An analytical prediction model for residual stress distribution and plastic deformation depth in ultrasonic-assisted single ball burnishing process

Abstract

Mechanical and metallurgical characteristics of the surface layers are modified as the material is subjected to the burnishing process. Plastic deformation is known as a major reason for property enhancement of the surface and subsurface layers. Residual stress distribution and influenced depth of plastic deformation provide useful information regarding the functionality and life cycles of the burnished sample. In the present study, a novel analytical approach was presented to predict residual stress distribution and the depth of plastic defamation in the ultrasonic-assisted ball burnishing process. The burnishing process was firstly analyzed using the contact mechanic of an elastic sphere with semi-infinite body theorem. Then, the plastic deformation and residual stress were modeled using the McDowell algorithm. The model could incorporate effects of vibration amplitude and frequency, static pressure, feed rate, and ball dimensions. A series of ultrasonic-assisted ball burnishing experiments have been carried out on aluminum 6061-T6 and AISI 1045 steel to confirm the proposed model prediction results. The prediction accuracy of the proposed model was further verified by residual stress distributions of AISI 304, Ti-6Al-4V, and Inconel 718 from other literatures. The research findings in this study indicated that the developed model could be used for a variety of engineering materials in the prediction of residual stress with adequate precision.

Appendices

Appendix 1. Calculation of maximum static and dynamic contact pressure

1.1 Analysis of loading

Static load

Based on Hertizan contact mechanic problems, contact or impact of two spheres is a general form of the contact between a sphere and flat surface. While the radius of one sphere tends to be infinite, the model would be identical. Considering the ball as rigid material and infinter radius of flat surface, the total displacement through radius (as shown in Fig. 19) is determined:

$$ {\overline{u}}_z(r)=\delta -\left(\frac{1}{2R}\right){r}^2 $$

(37)

where U_z(r) is radius-dependent axial displacement, δ is the maximum displacement in the contact center, and R is the radius of the sphere.

In the above equation, u_z(r) is the vertical displacement of the ball in sheet material in radial direction, δ is the displacement of the ball in the sheet at the center of contact, and R is the ball diameter.

Using Hertz contact, the radius-dependent pressure distribution and pressure-dependent displacement can be obtained as [16]

$$ P(r)={P}_0{\left[1-{\left(\frac{r}{r_{\mathrm{e}}}\right)}^2\right]}^{\frac{1}{2}} $$

(38)

$$ {\overline{u}}_z(r)=\frac{\pi {P}_0}{4{E}_{\mathrm{e}\mathrm{q}}{r}_{\mathrm{e}}}\left(2{r_{\mathrm{e}}}^2-{r}^2\right),\kern0.5em r\le {r}_{\mathrm{e}} $$

(39)

where P(r) is through radius pressure distribution, P₀ is the maximum pressure at contact midpoint, and r_e is the maximum elastic radius. E_eq is the equivalent Young modulus of ball and work material that depends on the Young modulus of the workpiece and ball (i.e., E₁ and E₂) as well their Poisson ratio (ϑ₁ and ϑ₂).

$$ \frac{1}{E_{\mathrm{eq}}}=\frac{1-{\vartheta_1}^2}{E_1}+\frac{1-{\vartheta_2}^2}{E_2} $$

(40)

Considering Eqs. 39 and 37 as identical, some geometrical characteristics of contact are identified. The boundary condition is that the displacement in the maximum elastic radius is zero. (r = re → U_z(r) = 0).

$$ \delta =\frac{\pi {r}_{\mathrm{e}}{P}_0}{2{E}_{\mathrm{e}\mathrm{q}}} $$

(41)

$$ {r}_{\mathrm{e}}=\frac{\pi {P}_0R}{2{E}_{\mathrm{e}\mathrm{q}}} $$

(42)

$$ {r}_{\mathrm{e}}=\sqrt{\delta R} $$

(43)

Considering the circular shape of the contact, the applied force which leads to pressure distribution is obtained by

$$ F=\underset{0}{\overset{r_e}{\int }}P(r)2\pi rdr=\frac{2}{3}\pi {r_{\mathrm{e}}}^2{P_0}^s $$

(44)

By substituting Eq. 42 in 44 and elimination of r_e, the relationship for obtaining maximum pressure as a result of static force is obtained.

$$ {P_0}^s=\frac{1}{\pi }{\left(\frac{6F{E_{\mathrm{eq}}}^2}{R^2}\right)}^{\frac{1}{3}} $$

(45)

Dynamic load

The dynamic load in ultrasonic-assisted burnishing is due to high-frequency vibration of the tool which is exerted on the surface. The displacement and velocity of the tool as a result of ultrasonic oscillation are obtained by

$$ {\displaystyle \begin{array}{l}X=A\sin \left(2\pi ft\right)\\ {}V=\frac{dX}{dt}=2\pi fA\cos \left(2\pi ft\right)\end{array}} $$

(46)

where X and V are the displacement and velocity, respectively. Also, vibration amplitude and frequency are denoted by A and f, respectively.

During the ultrasonic-assisted burnishing, the impact of the vibratory ball and the surface of the sample occurs when the velocity is maximum, i.e., V_max = 2πAf.

By writing work and energy principle, the displacement as a result of dynamic load is obtained.

$$ {\displaystyle \begin{array}{c}\frac{1}{2}m{V_{\mathrm{max}}}^2=\underset{0}{\overset{\delta^d}{\int }} Fd\delta =\underset{0}{\overset{\delta^{\ast }}{\int }}\left(\frac{4{E}_{\mathrm{eq}}}{3}\right){R}^{\frac{1}{2}}{\delta}^{\frac{3}{2}} d\delta \\ {}{\delta}^d=\frac{D}{2}{\left[\frac{5\rho {\pi}^3{A}^2{f}^2}{E_{\mathrm{eq}}}\right]}^{\frac{2}{5}}\end{array}} $$

(47)

where D is the ball diameter and ρ is the ball density.

The relationship between maximum contact pressure, maximum contact radius, and maximum contact depth which were used for static load can be used for dynamic load. This by substituting Eq. 47 into Eq. 41 to 43, the pressure, radius, and depth as a result of dynamic load are also established.

$$ {r_{\mathrm{e}}}^d=\frac{D}{2}{\left[\frac{5\rho {\pi}^3{A}^2{f}^2}{E_{\mathrm{e}\mathrm{q}}}\right]}^{\frac{1}{5}} $$

(48)

$$ {P_0}^d=\frac{4}{\pi }{\left[5\rho {\pi}^3{E}_{\mathrm{eq}}{A}^2{f}^2\right]}^{\frac{1}{5}} $$

(49)

Appendix 2. Identifying principal strains

The principal strains are the roots of cubic equations as follows:

$$ {\varepsilon}^3-{I}_1{\varepsilon}^2-{I}_2\varepsilon -{I}_3=0 $$

(50)

where the I₁, I₂, and I₃ are the stress invariants which are calculated by the following equations:

$$ {\displaystyle \begin{array}{c}{I}_1={\varepsilon}_{xx}+{\varepsilon}_{yy}+{\varepsilon}_{zz}\\ {}{I}_2={\gamma}_{xy}^2+{\gamma}_{yz}^2+{\gamma}_{zx}^2-\left({\varepsilon}_{xx}{\varepsilon}_{yy}+{\varepsilon}_{yy}{\varepsilon}_{zz}+{\varepsilon}_{zz}{\varepsilon}_{xx}\right)\\ {}{I}_3={\varepsilon}_{xx}{\varepsilon}_{yy}{\varepsilon}_{zz}+2{\gamma}_{xy}{\gamma}_{yz}{\gamma}_{zx}-\left({\varepsilon}_{xx}{\gamma}_{yz}^2+{\varepsilon}_{yy}{\gamma}_{zx}^2+{\varepsilon}_{zz}{\gamma}_{xy}^2\right)\end{array}} $$

(51)

According to the plane strain condition, the constant values are modified as follows:

$$ {\displaystyle \begin{array}{c}{I}_1={\varepsilon}_{xx}+{\varepsilon}_{zz}\\ {}{I}_2={\gamma}_{zx}^2-{\varepsilon}_{zz}{\varepsilon}_{xx}\\ {}{I}_3=0\end{array}} $$

(52)

Fig. 19

Schematic illustration of elastic contact of two spheres including geometrical characteristics [16]