Micromechanical mechanism-based anisotropic strength criteria for regularly arranged elliptical particle assembly

Abstract

Using the microstructural mechanics approach, the macroscopic anisotropic failure criteria expressed in closed form by the microscopic parameters are derived for regularly arranged elliptical particle assemblies. The close-packed, elliptical particle assembly is equivalent to a lattice model that is described by a lattice beam system, and the Mohr–Coulomb-like shear failure criterion is applied to describe the beam breakages. Through mechanical analysis of the instability state of the unit cell of the lattice beam system, macroscopic anisotropic strength criteria and the corresponding strength parameters, which are expressed by microscopic parameters and the tilting angle, are proposed. The macroscopic failure characteristics and strength parameters are qualitatively consistent with the results of the discrete element method (DEM), and the influences of the microscopic parameters and tilting angle on the macroscopic strength parameters are investigated in detail according to the proposed theoretical results. The applicability of the theoretical results to randomly distributed particles is explored by DEM simulation of irregularly arranged particles, and the deviation are also systematically discussed.

Appendices

Appendix A

Fig. A1

Force analysis of the unit cell

Figure A1 shows the force analysis of the unit cell in Fig. 2d, where F₁, F₂ and F₃ are the redundant axial force, shear force and bending moment, respectively, of rigid beams 4 and 7. According to the symmetric characteristic of the unit, F_i (i = 1, 2 and 3) in beams 4 and 7 are equal.

$\delta_{ij}$ and $\Delta_{{i{\text{P}}}}$ represent the displacement in the F_i (= 1, 2 and 3) direction induced by the unit redundant force $\overline{F}_{j} (j = {1, 2, 3})$ and external forces F_x, F_y and T_xy. They can be calculated by the principle of virtual work as follows:

$$\delta_{ij} = \sum\limits_{1}^{n} {\int\limits_{{l_{h} }} {\frac{{\overline{M}_{ih} \overline{M}_{jh} {\text{d}}s}}{{l_{h} k_{mh} }}} + } \sum\limits_{1}^{n} {\int\limits_{{l_{h} }} {\frac{{\overline{F}_{{{\text{N}}ih}} \overline{F}_{{{\text{N}}jh}} {\text{d}}s}}{{l_{h} k_{nh} }}} } + \sum\limits_{1}^{n} {\int\limits_{{l_{h} }} {\frac{{k\overline{F}_{{{\text{S}}ih}} \overline{F}_{{{\text{S}}jh}} {\text{d}}s}}{{l_{h} k_{sh} }}} } ,$$

(A-1)

$$\Delta_{{i{\text{P}}}} = \sum\limits_{1}^{n} {\int\limits_{{l_{h} }} {\frac{{\overline{M}_{ih} M_{{{\text{P}}h}} {\text{d}}s}}{{l_{h} k_{mh} }}{ + }\sum\limits_{1}^{n} {\int\limits_{{l_{h} }} {\frac{{\overline{F}_{{{\text{N}}ih}} \overline{F}_{{{\text{NP}}h}} {\text{d}}s}}{{l_{h} k_{nh} }}{ + }\sum\limits_{1}^{n} {\int\limits_{{l_{h} }} {\frac{{k\overline{F}_{{{\text{S}}ih}} \overline{F}_{{{\text{SP}}h}} {\text{d}}s}}{{l_{h} k_{sh} }}} } } } } } ,$$

(A-2)

where $k_{ih}$(i = n, s, m) and l_h are the contact stiffness along i direction and length of beam h, respectively; k is a parameter related to the shape of the cross section; and n = 9 represents the number of rigid beams. Additionally, $\overline{F}_{{{\text{N}}i}}^{h}$, $\overline{F}_{{{\text{S}}i}}^{h}$ and $\overline{M}_{i}^{h}$ ($F_{{{\text{NP}}}}^{h}$, $F_{{{\text{SP}}}}^{h}$ and $M_{{\text{P}}}^{h}$) are the axial force, shear force and bending moment, respectively, of beam h induced by the unit redundant force $\overline{F}_{i}$ (external forces F_x, F_y and T_xy) of the rigid beam (beams 4 and 7) force. Note that to keep consistent with the theoretical derivation, the material constants of original principle of virtual work [24] are replaced by contact stiffesses $k_{i}$ in this study according to the relations between material constants and contact stiffesses [3]. According to the typical equations of the force method in structure mechanics, $\delta_{ij} F_{i} + \Delta_{ip} = 0$, the redundant force F_i of rigid beams 4 and 7 can be expressed as:

$$\left\{ {\begin{array}{*{20}l} {F_{1} = \frac{{\delta_{33} \Delta_{1p} - \delta_{13} \Delta_{3p} }}{{\delta_{13}^{2} - \delta_{11} \delta_{33} }} = AF_{x} + BF_{y} } \hfill \\ {F_{2} = \frac{{ - \Delta_{2p} }}{{\delta_{22} }} = ET_{xy} } \hfill \\ {F_{3} = \frac{{\delta_{11} \Delta_{3p} - \delta_{13} \Delta_{1p} }}{{\delta_{13}^{2} - \delta_{11} \delta_{33} }} = CF_{x} + DF_{y} } \hfill \\ \end{array} ,} \right.$$

(A-3)

where

$$A{ = }\frac{1}{{\frac{{l_{2}^{2} \cos^{2} \alpha }}{6} \cdot \frac{\gamma }{\xi } + 2\left[ {1 + \gamma \left( {\sin^{2} \alpha + \frac{1}{\lambda }\cos^{2} \alpha } \right)} \right]}},\;B{ = }\frac{{\left[ {2\left( {\frac{{l_{2}^{2} }}{3}\frac{1}{\xi } - 1 + \frac{1}{\lambda }} \right) - \frac{{l_{2}^{2} }}{2}\frac{1}{\xi }} \right]\sin \alpha \cos \alpha }}{{\frac{1}{3}l_{2}^{2} \cos^{2} \alpha \frac{1}{\xi }{ + }4\left( {\frac{1}{\gamma } + \sin^{2} \alpha + \cos^{2} \alpha \frac{1}{\lambda }} \right)}},$$

(A-4)

$$C{ = } - \frac{{l_{2} \cos \alpha /\gamma }}{{\frac{1}{3}l_{2}^{2} \cos^{2} \alpha \frac{1}{\xi } + 4\left( {\frac{1}{\gamma } + \sin^{2} \alpha + \frac{1}{\lambda }\cos^{2} \alpha } \right)}},\;D{ = }\frac{{l_{2} \sin \alpha \left( {1 + 1/\gamma } \right)}}{{\frac{1}{3}l_{2}^{2} \cos^{2} \alpha \frac{1}{\xi } + 4\left( {\frac{1}{\gamma } + \sin^{2} \alpha + \frac{1}{\lambda }\cos^{2} \alpha } \right)}},$$

(A-5)

$$E = - \frac{{ - \frac{2}{3}\cos^{2} \alpha + \frac{2}{\gamma \lambda } + \frac{{2m^{2} \sin^{2} \alpha }}{\lambda } + \frac{1}{\xi }\frac{{a(2l_{1} + l_{4} )}}{12} + \frac{1}{\gamma \xi }\frac{{al_{1} }}{12}\frac{{m^{2} + 1}}{{m^{2} }}}}{{4\cos^{2} \alpha + \frac{4}{\gamma \lambda } + \frac{4}{\lambda }\sin^{2} \alpha + \frac{{l_{1} (l_{1} + 2l_{4} )}}{3\xi } + \frac{{l_{1}^{2} }}{3\gamma \xi }}}.$$

(A-6)

Substituting Eq. (1) and the lengths of beams in Table 1 into Eqs. (A-4) and (A-6), we obtain the dimensionless strength coefficients shown in Eqs. (13–15). After the redundant force of rigid beams 4 and 7 is determined, the internal forces of all remaining beams can be obtained in Eqs. (10–12) by force analysis of the statically determinate structure.

Appendix B

O, P, Q in Eq. (29) can be expressed as:

$$\begin{aligned} O\; = \; & (a_{1}^{2} b_{2}^{2} + 4a_{1}^{2} c_{2}^{2} - 2a_{1} a_{2} b_{1} b_{2} - 8a_{1} a_{2} c_{1} c_{2} - 2a_{1} b_{1} b_{2} c_{2} + 2a_{1} b_{2}^{2} c_{1} + a_{2}^{2} b_{1}^{2} + 4a_{2}^{2} c_{1}^{2} + 2a_{2} b_{1}^{2} c_{2} - 2a_{2} b_{1} b_{2} c_{1} + b_{1}^{2} c_{2}^{2} \\ & - 2b_{1} b_{2} c_{1} c_{2} + b_{2}^{2} c_{1}^{2} ) \times \sigma_{3}^{2} + (4d_{2} a_{1}^{2} c_{2} - 4d_{2} a_{1} a_{2} c_{1} - 4d_{1} a_{1} a_{2} c_{2} - 2d_{2} a_{1} b_{1} b_{2} + 2d_{1} a_{1} b_{2}^{2} - 4d_{2} a_{1} c_{1} c_{2} + 4d_{1} a_{1} c_{2}^{2} \\ & + 4d_{1} a_{2}^{2} c_{1} + 2d_{2} a_{2} b_{1}^{2} - 2d_{1} a_{2} b_{1} b_{2} + 4d_{2} a_{2} c_{1}^{2} - 4d_{1} a_{2} c_{1} c_{2} + 2d_{2} b_{1}^{2} c_{2} - 2d_{2} b_{1} b_{2} c_{1} - 2d_{1} b_{1} b_{2} c_{2} + 2d_{1} b_{2}^{2} c_{1} ) \times \sigma_{3} \\ & + b_{1}^{2} d_{2}^{2} + b_{2}^{2} d_{1}^{2} - 4a_{1} c_{1} d_{2}^{2} - 4a_{2} c_{2} d_{1}^{2} + 4a1c_{2} d_{1} d_{2} + 4a_{2} c_{1} d_{1} d_{2} - 2b_{1} b_{2} d_{1} d_{2} , \\ \end{aligned}$$

$$P = \left( {2a_{1} c_{2} - 2a_{2} c_{1} } \right) \times \sigma_{3} + 2c_{2} d_{1} - 2c_{1} d_{2} ,\;\;Q{ = }\left( {a_{1} b_{2} - a_{2} b_{1} - b_{1} c_{2} + b_{2} c_{1} } \right) \times \sigma_{3} ,$$

where a_i, b_i, c_i and d_i (i = 1, 2) for δ_ij can be expressed as:

δ₁₂: $a_{1} = 2mB$, $a_{2} = \left[ {\left( {\frac{1}{2} - \sqrt 3 mB} \right) - \mu_{p} \left( {B + \frac{\sqrt 3 m}{2}} \right)} \right]$, $b_{1} = 0$, $b_{2} = - \frac{\sqrt 3 }{m}\left[ {C + \frac{{m^{2} }}{2} + \sqrt 3 m\mu_{p} \left( {\frac{1}{6} - E} \right)} \right]$,

$c_{1} = \sqrt 3 (2A - 1)$, $c_{2} = - A\left( {3 + \frac{{\sqrt 3 \mu_{p} }}{m}} \right)$, $d_{1} = - mR_{t} /a$, $d_{2} = - \sqrt {3m^{2} + 1} c_{p} /a$;

δ₂₃: $a_{1} = \left[ {\left( {\frac{1}{2} - \sqrt 3 mB} \right) - \mu_{p} \left( {B + \frac{\sqrt 3 m}{2}} \right)} \right]$, $a_{2} = \left( {B + \frac{\sqrt 3 m}{2}} \right)$, $b_{1} = - \frac{\sqrt 3 }{m}\left[ {C + \frac{{m^{2} }}{2} + \sqrt 3 m\mu_{p} \left( {\frac{1}{6} - C} \right)} \right]$, $b_{2} = 3\left( {\frac{1}{6} - C} \right)$, $c_{1} = - A\left( {3 + \frac{{\sqrt 3 \mu_{p} }}{m}} \right)$, $c_{2} = \frac{\sqrt 3 A}{m}$ $d_{1} = - \sqrt {3m^{2} + 1} c_{p} /a$, $d_{2} = \sqrt {3m^{2} + 1} R_{t} /a$;

δ-₂₃: $a_{1} = \left[ {\left( {\frac{1}{2} - \sqrt 3 mB} \right){ + }\mu_{p} \left( {B + \frac{\sqrt 3 m}{2}} \right)} \right]$, $a_{2} = \left( {B + \frac{\sqrt 3 m}{2}} \right)$, $b_{1} = - \frac{\sqrt 3 }{m}\left[ {C + \frac{{m^{2} }}{2} - \sqrt 3 m\mu_{p} \left( {\frac{1}{6} - C} \right)} \right]$, $b_{2} = 3\left( {\frac{1}{6} - C} \right)$, $c_{1} = - A\left( {3 - \frac{{\sqrt 3 \mu_{p} }}{m}} \right)$, $c_{2} = \frac{\sqrt 3 A}{m}$ $d_{1} = \sqrt {3m^{2} + 1} c_{p} /a$, $d_{2} = \sqrt {3m^{2} + 1} R_{t} /a$;

δ₃₄: $a_{1} = \left[ {\left( {\sqrt 3 Bm - \frac{1}{2}} \right) - \mu_{p} \left( {B + \frac{\sqrt 3 m}{2}} \right)} \right]$, $a_{2} = \left( {B + \frac{\sqrt 3 m}{2}} \right)$, $b_{1} = - \frac{\sqrt 3 }{m}\left[ {C + \frac{{m^{2} }}{2} + \sqrt 3 m\mu_{p} \left( {C - \frac{1}{6}} \right)} \right]$, $b_{2} = 3\left( {\frac{1}{6} - C} \right)$, $c_{1} = A\left( {3 - \frac{{\sqrt 3 \mu_{p} }}{m}} \right)$, $c_{2} = \frac{\sqrt 3 A}{m}$ $d_{1} = - \sqrt {3m^{2} + 1} c_{p} /a$, $d_{2} = \sqrt {3m^{2} + 1} R_{t} /a$;

δ₂₄: $a_{1} = \left[ {\left( {\frac{1}{2} - \sqrt 3 mB} \right) - \mu_{p} \left( {B + \frac{\sqrt 3 m}{2}} \right)} \right]$, $a_{2} = \left[ {\left( {\sqrt 3 Bm - \frac{1}{2}} \right) - \mu_{p} \left( {B + \frac{\sqrt 3 m}{2}} \right)} \right]$, $c_{1} = - A\left( {3 + \frac{{\sqrt 3 \mu_{p} }}{m}} \right)$.

$b_{1} = b_{2} = - \frac{\sqrt 3 }{m}\left[ {C + \frac{{m^{2} }}{2} + \sqrt 3 m\mu_{p} \left( {\frac{1}{6} - C} \right)} \right]$, $c_{2} = A\left( {3 - \frac{{\sqrt 3 \mu_{p} }}{m}} \right)$, $d_{1} = d_{2} = - \sqrt {3m^{2} + 1} c_{p} /a$.