On the observation of extended microstructures and localized deformations in granular materials - A feature of excat quasi-convex energy envelope

Khan, Muhammad Sabeel

doi:10.1590/1679-78255419

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Latin American Journal of Solids and Structures

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ORIGINAL ARTICLE • Lat. Am. j. solids struct. 16 (04) • 2019 • https://doi.org/10.1590/1679-78255419 copy

On the observation of extended microstructures and localized deformations in granular materials - A feature of excat quasi-convex energy envelope

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

In this paper a most promising scope of the exact quasiconvex energy envelope in modeling the granular materials with microstructures is presented. This study shows that it is possible to observe both the extended microstructures and localized deformations in granular materials using a variational model based on the mathematical relaxation theory. The variational model is derived within the framework of Cosserat continuum. The computational algorithm based on finite element method is used to carry out numerical computations. The features of the proposed model are studied for three representative examples: the Couette shear cell, the rectangular specimen in compression and the indentation of a granular structure. The obtained results demonstrate on the possible applications and features of exact quasiconvex energy envelops in modeling the granular materials.

Keywords:
Granular materials; energy methods; extended microstructures; localized deformations; shear bands; relaxed Cosserat energy; and Finite element method

Figure	$E$	$ν$	$μ_{c}$	$α$	$β$	$\bar{μ}$	${\bar{μ}}_{c}$
-	(MPa)	-	(MPa)	(N.mm²)	mm^-1	(N)	(N)
2(a)	2.0x10²	0.3	2.0x10⁰	2.0x10⁵	5.8x10^-1	8.0x10¹	5.0x10¹
2(b)	2.0x10²	0.3	2.0x10²	2.0x10⁵	5.8x10^-1	8.0x10¹	5.0x10¹
2(c)	2.0x10²	0.3	2.0x10²	2.0x10⁵	4.0x10^-1	8.0x10¹	5.0x10¹

-	$E$	$ν$	$μ_{c}$	$α$	$β$	$\bar{μ}$	${\bar{μ}}_{c}$
-	(MPa)	-	(MPa)	(N.mm²)	mm^-1	(N)	(N)
Mesh	2.0x10⁵	0.3	2.0x10¹	5.0x10¹	1.5	7.0	2.0x10¹
Weak element	2.0x10³	0.3	2.0x10¹	1.0x10³	1.5	3.0x10²	4.0

-	$E$	$ν$	$μ_{c}$	$α$	$β$	$\bar{μ}$	${\bar{μ}}_{c}$
-	(N/cm²)	-	(N/cm²)	(N.cm²)	cm^-1	(N)	(N)
Granular medium	2.0x10⁴	0.3	2.0	5.0x10⁴	0.5	7.0x10³	2.0x10²
Indenter	2.0x10¹²	0.3	2.0	5.0x10³	0.5	7.0x10³	2.0x10²

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Step 1	Define material constants $E, ν, λ, μ, \bar{μ}, \bar{λ}, μ_{\circ} = \min (μ, \bar{μ}), μ_{c}, {\bar{μ}}_{c}, α$ and $β .$
Step 2	For each quasi-time increment: Divide the loading into tnts time steps as $t_{n + 1} = t_{n} + Δ t_{n + 1}$ where $n \in {0,1,2,3,..., t n t s} .$
Step 3	Initialize the field varialbles $u^{i} = 0, φ^{i} = 0.$
Step 4	Compute Cosserat strains $\nabla u^{i}$ and coupled strain $\nabla φ^{i}$
Step 5	Compute the term $θ (\nabla u^{i}, \nabla φ^{i}) = {‖ \nabla φ^{i} ‖}^{2} - β^{2} {‖ dev sym \nabla u^{i} ‖}^{2}$ that characterize microstructural and non-microstructural material phase
Step 6	Calculate the material tensors $K_{_{e e}}^{i}, K_{_{e k}}^{i}, K_{_{k e}}^{i}$ and $K_{_{k k}}^{i}$ such that if $θ (\nabla u^{i}, \nabla φ^{i}) \geq \frac{μ}{2 α β^{2}}$ then use equations (43) and (44), if $- \frac{μ_{\circ}}{2 α} \leq θ (\nabla u^{i}, \nabla φ^{i}) \leq \frac{μ}{2 α β^{2}}$ then use equations (45) and (46), if $θ (\nabla u^{i}, \nabla φ^{i}) \leq - \frac{μ_{\circ}}{2 α}$ then use equations (47) and (48).
*Step 7*	Determine the stiffness matrices $K_{u u}, K_{u φ}, K_{φ u}$ and $K_{φ φ}$ from equations (38), (40), (41) and (39) respectively.
*Step 8*	Update the solution $u_{n + 1}^{i + 1} {= u}_{n + 1}^{i} - {(j_{n + 1}^{i} (u_{n + 1}^{i}))}^{- 1} R_{n + 1}^{i} (u_{n + 1}^{i})$ (56) where $u_{n + 1}^{i} = [\begin{array}{l} u_{n + 1}^{i} \\ φ_{n + 1}^{i} \end{array}], R_{n + 1}^{i} (u_{n + 1}^{i}) = [\begin{array}{l} π_{l i n}^{i} (u_{n + 1}^{i}, φ_{n + 1}^{i}) \\ π_{a n g}^{i} (u_{n + 1}^{i}, φ_{n + 1}^{i}) \end{array}]$ (57) and $J_{n + 1}^{i} (u_{n + 1}^{i}) = {[\begin{array}{l} K_{u u} K_{u φ} \\ K_{φ u} K_{φ φ} \end{array}]}_{n + 1}^{i}$ (58)
*Step 9*	Check convergence, if $\| π_{l i n}^{} (u_{n + 1}^{i + 1}) \| < t o l$ ane $\| π_{a n g}^{} (u_{n + 1}^{i + 1}) \| < t o l$ then Update the load increment to time $t_{n} \leftarrow t_{n + 1}$ and go to Step 2 else Update the iteration counter $i \leftarrow i + 1$ and go to Step 3