Strength criteria at anisotropic principal directions expressed in closed form by interparticle parameters for elliptical particle assembly

Abstract

Using a microstructural mechanics approach, the close-packed elliptical particle assembly is first represented by a lattice model that is described by a beam system, and the Mohr–Coulomb (MC) shear failure criterion is used to describe the contact (beam) breakage. Through structural mechanics analysis of the unit cell, macroscopic strength criteria and associated parameters at two anisotropic principal directions are expressed in closed form with respect to the particle shape and microscopic strength parameters while considering the successive breakage of contacts. The macroscopic failure characteristics of elliptical particle assemblies are described by the MC failure criterion, and the macro–micro relations of strength parameters can be directly determined by the proposed expressions. The proposed analytical expressions are confirmed to be accurate due to the good consistency between theoretical and discrete element method (DEM) results based on the regular arrangement. The introduction of the fabric anisotropy parameter extends the applicability of the theoretical results to irregular arrangement cases, and the deviation with more realistic particle assemblies is investigated. The proposed expressions of the macroscopic strength parameters can help us better understand the influence of inherent anisotropy and microscopic properties and provide an alternative approach to determine the microscopic parameters in DEM simulations.

Appendix 1

Figure 

Fig. 15

Force analysis of the unit cell

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

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

$$\delta_{ij} = \sum\limits_{1}^{n} {\int\limits_{0}^{{l_{h} }} {\frac{{\overline{M}_{ih} \overline{M}_{jh} {\text{d}}l}}{{l_{h} k_{mh} }}} + } \sum\limits_{1}^{n} {\int\limits_{0}^{{l_{h} }} {\frac{{\overline{F}_{{{\text{N}}ih}} \overline{F}_{{{\text{N}}jh}} {\text{d}}l}}{{l_{h} k_{nh} }}} } + \sum\limits_{1}^{n} {\int\limits_{0}^{{l_{h} }} {\frac{{k\overline{F}_{{{\text{S}}ih}} \overline{F}_{{{\text{S}}jh}} {\text{d}}l}}{{l_{h} k_{sh} }}} }$$

(46)

$$\Delta_{{i{\text{P}}}} = \sum\limits_{1}^{n} {\int\limits_{0}^{{l_{h} }} {\frac{{\overline{M}_{ih} \,M_{Ph} dl}}{{l_{h} k_{mh} }}} } + \sum\limits_{1}^{n} {\int\limits_{0}^{{l_{h} }} {\frac{{\overline{F}_{Nih}\, \overline{F}_{NPh} dl}}{{l_{h} k_{nh} }}} } + \sum\limits_{1}^{n} {\int\limits_{0}^{{l_{h} }} {\frac{{k\overline{F}_{Sih}\, \overline{F}_{SPh} dl}}{{l_{h} k_{sh} }}} }$$

(47)

where $k_{ih}$(i = n, s, m) and l_h are the contact stiffness along the i direction and length of beam h, respectively; k is a parameter related to the shape of the cross-section; and n = 7 is the number of rigid beams. Additionally, $\overline{F}_{{{\text{N}}ih}}$,$\overline{F}_{{{\text{S}}ih}}$ and $\overline{M}_{ih}$ ($F_{{{\text{NP}}h}}$, $F_{{{\text{SP}}h}}$ and $M_{{{\text{P}}h}}$) are the axial force, shear force, and bending moment of beam h induced by the unit redundant force $\overline{F}_{i}$ (external forces F_x and F_y), respectively, which are the function of normal direction in the local coordinate system of beam h. dl is the differential of length of beam h. Noted that, to keep consistent with the theoretical derivation, the material constants of original principle of virtual work [67] are replaced by contact stiffesses $k_{i}$ in this study through the relations between material constants and contact stiffesses according to the beam model. Therefore, $\delta_{ij}$ and $\Delta_{{i{\text{P}}}}$ can be expressed as:

$$\begin{aligned} \delta_{11} &= \frac{4}{{k_{n1} }}{ + }\frac{{4\sin^{2} \alpha }}{{k_{n2} }} + \frac{{4\cos^{2} \alpha }}{{k_{s2} }} + \frac{{4l_{2}^{2} \cos^{2} \alpha }}{{3k_{m2} }} \hfill \\ \delta_{22} &= \frac{{4\cos^{2} \alpha }}{{k_{n2} }} + \frac{{4\sin^{2} \alpha }}{{k_{s2} }} + \frac{{4l_{1}^{2} }}{{3k_{m1} }} + \frac{{l_{4}^{2} }}{{3k_{m2} }} + \frac{{4l_{2}^{2} \sin^{2} \alpha }}{{k_{m2} }} \hfill \\ \delta_{33} &= \frac{4}{{k_{m2} }}\;\delta_{13} { = }\delta_{31} { = }\frac{{2l_{2} \cos \alpha }}{{k_{m2} }}\;\delta_{12} { = }\delta_{21} = \delta_{23} { = }\delta_{32} { = }0 \hfill \\ \Delta_{{{\text{1P}}}} &= - \frac{2}{{k_{n1} }}F_{x} + 2\sin \alpha \cos \alpha \left( - \frac{{l_{2}^{2} }}{{3k_{m2} }} + \frac{1}{{k_{n2} }} - \frac{1}{{k_{s2} }}\right)F_{y}, \\ \Delta_{{2{\text{P}}}} & = 0,\\ \Delta_{{{\text{3P}}}} & = - \frac{{l_{2} \sin \alpha }}{{k_{m2} }}F_{y} . \hfill \\ \end{aligned}$$

(48)

Substituting Eq. (48) into the typical equations of the force method in structure mechanics $\delta_{ij} F_{i} + \Delta_{ip} = 0$ [67], the redundant force F_i of rigid beams 4 and 7 can be expressed as:

$$\left\{ \begin{gathered} F_{1} = \frac{{\delta_{33} \Delta_{1p} - \delta_{13} \Delta_{3p} }}{{\delta_{13}^{2} - \delta_{11} \delta_{33} }} = AF_{x} + BF_{y} \hfill \\ F_{2} = 0 \hfill \\ F_{3} = \frac{{\delta_{11} \Delta_{3p} - \delta_{13} \Delta_{1p} }}{{\delta_{13}^{2} - \delta_{11} \delta_{33} }} = CF_{x} + DF_{y} \hfill \\ \end{gathered} \right.$$

(49)

where:

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

(50)

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

(51)

Substituting Eqs. (1–3) into Eq. (50) and considering the relation of m, we obtain the dimensionless strength coefficients in Eq. (12), which are expressed by microscopic parameters, including the major-minor axis ratio m and stiffness ratios λ, ξ, γ. After the redundant force of rigid beams 4 and 7 is determined, the internal forces of all remaining beams can be obtained by force analysis of the statically determinate structure, and the axial and shear forces of beams 1 and 2, 3, 5 and 6 can be expressed by strength coefficients A and B as follows:

$$\begin{gathered} {\text{Axial force}}:f_{n1} = 2BF_{y} - (1 - 2A)F_{x} \hfill \\ {\text{Shear force}}:f_{s1} = 0 \hfill \\ \end{gathered}$$

(52)

and:

$$\begin{gathered} {\text{Axial force}}:f_{n2} { = }A\sin \alpha F_{x} + (B\sin \alpha + \frac{\cos \alpha }{2})F_{y} \hfill \\ {\text{Shear force}}:f_{s2} = A\cos \alpha F_{x} + (B\cos \alpha - \frac{\sin \alpha }{2})F_{y} . \hfill \\ \end{gathered}$$

(53)

Letting f_x or f_y = 0, we can obtain the expressions of axial and shear forces of beams 2, 3, 5 and 6 and 1 for the uniaxial tension loading situations in the y or x directions in Eqs. (28) and (31), respectively.