Numerical/analytical solutions to the elastic response of arbitrarily functionally graded polar orthotropic rotating discs

Abstract

A substantial elastic analysis of uniform rotating discs made of radially functionally graded (FG) polar orthotropic materials is managed with both analytical and numerical methods by imposing possible boundary conditions and frequently used material grading rules such as a simple power and an exponential patterns. The complementary functions method (CFM) is originally chosen as a numerical technique to solve the governing equation having variable coefficients. Before applying CFM on the current two-point boundary value problem, the governing equation derived by the transformed on-axis in-plane stiffness terms is transformed into the initial value problem. To show the effectiveness of the method, as an additional study, some closed-form formulas are obtained to cover the rotating uniform discs made of functionally simple power-law graded polar orthotropic materials under the same constraints. It was shown that both analytical and numerical results display a perfect harmony. An extensive parametric study which considers physically or hypothetically exist different material types, several inhomogeneity indexes, anisotropy degrees varying in a wide range of 0.3–5, three types of boundary conditions, is conducted with the help of both analytical formulas and numerical solutions. To the best of the author’s knowledge, although especially the anisotropy effects have been worked through for ordinary polar orthotropic materials, investigations on the anisotropy degrees on the elastic behaviour of discs made of an advanced material having varying properties along the desired directions such as a FG anisotropic material is still so limited and requires vast knowledge. Since different transformed stiffness terms through the radial direction may be developed by considering different fibre orientations, present results comprising anisotropy effects may also be interpreted as the results of such discs made of specially orthotropic materials such as cross-ply or balanced symmetrical laminates. It is mainly concluded that the use of a radially FG anisotropic material having an anisotropy degree less than the unit proposes admissible circumferential stresses along with the effective properties which continuously radially increase from the inner surface towards the outer under centrifugal forces. On the other hand, anisotropy degrees greater than the unit with increasing material properties from the inner surface to the outer mitigates the radial stresses. Negative inhomogeneity indexes with smaller anisotropy degrees are preferable to get smaller radial displacements.

Appendix

CFM solution to a second-order differential equation with variable coefficients

	Given BVP
	$y^{\prime\prime}\left( x \right) + P\left( x \right) y^{\prime}\left( x \right) + Q\left( x \right) y\left( x \right) = R\left( x \right)$ (i) Solution interval: [a, b] Physical boundary conditions: $y\left( a \right) + y^{\prime } \left( a \right) = 0$ and $y\left( b \right) + y^{\prime } \left( b \right) = 0$
	Transformation of BVP into IVP
	Convert given BVP into IVP by simply letting $z_{1} \left( x \right) = y\left( x \right)$ (ii) $z_{2} \left( x \right) = y^{{\prime }} \left( x \right) = z_{1}^{{\prime }} \left( x \right)$ $z_{2}^{{\prime }} \left( x \right) = y^{{\prime \prime }} \left( x \right)$ Then obtain the following from Eq. (i) $z_{2}^{{\prime }} \left( x \right) = - P\left( x \right)z_{2} \left( x \right) - Q\left( x \right)z_{1} \left( x \right) + R\left( x \right)$ (iii) Using $z_{1}^{{\prime }} \left( x \right)$ and $z_{2}^{{\prime }} \left( x \right)$ in Eqs. (ii) and (iii), get the following IVP $\left\{ {\begin{array}{*{20}c} {z_{1}^{{\prime }} (x)} \\ {z_{2}^{{\prime }} (x)} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - Q(x)} & { - P(x)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {z_{1} (x)} \\ {z_{2} (x)} \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} 0 \\ {R(x)} \\ \end{array} } \right\}$ (iv) CFM solution of Eq. (iv) is given by: $y\left( x \right) = y_{0} \left( x \right) + b_{1} y_{1} \left( x \right) + b_{2} y_{2} \left( x \right)$ (v) Determine all the unknowns ($y_{0} \left( x \right),y_{1} \left( x \right),y_{2} \left( x \right), b_{1} , \, and \, b_{2} )$ in the solution, Eq. (v), by using related prescribed and physical boundary conditions as follows

CFM stages	Solve the following non-homogeneous differential equation	Under prescribed zero initial conditions	To get particular solution, $y_{0} \left( x \right)$
1	$\left\{ {\begin{array}{*{20}c} {z_{1}^{{\prime }} = y_{o}^{{\prime }} (x)} \\ {z_{2}^{{\prime }} = y_{o}^{{\prime \prime }} (x)} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - Q(x)} & { - P(x)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {z_{1} (x) = y_{o} (x)} \\ {z_{2} (x) = y_{o} '(x)} \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} 0 \\ {R(x)} \\ \end{array} } \right\}$	$\left\{ {\begin{array}{*{20}c} {z_{1} (a) = y_{o} (a) = 0} \\ {z_{2} (a) = y_{o}^{{\prime }} (a) = 0} \\ \end{array} } \right\}$	$\left\{ {\begin{array}{*{20}c} {z_{1} (x) = y_{o} (x)} \\ {z_{2} (x) = y_{o} '(x)} \\ \end{array} } \right\}$
	Solve the following homogeneous differential equation	Under prescribed initial conditions	To get the first homogeneous solution, $y_{1} \left( x \right)$
2	$\left\{ {\begin{array}{*{20}c} {z_{1}^{{\prime }} = y_{1}^{{\prime }} (x)} \\ {z_{2}^{{\prime }} = y_{1}^{{\prime \prime }} (x)} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - Q(x)} & { - P(x)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {z_{1} (x) = y_{1} (x)} \\ {z_{2} (x) = y_{1}^{{\prime }} (x)} \\ \end{array} } \right\}$	$\left\{ {\begin{array}{*{20}c} {z_{1} (a) = y_{1} (a) = 1} \\ {z_{2} (a) = y_{1}^{{\prime }} (a) = 0} \\ \end{array} } \right\}$	$\left\{ {\begin{array}{*{20}c} {z_{1} (x) = y_{1} (x)} \\ {z_{2} (x) = y_{1} '(x)} \\ \end{array} } \right\}$
	Solve the following homogeneous differential equation	Under prescribed initial conditions	To get the second homogeneous solution, $y_{2} \left( x \right)$
3	$\left\{ {\begin{array}{*{20}c} {z_{1}^{{\prime }} = y_{2}^{{\prime }} (x)} \\ {z_{2}^{{\prime }} = y_{1}^{{\prime \prime }} (x)} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - Q(x)} & { - P(x)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {z_{1} (x) = y_{2} (x)} \\ {z_{2} (x) = y_{2}^{{\prime }} (x)} \\ \end{array} } \right\}$	$\left\{ {\begin{array}{*{20}c} {z_{1} (a) = y_{2} (a) = 0} \\ {z_{2} (a) = y_{2} '(a) = 1} \\ \end{array} } \right\}$	$\left\{ {\begin{array}{*{20}c} {z_{1} (x) = y_{2} (x)} \\ {z_{2} (x) = y_{2} '(x)} \\ \end{array} } \right\}$
	After determining $y_{o} (x)$, $y_{1} (x)$, and $y_{2} (x)$, form the following CFM solution and its first derivative	By applying physical boundary conditions	To get the remaining unknown constants
4	$y\left( x \right) = y_{0} \left( x \right) + b_{1} y_{1} \left( x \right) + b_{2} y_{2} \left( x \right)$ $y^{{\prime }} \left( x \right) = y_{0}^{{\prime }} \left( x \right) + b_{1} y_{1}^{{\prime }} \left( x \right) + b_{2} y_{2}^{{\prime }} \left( x \right)$	$y\left( a \right) + y^{{\prime }} \left( a \right) = 0$ $y\left( b \right) + y^{{\prime }} \left( b \right) = 0$	$b_{1}$ and $b_{2}$
	as follows $\left[ {\begin{array}{*{20}c} {y_{1} (a) + y_{1}^{{\prime }} (a)} & {y_{2} (a) + y_{2}^{{\prime }} (a)} \\ {y_{1} (b) + y_{1}^{{\prime }} (b)} & {y_{2} (b) + y_{2}^{{\prime }} (b)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {b_{1} } \\ {b_{2} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} { - y_{o} (a) - y_{o}^{{\prime }} (a)} \\ { - y_{o} (b) - y_{0}^{{\prime }} (b)} \\ \end{array} } \right\}$ $b_{1} = \frac{{\left| {\begin{array}{*{20}c} { - y_{o} (a) - y_{o}^{{\prime }} (a)} & {y_{2} (a) + y_{2}^{{\prime }} (a)} \\ { - y_{o} (b) - y_{0}^{{\prime }} (b)} & {y_{2} (b) + y_{2}^{{\prime }} (b)} \\ \end{array} } \right|}}{{\left| {\begin{array}{*{20}c} {y_{1} (a) + y_{1}^{{\prime }} (a)} & {y_{2} (a) + y_{2}^{{\prime }} (a)} \\ {y_{1} (b) + y_{1}^{{\prime }} (b)} & {y_{2} (b) + y_{2}^{{\prime }} (b)} \\ \end{array} } \right|}};\quad b_{2} = \frac{{\left| {\begin{array}{*{20}c} {y_{1} (a) + y_{1}^{{\prime }} (a)} & { - y_{o} (a) - y_{o}^{{\prime }} (a)} \\ {y_{1} (b) + y_{1}^{{\prime }} (b)} & { - y_{o} (b) - y_{0}^{{\prime }} (b)} \\ \end{array} } \right|}}{{\left| {\begin{array}{*{20}c} {y_{1} (a) + y_{1}^{{\prime }} (a)} & {y_{2} (a) + y_{2}^{{\prime }} (a)} \\ {y_{1} (b) + y_{1}^{{\prime }} (b)} & {y_{2} (b) + y_{2}^{{\prime }} (b)} \\ \end{array} } \right|}}$