \phi, $$ where the comma in the subscripts denotes a partial derivative with respect to the variable that follows.

The script outputs the inner and outer radii of the spherical shell (R1 and R2) as well as information needed to build a polynomial interpolant (via barycentric interpolation) of the solution. The full output of the script is given in shell_data.m.

To convert from the spherical to Cartesian system the following transforms of the radial displacement $u_{r}$ and the derivative $u_{r,r}$ can be used (from which the stresses and/or strains can be defined)

$$ u_{x} = \frac{x}{r} u_{r}, \quad u_{y} = \frac{y}{r} u_{r}, \quad u_{z} = \frac{z}{r} u_{r}, $$

$$ u_{x,x} = \frac{x^2 r u_{r,r} + y^2 u_{r} + z^2 u_{r}}{r^3}, \quad u_{x,y} = \frac{x y (r u_{r,r} - u_r)}{r^3}, \quad u_{x,z} = \frac{x z (r u_{r,r} - u_r)}{r^3}, $$

$$ u_{y,x} = \frac{y y (r u_{r,r} - u_r)}{r^3}, \quad u_{y,y} = \frac{y^2 r u_{r,r} + x^2 u_{r} + Z^2 u_{r}}{r^3}, \quad u_{y,z} = \frac{y z (r u_{r,r} - u_r)}{r^3}, $$

$$ u_{z,x} = \frac{z y (r u_{r,r} - u_r)}{r^3}, \quad u_{z,y} = \frac{z y (r u_{r,r} - u_r)}{r^3}, \quad u_{z,z} = \frac{z^2 r u_{r,r} + x^2 u_{r} + y^2 u_{r}}{r^3}. $$

Plots of the non-zero components of the solution and the material properties are shown in the figure

Dependencies

• MATLAB
• Chebfun

License

Creative Commons Zero v1.0 Universal