Elastic-plastic stresses, strains, and displacements have been obtained for a thin rotating annular disk with exponentially variable thickness and exponentially variable density with nonlinear strain hardening material by finite difference method using Von-Mises' yield criterion. Results have been computed numerically and depicted graphically. From the numerical results, it can be concluded that disk whose thickness decreases radially and density increases radially is on the safer side of design as compared to the disk with exponentially varying thickness and exponentially varying density as well as to flat disk.
Due to wide applications of rotating disk, circular disk, spherical shells, cylinders, and shafts in engineering, elastic-plastic analysis of rotating disk, becoming more and more active topic in the field of solid mechanics. The research on them is always an important topic, and their benefits have been included in some books [
The obvious advantage using a linear strain hardening stress-plastic strain is that a closed form solution can be obtained for annular disks with constant thickness and some especially variable thickness functions. However, most of the materials exhibit nonlinear strain hardening behavior; thus, this nonlinearity is obvious in the transition region from elastic to plastic parts of stress-strain curve. Due to the previously reason, a polynomial stress-strain relation of nonlinear strain hardening material is proposed in the papers of You et al. [
In this paper, we proposed a more straightforward and most effective numerical method such as finite difference method which is most celebrated method to solve boundary value problems. The proposed method is used to analyze the stresses, strains, and displacements for annular disk having exponential variable thickness and exponential variable density with nonlinear strain hardening material behavior.
Assuming that the stresses vary over the thickness of the disk, the theory of the disks of variable thickness can give good result as that of the disks of constant thickness as long as they meet the assumption of plane stress. For disk profile, it is assumed that disk is symmetric with respect to the mid plane. This profile is defined by the thickness function
For rotating disk with variable thickness and variable density, the governing equilibrium equation is
The strains and radial displacement are
The equation of compatibility can be derived from (
For plastic deformation, the relation between the stresses and plastic strains can be determined according to the deformation theory of plasticity [
The Von-Mises yield criterion is given by
The total strains are the sum of elastic and plastic strain
By the substitution of (
By the nonlinear strain-hardening material model proposed by You and Zhang [
Substitution of the second equation of (
The governing equation in the plastic region of the rotating disks in terms of stresses and stress function can be obtained by substituting (
The boundary conditions for the rotating annular disks are
To determine the elastic-plastic stresses, strains, and displacement in thin rotating disks with a nonlinear strain hardening material, we have to solve the second-order nonlinear differential equation (
The second-order differential equation ( First, partition the domain To express the differential operators With After simplifying and collecting coefficients of The solution of the
A two-dimensional plane stress analysis of rotating disk with nonuniform thickness and nonuniform density are carried out using finite difference method. The radius of the rotating disk is taken to be
It has been observed from Figures
Radial stress (
Radial stress (
Radial stress (
From all previous analysis, we can conclude that disk whose thickness decreases radially and density increases radially is on the safer side of the design as compared to other thickness and density parameters, because circumferential stress is less for the disk whose thickness decreases radially and density increases radially as compared to other thickness and density parameters.
It has been observed from Figures
Radial stress (
Radial stress (
Radial stress (
From the previous analysis, we observed that disk with high density and less thickness is on the safer side of the design as compared to other parameters because circumferential stress is less for previous case as compared to other disk profiles. The results calculated for variable thickness and variable density
It has been observed from Figures
Radial stress (
After analyzing all the three disk profiles, it can be concluded that circumferential stresses are maximum at internal surface. It is also concluded that disk whose thickness decreases radially and density increases radially is on the safer side of design as compared to the disk with exponentially varying thickness and exponentially varying density as well as to flat disk. This is because circumferential stresses for the disks whose thickness decreases and density increases radially are less as compared to the disk with exponentially varying thickness and exponentially varying density as well as to flat disk.