tadd system of equations for 3d beam - slidergrid - grid of elastic sliders on … | |
git clone git://src.adamsgaard.dk/slidergrid | |
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--- | |
commit 897289bb9cc3233e556ccd4b863cdfaf4dea29b0 | |
parent 01b81b008518de59f62d4471b397364dd6b78f9a | |
Author: Anders Damsgaard Christensen <[email protected]> | |
Date: Mon, 2 May 2016 09:58:42 -0700 | |
add system of equations for 3d beam | |
Diffstat: | |
M doc/doc.pdf | 0 | |
M doc/doc.tex | 295 ++++++++++++++++++++++++++++-… | |
M slidergrid/slider.h | 4 ++++ | |
3 files changed, 275 insertions(+), 24 deletions(-) | |
--- | |
diff --git a/doc/doc.pdf b/doc/doc.pdf | |
Binary files differ. | |
diff --git a/doc/doc.tex b/doc/doc.tex | |
t@@ -1,4 +1,4 @@ | |
-\documentclass[11pt]{article} | |
+\documentclass[11pt,a4paper]{article} | |
\usepackage{a4wide} | |
t@@ -14,6 +14,7 @@ | |
%\usepackage{subfig} | |
%\usepackage{rotating} | |
\usepackage{amsmath} | |
+\setcounter{MaxMatrixCols}{20} % allow more than 10 matrix columns | |
\usepackage[T1]{fontenc} % Font encoding | |
\usepackage{charter} % Serif body font | |
t@@ -37,7 +38,7 @@ maxcitenames=2, backend=bibtex8]{biblatex} | |
\begin{document} | |
\title{Lagrangian model of the elastic, viscous and plastic deformation of a | |
- series of bonded points moving on a frictional surface} | |
+ series of bonded nodes moving on a frictional surface} | |
\author{Anders Damsgaard} | |
\date{{\small Institute of Geophysics and Planetary Physics\\Scripps Instituti… | |
t@@ -46,54 +47,300 @@ of Oceanography\\University of California, San Diego}\\[3… | |
\maketitle | |
- | |
\section{Methods} | |
-The Lagrangian points are connected with visco-elastic beams which are resisti… | |
-to relative translational or rotational movement between a pair of bonded | |
-points. At the beginning of each time step the accumulated strain on each | |
-inter-point bond is determined by considering the relative motion of the bonde… | |
-points. The bond deformation is decomposed per kinematic degree of freedom, | |
-andis determined by an incremental method derived from \citet{Potyondy2004}. | |
-The strain can be decomposed into bond tension and compression, bond shearing, | |
-bond twisting, and bond bending. The accumulated strains are used to determin… | |
-the magnitude of the forces and torques resistive to the deformation. | |
+The method is derived from \citet{Schlangen1996}, \citet{Radjai2011} and | |
+\citet{Potyondy2004} but is, relative to the cited works, adapted for three | |
+spatial dimensions and non-linear properties. | |
+ | |
+The Lagrangian nodes are connected with visco-elastic beam elements which are | |
+resistive to relative translational or rotational movement. The kinematic | |
+degrees of freedom are determined by explicit integration of Newton's second l… | |
+of motion for translation and rotation. For a point $i$ with bonded | |
+interactions to nodes $j\in N_c$, the translational accelerations | |
+($\boldsymbol{a}$) are found from the sums of forces: | |
+\begin{equation} | |
+ \boldsymbol{a}_i = | |
+ \frac{ | |
+ \boldsymbol{f}_i^\text{d} | |
+ + \boldsymbol{f}_i^\text{f} | |
+ + \sum^{N_c}_j \left[ | |
+ \boldsymbol{f}_{i,j}^\text{p} + | |
+ \boldsymbol{f}_{i,j}^\text{s} | |
+ \right] | |
+}{m_i} | |
+ + \boldsymbol{g} | |
+\label{eq:n2-tran} | |
+\end{equation} | |
+where $\boldsymbol{f}_i^\text{d}$ is the gravitational driving stress due to | |
+surface slope, $\boldsymbol{g}$ is the gravitational acceleration, and | |
+$\boldsymbol{f}_i^\text{f}$ is the frictional force provided if the point is | |
+resting on the lower surface. Bonded interaction with another point $j$ | |
+contributes to translational acceleration through bond-parallel and bond-norma… | |
+shear forces, $\boldsymbol{f}_{i,j}^\text{p}$ and | |
+$\boldsymbol{f}_{i,j}^\text{s}$, respectively. | |
+ | |
+The angular accelerations ($\boldsymbol{\alpha}$) are found from the sums of | |
+torques: | |
+\begin{equation} | |
+ \boldsymbol{\alpha}_i = | |
+ \sum^{N_c}_j | |
+ \left[ | |
+ \frac{\boldsymbol{t}^\text{s}_{i,j}}{I_i} + | |
+ \frac{\boldsymbol{t}^\text{t}_{i,j}}{J_{i,i}} | |
+ \right] | |
+\label{eq:n2-ang} | |
+\end{equation} | |
+here, $\boldsymbol{t}^\text{s}$ is the torque resulting from shearing motion o… | |
+the bond, while the torque $\boldsymbol{t}^{t}$ results from relative twisting. | |
+$I_i$ is the local moment of inertia at the point, and $J_{i,j}$ is polar mome… | |
+of inertia of the bond. | |
+ | |
+ | |
+At the beginning of each time step the accumulated strain on each inter-point | |
+bond is determined by considering the relative motion of the bonded nodes. Th… | |
+bond deformation is decomposed per kinematic degree of freedom, andis determin… | |
+by an incremental method derived from \citet{Potyondy2004}. The strain can be | |
+decomposed into bond tension and compression, bond shearing, bond twisting, an… | |
+bond bending. The accumulated strains are used to determine the magnitude of | |
+the forces and torques resistive to the deformation. | |
The deformation and reactive forces are determined relative to the orientation | |
of the bond. Common geometrical vectors include the inter-distance vector | |
-$\boldsymbol{d}$ between points $\boldsymbol{p}_i$ and $\boldsymbol{p}_j$: | |
+$\boldsymbol{d}$ between nodes $\boldsymbol{p}_i$ and $\boldsymbol{p}_j$: | |
\begin{equation} | |
\boldsymbol{d}_{i,j} = \boldsymbol{p}_i - \boldsymbol{p}_j | |
\end{equation} | |
-which in normalized form constitutes the bond-parallel normal vector: | |
+which in normalized form constitutes the bond-parallel unit vector: | |
\begin{equation} | |
\boldsymbol{n}_{i,j} = \frac{\boldsymbol{d}_{i,j}}{||\boldsymbol{d}_{i,j}|… | |
\end{equation} | |
-The points are moving by translational and rotational velocities. The combine… | |
-relative velocity between the points is found as \citep{Hinrichsen2004, | |
+The nodes move by translational and rotational velocities. The combined | |
+relative velocity between the nodes is found as \citep[e.g.][]{Hinrichsen2004, | |
Luding2008}: | |
\begin{equation} | |
\boldsymbol{v}_{i,j} = \boldsymbol{v}_i - \boldsymbol{v}_j + | |
- \frac{d_{i,j}}{2} \times \omega_i + | |
- \frac{d_{i,j}}{2} \times \omega_j | |
+ \frac{\boldsymbol{d}_{i,j}}{2} \times \boldsymbol{\omega}_i + | |
+ \frac{\boldsymbol{d}_{i,j}}{2} \times \boldsymbol{\omega}_j | |
+\end{equation} | |
+The velocity can be decomposed into spatial components relative to the bond | |
+orientation, e.g.\ the bond-parallel and bond-shear velocity, respectively: | |
+\begin{equation} | |
+ v^\text{p}_{i,j} = \boldsymbol{v}_{i,j} \cdot \boldsymbol{n}_{i,j} | |
+\end{equation} | |
+\begin{equation} | |
+ \boldsymbol{v}^\text{s}_{i,j} = \boldsymbol{v}_{i,j} - \boldsymbol{n}_{i,j} | |
+ \left( | |
+ \boldsymbol{v}_{i,j} | |
+ \cdot | |
+ \boldsymbol{n}_{i,j} | |
+ \right) | |
\end{equation} | |
+The axial strain is the bond-parallel deformation and is determined as the | |
+change in inter-point length relative to the initial distance: | |
+\begin{equation} | |
+ \epsilon_a = \frac{ | |
+ (\boldsymbol{d}_{i,j} - \boldsymbol{d}^0_{i,j}) \cdot n_{i,j}} | |
+ {||\boldsymbol{d}^0_{i,j}||} | |
+\end{equation} | |
+The cross-sectional area of a bond ($A_{i,j}$) varies with axial strain | |
+($\epsilon_a$) scaled by Poissons ratio $\nu$: | |
+\begin{equation} | |
+ A_{i,j} = A^0_{i,j} | |
+ - A^0_{i,j} | |
+ \left( | |
+ 1 - | |
+ \left( | |
+ 1 + \epsilon_a | |
+ \right)^{-\nu} | |
+ \right) | |
+\end{equation} | |
+The mass of point $i$ is defined as the half of the mass of each of its bonds: | |
+\begin{equation} | |
+ m_i = \frac{\rho}{2} \sum^{N_c}_j A^0_{i,j} ||\boldsymbol{d}^0_{i,j}|| | |
+\end{equation} | |
+The density ($\rho$) is adjusted so that the total mass of all nodes matches t… | |
+desired value. | |
- | |
-\subsection{Bond tension and compression} | |
+\subsection{Resistance to tension and compression} | |
+Bond tension and compression takes place when the relative translational | |
+distance between a pair of bonded nodes changes, and is the most important | |
+deformational mode in this model. The current axial strain is determined with… | |
+second-order central difference scheme. It is determined from the previous | |
+point positions and projected future positions: | |
+\begin{equation} | |
+ \Delta d^t_{i,j} = \frac{d_{i,j}^{*,t+\Delta t} - d_{i,j}^{t-\Delta t}}{2} | |
+\end{equation} | |
+The future point distance in the above ($d_{i,j}^{*,t+\Delta}$) is found by | |
+applying a second-order Taylor expansion: | |
+\begin{equation} | |
+ \boldsymbol{p}_i^{*,t+\Delta t} = | |
+ \boldsymbol{p}_i^{t} + | |
+ \boldsymbol{v}_i^{t} \Delta t + | |
+ \frac{1}{2}\boldsymbol{a}_i^{t} \Delta t^2 | |
+\end{equation} | |
-\subsection{Bond shear} | |
-\subsection{Bond twist} | |
+The bond-parallel force is determined from Young's modulus ($E$) and the | |
+cross-sectional area ($A_{i,j}$) of the bond: | |
+\begin{equation} | |
+ \boldsymbol{f}^{i,j}_\text{p} = | |
+ \frac{E A_{i,j}}{|| \boldsymbol{d}^0_{i,j} ||} | |
+ \left( | |
+ \boldsymbol{d}_{i,j} - | |
+ \boldsymbol{d}^0_{i,j} | |
+ \right) | |
+\end{equation} | |
-\subsection{Bond bend} | |
+\subsection{Shear resistance} | |
+The bond-shear force is determined incrementally for the duration of the | |
+interaction: | |
+\begin{equation} | |
+ \boldsymbol{f}^{i,j}_\text{s} = \int^t \Delta \boldsymbol{f}^{i,j}_\text{s… | |
+ %\, dt | |
+\end{equation} | |
+where the increment in shear force is determined from the shear modulus ($G$), | |
+the cross-sectional area ($A_{i,j}$) of the bond, and the | |
+\begin{equation} | |
+ \Delta \boldsymbol{f}^{i,j}_\text{s} = | |
+ \frac{G A_{i,j}}{||\boldsymbol{d}^{i,j}||} | |
+ \Delta \boldsymbol{d}^{i,j}_\text{s} | |
+\end{equation} | |
-\subsection{Temporal integration} | |
+\subsection{Twisting resistance} | |
+\subsection{Bending resistance} | |
+\subsection{Temporal integration} | |
+Once the force and torque sum components at time $t$ have been determined, the | |
+kinematic degrees of freedom at time $t+\Delta t$ can be found by explicit | |
+temporal integration of moment balance equations~\ref{eq:n2-tran} | |
+and~\ref{eq:n2-ang}. | |
+We use an integration scheme based on the third-order Taylor expansion, which | |
+results in a truncation error on the order of $O(\Delta t^4)$ for positions an… | |
+$O(\Delta t^3)$ for velocities. This scheme includes changes in acceleration … | |
+the highest order term, which are approximated by backwards differences. For | |
+the translational degrees of freedom: | |
+\begin{equation} | |
+ \boldsymbol{p}^i_{t+\Delta t} = | |
+ \boldsymbol{p}^i_{t} + | |
+ \boldsymbol{v}^i_{t} \Delta t + | |
+ \frac{1}{2} \boldsymbol{a}^i_{t} \Delta t^2 + | |
+ \frac{1}{6} \frac{\boldsymbol{a}^i_{t} | |
+ - \boldsymbol{a}^i_{t - \Delta t}}{\Delta t} \Delta t^3 | |
+\end{equation} | |
+\begin{equation} | |
+ \boldsymbol{v}^i_{t+\Delta t} = | |
+ \boldsymbol{v}^i_{t} + | |
+ \boldsymbol{a}^i_{t} \Delta t + | |
+ \frac{1}{2} \frac{\boldsymbol{a}^i_{t} | |
+ - \boldsymbol{a}^i_{t - \Delta t}}{\Delta t} \Delta t^2 | |
+\end{equation} | |
+At $t=0$ the acceleration change term is defined as zero. The angular degrees | |
+of freedom are found correspondingly: | |
+\begin{equation} | |
+ \boldsymbol{\Omega}^i_{t+\Delta t} = | |
+ \boldsymbol{\Omega}^i_{t} + | |
+ \boldsymbol{\omega}^i_{t} \Delta t + | |
+ \frac{1}{2} \boldsymbol{\alpha}^i_{t} \Delta t^2 + | |
+ \frac{1}{6} \frac{\boldsymbol{\alpha}^i_{t} | |
+ - \boldsymbol{\alpha}^i_{t - \Delta t}}{\Delta t} \Delta t^3 | |
+\end{equation} | |
+\begin{equation} | |
+ \boldsymbol{\omega}^i_{t+\Delta t} = | |
+ \boldsymbol{\omega}^i_{t} + | |
+ \boldsymbol{\alpha}^i_{t} \Delta t + | |
+ \frac{1}{2} \frac{\boldsymbol{\alpha}^i_{t} | |
+ - \boldsymbol{\alpha}^i_{t - \Delta t}}{\Delta t} \Delta t^2 | |
+\end{equation} | |
+The numerical time step $\Delta t$ is found by considering the largest elastic | |
+stiffness in the system relative to the smallest mass: | |
+\begin{equation} | |
+ \Delta t = | |
+ \epsilon | |
+ \left[ | |
+ \min (m_i)^{-1} | |
+ \max \left( | |
+ \max \left( | |
+ \frac{E A_{i,j}}{||\boldsymbol{d}_{0}^{i,j}||} | |
+ \right) | |
+ , | |
+ \max \left( | |
+ \frac{G A_{i,j}}{||\boldsymbol{d}^{i,j}||} | |
+ \right) | |
+ \right) | |
+ \right]^{-1/2} | |
+\end{equation} | |
+where $\epsilon$ is a safety factor related to the geometric structure of the | |
+bonded network. We use $\epsilon = 0.07$. | |
+ | |
+The total force ($\boldsymbol{f}$) and torque ($\boldsymbol{t}$) on two nodes | |
+($i$ and $j$) with translational ($\boldsymbol{p}$) and angular | |
+($\boldsymbol{\Omega}$) positions interconnected with a three-dimensional | |
+elastic beam can be expressed as the following set of equations. The | |
+interaction accounts for resistance to tension and compression, shear, torsion… | |
+and bending. The symmetrical matrix on the right hand side constitutes the | |
+\emph{stiffness matrix} \citep{Schlangen1996, Austrell2004}: | |
+\begin{equation} | |
+ \begin{bmatrix} | |
+ f_\text{x}^i\\[0.6em] | |
+ f_\text{y}^i\\[0.6em] | |
+ f_\text{z}^i\\[0.6em] | |
+ t_\text{x}^i\\[0.6em] | |
+ t_\text{y}^i\\[0.6em] | |
+ t_\text{z}^i\\[0.6em] | |
+ f_\text{x}^j\\[0.6em] | |
+ f_\text{y}^j\\[0.6em] | |
+ f_\text{z}^j\\[0.6em] | |
+ t_\text{x}^j\\[0.6em] | |
+ t_\text{y}^j\\[0.6em] | |
+ t_\text{z}^j\\ | |
+ \end{bmatrix} | |
+ = | |
+ \begin{bmatrix} | |
+ \frac{EA}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-EA}{L} & 0 & 0 & 0 & 0 & 0\\… | |
+ 0 & \frac{12EI_\text{z}}{L^3} & 0 & 0 & 0 & \frac{6EI_\text{z}}{L^2} &… | |
+ 0 & 0 & \frac{12EI_\text{y}}{L^3} & 0 & \frac{-6EI_\text{y}}{L^2} & 0 … | |
+ 0 & 0 & 0 & \frac{GK_\text{v}}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-GK_\tex… | |
+ 0 & 0 & \frac{-6EI_\text{y}}{L^2} & 0 & \frac{4EI_\text{y}}{L} & 0 & 0… | |
+ 0 & \frac{6EI_\text{z}}{L^2} & 0 & 0 & 0 & \frac{4EI_\text{z}}{L} & 0 … | |
+ \frac{-EA}{L} & 0 & 0 & 0 & 0 & 0 & \frac{EA}{L} & 0 & 0 & 0 & 0 & 0\\… | |
+ 0 & \frac{-12EI_\text{z}}{L^3} & 0 & 0 & 0 & \frac{-6EI_\text{z}}{L^2}… | |
+ 0 & 0 & \frac{-12EI_\text{y}}{L^3} & 0 & \frac{-6EI_\text{y}}{L^2} & 0… | |
+ 0 & 0 & 0 & \frac{-GK_\text{v}}{L} & 0 & 0 & 0 & 0 & 0 & \frac{GK_\tex… | |
+ 0 & 0 & \frac{-6EI_\text{y}}{L^2} & 0 & \frac{2EI_\text{y}}{L} & 0 & 0… | |
+ 0 & \frac{6EI_\text{z}}{L^2} & 0 & 0 & 0 & \frac{2EI_\text{z}}{L} & 0 … | |
+ \end{bmatrix} | |
+ \begin{bmatrix} | |
+ p_\text{x}^i\\[0.6em] | |
+ p_\text{y}^i\\[0.6em] | |
+ p_\text{z}^i\\[0.6em] | |
+ \Omega_\text{x}^i\\[0.6em] | |
+ \Omega_\text{y}^i\\[0.6em] | |
+ \Omega_\text{z}^i\\[0.6em] | |
+ p_\text{x}^j\\[0.6em] | |
+ p_\text{y}^j\\[0.6em] | |
+ p_\text{z}^j\\[0.6em] | |
+ \Omega_\text{x}^j\\[0.6em] | |
+ \Omega_\text{y}^j\\[0.6em] | |
+ \Omega_\text{z}^j\\ | |
+ \end{bmatrix} | |
+\end{equation} | |
+$E$ is Young's modulus, $G$ is the shear stiffnes, $A$ is the beam | |
+cross-sectional area, and $L$ is the original beam length. $I_\text{y}$ is the | |
+moment of inertia normal to the beam in the $\bar{y}$-direction, and | |
+$I_\text{z}$ is the moment of inertia normal to the beam in the | |
+$\bar{z}$-direction. $K_\text{v}$ is the Saint-Venant torsional stiffness. | |
+ | |
+% Torsional constant: | |
+% https://en.wikipedia.org/wiki/Torsion_constant | |
+% http://mathworld.wolfram.com/TorsionalRigidity.html | |
+% http://physics.stackexchange.com/questions/83148/where-i-can-find-a-torsiona… | |
+% St Venant torsion: K_v = 1/G (Austrell et al. 2004, table 3) Does it make s… | |
diff --git a/slidergrid/slider.h b/slidergrid/slider.h | |
t@@ -31,6 +31,10 @@ typedef struct { | |
// moment of inertia [kg m*m] | |
Float moment_of_inertia; | |
+ // Macroscopic mechanical properties | |
+ Float youngs_modulus; | |
+ Float shear_modulus; | |
+ | |
// inter-slider bond-parallel Kelvin-Voigt contact model parameters | |
Float bond_parallel_kv_stiffness; // Hookean elastic stiffness [N/m] | |
Float bond_parallel_kv_viscosity; // viscosity [N/(m*s)] |