(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 35256, 980] NotebookOptionsPosition[ 28458, 801] NotebookOutlinePosition[ 32708, 910] CellTagsIndexPosition[ 31919, 892] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell["Stress Strain, Principal Axes, and Mohr's Circle", "Title"], Cell[CellGroupData[{ Cell["General Stress State Rotated from the Principal Axes System", \ "Subtitle", CellChangeTimes->{{3.398521341572761*^9, 3.3985213497631283`*^9}, 3.398521386031128*^9}, 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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RowBox[{ RowBox[{"\[Sigma]rotalt", " ", "=", RowBox[{"Collect", "[", RowBox[{ RowBox[{"\[Sigma]rot", "//", "TrigReduce"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", RowBox[{"2", "\[Theta]"}], "]"}], ",", RowBox[{"Sin", "[", RowBox[{"2", " ", "\[Theta]"}], "]"}]}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[Sigma]rotalt", "//", "MatrixForm"}]}], "Input", CellTags->{"mtag:10:Collect[]", "mtag:10:TrigReduce[]"}], Cell["\<\ Naming the coefficients of the rotated two-dimensional state:\ \>", "Text", CellChangeTimes->{{3.398521719741212*^9, 3.398521728290906*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"\[Sigma]labMat", "=", RowBox[{ RowBox[{"(", "\[NoBreak]", GridBox[{ { SubscriptBox["\[Sigma]lab", "xx"], SubscriptBox["\[Sigma]lab", "xy"], SubscriptBox["\[Sigma]lab", "xz"]}, { SubscriptBox["\[Sigma]lab", "xy"], SubscriptBox["\[Sigma]lab", "yy"], SubscriptBox["\[Sigma]lab", "yz"]}, { SubscriptBox["\[Sigma]lab", "xz"], SubscriptBox["\[Sigma]lab", "yz"], SubscriptBox["\[Sigma]lab", "zz"]} }], "\[NoBreak]", ")"}], "=", " ", "\[Sigma]rotalt"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[Sigma]labMat", "//", "MatrixForm"}]}], "Input", CellTags->{ "mtag:10:assignment_of_matrix_elements__example", "mtag:10:matrices__example__assigning_element_by_element"}], Cell["\<\ Looking at the x-y components of stress (i.e, the upper-left 2\[Times]2 \ submatrix), notice that there are two invariants of the generalized \ two-dimensional stress state: The trace and the determinant:\ \>", "Text", CellChangeTimes->{{3.398521093315215*^9, 3.398521177858827*^9}, { 3.398521606292947*^9, 3.398521615330989*^9}}, CellTags->"mtag:10:rotation_invariants"], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ SubscriptBox["\[Sigma]lab", "xx"], " ", "+", " ", SubscriptBox["\[Sigma]lab", "yy"]}], " ", "]"}]], "Input"], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ SubscriptBox["\[Sigma]lab", "xx"], " ", SubscriptBox["\[Sigma]lab", "yy"]}], " ", "-", RowBox[{ RowBox[{"(", SubscriptBox["\[Sigma]lab", "xy"], ")"}], "^", "2"}]}], "]"}]], "Input"], Cell["\<\ Do not depend on \[Theta]; thus illustrating the invariance of these \ quantities under rotation of coordinate rotations.\ \>", "Text", CellChangeTimes->{{3.398521267819919*^9, 3.398521299123801*^9}, { 3.398521631675811*^9, 3.398521633363654*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Two-Dimensional Stress under Rotation: Mohr's Circle", "Subtitle", CellChangeTimes->{{3.398521399115279*^9, 3.398521460707005*^9}}, CellTags->"mtag:10:principal_axes"], Cell[BoxData[ RowBox[{ StyleBox["1.", "Text"], StyleBox[" ", "Text"], StyleBox[ SubscriptBox["\[Sigma]lab", "xx"], "Text"], StyleBox[" ", "Text"], StyleBox["in", "Text"], StyleBox[" ", "Text"], StyleBox["laboratory", "Text"], StyleBox[" ", "Text"], StyleBox["system", "Text"], StyleBox[" ", "Text"], StyleBox["rotated", "Text"], StyleBox[" ", "Text"], StyleBox["by", "Text"], StyleBox[" ", "Text"], StyleBox["\[Theta]", "Text"], StyleBox[" ", 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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.\ \>", "Text", CellChangeTimes->{{3.398520428141933*^9, 3.3985204579630957`*^9}}, CellTags->"mtag:10:stress__example__Mohrs_circle"], Cell[TextData[{ "Example of Mohr's circle for two-dimensional body in uniaxial tension with \ ", Cell[BoxData[ SubscriptBox["\[Sigma]princ", "xx"]]], " = 10 MPa and all other stress components equal to zero" }], "Text"], Cell[BoxData[ RowBox[{"uniaxial10", "=", " ", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["\[Sigma]princ", "xx"], "->", "10"}], ",", RowBox[{ SubscriptBox["\[Sigma]princ", "yy"], "->", "0"}]}], "}"}]}]], "Input"], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["\[Sigma]lab", "xx"], ",", SubscriptBox["\[Sigma]lab", "xy"]}], "}"}], "/.", "uniaxial10"}], "\[IndentingNewLine]", ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", "\[Pi]"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", RowBox[{"AspectRatio", "\[Rule]", "1"}], ",", RowBox[{ "PlotLabel", "\[Rule]", "\"\< \\t \\t Mohr Circle for 10 MPa Uniaxial Tension\>\""}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Thickness", "[", "0.01", "]"}], ",", RowBox[{"Hue", "[", "1", "]"}]}], "}"}]}]}], "]"}]], "Input", PageBreakAbove->True, CellTags->"mtag:10:stress__example__Mohrs_circle_plotted"], Cell[BoxData[ RowBox[{"uniaxialother", "=", " ", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox["\[Sigma]princ", "xx"], "->", "30"}], ",", RowBox[{ SubscriptBox["\[Sigma]princ", "yy"], "->", "10"}]}], "}"}]}]], "Input"], Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ SubscriptBox["\[Sigma]lab", "xx"], ",", SubscriptBox["\[Sigma]lab", "xy"]}], "}"}], "/.", "uniaxialother"}], "\[IndentingNewLine]", ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", "\[Pi]"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", RowBox[{"AspectRatio", "\[Rule]", "1"}], ",", RowBox[{"PlotRange", "->", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "40"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "20"}], ",", "20"}], "}"}]}], "}"}]}], ",", RowBox[{ "PlotLabel", "\[Rule]", "\"\< \\t \\t Mohr Circle for \!\(\*SubscriptBox[\(\[Sigma]princ\), \ \(xx\)]\)= 30 \!\(\*SubscriptBox[\(\[Sigma]princ\), \(yy\)]\)=10\>\""}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Thickness", "[", "0.01", "]"}], ",", RowBox[{"Hue", "[", "1", "]"}]}], "}"}]}]}], "]"}]], "Input", PageBreakAbove->True, CellTags->"mtag:10:stress__example__Mohrs_circle_plotted"], Cell["\<\ Comparing this plot with Figure 10-3 in the lecture notes, we see that the \ maximum and minimum tensile stresses are 10 and 30 MPa (from intercepts with \ x axis), as expected, and the maximum shear stress is \[PlusMinus]5 MPa and \ it is experienced on a plane oriented at 2\[Theta] = 90\[Degree] or \[Theta] \ = 45\[Degree] to the tensile axis (remember that angles on Mohr's circle \ plots are twice the angle in the body).\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Visualization Example: Graphics for Mohr's Circle", "Subtitle", CellChangeTimes->{{3.398519569309819*^9, 3.398519574378683*^9}, { 3.398592830662013*^9, 3.398592846324191*^9}}, CellTags->"mtag:10:principal_axes"], Cell["\<\ This is a straightforward, but fairly lengthy, example of using mouse \ interaction to visualize Mohr's circle of stress. 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