(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[BoxData[ \(Simplify[\[Sigma]lab\_xx\ + \ \[Sigma]lab\_yy\ ]\)], "Input"], Cell["And the determinant", "Text"], Cell[BoxData[ \(Simplify[\[Sigma]lab\_xx\ \ \[Sigma]lab\_yy\ - \ \((\[Sigma]lab\_xy)\)^2]\)], "Input"], 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[BoxData[ \(uniaxial10 = \ {\[Sigma]princ\_xx -> 10, \[Sigma]princ\_yy -> 0}\)], "Input"], Cell[BoxData[ \(ParametricPlot[{\[Sigma]lab\_xx, \[Sigma]lab\_xy} /. uniaxial10\[IndentingNewLine], {\[Theta], 0, \[Pi]}, AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] 1, PlotLabel \[Rule] "\< \t \t Mohr Circle for 10 MPa Uniaxial Tension\>", PlotStyle \[Rule] {Thickness[0.01], Hue[1]}]\)], "Input", PageBreakAbove->True, CellTags->"mtag:10:stress__example__Mohrs_circle_plotted"], Cell[BoxData[ \(uniaxialother = \ {\[Sigma]princ\_xx -> 30, \[Sigma]princ\_yy -> 10}\)], "Input"], Cell[BoxData[ \(ParametricPlot[{\[Sigma]lab\_xx, \[Sigma]lab\_xy} /. uniaxialother\[IndentingNewLine], {\[Theta], 0, \[Pi]}, AxesLabel \[Rule] {"\", "\"}, AspectRatio \[Rule] 1, PlotRange -> {{0, 40}, {\(-20\), 20}}, PlotLabel \[Rule] \*"\"\< \\t \\t Mohr Circle for \!\(\[Sigma]princ\_xx\ \)= 30 \!\(\[Sigma]princ\_yy\)=10\>\"", PlotStyle \[Rule] {Thickness[0.01], 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"] }, Open ]] }, FrontEndVersion->"5.2 for Macintosh", ScreenRectangle->{{4, 1280}, {0, 832}}, ScreenStyleEnvironment->"Presentation", PrintingStyleEnvironment->"Printout", CellGrouping->Manual, WindowSize->{1218, 845}, WindowMargins->{{Automatic, 162}, {Automatic, 63}}, WindowTitle->"Lecture 10 MIT 3.016 (Fall 2006) \[Copyright] W. 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