172 lines
8.4 KiB
TeX
172 lines
8.4 KiB
TeX
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\subsubsection{Continuum Description.}
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Continuum description assumes a hypothetical continuous medium to describe behavior of a fluid/solid body. This allows description of material using continuous mathematical functions.
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\subsubsection{Mechanics.}
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Mechanics is classified into statics and dynamics. In crystal plasticity we study dynamics of plastic deformation. Dynamics can be further classified into Kinematics and Kinetics.
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\begin{itemize}
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\item Kinematics: Study of deformation in the continuum element under consideration. Quantities measured are displacements, strains, velocities and strain rates.
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\item Kinetics: Study of forces and moments acting on the continuum object and how it affects displacements/strains or velocities/strain-rates. It involves description of material behaviour by quantification of traction/stresses and body forces using constitutive laws, material properties and conservation laws.
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\end{itemize}
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\section{Continuum Mechanics description of a body undergoing deformation.}
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The figure \ref{Continuum_body_deformation} shows the undeformed continuum body $\mathcal{B}_0$ in reference configuration and deformed body $\mathcal{B}_t$ in current configuration. Reference configuration is time independent and current configuration is time dependent. If the deformation is described in reference/material configuration it is called Lagrangian description and if it is described in current/spatial configuration it is called Eulerian description.
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Displacement at a given deformation state (at time = t) is given by,
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\begin{equation}
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\label{eq:A1}
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u(X) = x(X) - X
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\end{equation}
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A line segment $dX$ in reference configuration is transformed transformed to current configuration as,
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\begin{equation}
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\label{eq:A2}
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x(X) + dx = x(X) + \frac{\partial x}{\partial X}.\textrm{d}X + \textrm{O} (\textrm{d}X^2)
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\end{equation}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.9\textwidth]{images/Continuum_Mechanics_configurations.pdf}
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\caption{Deformation of a continuum body.}
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\label{Continuum_body_deformation}
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\end{figure}
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Neglecting higher order terms $\textrm{O} (\textrm{d}X^2)$,
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\begin{equation}
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\label{eq:A3}
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dx = \frac{\partial x}{\partial X}.\textrm{d}X = F.\textrm{d}X
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\end{equation}
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where $F:=\frac{\partial x}{\partial X}$ is called deformation gradient, a second order tensor. The deformation gradient maps the vector $\textrm{d}X$ at $X$ in the reference configuration to the vector $\textrm{d}x$ at $x$ in the current configuration. It is a 2-leg tensor because has one base at reference and one at current configuration.
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From equations \ref{eq:A1} and \ref{eq:A3},
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\begin{equation}
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\label{eq:A4}
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F = \frac{\partial u + X}{\partial X}
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\end{equation}
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\begin{equation}
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\label{eq:A5}
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F = I + \frac{\partial u}{\partial X}
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\end{equation}
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where $U:=\frac{\partial u}{\partial X}$ is called displacement gradient. Displacement gradient and deformation gradient are used to describe deformation of a body. Both are 2-leg tensors.
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\section{Strain Measures}
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It is possible to describe deformation only in the reference configuration as follows:
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\begin{align*}
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\textrm{d}x.\textrm{d}x - \textrm{d}X.\textrm{d}X &= F.\textrm{d}X.F.\textrm{d}X -\textrm{d}X.\textrm{d}X \\
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&= \textrm{d}X.(F^T . F).\textrm{d}X - \textrm{d}X.\textrm{d}X \\
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&= \textrm{d}X.(F^T . F - I).\textrm{d}X \\
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&= \textrm{d}X.(2E_0).\textrm{d}X
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\end{align*}
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where $E_0 = \frac{1}{2}(F^T . F - I)$ is called Green-Lagrange strain tensor which is defined only in reference configuration.
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Similarly, it is also possible to describe deformation in current configuration
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\begin{align*}
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\textrm{d}x.\textrm{d}x - \textrm{d}X.\textrm{d}X &= \textrm{d}x.\textrm{d}x - F^{-1}.\textrm{d}x.F^{-1}.\textrm{d}x\\
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&= \textrm{d}x.\textrm{d}x - \textrm{d}x.(F^{-T} . F^{-1}).\textrm{d}x\\
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&= \textrm{d}x.(I - F^{-T} . F^{-1}).\textrm{d}x \\
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&= \textrm{d}X.(2E_t).\textrm{d}X
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\end{align*}
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this leads to definition of Euler-Almansi strain tensor $E_t = \frac{1}{2}(I - F^{-T} . F^{-1})$ which is defined only in current configuration.
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Cauchy strain can be defined in terms of displacement gradient as,
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\begin{equation}
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\epsilon = \frac{1}{2} \left( \nabla u + ( \nabla u)^T \right)
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\end{equation}
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For ONE-DIMENSION cases without lateral contraction the deformation can be described in terms of two variables: length in reference configuration $l_0$ and length in current configuration $l_t$ . By defining the ratio of these 2 variables as 'stretch ratio' we can compare the strain measures as given in the table \ref{Strain_measures_def}
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\begin{table}[H]
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\centering
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\caption{Definition of strain measures for 1D cases}
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\begin{tabular}{c|cc}
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configuration & strain measure & definition in 1D \\
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\hline
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reference & Green-Lagrange strain & $E_{0,1dim = \frac{1}{2}(\lambda^2-1)}$ \\
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current & Euler-Almansi strain & $E_{t,1dim = \frac{1}{2}(1 - \frac{1}{\lambda^2})}$ \\
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2-leg & Cauchy strain & $\epsilon_{1dim} = \lambda - 1$
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\end{tabular}
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\label{Strain_measures_def}
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\end{table}
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\section{Stress Measures}
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Stress is a second order tensor defined using force and area vectors which can be defined in reference or current configuration. As a result different stress measures exist.
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The stress measures along with strain measures are summarized in the table \ref{Stress_Measures_Configurations}.
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\begin{table}[H]
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\centering
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\caption{Stress and Strain measures in different configurations.}
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\renewcommand\arraystretch{1.4}
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\renewcommand\baselinestretch{1.4}
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\begin{tabular}{c|ccc}
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configuration & stress & strain & symmetry \\
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\hline
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current/spatial & Cauchy Stress & Euler-Almansi strain & symmetric \\
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2-leg & 1st Piola-Kirchhoff stress & Displacement Gradient & non-symmetric\\
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reference/material & 2nd Piola-Kirchhoff stress & Green-Lagrange strain & symmetric\\
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\end{tabular}
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\label{Stress_Measures_Configurations}
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\end{table}
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The conversion between different stress measures used in DAMASK is given in table \ref{Stress_measures}
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\begin{table}[H]
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\centering
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\caption{Conversion between different stress measures used in DAMASK.}
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\renewcommand\arraystretch{1.4}
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\renewcommand\baselinestretch{1.4}
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\begin{tabular}{c|c|c|c}
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& $P$ (PK1)) & $S$ (PK2) & $\sigma$ (Cauchy) \\
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\hline
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$P$ & $P$ & det$F_pF_eF_iSF_p^{-T}$ & det$F\sigma F^{-T}$ \\
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$S$ & $\frac{1}{\mbox{det} F_p}F_i^{-1}F_e^{-1}PF_p^T$ & $S$ & det$(F_eF_i)F_i^{-1}F_e^{-1}\sigma F_e^{-T}F_i^{-T}$\\
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$\sigma$ & $\frac{1}{\mbox{det}F}PF^T$ & $\frac{1}{\mbox{det} (F_eF_i)}F_eF_iSF_i^{T}F_e^{T}$ & $\sigma$\\
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\end{tabular}
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\label{Stress_measures}
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\end{table}
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\section{Constitutive relation for Linear Elasticity}
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Each stress components can be expressed as a linear combination of strain components in linear elasticity theory or Hooke's law, which is written as:
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\begin{equation*}
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\sigma_{ij} = c_{ijkl} \epsilon_{kl}
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\end{equation*}
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where $c_{ijkl}$ are components of the elastic stiffness tensor. The symmetries in the strain and the stress reduce 81 different entries in the elastic tensor to 36 independent elements.
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In the linear elasticity, the potential energy must be quadratic function of elastic strain. This further reduces the number of independent elements to 21.
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Various crystallographic symmetries in the lattice structure reduces the number of independent elastic constants further as given in the table \ref{Independent_Cijkl}. To simplify representation of elastic constants Vogit notation is used in the below table where indices are mapped as: $11\to 1, \ 22 \to 2, \ 33 \to 3, \ 23\&32 \to 4, \ 13\&31 \to 4, \ 12\&21 \to 6$
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\begin{table}[H]
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\centering
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\caption{Independent elastic constants for various crystal symmetries.}
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\begin{tabular}{c m{4em} c}
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\hline
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Crystal Class & Independent $C_{ij}$ & List of independent $C_{ij}$ \\
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\hline
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Triclinic & 21 & All possible combinations \\
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Monoclinic & 13 & $C_{11}, C_{12}, C_{13}, C_{16}, C_{22}, C_{23}, C_{26}, C_{33}, C_{36}, C_{44}, C_{45}, C_{55}, C_{66}$ \\
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Orthorhombic & 9 & $C_{11}, C_{12}, C_{13}, C_{22}, C_{23}, C_{33}, C_{44}, C_{55}, C_{66}$ \\
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Trigonal & 6 or 7 & $C_{11}, C_{12}, C_{13}, C_{14}, C_{25}, C_{33}, C_{44}$ \\
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Tetragonal & 6 & $C_{11}, C_{12}, C_{13}, C_{33}, C_{44}, C_{66}$ \\
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Hexagonal & 5 & $C_{11}, C_{33}, C_{44}, C_{12}, C_{14}$ \\
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Cubic & 3 & $C_{11}, C_{12}, C_{44}$ \\
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Isotropic & 2 & $C_{11}, C_{44}$ \\
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\hline
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\end{tabular}
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\label{Independent_Cijkl}
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\end{table}
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