Theoretical treatment of disc cutters subjected to general cutting forces

Abstract

This paper constructs a theoretical framework to investigate the stress and deformation state within the body of a roller disc cutter from the cutter characteristics (radii of the shaft and disc and the length of the shaft), the characteristics of the cut (depth of cut), the mechanical properties of the constructive material (Poisson’s ratio and Young modulus) and the boundary conditions involved during the disc cutting process. To pursue the objectives, mathematical formulations and calculations required for the analysis of elastic variables induced in the disc and shaft parts of a roller disc cutter subjected to general cutting forces are investigated. Assuming elastic material behaviour, the disc cutter in contact with its shaft is considered as a partially loaded annulus, and the shaft inside the disc as a laterally loaded cylinder subjected to stresses corresponding to different cutting configurations. The Airy stress function, Michell’s method and Papkovich–Neuber solutions are employed in two- and three-dimensional elastic domains using a double Fourier series technique.

Appendices

Appendix: Elastic field variables in partially loaded hollow discs with $\alpha =\beta$

Stress components for a particular case of a partially loaded annulus when $\alpha =\beta$ discussed in [23] are further verified as follows:

$$\begin{aligned} \sigma _{rr}= & {} -\left( \frac{ba}{r^2}+1\right) \frac{ap\,\beta }{(a+b)\,\pi }\nonumber \\&-\,{\frac{ ( \kappa -1 ) {r}^{4}+ ( {a}^{2}+{b}^{2} ) ( 3+\kappa ) {r}^{2}+{b}^{2}{a}^{2} ( \kappa -1 ) }{\pi \,{r}^{3} ( {a}^{2}+{b}^{2} ) ( 1+\kappa ) } \,ap \sin ( \beta ) \cos ( \theta ) } \nonumber \\&-\sum _{n=2}^ \infty \frac{A^{\prime \prime }(n-2) r^{2n+2}- B^{\prime \prime }n r^{2n}+ C^{\prime \prime }(n+2) r^2+D^{\prime \prime }n}{\pi \, F^{\prime \prime } \, n\, r^{n+2}}\,ap \sin (n \beta ) \cos ( n \theta ),\quad \quad \end{aligned}$$

(72)

$$\begin{aligned} \sigma _{\theta \theta }= & {} \left( \frac{ba}{r^2}-1\right) \frac{ap\,\beta }{(a+b)\,\pi }\nonumber \\&-\,{\frac{ ( {a}^{2}+{b}^{2} ) {r}^{2}+{b}^{2}{a}^{2} -3\, {r}^{4}}{\pi \,{r}^{3} ( {a}^{2}+{b}^{2} ) ( 1+\kappa ) }( 1-\kappa ) \,ap \sin ( \beta )\cos ( \theta )} \nonumber \\&+\sum _{n=2}^ \infty \frac{A^{\prime \prime } (n+2)r^{2n+2}-B^{\prime \prime }n r^{2n}+ C^{\prime \prime }(n-2) r^2+D^{\prime \prime }n}{\pi \, F^{\prime \prime } \, n\,r^{n+2}}\,ap \sin (n \beta ) \cos ( n \theta ),\quad \quad \end{aligned}$$

(73)

$$\begin{aligned} \tau _{r\theta }= & {} {\frac{ (b^2-r^2)(a^2-r^2) }{\pi \,{r}^{3} ( {a}^{2}+{b}^{2} ) ( 1+\kappa ) } ( 1-\kappa )\,ap\sin ( \beta )\sin ( \theta ) } \nonumber \\&+\sum _{n=2}^ \infty \frac{A^{\prime \prime } r^{2n+2}-B^{\prime \prime } r^{2n}-C^{\prime \prime } r^2-D^{\prime \prime }}{\pi \, F^{\prime \prime } \, r^{n+2}}\,a p \sin (n \beta ) \sin ( n \theta ),\quad \quad \quad \quad \quad \quad \quad \quad \quad \end{aligned}$$

(74)

where

$$\begin{aligned} A^{\prime \prime }= & {} b^{n-1}-a^{n-1}, \nonumber \\ B^{\prime \prime }= & {} b^{n+1}-a^{n+1},\nonumber \\ C^{\prime \prime }= & {} a^{2n}b^{n-1}-a^{n-1}b^{2n},\nonumber \\ D^{\prime \prime }= & {} a^{n+1}b^{2n}-a^{2n}b^{n+1},\nonumber \\ F^{\prime \prime }= & {} a^{2n}+n\,a^{n-1}b^{n-1}(a^2-b^2)-b^{2n}. \end{aligned}$$

(75)

Tensor calculus in general curvilinear coordinates

In a generalised curvilinear coordinate system, $\text {Y}$ identified by its covariant and contravariant local basis vectors $\varvec{e}_{i}$ and $\varvec{e}^{i}$, respectively, the process of the divergence ($\nabla \cdot$), del ($\nabla$) and Laplacian ($\nabla ^2$) operators, operating possibly on either a scalar ($\phi$) or a vector ($\varvec{u}$), is guaranteed if they satisfy the following generic rules and relations: [31–35]:

$$\begin{aligned} \nabla \phi= & {} \varvec{e}^{j} \delta _{j}^{i} D_{i} \phi =\varvec{e}^{i} D_{i} \phi =g^{ij}\varvec{e}_jD_i\phi , \end{aligned}$$

(76)

$$\begin{aligned} \nabla \varvec{u}= & {} \varvec{e}^{i} D_{i} \big (\varvec{e}_{j} u^{j} \big )=\varvec{e}^{i} \varvec{e}_{j} D_{i} u^{j} +\varvec{e}^{i} u^{j} D_{i} \varvec{e}_{j}=\varvec{e}^{i} \varvec{e}_{j} \big (D_{i} u^{j} +u^{k} \Gamma _{ik}^{j} \big )=\varvec{e}^{i} \varvec{e}^{j} \big (D_{i} u_{j} -u_{k} \Gamma _{ij}^{k}\big ), \end{aligned}$$

(77)

$$\begin{aligned} \nabla \cdot \varvec{u}= & {} \varvec{e}^{i}\cdot \varvec{e}_{j} \big (D_{i} u^{j} +u^{k} \Gamma _{ik}^{j} \big )=\delta _{j}^{i} \big (D_{i} u^{j} +u^{k} \Gamma _{ik}^{j} \big ) =\big (D_{i} +\Gamma _{ik}^{k}\big )u^{i} =\big (D_{i} +\frac{1}{\sqrt{g}} D_{i} \sqrt{g} \big )u^{i} \nonumber \\= & {} \frac{1}{\sqrt{g}}D_i\big (g^{ij}\sqrt{g}u_j\big ), \end{aligned}$$

(78)

$$\begin{aligned} \nabla ^{2} \phi= & {} \nabla \cdot \nabla \phi =\nabla \cdot \varvec{u}=\big (D_{i} +\frac{1}{\sqrt{g}} D_{i} \sqrt{g} \big )g^{ij} u_{j} =\frac{1}{\sqrt{g} } D_{i} \big (g^{ij} \sqrt{g} D_{j} \phi \big ), \end{aligned}$$

(79)

$$\begin{aligned} \nabla ^{2} \varvec{u}= & {} \nabla \cdot \nabla \varvec{u}=\varvec{e}^{i} D_{i} \cdot \big (\varvec{e}^{j} \varvec{e}_{k} D_{j} u^{k} +\varvec{e}^{j} u^{k} D_{j} \varvec{e}_{k} \big )=\varvec{e}^{i} D_{i} \cdot \big [\varvec{e}^{j} \varvec{e}_{k} \big (D_{j} u^{k} +u^{l} \Gamma _{jl}^{k} \big )\big ], \end{aligned}$$

(80)

$$\begin{aligned} \nabla ^{2} \varvec{u}= & {} \big (\varvec{e}^{i} D_{i} \varvec{e}^{j} \big ) \cdot \big (\varvec{e}_{k} D_{j} u^{k} +u^{k} D_{j} \varvec{e}_{k}\big )+\varvec{e}^{i} \cdot \varvec{e}^{j} \big (D_{i} \varvec{e}_{k} D_{j} u^{k} +\varvec{e}_{k} D_{i} D_{j} u^{k}+D_{i} u^{k} D_{j} \varvec{e}_{k} +u^{k} D_{i} D_{j} \varvec{e}_{k} \big ), \end{aligned}$$

(81)

$$\begin{aligned} \nabla ^{2} \varvec{u}= & {} \varvec{e}^{i} \cdot \big (D_{i} \varvec{e}^{j}\big ) \varvec{e}_{k} \big (D_{j} u^{k} +u^{l} \Gamma _{jl}^{k}\big )+g^{ij} \big (D_{i} \varvec{e}_{k} \big )\big (D_{j} u^{k} + u^{l} \Gamma _{jl}^{k}\big )+g^{ij} \varvec{e}_{k} D_{i} \big (D_{j} u^{k} +u^{l} \Gamma _{jl}^{k}\big ), \end{aligned}$$

(82)

$$\begin{aligned} \nabla ^{2}\varvec{u}= & {} \big [g^{ij} D_{i} \big (D_{j} u^{k} +u^{l} \Gamma _{jl}^{k} \big ) + g^{ij} \Gamma _{im}^{k} \big (D_{j} u^{m} +u^{l} \Gamma _{jl}^{m}\big )- g^{mi} \Gamma _{im}^{j} \big (D_{j} u^{k} +u^{l} \Gamma _{jl}^{k}\big )\big ]\,\varvec{e}_{k}, \end{aligned}$$

(83)

where $\delta ^{i}_{j}$ is the Kronecker delta, $\Gamma$ is the familiar Christoffel symbol, $g_{ij}$ are the components of the coordinate covariant metric tensor, $g=\text {det}[g_{ij}]$, $i,j,k,l,m=1,2,3$, and the Einstein’s summation convention is applied. However, it should be noted that $\varvec{e}^i$ (or $\varvec{e}_i$) in Eqs. (76)–(83) does not generally result in basis vectors with unitary length, as occurs in Cartesian coordinates. Coordinate scale factors ($h_i$) are thus to be used to normalise both covariant and contravariant vector components independent of their length, so that $\hat{\varvec{e}}$ is the normalised unit vector form of $\varvec{e}$, viz. Therefore, the gradient Eq. (76) and divergence Eq. (78) can also take the following forms:

$$\begin{aligned} \nabla \phi= & {} g^{ij}h_j\varvec{\hat{e}}_jD_i\phi , \end{aligned}$$

(84)

$$\begin{aligned} \nabla \cdot \varvec{u}= & {} \frac{1}{\sqrt{g}}D_i\Big (\frac{\sqrt{g}}{h_i}\hat{u}^{i}\Big ). \end{aligned}$$

(85)

Solutions of Laplace’s equation in cylindrical polar coordinates

In order to find the general solution of Laplace’s relation in Eqs. (38) and (39), the method of separation of variables can be employed to reduce the resultant PDEs to ODEs [34, 36]. The Laplacian of a harmonic scalar $\phi$ in cylindrical coordinates is obtained by solving the following linear partial differential equation:

$$\begin{aligned} \phi _{,rr}+\phi _{,zz}+\frac{\phi _{,r}}{r}+\frac{\phi _{,\theta \theta }}{r^2}=0. \end{aligned}$$

(86)

It follows that $\phi$ takes either the form

$$\begin{aligned} \phi (r,\theta , z)= \sum _{m=0}^\infty \sum _{n=0}^\infty \big (A_\mathrm{n}\sin (\zeta _n z)+ B_\mathrm{n}\cos (\zeta _n z)\big )\big (A_m\sin (m\theta )+ B_m\cos (m\theta )\big )\big ( a_{mn}I_m(\zeta _n r)+ b_{mn}K_m(\zeta _n r)\big ),\nonumber \\ \end{aligned}$$

(87)

or

$$\begin{aligned} \phi (r,\theta , z)= \sum _{m=0}^\infty \sum _{n=0}^\infty \left( A_\mathrm{n} e^{\zeta _n z}+ B_n e^{-\zeta _n z}\right) \big (A_m\sin (m\theta )+ B_m\cos (m\theta )\big ) \big (a_{mn}J_m\left( \zeta _n r\right) + b_{mn}Y_m\left( \zeta _n r\right) \big ). \end{aligned}$$

(88)

Here, $I_m(\zeta _n r)$ and $K_m(\zeta _n r)$ denote the modified Bessel function of the first and the second kinds of order m and the non-negative argument $\zeta _n r$, respectively; $J_m(\zeta _n r)$ is the Bessel function of the first kind; $Y_m(\zeta _n r)$ is the Bessel function of the second kind; and $A_{n}$, $A_m,\ldots$ are the undetermined integral constants for each m and n. It should be pointed out that the terms $K_m(\zeta _n r)$ and $Y_m(\zeta _n r)$ are singular at the origin and must be discarded for bounded domains. However, for problems in which the boundary conditions are given by Fourier series, e.g. in the form of Eq. (52), the former general solution in Eq. (87) to the Laplace equation (86) is eventually accepted. For special cases of a solid cylinder with the origin included, this becomes

$$\begin{aligned} \phi (r,\theta , z)= & {} \sum _{n=1}^\infty A_{0n}\, I_0(\zeta _n r)\sin (\zeta _n z)+ B_{0n} \,I_0(\zeta _n r) \cos (\zeta _n z) \\&+\sum _{m=1}^\infty \sum _{n=1}^\infty A_{mn}\, I_m(\zeta _n r)\sin (\zeta _n z)\sin (m\theta )+ B_{mn}\, I_m(\zeta _n r)\sin (\zeta _n z)\cos (m\theta )\nonumber \\&+\sum _{m=1}^\infty \sum _{n=1}^\infty C_{mn}\, I_m(\zeta _n r)\cos (\zeta _n z)\sin (m\theta )+D_{mn}\, I_m(\zeta _n r)\cos (\zeta _n z)\cos (m\theta ).\nonumber \end{aligned}$$

(89)

In the same manner and remembering that the Cartesian components $\psi _x$, $\psi _y$ and $\psi _z$ of a harmonic vector $\varvec{\psi }{(x,y,z)}$ are all harmonic [21, 25, 37], the Laplace operator on non-harmonic components, $\psi _r$ and $\psi _\theta$, in cylindrical coordinates is related to the appropriate Laplace equation on Cartesian components according to the geometric relationships between the two systems and the classical principles of conversion of a vector from rectangular to cylindrical coordinates. This means,

$$\begin{aligned} \nabla ^2\psi _x= & {} \nabla ^2\big (\psi _r\cos (\theta )-\psi _\theta \sin (\theta )\big )=0, \end{aligned}$$

(90)

$$\begin{aligned} \nabla ^2\psi _y= & {} \nabla ^2\big (\psi _r\sin (\theta )+\psi _\theta \cos (\theta )\big )=0. \end{aligned}$$

(91)

With Eqs. (90) and (91) in mind, acceptable trial functions of $\psi _r$ and $\psi _\theta$ can be investigated to satisfy the distribution of any prescribed field variables given for a problem. For instance, if the hoop stress ($\sigma _{\theta \theta }$) of a target problem has only $\sin (\zeta _n z)\sin (m\theta )$ terms at $r=r_0$, suitable choices of $\psi _r$ and $\psi _\theta$ would be

$$\begin{aligned} \psi _r= & {} \sum _{m=1}^\infty \sum _{n=1}^\infty \big ( f(r)_{mn}+g(r)_{mn}\big )\sin (\zeta _n z)\sin (m\theta ), \end{aligned}$$

(92)

$$\begin{aligned} \psi _\theta= & {} \sum _{m=1}^\infty \sum _{n=1}^\infty \big (f(r)_{mn}-g(r)_{mn}\big )\sin (\zeta _n z)\cos (m\theta ), \end{aligned}$$

(93)

where $f(r)_{mn}$, $g(r)_{mn}$ are only functions of r for each m and n. The reason for selecting such trial functions is seen in Eq. (42), in which $\psi _r$ and its first derivative with respect to r ($\psi _{r,\,r}$) and its second derivative with respect to $\theta$ ($\psi _{r,\,\theta \theta }$) all participate in calculating the hoop stress. In contrast, only the first derivative of $\psi _\theta$ with respect to $\theta$ contributes to the hoop stress formulation. Consequently, if the trial functions of $\psi _r$ and $\psi _\theta$ defined in Eqs. (92) and (93) are substituted in Eq. (42), it can be readily checked that the same target trigonometric terms of $\sin (\zeta _n z)\sin (m\theta )$ are obtained. Equation (90) therefore yields

$$\begin{aligned} \nabla ^2\psi _x= & {} \nabla ^2\Bigg [\sum _{m=1}^\infty \sum _{n=1}^\infty \big (f(r)_{mn}+g(r)_{mn}\big )\sin (\zeta _n z)\sin (m\theta )\cos (\theta )\nonumber \\&-\big (f(r)_{mn}-g(r)_{mn}\big )\sin (\zeta _n z)\cos (m\theta )\sin (\theta )\Bigg ]=0. \end{aligned}$$

(94)

Since $\nabla ^2\psi _x=0$, the solution to Eq. (94) is recognised as various forms of modified Bessel’s equation leading to

$$\begin{aligned} f(r)_{mn}= & {} \sum _{m=1}^\infty \sum _{n=1}^\infty A^{\prime \prime }_{mn}\,I_{m-1}(\zeta _n r)+a^{\prime \prime }_{mn}\,K_{m-1}(\zeta _n r), \end{aligned}$$

(95)

$$\begin{aligned} g(r)_{mn}= & {} \sum _{m=1}^\infty \sum _{n=1}^\infty A^{\prime }_{mn}\,I_{m+1}(\zeta _n r)+a^{\prime }_{mn}\,K_{m+1}(\zeta _n r), \end{aligned}$$

(96)

where $A^{\prime }_{mn}$, $A^{\prime \prime }_{mn},\cdots$ are yet unknown arbitrary integration constants to be determined from a given boundary condition, and $K(\zeta _nr)$ should be excluded for the case of a bounded problem. Substituting Eqs. (95) and (96) into Eqs. (92) and (93) gives

$$\begin{aligned} \psi _r = \sum _{m=1}^\infty \sum _{n=1}^\infty \left[ \left( A^{\prime }_{mn} \, I_{m+1}\left( \zeta _n r\right) +A^{\prime \prime }_{mn} \, I_{m-1}\left( \zeta _n r\right) \right) +\left( a^{\prime }_{mn} \, K_{m+1}\left( \zeta _n r\right) +a^{\prime \prime }_{mn} \, K_{m-1}\left( \zeta _n r\right) \right) \right] \sin \left( \zeta _n z\right) \sin (m\theta ), \nonumber \\\end{aligned}$$

(97)

$$\begin{aligned} \psi _\theta = \sum _{m=1}^\infty \sum _{n=1}^\infty \left[ \left( A^{\prime \prime }_{mn} \,I_{m-1}\left( \zeta _n r\right) -A^{\prime }_{mn} \, I_{m+1}\left( \zeta _n r\right) \right) +\left( a^{\prime \prime }_{mn} \, K_{m-1}\left( \zeta _n r\right) -a^{\prime }_{mn} \, K_{m+1}\left( \zeta _n r\right) \right) \right] \sin \left( \zeta _n z\right) \cos (m\theta ). \nonumber \\ \end{aligned}$$

(98)

In a similar fashion, acceptable vector components $\psi _r$ and $\psi _\theta$ for bounded problems are determined in the general forms of Eqs. (99) and (100).

$$\begin{aligned} \psi _r= & {} \sum _{n=1}^\infty A^{\prime }_{0n}\,I_1(\zeta _n r)\sin (\zeta _n z)+B^{\prime }_{0n}\,I_1(\zeta _n r)\cos (\zeta _n z) \nonumber \\&+\sum _{m=1}^\infty \sum _{n=1}^\infty \Big (A^{\prime }_{mn} \, I_{m+1}(\zeta _n r)+A^{\prime \prime }_{mn} \, I_{m-1}(\zeta _n r)\Big )\sin (\zeta _n z)\sin (m\theta ) \nonumber \\&+ \sum _{m=1}^\infty \sum _{n=1}^\infty \Big (B^{\prime }_{mn} \, I_{m+1}(\zeta _n r)+B^{\prime \prime }_{mn} \, I_{m-1}(\zeta _n r)\Big )\sin (\zeta _n z)\cos (m\theta )\nonumber \\&+ \sum _{m=1}^\infty \sum _{n=1}^\infty \Big (C^{\prime }_{mn} \, I_{m+1}(\zeta _n r)+C^{\prime \prime }_{mn} \, I_{m-1}(\zeta _n r) \Big )\cos (\zeta _n z)\sin (m\theta )\nonumber \\&+ \sum _{m=1}^\infty \sum _{n=1}^\infty \Big (D^{\prime }_{mn} \, I_{m+1}(\zeta _n r)+D^{\prime \prime }_{mn} \, I_{m-1}(\zeta _n r) \Big )\cos (\zeta _n z)\cos (m\theta ), \end{aligned}$$

(99)

$$\begin{aligned} \psi _\theta= & {} \sum _{n=1}^\infty A^{\prime \prime }_{0n}\,I_1(\zeta _n r)\sin (\zeta _n z)+B^{\prime \prime }_{0n}\,I_1(\zeta _n r)\cos (\zeta _n z) \nonumber \\&+\sum _{m=1}^\infty \sum _{n=1}^\infty \Big (B^{\prime }_{mn} \, I_{m+1}(\zeta _n r)-B^{\prime \prime }_{mn} \, I_{m-1}(\zeta _n r) \Big )\sin (\zeta _n z)\sin (m\theta ) \nonumber \\&+ \sum _{m=1}^\infty \sum _{n=1}^\infty \Big (A^{\prime \prime }_{mn} \, I_{m-1}(\zeta _n r)-A^{\prime }_{mn} \, I_{m+1}(\zeta _n r) \Big )\sin (\zeta _n z)\cos (m\theta )\nonumber \\&+ \sum _{m=1}^\infty \sum _{n=1}^\infty \Big (D^{\prime }_{mn} \, I_{m+1}(\zeta _n r)-D^{\prime \prime }_{mn} \, I_{m-1}(\zeta _n r) \Big )\cos (\zeta _n z)\sin (m\theta )\nonumber \\&+ \sum _{m=1}^\infty \sum _{n=1}^\infty \Big (C^{\prime \prime }_{mn} \, I_{m-1}(\zeta _n r) -C^{\prime }_{mn} \, I_{m+1}(\zeta _n r)\Big )\cos (\zeta _n z)\cos (m\theta ). \end{aligned}$$

(100)