tdoc.tex - slidergrid - grid of elastic sliders on a frictional surface
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tdoc.tex (14294B)
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1 \documentclass[11pt,a4paper]{article}
2
3 \usepackage{a4wide}
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5 %\usepackage[german, english]{babel}
6 %\usepackage{tabularx}
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9 %\usepackage{supertabular}
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12 %\usepackage{amsthm}
13 %\usepackage{float}
14 %\usepackage{subfig}
15 %\usepackage{rotating}
16 \usepackage{amsmath}
17 \setcounter{MaxMatrixCols}{20} % allow more than 10 matrix columns
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19 \usepackage[T1]{fontenc} % Font encoding
20 \usepackage{charter} % Serif body font
21 \usepackage[charter]{mathdesign} % Math font
22 \usepackage[scale=0.8]{sourcecodepro} % Monospaced fontenc
23 \usepackage[lf]{FiraSans} % Sans-serif font
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25 \usepackage{listings}
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27 \usepackage{hyperref}
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29 \usepackage{soul} % for st strikethrough command
30
31 %\usepackage[round]{natbib}
32 \usepackage[natbib=true, style=authoryear, bibstyle=authoryear-comp,
33 maxbibnames=10,
34 maxcitenames=2, backend=bibtex8]{biblatex}
35 \bibliography{/home/ad/articles/own/BIBnew.bib}
36
37
38 \begin{document}
39
40 \title{Lagrangian model of the elastic, viscous and plastic deformation of a
41 series of bonded nodes moving on a frictional surface}
42
43 \author{Anders Damsgaard}
44 \date{{\small Institute of Geophysics and Planetary Physics\\Scripps Institution
45 of Oceanography\\University of California, San Diego}\\[3mm] Last revision:
46 \today}
47
48 \maketitle
49
50 \section{Methods}
51 Our approach treats the short-temporal scale behavior of stick-slip as a
52 rigid-body dynamics problem. The material is represented as a discrete number
53 of Lagrangian points (\emph{nodes}) which are mechanically interacting with each
54 other and the boundary conditions.
55
56 The Lagrangian nodes are connected with visco-elastic beam elements.
57 The bonds are resistive to tension and compression, shearing, twisting, and
58 bending, which ensures elastic uniformity regardless of geometric node
59 arrangement \citep{Bolander1998, Radjai2011}. Alternatively, the node
60 interaction could be parameterized as simple springs which exclusively provide
61 resistance to tension and compression, and resistance to shearing, bending and
62 twisting would be introduced by discretizing the elastic material into an
63 irregular network of many more nodes \citep[e.g.][]{Topin2007, Topin2009}. Here
64 we chose the former approach which allows us to keep the number of nodes low.
65
66 The kinematic degrees of freedom are determined by explicit integration of
67 Newton's second law of motion for translation and rotation. For a point $i$
68 with bonded interactions to nodes $j\in N_c$, the translational accelerations
69 ($\boldsymbol{a}$) are found from the sums of forces:
70 \begin{equation}
71 \boldsymbol{a}_i =
72 \frac{
73 \boldsymbol{f}_i^\text{d}
74 + \boldsymbol{f}_i^\text{f}
75 + \sum^{N_c}_j \left[
76 \boldsymbol{f}_{i,j}^\text{p} +
77 \boldsymbol{f}_{i,j}^\text{s}
78 \right]
79 }{m_i}
80 + \boldsymbol{g}
81 \label{eq:n2-tran}
82 \end{equation}
83 where $\boldsymbol{f}_i^\text{d}$ is the gravitational driving stress due to
84 surface slope, $\boldsymbol{g}$ is the gravitational acceleration, and
85 $\boldsymbol{f}_i^\text{f}$ is the frictional force provided if the point is
86 resting on the lower surface. Bonded interaction with another point $j$
87 contributes to translational acceleration through bond-parallel and bond-normal
88 shear forces, $\boldsymbol{f}_{i,j}^\text{p}$ and
89 $\boldsymbol{f}_{i,j}^\text{s}$, respectively.
90
91 The angular accelerations ($\boldsymbol{\alpha}$) are found from the sums of
92 torques:
93 \begin{equation}
94 \boldsymbol{\alpha}_i =
95 \sum^{N_c}_j
96 \left[
97 \frac{\boldsymbol{t}^{i,j}_{\bar{x}}}{I^\text{p}_{i,j}} +
98 \frac{\boldsymbol{t}^{i,j}_{\bar{y}}}{I^\text{n}_{i,j}} +
99 \frac{\boldsymbol{t}^{i,j}_{\bar{z}}}{I^\text{n}_{i,j}}
100 \right]
101 \label{eq:n2-ang}
102 \end{equation}
103 here, $\boldsymbol{t}^\text{s}$ is the torque resulting from shearing motion of
104 the bond, while the torque $\boldsymbol{t}^{t}$ results from relative twisting.
105 $I^\text{n}_{i,j}$ is the bond-normal mass moment of inertia at the point, and
106 $I^\text{p}$ is polar mass moment of inertia of the bond. The above equation
107 implies the simplifying assumption that the nodes are bonded in a configuration
108 with geometric symmetry, which is a good approximation inside the grid but
109 slightly worse at the grid edges.
110
111
112 \subsection{Visco-elastic interaction between nodes}
113 The total force ($\boldsymbol{f}$) and torque ($\boldsymbol{t}$) on two nodes
114 ($i$ and $j$) is found by determining the stress response of a three-dimensional
115 elastic Timoshenko beam to strain \citet{Schlangen1996, Austrell2004,
116 Aastroem2013}.
117 In the following the forces and torques are described for node $i$ but are
118 of equal magnitude and with opposite sign for node $j$. The interaction
119 accounts for resistance to tension and compression, shear, torsion, and bending.
120 The equations below are derived from the stiffness matrix in
121 \citet{Austrell2004}. The components for the three-dimensional force vector on
122 node $i$ are:
123 \begin{equation}
124 \begin{split}
125 f_{\bar{x}}^i & = \frac{EA}{L}
126 \left( p_{\bar{x}}^{*,i} - p_{\bar{x}}^{*,j} \right)\\
127 %
128 f_{\bar{y}}^i & = \frac{12EI_A}{L^3}
129 \left( p_{\bar{y}}^{*,i} - p_{\bar{y}}^{*,j} \right)
130 + \frac{6EI_A}{L^2}
131 \left( \Omega_{\bar{z}}^{*,i} + \Omega_{\bar{z}}^{*,j} \right)\\
132 %
133 f_{\bar{z}}^i & = \frac{12EI_A}{L^3}
134 \left( p_{\bar{z}}^{*,i} - p_{\bar{z}}^{*,j} \right)
135 - \frac{6EI_A}{L^2}
136 \left( \Omega_{\bar{y}}^{*,i} + \Omega_{\bar{y}}^{*,j} \right)
137 \end{split}
138 \end{equation}
139 where $\bar{x}, \bar{y}, \bar{z}$ are the bond-relative coordinates.
140 The linear and angular relative displacement of the nodes is described by $p^*$
141 and $\Omega^*$. $E$ is Young's modulus, $G$ is the shear modulus, $A$ is the
142 beam cross-sectional area, and $L$ is the original beam length. $I_A$ is the
143 area moment of inertia of the beam ($I_A = a^4 12^{-1}$ where $a$ is the beam
144 side length). The torque on node $i$ is:
145 \begin{equation}
146 \begin{split}
147 t_{\bar{x}}^i & = \frac{GJ}{L}
148 \left( \Omega_{\bar{x}}^{*,i} + \Omega_{\bar{x}}^{*,j} \right)\\
149 %
150 t_{\bar{y}}^i & = \frac{6EI_A}{L^2}
151 \left(p_{\bar{z}}^{*,j} - p_{\bar{z}}^{*,i} \right)
152 + \frac{4EI_A}{L}
153 \left( \Omega_{\bar{y}}^{*,i} + \frac{\Omega_{\bar{y}}^{*,j}}{2}
154 \right)\\
155 %
156 t_{\bar{z}}^i & = \frac{6EI_A}{L^2}
157 \left(p_{\bar{y}}^{*,i} - p_{\bar{y}}^{*,j} \right)
158 + \frac{4EI_A}{L}
159 \left( \Omega_{\bar{z}}^{*,i} + \frac{\Omega_{\bar{z}}^{*,j}}{2}
160 \right)\\
161 \end{split}
162 \end{equation}
163 $GJ$ is the Saint-Venant torsional stiffness, where $J$ called the torsional
164 rigidity multiplier or torsion constant. For a beam with a solid square
165 cross-sectional shape it can be approximated as $J \approx 0.140577 a^4$ where
166 $a$ is the side length \citep{Timoshenko1951, Roark1954, Weisstein2016}.
167
168 % Torsional constant:
169 % https://en.wikipedia.org/wiki/Torsion_constant
170 % http://mathworld.wolfram.com/TorsionalRigidity.html (K: K_v)
171 % http://physics.stackexchange.com/questions/83148/where-i-can-find-a-torsional-stiffness-table-for-different-types-of-stainless-st
172 % St Venant torsion: K_v = 1/G (Austrell et al. 2004, table 3) Does it make sense?
173 % chapter4.pdf: K_v = J
174 % https://skyciv.com/free-moment-of-inertia-calculator/
175
176
177 The deformation and reactive forces are determined relative to the orientation
178 of the bond. Common geometrical vectors include the inter-distance vector
179 $\boldsymbol{d}$ between nodes $\boldsymbol{p}_i$ and $\boldsymbol{p}_j$:
180 \begin{equation}
181 \boldsymbol{d}_{i,j} = \boldsymbol{p}_i - \boldsymbol{p}_j
182 \end{equation}
183 which in normalized form constitutes the bond-parallel unit vector:
184 \begin{equation}
185 \boldsymbol{n}_{i,j} = \frac{\boldsymbol{d}_{i,j}}{||\boldsymbol{d}_{i,j}||}
186 \end{equation}
187
188 The nodes move by translational and rotational velocities. The combined
189 relative velocity between the nodes is found as \citep[e.g.][]{Hinrichsen2004,
190 Luding2008}:
191 \begin{equation}
192 \boldsymbol{v}_{i,j} = \boldsymbol{v}_i - \boldsymbol{v}_j +
193 \frac{\boldsymbol{d}_{i,j}}{2} \times \boldsymbol{\omega}_i +
194 \frac{\boldsymbol{d}_{i,j}}{2} \times \boldsymbol{\omega}_j
195 \end{equation}
196 The velocity can be decomposed into spatial components relative to the bond
197 orientation, e.g.\ the bond-parallel and bond-shear velocity, respectively:
198 \begin{equation}
199 v^\text{p}_{i,j} = \boldsymbol{v}_{i,j} \cdot \boldsymbol{n}_{i,j}
200 \end{equation}
201 \begin{equation}
202 \boldsymbol{v}^\text{s}_{i,j} = \boldsymbol{v}_{i,j} - \boldsymbol{n}_{i,j}
203 \left(
204 \boldsymbol{v}_{i,j}
205 \cdot
206 \boldsymbol{n}_{i,j}
207 \right)
208 \end{equation}
209
210
211
212 \subsection{Temporal integration}
213 Once the force and torque sum components at time $t$ have been determined, the
214 kinematic degrees of freedom at time $t+\Delta t$ can be found by explicit
215 temporal integration of moment balance equations~\ref{eq:n2-tran}
216 and~\ref{eq:n2-ang}.
217 We use an integration scheme based on the third-order Taylor expansion, which
218 results in a truncation error on the order of $O(\Delta t^4)$ for positions and
219 $O(\Delta t^3)$ for velocities. This scheme includes changes in acceleration as
220 the highest order term, which are approximated by backwards differences. For
221 the translational degrees of freedom:
222 \begin{equation}
223 \boldsymbol{p}^i_{t+\Delta t} =
224 \boldsymbol{p}^i_{t} +
225 \boldsymbol{v}^i_{t} \Delta t +
226 \frac{1}{2} \boldsymbol{a}^i_{t} \Delta t^2 +
227 \frac{1}{6} \frac{\boldsymbol{a}^i_{t}
228 - \boldsymbol{a}^i_{t - \Delta t}}{\Delta t} \Delta t^3
229 \end{equation}
230 \begin{equation}
231 \boldsymbol{v}^i_{t+\Delta t} =
232 \boldsymbol{v}^i_{t} +
233 \boldsymbol{a}^i_{t} \Delta t +
234 \frac{1}{2} \frac{\boldsymbol{a}^i_{t}
235 - \boldsymbol{a}^i_{t - \Delta t}}{\Delta t} \Delta t^2
236 \end{equation}
237 At $t=0$ the acceleration change term is defined as zero. The angular degrees
238 of freedom are found correspondingly:
239 \begin{equation}
240 \boldsymbol{\Omega}^i_{t+\Delta t} =
241 \boldsymbol{\Omega}^i_{t} +
242 \boldsymbol{\omega}^i_{t} \Delta t +
243 \frac{1}{2} \boldsymbol{\alpha}^i_{t} \Delta t^2 +
244 \frac{1}{6} \frac{\boldsymbol{\alpha}^i_{t}
245 - \boldsymbol{\alpha}^i_{t - \Delta t}}{\Delta t} \Delta t^3
246 \end{equation}
247 \begin{equation}
248 \boldsymbol{\omega}^i_{t+\Delta t} =
249 \boldsymbol{\omega}^i_{t} +
250 \boldsymbol{\alpha}^i_{t} \Delta t +
251 \frac{1}{2} \frac{\boldsymbol{\alpha}^i_{t}
252 - \boldsymbol{\alpha}^i_{t - \Delta t}}{\Delta t} \Delta t^2
253 \end{equation}
254
255 The numerical time step $\Delta t$ is found by considering the largest elastic
256 stiffness in the system relative to the smallest mass:
257 \begin{equation}
258 \Delta t =
259 \epsilon
260 \left[
261 \min (m_i)^{-1}
262 \max \left(
263 \max \left(
264 \frac{E A_{i,j}}{||\boldsymbol{d}_{0}^{i,j}||}
265 \right)
266 ,
267 \max \left(
268 \frac{G A_{i,j}}{||\boldsymbol{d}^{i,j}||}
269 \right)
270 \right)
271 \right]^{-1/2}
272 \end{equation}
273 where $\epsilon$ is a safety factor related to the geometric structure of the
274 bonded network. We use $\epsilon = 0.07$.
275
276
277 \appendix
278 \section{Stiffness matrix}
279 \begin{equation}
280 \begin{bmatrix}
281 f_{\bar{x}}^i\\[0.6em]
282 f_{\bar{y}}^i\\[0.6em]
283 f_{\bar{z}}^i\\[0.6em]
284 t_{\bar{x}}^i\\[0.6em]
285 t_{\bar{y}}^i\\[0.6em]
286 t_{\bar{z}}^i\\[0.6em]
287 f_{\bar{x}}^j\\[0.6em]
288 f_{\bar{y}}^j\\[0.6em]
289 f_{\bar{z}}^j\\[0.6em]
290 t_{\bar{x}}^j\\[0.6em]
291 t_{\bar{y}}^j\\[0.6em]
292 t_{\bar{z}}^j\\
293 \end{bmatrix}
294 =
295 \begin{bmatrix}
296 \frac{EA}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-EA}{L} & 0 & 0 & 0 & 0 & 0\\[0.5em]
297 0 & \frac{12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 & \frac{6EI_{\bar{z}}}{L^2} &
298 0 & \frac{-12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 &
299 \frac{6EI_{\bar{z}}}{L^2}\\[0.5em]
300 0 & 0 & \frac{12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}}{L^2} & 0
301 & 0 & 0 & \frac{-12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}}{L^2} &
302 0\\[0.5em]
303 0 & 0 & 0 & \frac{GJ}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-GJ}{L} & 0 &
304 0\\[0.5em]
305 0 & 0 & \frac{-6EI_{\bar{y}}}{L^2} & 0 & \frac{4EI_{\bar{y}}}{L} & 0 & 0
306 & 0 & \frac{6EI_{\bar{y}}}{L^2} & 0 & \frac{2EI_{\bar{y}}}{L} &
307 0\\[0.5em]
308 0 & \frac{6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{4EI_{\bar{z}}}{L} & 0
309 & \frac{-6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 &
310 \frac{2EI_{\bar{z}}}{L}\\[0.5em]
311 \frac{-EA}{L} & 0 & 0 & 0 & 0 & 0 & \frac{EA}{L} & 0 & 0 & 0 & 0 & 0\\[0.5em]
312 0 & \frac{-12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 & \frac{-6EI_{\bar{z}}}{L^2}
313 & 0 & \frac{12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 &
314 \frac{-6EI_{\bar{z}}}{L^2}\\[0.5em]
315 0 & 0 & \frac{-12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}}{L^2} & 0
316 & 0 & 0 & \frac{12EI_{\bar{y}}}{L^3} & 0 & \frac{6EI_{\bar{y}}}{L^2} &
317 0\\[0.5em]
318 0 & 0 & 0 & \frac{-GJ}{L} & 0 & 0 & 0 & 0 & 0 & \frac{GJ}{L} & 0 &
319 0\\[0.5em]
320 0 & 0 & \frac{-6EI_{\bar{y}}}{L^2} & 0 & \frac{2EI_{\bar{y}}}{L} & 0 & 0
321 & 0 & \frac{6EI_{\bar{y}}}{L^2} & 0 & \frac{4EI_{\bar{y}}}{L} &
322 0\\[0.5em]
323 0 & \frac{6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{2EI_{\bar{z}}}{L} & 0
324 & \frac{-6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{4EI_{\bar{z}}}{L}\\
325 \end{bmatrix}
326 \begin{bmatrix}
327 p_{\bar{x}}^i\\[0.6em]
328 p_{\bar{y}}^i\\[0.6em]
329 p_{\bar{z}}^i\\[0.6em]
330 \Omega_{\bar{x}}^i\\[0.6em]
331 \Omega_{\bar{y}}^i\\[0.6em]
332 \Omega_{\bar{z}}^i\\[0.6em]
333 p_{\bar{x}}^j\\[0.6em]
334 p_{\bar{y}}^j\\[0.6em]
335 p_{\bar{z}}^j\\[0.6em]
336 \Omega_{\bar{x}}^j\\[0.6em]
337 \Omega_{\bar{y}}^j\\[0.6em]
338 \Omega_{\bar{z}}^j\\
339 \end{bmatrix}
340 \end{equation}
341
342
343
344 \printbibliography{}
345
346 \end{document}