(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 40346, 1069] NotebookOptionsPosition[ 33395, 885] NotebookOutlinePosition[ 37726, 996] CellTagsIndexPosition[ 36937, 978] WindowTitle->Lecture 10 MIT 3.016 (Fall 2011) \251 W. Craig Carter \ 2003--2011 WindowFrame->Normal*) (* 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"], Cell[BoxData[{ RowBox[{ RowBox[{"\[Sigma]tensordiag", "=", RowBox[{"(", GridBox[{ { SubscriptBox["\[Sigma]princ", "xx"], "0", "0"}, {"0", SubscriptBox["\[Sigma]princ", "yy"], "0"}, {"0", "0", SubscriptBox["\[Sigma]princ", "zz"]} }, GridBoxDividers->{ "Columns" -> {{True}}, "ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[Sigma]tensordiag", "//", "MatrixForm"}]}], "Input", CellTags->{ "mtag:10:stress_tensor__principal_axes", "mtag:10:stress_tensor_2D__principal_axes"}], Cell[CellGroupData[{ Cell[TextData[{ "Rotation about ", StyleBox["z", FontSlant->"Italic"], "-axis by angle \[Theta]" }], "Section"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"rotmat", "[", "\[Theta]_", "]"}], " ", ":=", RowBox[{"(", GridBox[{ { RowBox[{"Cos", "[", "\[Theta]", "]"}], RowBox[{"-", RowBox[{"Sin", "[", "\[Theta]", "]"}]}], "0"}, { RowBox[{"Sin", "[", "\[Theta]", "]"}], RowBox[{"Cos", "[", "\[Theta]", "]"}], "0"}, {"0", "0", "1"} }, GridBoxDividers->{ "Columns" -> {{True}}, "ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}], ")"}]}], ";"}], "\n", RowBox[{ RowBox[{"rotmat", "[", "\[Theta]", "]"}], "//", "MatrixForm"}]}], "Input", CellTags->"mtag:10:rotation_about_z_axis"] }, Open ]], Cell["\<\ Transformation to general two-dimensional stress state coordinate system by \ rotating the principal system by \[Theta] around z-axis\ \>", "Section"], Cell[BoxData[{ RowBox[{ RowBox[{"\[Sigma]rot", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"Transpose", "[", RowBox[{"rotmat", "[", "\[Theta]", "]"}], "]"}], ".", "\[Sigma]tensordiag", ".", RowBox[{"rotmat", "[", "\[Theta]", "]"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"\[Sigma]rot", "//", "MatrixForm"}]}], "Input", CellTags->{ "mtag:10:similarity_transformation_example", "mtag:10:stress__in_rotated_coordinate_system", "mtag:10:Transpose[]"}], Cell["Writing the same equation in a slightly different way...", "Text"], Cell[BoxData[{ 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[ StyleBox[ RowBox[{"1.", " ", SubscriptBox["\[Sigma]lab", "xx"], " ", "in", " ", "laboratory", " ", "system", " ", "rotated", " ", "by", " ", "\[Theta]", " ", "from", " ", "principal", " ", "axis", " ", "system"}], "Text"]], "Text"], Cell[BoxData[ SubscriptBox["\[Sigma]lab", "xx"]], "Input"], Cell[BoxData[ StyleBox[ RowBox[{"2.", " ", SubscriptBox["\[Sigma]lab", "yy"], " ", "in", " ", "laboratory", " ", "system", " ", "rotated", " ", "by", " ", "\[Theta]", " ", "from", " ", "principal", " ", "axis", " ", "system"}], "Text"]], "Text"], Cell[BoxData[ SubscriptBox["\[Sigma]lab", "yy"]], "Input"], Cell[BoxData[ StyleBox[ RowBox[{"3.", " ", SubscriptBox["\[Sigma]lab", "xy"], " ", "in", " ", "laboratory", " ", "system", " ", "rotated", " ", "by", " ", "\[Theta]", " ", "from", " ", "principal", " ", "axis", " ", "system"}], "Text"]], "Text"], Cell[BoxData[ SubscriptBox["\[Sigma]lab", "xy"]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"All", " ", "z"}], "-", RowBox[{ "components", " ", "remain", " ", "zero", " ", "except", " ", "the", " ", "original", " ", "diagonal", " ", "term", " ", SubscriptBox["\[Sigma]lab", "zz"]}]}]], "Text"], Cell["\<\ The above three equations can be graphically represented 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.\ \>", "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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