1. Introduccian

2. Equations of Motion

there is a hollow disk made of functionally graded materials, which is sufficiently thin and large, with inner radius a and outer radius b, rotating at an angular velocity $\omega$. Considering the geometry of the problem, the formulation and analysis are performed in cylindrical coordinates $(z,\theta ,r)$. The thickness of the disk section, elastic modulus, density, and yield stress are assumed to be power functions (1) of the radial coordinate:

(1)

In order,$h_0$ the values of thickness, Young’s modulus $E_0$ are given by equations (1). ${\rho }_0$density, and ${\sigma }_{Y_0}$ yield stress at the outer radius $(r=b)$ are also provided. The parameters $n_t$,$n_E$,$n_{\rho }$ and $n_{\sigma }$ represent the geometric and material properties and are defined as per equations (2).

(2)

The equation of motion for the rotating disk, considering the effect of thickness, is expressed as equation (3).

(3)

The components of radial and hoop stresses, ${\sigma }_r$ and ${\sigma }_{\theta }$, in equation (3) are represented. It should be noted that the volumetric force due to weight ($\rho g$) has been neglected. Now, the radial and hoop displacements, u and v, are considered. Due to axial symmetry, there is no hoop displacement, and in other words, $v=0$. Therefore, the stress equations in cylindrical coordinates are as follows:

(4)

Using Hooke’s law for the state of plane stress and displacement, strain equations (4) are employed, and the stress-displacement relationship is obtained in the form of equations (5).

(5)

By substituting the stress components into equation (3), the equation of motion the disk is transformed into the form of equation (6).

(6)

The analytical solution of the second-order differential equation (6) has a general solution in the form of equation (7):

(7)

In which $C_1$ and $C_2$ are integration constants. Also, the constant parameters in equation (7) are defined as equations (8).

(8)

By substituting the displacement equation (7) into the strain equations (5), the radial and hoop stresses are obtained in terms of the constants $C_1$ and $C_2$ as equations (9).

(9)

The constants $C_1$ and $C_2$ are obtained. For a hollow disk, the radial stress at the inner boundaries ${\left({\sigma }_r\right)}_{r=a}$ and the exterior ${\left({\sigma }_r\right)}_{r=b}$ is zero. Therefore, by applying the boundary conditions for the disk, the constants $C_1$ and $C_2$ are obtained in equation (10).

(10)

In which R is a constant parameter defined in equation (10) as follows:

(11)

In order to obtain general solutions, the derived equations are made dimensionless using equations (12).

(12)

3. Analyze yielding state

In order to determine the angular velocities corresponding to the yielding threshold and examine the yielding conditions, the Tresca yielding criterion has been employed. The utilization of the Tresca yielding criterion necessitates establishing the order of principal stresses ${\sigma }_r$ and ${\sigma }_{\theta }$. On the other hand, the order of principal stresses is dependent on the numerical values of the power parameters ($n_t$, $n_p$,$n_E$ and $n_{\sigma }$) and the ratio of radii $(r/b)$. Therefore, for monitoring the initiation of yielding, a dimensionless variable $\Psi$ based on the Tresca criterion is utilized. This variable is defined by the equation (13).

(13)

3.1. State 1: Initiating yielding from the inner radius

Yielding occurs when $\Psi \left(\overline{a}\right)=1$ and the function $\Psi$ has its maximum absolute value at the inner radius. By utilizing the stress equations (9) and applying the boundary condition ${\left({\overline{\sigma }}_r\right)}_{\overline{r}=\overline{a}=0}$, the dimensionless terminal rotational velocity ${\Omega }_{e1}$ can be defined as per the equation (14):

(14)

The constant S is defined as equation (15).

(15)

Where H is expressed as equation (16).

(16)

3.2. State 2: Initiating yielding from the outer radius

In this case, yielding occurs when $\Psi \left(1\right)=1$ and the function $\Psi$ has its maximum absolute value at the outer radius. By using the stress equations (9) and applying the boundary condition ${\left({\overline{\sigma }}_r\right)}_{\overline{r}=\overline{b}}=0$, the dimensionless terminal rotational velocity ${\omega }_{e_2}$ is defined as an equation (17):

(17)

In which the constant $S_b$ is defined as equation (18):

(18)

3.3. State 3: Simultaneous yielding initiation from both inner and outer radii.

In this case, yielding commences simultaneously from the inner and outer radii when both $\Psi \left(\overline{a}\right)=1$ and $\Psi \left(1\right)=1$ are satisfied, and the function $\Psi$ must have maximum absolute values at both the inner and outer radii. Considering the boundary conditions of the disc and using equations (14) and(17), the critical dimensionless angular velocity ${\omega }_{c_r}$ and critical power parameter $n_{c_r}$ can be obtained by solving the system of equations (19).

(19)

3.4. State 4: Yielding initiation from a radius between the inner and outer radii.

To initiate yielding from a radius between the inner and outer radii, the dimensionless function $\Psi$ at the yielding initiation point $r_{ep}$ must be equal to 1 and have the maximum absolute value at this point. For this reason, equation (20) must hold true to initiate yielding from the boundary between the inner and outer radii.

(20)

Equation (20) is used to calculate the required terminal rotational velocity for initiating plastic flow and the location of yielding initiation point ${\overline{r}}_{ep}$ for a specific power parameter n.

4. Conclusions

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