Strain Energy During Mechanical Milling: Part I. Mathematical Modeling

Abstract

In this study, we formulate a mathematical model that can be implemented to calculate the amount of strain energy both introduced to (U _i) and stored in (U _s) metal powders during mechanical milling. The theoretical analysis presented in this study proposes that the strain energy is primarily induced by normal and shear strains, and moreover, that the contributions from torsion can be neglected. This theoretical framework was implemented to evaluate the influence of various mechanical milling processing parameters on U _i and U _s. The calculated results show that the magnitude of U _i increases with increases in the following processing parameters: attritor diameter, impeller’s rotational frequency, and ball-to-powder mass ratio, and the magnitude of U _i increases with a decrease in diameter of the milling media. The percentage of the shear strains’ contribution to the total U _i is insensitive to the mechanical milling processing parameters, varying within the range of 35 pct to 42 pct. The calculated magnitude of U _s ranges from a few to a few tens of joules per gram, which is three to four orders of magnitude lower than that of the calculated U _i. Although with respect to the mechanical milling processing parameters, the calculated U _s has trends similar to those for U _i, the changing rates of U _s are much lower than those for U _i.

Appendices

Appendix A

1.1 The Strain Energy Contributed by the Normal Stresses σ _z and σ _r

A small unit cell is taken from the entrapped powder aggregate, which is subjected to three-dimensional compressive stresses σ _z , σ _r , and σ _θ,[23] as shown in Figure 2. Assuming axisymmetric flow, ε _r = ε _θ , where ε _r and ε _θ are normal strains corresponding to σ _r and σ _θ , respectively, and hence, σ _r  = σ _θ . Using the von Mises yield criterion, σ _r  − σ _z  = Y. According to Reference 23

$$ \sigma_{z} = Y\exp \left[ {\frac{2\mu }{h}(r_{\text{E}} - r)} \right] $$

(A1)

where r is the distance away from the center of the entrapped powder aggregate that is assumed as cylindrical geometry. The strain energy introduced by σ _z is analyzed by considering a ring of r in the distance away from the center, dr in width and h in thickness. The work on the ring contributed by σ _z is σ _z  · 2πrdr · Δh. Therefore, the strain energy introduced to the entrapped powders by σ _z can be expressed as

$$ u_{{{\text{E}},\sigma_{z} }} = \int_{0}^{{r_{E} }} {2\pi r\Updelta h\sigma_{z} dr} = 2\pi \Updelta hY\left\{ { - \frac{h}{2\mu }r\exp \left[ {\frac{2\mu }{h}\left( {r_{\text{E}} - r} \right)} \right] - \left( {\frac{h}{2\mu }} \right)^{2} \exp \left[ {\frac{2\mu }{h}\left( {r_{\text{E}} - r} \right)} \right]} \right\}_{0}^{{r_{{\text{E}}} }} = 2\pi \Updelta hY\left\{ { - \frac{h}{2\mu }r_{\text{E}} - \left( {\frac{h}{2\mu }} \right)^{2} + \left( {\frac{h}{2\mu }} \right)^{2} \exp \left( {\frac{2\mu }{h}r_{\text{E}} } \right)} \right\} $$

(A2)

The strain energy introduced by σ _r is analyzed by considering the aforementioned ring. The elongation along radial direction per unit length is $\sqrt {h/(h - \Updelta h)} - 1$. Thus, the work on the ring contributed by σ _r is $\sigma_{r} \cdot 2\pi rh \cdot [\sqrt {h/(h - \Updelta h)} - 1]dr$, and the strain energy introduced by σ _r can be evaluated

$$ u_{{{\rm E},\sigma_{r} }} = \int_{0}^{{r_{\text{E}} }} {2\pi rh} \left[ {\sqrt {h/(h - \Updelta h)} - 1} \right]\sigma_{r} dr = 2\pi hY\left[ {\sqrt {h/(h - \Updelta h)} - 1} \right]\left\{ { - \frac{h}{2\mu }r\exp \left[ {\frac{2\mu }{h}\left( {r_{\text{E}} - r} \right)} \right] - \left( {\frac{h}{2\mu }} \right)^{2} \exp \left[ {\frac{2\mu }{h}\left( {r_{\text{E}} - r} \right)} \right] - \frac{{r^{2} }}{2}} \right\}_{0}^{{r_{\text{E}} }} = 2\pi hY\left[ {\sqrt {h/(h - \Updelta h)} - 1} \right]\left\{ { - \frac{h}{2\mu }r_{\text{E}} - \left( {\frac{h}{2\mu }} \right)^{2} + \left( {\frac{h}{2\mu }} \right)^{2} \exp \left( {\frac{2\mu }{h}r_{\text{E}} } \right) - \frac{{r_{\text{E}}^{2} }}{2}} \right\} $$

(A3)

Appendix B

2.1 Analysis of Shear Strains during an Individual Collision Between Balls

A relationship between a principal normal strain of a point ε and the normal and shear strains of the point in a given orthogonal coordinate system ε _x , ε _y, ε _z , γ _xy  = γ _yx , γ _xz  = γ _zx and γ _yz  = γ _zy is given by[41]

$$ \left. \begin{gathered} (\varepsilon_{x} - \varepsilon )l + \frac{1}{2}\gamma_{yx} m + \frac{1}{2}\gamma_{zx} n = 0 \hfill \\ (\varepsilon_{y} - \varepsilon )m + \frac{1}{2}\gamma_{xy} l + \frac{1}{2}\gamma_{zy} n = 0 \hfill \\ (\varepsilon_{z} - \varepsilon )n + \frac{1}{2}\gamma_{xz} l + \frac{1}{2}\gamma_{zy} m = 0 \hfill \\ \end{gathered} \right\} $$

(B1)

where l, m, and n are directional cosines of the principal normal strain (l ² + m ² + n ² = 1). The shear strains can be obtained from Eq. [B1]:

$$ \left. \begin{gathered} \gamma_{xy} = \gamma_{yx} = (\varepsilon_{z} - \varepsilon )n^{2} /lm - (\varepsilon_{x} - \varepsilon )l/m - (\varepsilon_{y} - \varepsilon )m/l \hfill \\ \gamma_{yz} = \gamma_{zy} = (\varepsilon_{x} - \varepsilon )l^{2} /mn - (\varepsilon_{y} - \varepsilon )m/n - (\varepsilon_{z} - \varepsilon )n/m \hfill \\ \gamma_{xz} = \gamma_{zx} = (\varepsilon_{y} - \varepsilon )m^{2} /nl - (\varepsilon_{x} - \varepsilon )l/n - (\varepsilon_{z} - \varepsilon )n/l \hfill \\ \end{gathered} \right\} $$

(B2)

According to Eq. [8], normal strain is a function of properties of ball and powder, in addition to the component of the two balls’ velocity difference along the normal strain, which is described by

$$ \left. \begin{gathered} \varepsilon_{x} = 5.226C\Updelta v_{{{\text{n}}x}}^{0.8} \rho_{\text{P}} \rho_{\text{B}}^{ - 06} E_{\text{B}}^{ - 0.4} \hfill \\ \varepsilon_{y} = 5.226C\Updelta v_{{{\text{n}}y}}^{0.8} \rho_{\text{P}} \rho_{\text{B}}^{ - 06} E_{\text{B}}^{ - 0.4} \hfill \\ \varepsilon_{z} = 5.226C\Updelta v_{{{\text{n}}z}}^{0.8} \rho_{\text{P}} \rho_{\text{B}}^{ - 06} E_{\text{B}}^{ - 0.4} \hfill \\ \varepsilon_{p} = 5.226C\Updelta v_{{{\text{n}}p}}^{0.8} \rho_{\text{P}} \rho_{\text{B}}^{ - 06} E_{\text{B}}^{ - 0.4} \hfill \\ \end{gathered} \right\} $$

(B3)

where Δv _nx , Δv _ny, Δv _nz , and Δv _np are the components of the two balls’ velocity difference along three coordinate axes x, y, and z, and along the principal normal strain direction, respectively. Inserting Eq. [B3] into Eq. [B2] yields:

$$ \left. \begin{aligned} \gamma_{xy} & = \gamma_{yx} = 5.226C\rho_{\text{P}} \rho_{\text{B}}^{ - 06} E_{\text{B}}^{ - 0.4} \left[ {(\Updelta v_{{{\text{n}}z}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )n^{2} /lm - (\Updelta v_{{{\text{n}}x}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )l/m - (\Updelta v_{{{\text{n}}y}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )m/l} \right] \\ \gamma_{yz} & = \gamma_{zy} = 5.226C\rho_{\text{P}} \rho_{\text{B}}^{ - 06} E_{\text{B}}^{ - 0.4} \left[ {(\Updelta v_{{{\text{n}}x}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )l^{2} /mn - (\Updelta v_{{{\text{n}}y}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )m/n - (\Updelta v_{{{\text{n}}z}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )n/m} \right] \\ \gamma_{xz} & = \gamma_{zx} = 5.226C\rho_{\text{P}} \rho_{\text{B}}^{ - 06} E_{\text{B}}^{ - 0.4} \left[ {(\Updelta v_{{{\text{n}}y}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )m^{2} /nl - (\Updelta v_{{{\text{n}}x}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )l/n - (\Updelta v_{{{\text{n}}z}}^{0.8} - \Updelta v_{{{\text{n}}p}}^{0.8} )n/l} \right] \\ \end{aligned} \right\} $$

(B4)

Equation [B4] expresses that a shear strain is directly proportional to C, ρ _P, ρ ^−0.6_B , and E ^−0.4_B like normal strain. Based on Eq. [B4], the shear strain seems to be directly proportional to the components of the two balls’ velocity difference. Thus, the shear strains are assumed directly proportional to the components of the two balls’ velocity difference along the corresponding shear strain directions. It is challenging to determine the constant that correlates γ to Cρ _P ρ ^−0.6 _B E ^−0.4 _B Δv ^0.8 _s . To render this problem tractable, it is assumed that the constant is 5.226, a value determined for a normal strain (see Eq. [8]).