(************** Content-type: application/mathematica ************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 88200, 2254]*) (*NotebookOutlinePosition[ 88861, 2277]*) (* CellTagsIndexPosition[ 88817, 2273]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Corpo affine elastico vincolato", "Title"], Cell[TextData[StyleBox["v. 2.08 (23/6/2003) \[Copyright] A. Tatone \ [Universit\[AGrave] dell'Aquila]", FontFamily->"Arial", FontWeight->"Bold"]], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[CellGroupData[{ Cell["Inizializzazione", "Section"], Cell[BoxData[{ \(\(Off[General::"\"];\)\), "\[IndentingNewLine]", \(\(Off[General::"\"];\)\), "\[IndentingNewLine]", \(\(Off[Solve::"\"];\)\)}], "Input"], Cell["Prodotto tensoriale", "Text"], Cell[BoxData[ \(prt[u_, v_] := Transpose[{v}] . {u}\)], "Input"], Cell["Prodotto scalare", "Text"], Cell[BoxData[ \(prs[u_, v_] := Flatten[u] . Flatten[v]\)], "Input"], Cell["Traccia", "Text"], Cell[BoxData[ \(tr[m_] := m\_\(\(\[LeftDoubleBracket]\)\(1, 1\)\(\[RightDoubleBracket]\)\) + m\_\(\(\[LeftDoubleBracket]\)\(2, 2\)\(\[RightDoubleBracket]\)\) + m\_\(\(\[LeftDoubleBracket]\)\(3, 3\)\(\[RightDoubleBracket]\)\)\)], \ "Input"], Cell["Gradiente dello spostamento", "Text"], Cell[BoxData[ \(\(mH = Array[ug[#1, #2]\ &, {3, 3}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[mH]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(ug[1, 1]\), \(ug[1, 2]\), \(ug[1, 3]\)}, {\(ug[2, 1]\), \(ug[2, 2]\), \(ug[2, 3]\)}, {\(ug[3, 1]\), \(ug[3, 2]\), \(ug[3, 3]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["Rotazione infinitesima", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"m\[Theta]", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", \(-\[Theta][3]\), \(\[Theta][2]\)}, {\(\[Theta][3]\), "0", \(-\[Theta][1]\)}, {\(-\[Theta][2]\), 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\[Rule] 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Length[uvinc]\)], "Input"], Cell[BoxData[ \(5\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(wvinc = \(uvinc /. ug \[Rule] g\) /. {u01 \[Rule] w01, u02 \[Rule] w02, u03 \[Rule] w03}\)], "Input"], Cell[BoxData[ \({g[2, 2] \[Rule] w02\/L2, g[1, 3] \[Rule] 0, w01 \[Rule] w02, g[2, 1] \[Rule] 0, g[2, 3] \[Rule] 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mH /. uvinc // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(ug[1, 1]\), \(ug[1, 2]\), "0"}, {"0", \(u02\/L2\), "0"}, {\(ug[3, 1]\), \(ug[3, 2]\), \(ug[3, 3]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(u0 /. uvinc\)], "Input"], Cell[BoxData[ \({u02, u02, u03}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mG //. wvinc // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(g[1, 1]\), \(g[1, 2]\), "0"}, {"0", \(w02\/L2\), "0"}, {\(g[3, 1]\), \(g[3, 2]\), \(g[3, 3]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(w0 //. wvinc\)], "Input"], Cell[BoxData[ \({w02, w02, w03}\)], "Output"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Risultante e momento delle forze attive", "Section"], Cell[BoxData[ \(\(mMa = \[Integral]\_\(-\(L3\/2\)\)\%\(L3\/2\)\(\[Integral]\_\(-\(L2\/2\ \)\)\%\(L2\/2\)\((prt[\((faccia1p - x0)\), \(-p\)\ e1])\) \[DifferentialD]\[Zeta]2 \ \[DifferentialD]\[Zeta]3\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(1\/vol\) mMa // Simplify\) // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-p\), \(\(L2\ p\)\/\(2\ L1\)\), "0"}, {"0", "0", "0"}, {"0", "0", "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fa = \ \[Integral]\_\(-\(L3\/2\)\)\%\(L3\/2\)\(\[Integral]\_\(-\(L2\/2\)\)\%\(L2\/2\)\ \((\(-p\)\ e1)\) \[DifferentialD]\[Zeta]2 \[DifferentialD]\[Zeta]3\)\)], \ "Input"], Cell[BoxData[ \({\(-L2\)\ L3\ p, 0, 0}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Tensione", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(wpar = Join[Flatten[mG], w0]\)], "Input"], Cell[BoxData[ \({g[1, 1], g[1, 2], g[1, 3], g[2, 1], g[2, 2], g[2, 3], g[3, 1], g[3, 2], g[3, 3], w01, w02, w03}\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"mT", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\[Sigma][1, 1]\), \(\[Sigma][1, 2]\), \(\[Sigma][1, 3]\)}, {\(\[Sigma][1, 2]\), \(\[Sigma][2, 2]\), \(\[Sigma][2, 3]\)}, {\(\[Sigma][1, 3]\), \(\[Sigma][2, 3]\), \(\[Sigma][3, 3]\)} }], "\[NoBreak]", ")"}]}], ";"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[mT]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\[Sigma][1, 1]\), \(\[Sigma][1, 2]\), \(\[Sigma][1, 3]\)}, {\(\[Sigma][1, 2]\), \(\[Sigma][2, 2]\), \(\[Sigma][2, 3]\)}, {\(\[Sigma][1, 3]\), \(\[Sigma][2, 3]\), \(\[Sigma][3, 3]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(prs[fa, w0] //. wvinc\)], "Input"], Cell[BoxData[ \(\(-L2\)\ L3\ p\ w02\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(prs[fa, w0]\)], "Input"], Cell[BoxData[ \(\(-L2\)\ L3\ p\ w01\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((\((prs[fa, w0] + prs[mMa, mG])\) 1\/vol - prs[mT, mG])\) //. wvinc\)], "Input"], Cell[BoxData[ \(\(\(-L2\)\ L3\ p\ w02 - L1\ L2\ L3\ p\ g[1, 1] + 1\/2\ L2\^2\ L3\ p\ \ g[1, 2]\)\/\(L1\ L2\ L3\) - g[1, 1]\ \[Sigma][1, 1] - g[1, 2]\ \[Sigma][1, 2] - g[3, 1]\ \[Sigma][1, 3] - \(w02\ \[Sigma][2, 2]\)\/L2 - g[3, 2]\ \[Sigma][2, 3] - g[3, 3]\ \[Sigma][3, 3]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Complement[ Coefficient[\((\((prs[fa, w0] + prs[mMa, mG])\) 1\/vol - prs[mT, mG])\) //. wvinc, wpar], {0}] // Simplify\)], "Input"], Cell[BoxData[ \({\(-p\) - \[Sigma][1, 1], \(L2\ p\)\/\(2\ L1\) - \[Sigma][1, 2], \(-\[Sigma][1, 3]\), \(-\(p\/L1\)\) - \[Sigma][2, 2]\/L2, \(-\[Sigma][2, 3]\), \(-\[Sigma][3, 3]\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(tens = \(Solve[Map[\((# \[Equal] 0)\)\ &, %], \ Union[Flatten[mT]]]\)\_\(\(\[LeftDoubleBracket]\)\(1\)\(\[RightDoubleBracket]\ \)\)\)], "Input"], Cell[BoxData[ \({\[Sigma][1, 1] \[Rule] \(-p\), \[Sigma][1, 2] \[Rule] \(L2\ p\)\/\(2\ L1\), \[Sigma][1, 3] \[Rule] 0, \[Sigma][2, 2] \[Rule] \(-\(\(L2\ p\)\/L1\)\), \[Sigma][2, 3] \[Rule] 0, \[Sigma][3, 3] \[Rule] 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(mT /. tens // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-p\), \(\(L2\ p\)\/\(2\ L1\)\), "0"}, {\(\(L2\ p\)\/\(2\ L1\)\), \(-\(\(L2\ p\)\/L1\)\), "0"}, {"0", "0", "0"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Dilatazione infinitesima e rotazione infinitesima", "Section"], Cell[CellGroupData[{ Cell["Dalla tensione", "Subsection"], Cell[BoxData[ \(\(mE = \(1\/\(2 \[Mu]\)\) \((mT - \(\[Lambda]\/\(3 \[Lambda] + 2 \[Mu]\)\) tr[mT] mI)\) //. tens // Simplify;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[mE // Factor]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-\(\(p\ \((2\ L1\ \[Lambda] - L2\ \[Lambda] + 2\ L1\ \[Mu])\)\)\/\(2\ L1\ \[Mu]\ \((3\ \[Lambda] \ + 2\ \[Mu])\)\)\)\), \(\(L2\ p\)\/\(4\ L1\ \[Mu]\)\), "0"}, {\(\(L2\ p\)\/\(4\ L1\ \[Mu]\)\), \(\(p\ \((L1\ \[Lambda] - 2\ L2\ \[Lambda] - 2\ L2\ \[Mu])\)\)\/\(2\ L1\ \[Mu]\ \((3\ \[Lambda] + 2\ \[Mu])\)\)\), "0"}, {"0", "0", \(\(\((L1 + L2)\)\ p\ \[Lambda]\)\/\(2\ L1\ \[Mu]\ \((3\ \ \[Lambda] + 2\ \[Mu])\)\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Dalle condizioni di vincolo", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[mH]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(ug[1, 1]\), \(ug[1, 2]\), \(ug[1, 3]\)}, {\(ug[2, 1]\), \(ug[2, 2]\), \(ug[2, 3]\)}, {\(ug[3, 1]\), \(ug[3, 2]\), \(ug[3, 3]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[mH /. uvinc]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(ug[1, 1]\), \(ug[1, 2]\), "0"}, {"0", \(u02\/L2\), "0"}, {\(ug[3, 1]\), \(ug[3, 2]\), \(ug[3, 3]\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(u0 /. uvinc\)], "Input"], Cell[BoxData[ \({u02, u02, u03}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ 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