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Notebook[{
Cell["Stress Strain, Principal Axes, and Mohr's Circle", "Title"],

Cell[CellGroupData[{

Cell["General Stress State  in Principal Axes  System", "Section",
  CellTags->"mtag:10:principal_axes"],

Cell[TextData[{
  "This is a general state, we will rotate about the ",
  StyleBox["z",
    FontSlant->"Italic"],
  "-axis and compare the result to a general two-dimensional stress state."
}], "Text"],

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              {"0", \(\[Sigma]princ\_yy\), "0"},
              {"0", "0", \(\[Sigma]princ\_zz\)}
              },
            GridFrame->True,
            RowLines->True,
            ColumnLines->True], ")"}]}], 
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  CellTags->{
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    "mtag:10:stress_tensor_2D__principal_axes"}],

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          RowSpacings->1,
          ColumnSpacings->1,
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  CellTags->{
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    "mtag:10:stress_tensor_2D__principal_axes"}]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Rotation about ",
  StyleBox["z",
    FontSlant->"Italic"],
  "-axis by angle \[Theta]"
}], "Section"],

Cell[CellGroupData[{

Cell[BoxData[{
    RowBox[{
      RowBox[{\(rotmat[\[Theta]_]\), " ", ":=", 
        RowBox[{"(", GridBox[{
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              {\(Sin[\[Theta]]\), \(Cos[\[Theta]]\), "0"},
              {"0", "0", "1"}
              },
            GridFrame->True,
            RowLines->True,
            ColumnLines->True], ")"}]}], ";"}], "\n", \(rotmat[\[Theta]] // 
      MatrixForm\)}], "Input",
  CellTags->"mtag:10:rotation_about_z_axis"],

Cell[BoxData[
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          RowSpacings->1,
          ColumnSpacings->1,
          ColumnAlignments->{Left}], "\[NoBreak]", ")"}],
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        MatrixForm[ BoxForm`e$]]]], "Output",
  CellTags->"mtag:10:rotation_about_z_axis"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Transformation to general two-dimensional stress state coordinate \
system by rotating the principal system by \[Theta] around z-axis\
\>", \
"Section"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(\[Sigma]rot\  = \ 
        Simplify[
          Transpose[rotmat[\[Theta]]] . \[Sigma]tensordiag . 
            rotmat[\[Theta]]];\)\), "\[IndentingNewLine]", 
    \(\[Sigma]rot // MatrixForm\)}], "Input",
  CellTags->{
  "mtag:10:similarity_transformation_example", 
    "mtag:10:stress__in_rotated_coordinate_system", "mtag:10:Transpose[]"}],

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\[Sigma]princ\_yy)\)\), \(Sin[\[Theta]]\^2\ \[Sigma]princ\_xx + 
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  CellTags->{
  "mtag:10:similarity_transformation_example", 
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Cell["Writing the same equation in a slightly different way...", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(\[Sigma]rotalt\  = 
        Collect[\[Sigma]rot // TrigReduce, {Cos[2  \[Theta]], 
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                  2\ \[Theta]]\ \((\(-\[Sigma]princ\_xx\) + \
\[Sigma]princ\_yy)\)\), "0"},
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\[Sigma]princ\_yy)\)\), \(1\/2\ Cos[
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Cell["\<\
Naming the coefficients of the two-dimensional state in the rotated \
principal system\
\>", "Text"],

Cell[BoxData[
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\(\[Sigma]lab\_yz\)},
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  CellTags->{
  "mtag:10:assignment_of_matrix_elements__example", 
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Cell[BoxData[
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from\ principal\ axis\ system\)], "Text"],

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}, Open  ]],

Cell[BoxData[
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from\ principal\ axis\ system\)], "Text"],

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Cell[BoxData[
    \(\[Sigma]lab\_yy\)], "Input"],

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          2\ \[Theta]]\ \((\(-\[Sigma]princ\_xx\) + \[Sigma]princ\_yy)\) + 
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}, Open  ]],

Cell[BoxData[
    \(3. \ \[Sigma]lab\_xy\ \ in\ laboratory\ system\ rotated\ by\ \[Theta]\ \
from\ principal\ axis\ system\)], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[Sigma]lab\_xy\)], "Input"],

Cell[BoxData[
    \(1\/2\ Sin[
        2\ \[Theta]]\ \((\(-\[Sigma]princ\_xx\) + \[Sigma]princ\_yy)\)\)], \
"Output"]
}, Open  ]],

Cell[BoxData[
    \(All\ z - 
      components\ remain\ zero\ except\ the\ original\ diagonal\ term\ \(\(\
\[Sigma]lab\_zz\)\(:\)\)\)], "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\[Sigma]lab\_yz\), "\[IndentingNewLine]", 
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["\<\
Equations 1, 2, and 3 are the equations that express a circle known \
as Mohr's circle of stress (see accompanying class notes). The equations show \
the way in which the stress tensor components in a two-dimensional state of \
stress (a \"biaxial\" stress state) vary with orientation of the coordinate \
system in which the stresses are described.\
\>", "Subsubsection",
  CellTags->"mtag:10:stress__example__Mohrs_circle"],

Cell[BoxData[
    \(Notice\ that\ there\ are\ two\ invariants\ of\ the\ general\ \(\(stress\
\)\(:\)\)\)], "Text",
  CellTags->"mtag:10:rotation_invariants"],

Cell["The trace (or twice the offset of Mohr's Circle):", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[\[Sigma]lab\_xx\  + \ \[Sigma]lab\_yy\ ]\)], "Input"],

Cell[BoxData[
    \(\[Sigma]princ\_xx + \[Sigma]princ\_yy\)], "Output"]
}, Open  ]],

Cell["And the determinant", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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\((\[Sigma]lab\_xy)\)^2]\)], "Input"],

Cell[BoxData[
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}, Open  ]],

Cell["\<\
These last two results are precisely the trace and determinant of \
the x and y terms in the original diagonal form of the stress state, thus \
illustrating the invariance of these quantities under rotation of coordinate \
rotations.\
\>", "Text"],

Cell[TextData[{
  "Example of Mohr's circle for two-dimensional body in uniaxial tension with \
",
  Cell[BoxData[
      \(\[Sigma]princ\_xx\)]],
  " = 10 MPa and all other stress components equal to zero"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(uniaxial10 = \ {\[Sigma]princ\_xx -> 10, \[Sigma]princ\_yy -> 
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Cell[BoxData[
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Cell[CellGroupData[{

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      AxesLabel \[Rule] {"\<normal stress\>", "\<shear stress\>"}, 
      AspectRatio \[Rule] 1, 
      PlotLabel \[Rule] "\< \t \t Mohr Circle for 10 MPa Uniaxial Tension\>", 
      PlotStyle \[Rule] {Thickness[0.01], Hue[1]}]\)], "Input",
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