(************** 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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Ile du Saulcy, 57045 Metz Cedex 01 - France.\ \>", "TextAboutAuthor"], Cell["\<\ The present study deals with the analytic resolution of fundamental \ problems of the plane elasticity theory. Based on the general framework \ suggested by Kolossov and Muskhelishvili through the complex representation \ theory, the investigation focuses on the determination of the analytical \ expressions of stress and displacements fields inside an elastic plate \ containing elliptic cracks and subjected to specific boundary conditions. \ This analytic approach, initially difficult to perform, is associated to \ Mathematica to overcome computations difficulties arising when one attempts \ calculate exact analytical solutions. The obtained results precise the validity of asymptotic solutions, widely \ used in fracture mechanics. Some specific aspects of the fracture mechanics \ such as stress contourlines at the crack's vicinity and the stress intensity \ factor are analyzed. Moreover, the more complex case of two elliptic cracks \ is also treated and corresponding analytic solutions provided.\ \>", \ "Abstract", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Introduction" }], "Section"], Cell[TextData[StyleBox["To solve fundamental problems of the plane elasticity \ theory, some numerical techniques such as the finite elements methods, are \ often used. A few investigations has been devoted to analytic approaches of \ these particular problems but all them generally propose approximative \ semi-analytic solutions.\nFor the complete solution of equations of plane \ elasticity theory, the complex representation introduced by G.V. Kolossov [5] \ and developed by N. I. Muskhelishvili [6] has been applied by several authors \ to study problems of plate containing holes and to suggest numerical and \ semi-numerical results: Sneddon [8], Green and Zerna [4], Aifantis et al. \ [1]. These initial investigations give up, in front of the complexity of \ analytical calculations, to clarify the solutions and being generally \ satisfied to give partial or numerical results. ", CharacterEncoding->"WindowsANSI"]], "Text", TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". The plane elasticity theory" }], "Section"], Cell[TextData[{ "In the plane theory of elasticity and in the assumption of vanishing body \ forces, stress components can be expressed by means of the so-called Airy \ function ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalA](x, \ y)\)]], " defined by:" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[ StyleBox[\(\[Sigma]\_xx = \(\[ScriptCapitalA]\_\(\(,\)\(yy\)\(\ \ \)\(,\ \)\(\ \)\)\ \[Sigma]\_yy = \[ScriptCapitalA]\_\(\(,\)\(xx\)\)\)\ , \ \ \[Tau]\ \_xy = \(-\[ScriptCapitalA]\_\(\(,\)\(xy\)\)\)\), FontFamily->"Courier"], TextForm]], "NumberedEquation"], Cell[TextData[{ "The Airy function, which is biharmonic, have also to satisfy to boundary \ and compatibility conditions of the specific problem. According to the \ theorem of E. Almansi, any poly-harmonic function of ", Cell[BoxData[ \(TraditionalForm\`N\)]], " order is representable by ", Cell[BoxData[ \(TraditionalForm\`N\)]], " harmonic functions. The solution of plane elasticity problems therefore, \ rises from two functions (", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\)]], ") of the complex variable ", Cell[BoxData[ \(TraditionalForm\`z = x + i\ y\)]], ". " }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[ RowBox[{Cell[TextData[{ "\[ScriptCapitalA](z)= Re (", Cell[BoxData[ \(TraditionalForm\`z\&\(\(\ \)\(_\)\)\ \(\[CurlyPhi]( z)\) + \[Chi](z)\)]], ")" }]], Cell[""]}], TextForm]], "NumberedEquation"], Cell[TextData[{ "where Re denotes the real part and ", Cell[BoxData[ \(TraditionalForm\`z\&_ = x - i\ y\)]], " the conjugate of the compex variable ", Cell[BoxData[ \(TraditionalForm\`z\)]], ". Boundary conditions on the hole contour can be expressed as:" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ \(f \((x, y)\) = \(i\ \((X + i\ Y)\) + const = \[CurlyPhi] \((z)\) + \(z\[CurlyPhi]' \((z)\)\)\&_ + \ \(\[Psi] \((z)\)\)\&_\)\)]]]], TextForm]], "NumberedEquation"], Cell[TextData[{ "(", Cell[BoxData[ \(TraditionalForm\`X\)]], ", ", Cell[BoxData[ \(TraditionalForm\`Y\)]], ") are the components of the resultant force acting on hole's contours in \ the case of multiply connected regions and ", Cell[BoxData[ \(TraditionalForm\`\[Psi](z) = \[Chi]' \((z)\)\)]], ". Kolossov-Muskhelishvili potentials (", Cell[BoxData[ \(TraditionalForm\`\[CurlyPhi]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Chi]\)]], ") depend on the geometry of the selected region: finite or infinite, \ simply or multiply connected. Subsequent stress and displacements fields are \ deduced from the following equations:" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[{ FormBox[\(\[Sigma]\_xx + \[Sigma]\_yy = 2\ \(Re(\[Phi](z))\)\), TraditionalForm], "\[IndentingNewLine]", FormBox[\(\[Sigma]\_yy - \[Sigma]\_xx + 2\ i\ \[Tau]\_xy = 2\ \((z\&_\ \[Phi]' \((z)\) + \[Psi]' \((z)\))\)\), TraditionalForm], "\[IndentingNewLine]", FormBox[\(2\ \[Mu]\ \((u\_x + i\ u\_y)\) = \[Kappa]\ \(\[CurlyPhi]( z)\) - z\ \(\[Phi](z)\)\&_ - \[Psi](z)\), TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"\[Kappa]", "=", RowBox[{\(\(5\ \[Lambda] + 6\ \[Mu]\)\/\(3\ \[Lambda] + 2\ \[Mu]\)\), " ", "in", " ", StyleBox["a", FontSlant->"Plain"], " ", "plane", " ", "stress", " ", "state"}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"\[Kappa]", "=", RowBox[{\(\(\(\ \)\(\[Lambda] + 3\ \[Mu]\)\)\/\(\[Lambda] + \ \[Mu]\)\), " ", "in", " ", StyleBox["a", FontSlant->"Plain"], " ", "plane", " ", "strain", " ", "state"}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{""}], TraditionalForm]}]]]], TextForm]], "NumberedEquation"], Cell[TextData[{ "Where ", Cell[BoxData[ \(TraditionalForm\`\[Phi](z) = \[CurlyPhi]' \((z)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\((\[Mu], \ \[Lambda])\)\)]], ", the Lam\[EAcute] constants of the isotropic elastic body.\n\nLet's \ consider an elastic region ", Cell[BoxData[ \(TraditionalForm\`D\)]], " containing ", Cell[BoxData[ \(TraditionalForm\`n\)]], " elliptic cracks ", Cell[BoxData[ \(TraditionalForm\`L\_k = a\_k\ b\_k\)]], ", where ", Cell[BoxData[ \(TraditionalForm\`a\_k\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\_k\)]], " are the crack tips of the selected crack ", Cell[BoxData[ \(TraditionalForm\`L\_k\)]], " lying on the ", Cell[BoxData[ \(TraditionalForm\`x\)]], "-coordinate." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ StyleBox["Figure ", "SB"], StyleBox[ CounterBox["NumberedFigure"], "SB"], StyleBox[". 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Through the general framework suggested by Muskhilshvili, the \ complex stress functions associated to this problem are holomorphic on the \ region exterior to the considered cracks and the following formulas hold \ true: \ \>", "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[ Cell[TextData[Cell[ BoxData[{\(\[Phi] \((z)\) = \(-i\)\ \(\(X + i\ Y\)\/\(2\ \[Pi] \((\[Kappa] + 1)\)\)\) 1\/z + \[CapitalGamma]\_1 + 0 \((1\/z\^2)\)\), "\n", \(\[CapitalPsi] \((z)\) = \[Kappa] \(\( X - i\ Y\)\/\(2\ \[Pi] \((\[Kappa] + 1)\)\)\) 1\/z + \[CapitalGamma]\_2 + 0 \((1\/z\^2)\)\), "\n", FormBox[\(\[CapitalGamma]\_1 = \(N\_1 + N\_2\)\/4 + i\ \(2\ \[Mu]\ \[CurlyEpsilon]\_\[Infinity]\)\/\(\[Kappa] + 1\ \)\), "TraditionalForm"], "\n", \(\[CapitalGamma]\_2 = \(\(N\_2 - N\_1\)\/2\) e\^\(\(-2\) i\ \[Alpha]\)\)}]]]], TextForm]], "NumberedEquation"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\((N\_1, \ N\_2)\)\)]], " are the principal values of the remote stress applied at infinity, ", Cell[BoxData[ \(TraditionalForm\`\[Alpha] = \((N\_1, O\ x)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[CurlyEpsilon]\_\[Infinity]\)]], " the rotation at infinity which can be assumed to vanish without \ influencing the generality of results. Let's replace the function ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPsi](z)\)]], " by a new one, ", Cell[BoxData[ \(TraditionalForm\`\[CapitalOmega](z)\)]], " defined by:" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ \(\[CapitalOmega] \((z)\) = \(\(\[Phi]\&_\) \((z)\) + z\ \(\(\[Phi]'\)\&_\) \((z)\) + \(\[CapitalPsi]\&_\) \((z)\) \ = \(\[CapitalGamma]\_1\)\&_ + \(\[CapitalGamma]\_2\)\&_ + \[Kappa] \(\( X + i\ Y\)\/\(2\ \[Pi] \((\[Kappa] + 1)\)\)\) 1\/z + 0 \((1\/z\^2)\)\)\)]]]], TextForm]], "NumberedEquation"], Cell[TextData[{ "For the problem of infinite elastic region containing ", Cell[BoxData[ \(TraditionalForm\`n\)]], " elliptic cracks, following alternative expressions are suggested: " }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[{ \(\[Phi] \((z)\) = \(\(P\_n\) \((z)\)\)\/\(\[Chi] \((z)\)\) - \ \(1\/2\) \(\[CapitalGamma]\_2\)\&_\), "\n", \(\[CapitalOmega] \((z)\) = \(\(P\_n\) \((z)\)\)\/\(\[Chi] \ \((z)\)\) + \(1\/2\) \(\[CapitalGamma]\_2\)\&_\), "\n", \(\[Chi] \((z)\) = \[Product]\_\(k = 1\)\%n\@\(\((z - a\_k)\) \((z \ - b\_k)\)\), \ \ \(P\_n\) \((z)\) = \(\(c\_0\) z\^n + \(c\_1\) z\^\(n - 1\) + ... \) + c\_n\)}]]]], TextForm]], "NumberedEquation"], Cell["\<\ Boundary conditions at infinity lead to the following \ relations:\ \>", "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ \(\[CurlyPhi] \((\[Infinity])\) = \(\[CapitalGamma]\ \ \ \ \ \ \ \[DoubleLongRightArrow] c\_0 = \[CapitalGamma]\_1 + \(1\/2\) \ \(\[CapitalGamma]\_2\)\&_\)\)]]]], TextForm]], "NumberedEquation"], Cell["\<\ For simplicity, the elliptic cracks contours are assumed to be \ stress-free:\ \>", "Text"], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ \(X = \(Y = 0\ \ \ \ \ \ \[DoubleLongRightArrow]\(\[Integral]\_\(L\_k\)\(\(\ \(P\_n\) \((t)\)\)\/\(\[Chi] \((z)\)\)\) dt\)\); \ k = 1, 2, \ ... , \ n\)]]]], TextForm]], "NumberedEquation"], Cell[TextData[{ "The other parameters ", Cell[BoxData[ \(TraditionalForm\`c\_1, \ c\_2, \ ... , \ c\_n\)]], " of the function ", Cell[BoxData[ \(TraditionalForm\`\(P\_n\)(z)\)]], " are then calculated through the resolution of the equations (9). Stress \ parameters ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\_2\)]], " are known since the remote stress at infinity is selected (equations 5c \ and 5d). Moreover, the process of resolution can be resumed as the following:\ \n1. Calculation of ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\_2\)]], " through (5) according to the stress boundary conditions at infinity;\n2. \ Calculation of constants ", Cell[BoxData[ \(TraditionalForm\`c\_1, \ c\_2, \ ... , \ c\_n\)]], " from system (9);\n3. Expression of stress functions defined using (7);\n\ 4. Stress and displacement fields' expression by (4)." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[{ " ", Cell[BoxData[ \(\[Sigma]\_xx\)], FontFamily->"Times"], " + ", Cell[BoxData[ \(\[Sigma]\_yy\)], FontFamily->"Times"], " = 4 Re(\[Phi](z)),\n ", Cell[BoxData[ \(\[Sigma]\_yy\)], FontFamily->"Times"], " - ", Cell[BoxData[ \(\[Sigma]\_xx\)], FontFamily->"Times"], " = 2 Re((", Cell[BoxData[ \(z\&_\)], FontFamily->"Times"], " - z)\[Phi]'(z) + ", Cell[BoxData[ \(\[CapitalOmega]\&_\)], FontFamily->"Times"], "(z) - \[Phi](z))\n ", Cell[BoxData[ \(\[Sigma]\_xy\)], FontFamily->"Times"], " = Im((", Cell[BoxData[ \(z\&_\)], FontFamily->"Times"], " - z)\[Phi]'(z) + ", Cell[BoxData[ \(\[CapitalOmega]\&_\)], FontFamily->"Times"], "(z) - \[Phi](z))\n ", Cell[BoxData[ \(2 \[Mu] \((u\_x + i\ u\_y)\) = \[Kappa]\ \[CurlyPhi] \((z)\) - z\ \(\(\[CurlyPhi]\^'\) \((z)\)\)\&_\ - \(\[Psi] \ \((z)\)\)\&_\)], FontFamily->"Times"] }]], TextForm]], "NumberedEquation"], Cell["\<\ The proposed procedure is applied to the problem of elastic \ infinite plate weakened by one or two elliptic cracks and subjected to \ particular remote laodings.\ \>", "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Subsection"], ". ", "Infinite plate with one elliptic crack sujected to a Mode I loading." }], "Subsection"], Cell[TextData[{ "For this classical case, boundary conditions at infinity (", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_yy = \[Sigma]\^\[Infinity], \ \[Sigma]\_xx \ = \(\[Tau]\_xy = 0\)\)]], ") lead to:" }], "Text"], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ \(N\_1 = 0\ , N\_2 = \[Sigma], \ \[Alpha] = \(0\ \ \ \ \ \ \ \ \[DoubleLongRightArrow]\[CapitalGamma]\_1 = \[Sigma]\^\[Infinity]\/4\), \(\(\ \[CapitalGamma]\_2\)\(=\)\(\[Sigma]\^\[Infinity]\/2\)\(\ \)\)\)]]]], TextForm]], "NumberedEquation"], Cell[TextData[{ "On the other hand, crack tips are defined by ", Cell[BoxData[ \(TraditionalForm\`a\_1 = 0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`b\_1 = \(-2\) a\)]], ", so:" }], "Text"], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ \(\[Chi] \((z)\) = \@\(z \((z + 2 a)\)\), and\ \ \ \(P\_1\) \((z)\) = \(c\_0\) z + c\_1\)]]]], TextForm]], "NumberedEquation"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`c\_0\)]], " is given by (8) and ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], " calculated from the equation ", Cell[BoxData[ \(\[Integral]\_0\%\(\(-2\) a\)\(\(\(C\_0\) t\ + \ C\_1\)\/\@\(t \((t + 2 a)\)\)\) \[DifferentialD]t\)], FontFamily->"Times", FontSize->10], StyleBox[" = 0 ", FontSize->10], "leading to ", Cell[BoxData[ \(TraditionalForm\`c\_1 = a\ c\_0\)]], ". Finally the stress functions expressions are:" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[{ RowBox[{ RowBox[{ StyleBox[\(\[Phi] \((z)\)\), FontSize->12], StyleBox[" ", FontSize->12], StyleBox["=", FontSize->12], StyleBox[" ", FontSize-> 12], \(\(\[Sigma]\^\[Infinity]\/4\) \((\(-1\) + \ \(2 \((z \ + a)\)\)\/\@\(z \((z + 2 a)\)\)\ )\)\)}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"\[CapitalOmega]", StyleBox[\((z)\), FontSize->12]}], StyleBox[" ", FontSize->12], StyleBox["=", FontSize->12], StyleBox[" ", FontSize-> 12], \(\(\[Sigma]\^\[Infinity]\/4\) \((1 + \ \(2 \((z + \ a)\)\)\/\@\(z \((z + 2 a)\)\)\ )\)\)}], ";"}]}], PageBreakAbove->Automatic]]], TextForm]], "NumberedEquation"], Cell["\<\ The analytical expressions of stress tensor components then rise \ for relations (4). The following function is defined to perform the \ calculation of the stress components:\ \>", "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(\(\(CrackStress[N1_, N2_, \[Alpha]_] := Module[{v2, x, y, c0, sol, \[CapitalGamma]1, \[CapitalGamma]2}, \ \[IndentingNewLine]polarForm[expr_] := expr /. z \[Rule] r\ Exp[I\ \[Theta]]; \[IndentingNewLine]conjugue[expr_] := expr /. Complex[x_, y_] \[Rule] Complex[x, \(-y\)]; \[IndentingNewLine]realPart[ expr_] := \((expr + conjugue[expr])\)/ 2; \[IndentingNewLine]imaginaryPart[ expr_] := \((expr - conjugue[expr])\)/\((2 I)\); \[IndentingNewLine]sol = Solve[Integrate[\(c1\ + c0\ z\)\/\@\(z \((z + 2\ a)\)\), {z, \ \(-2\)\ a, 0}, Assumptions \[Rule] a > 0] \[Equal] 0, c1] // Flatten; \[IndentingNewLine]\[CapitalGamma]1 = \ \(N1 + \ N2\)\/4; \[CapitalGamma]2 = \(\(N2 - N1\)\/2\) Exp[\(-2\)\ I\ \[Alpha]]; c0 = \ \[CapitalGamma]1 + conjugue[\[CapitalGamma]2]\/2; \[IndentingNewLine]\[Phi] = \ \((\(c0\ \ z + c1\)\/\(\(\ \)\(\@\(z \((z + 2\ a)\)\)\)\) - conjugue[\[CapitalGamma]2]\/2)\) /. sol; \[IndentingNewLine]\[CapitalOmega] = \((\(c0\ \ z + c1\)\/\ \(\(\ \)\(\@\(z \((z + 2\ a)\)\)\)\) + conjugue[\[CapitalGamma]2]\/2)\) /. sol; \[IndentingNewLine]\[CapitalPhi] = D[\[Phi], z]; \[CapitalPsi] = \[CapitalOmega] - \[Phi] - z\ \ \[CapitalPhi]; v2 = \ r\ Exp[\(-I\)\ \[Theta]]\ \ polarForm[\[CapitalPhi]] + polarForm[\[CapitalPsi]]; stress = LinearSolve[{{1, 1}, {\(-1\), 1}}, {4\ realPart[polarForm[\[Phi]]], 2\ realPart[ v2]}]; \[IndentingNewLine]\[Sigma]xx = \(\[Sigma]xx = ComplexExpand[stress[\([1]\)], TargetFunctions -> {Re, Im}]\); \[IndentingNewLine]\[Sigma]yy = ComplexExpand[stress[\([2]\)], TargetFunctions -> {Re, Im}]; \[Sigma]xy = ComplexExpand[imaginaryPart[v2], TargetFunctions -> {Re, Im}];\[IndentingNewLine]];\)\(\ \ \ \ \ \ \ \)\)\)], "Input"], Cell[TextData[{ "For a mode I loading, one can run CrackStress[0,", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_\[Infinity]\)]], ",0] and obtains after simplifications the following expressions: " }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ " ", Cell[BoxData[{ \(TraditionalForm\`\(\[Sigma]\_xx = \(1\/\(2\ \@r\ \((4\ a\^2 + 4\ r\ \ \(cos(\[Theta])\)\ a + r\^2)\)\^\(9/4\)\)\) \((\[Sigma]\_\[Infinity]\ \ \((\(-16\)\ a\ \((4\ a\^2 + r\^2)\)\ \(cos(\[Theta])\)\ \@\(4\ a\^2 + 4\ r\ \ \(cos(\[Theta])\)\ a + r\^2\)\%4\ r\^\(3/2\) - 32\ a\^2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\ \)\%4\ r\^\(5/2\) - 16\ a\^2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\ \)\%4\ \(cos(2\ \[Theta])\)\ r\^\(5/2\) - 2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\)\%4\ \ r\^\(9/2\) + 4\ \(cos(\[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^5 + 20\ a\ \(cos( 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\[Theta])\))\ \))\)\ r\^4 - 2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\)\ \ \(cos(\[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^4 + 16\ a\ \(cos( 2\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^4 - 8\ a\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\)\ \ \(cos(1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\[Theta])\))\ \))\)\ r\^3 + 32\ a\^2\ \(cos(\[Theta] + 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^3 + 80\ a\^2\ \(cos(\[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^3 - 6\ a\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\)\ \ \(cos(2\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^3 + 16\ a\^2\ \(cos( 3\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^3 + 128\ a\^3\ \(cos( 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\[Theta])\))\ \))\)\ r\^2 + 16\ a\^3\ \(cos( 1\/2\ \((4\ \[Theta] + \(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\)))\))\)\ r\^2 - 8\ a\^2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + \ r\^2\)\ \(cos(\[Theta] + 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^2 - 24\ a\^2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\ \)\ \(cos(\[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^2 + 80\ a\^3\ \(cos( 2\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^2 - 5\ a\^2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + \ r\^2\)\ \(cos(3\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^2 + a\^2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\)\ \ \(cos(5\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^2 - 24\ a\^3\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\ \)\ \(cos(1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\[Theta])\))\ \))\)\ r + 64\ a\^4\ \(cos(\[Theta] + 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r + 128\ a\^4\ \(cos(\[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r - 20\ a\^3\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\ \)\ \(cos(2\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r + 4\ a\^3\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + \ r\^2\)\ \(cos(4\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r - 32\ a\^4\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\ \)\%4\ \@r + 64\ a\^5\ \(cos( 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\[Theta])\))\ \))\) - 20\ a\^4\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\)\ \(cos(\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\) + 4\ a\^4\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + \ r\^2\)\ \(cos(3\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\))\))\);\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(TraditionalForm\`\(\[Sigma]\_yy = \[Sigma]\_\[Infinity]\ \ \((a\ \((r\ \ \((24\ \(cos(1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\ \[Theta])\))\))\)\ a\^2 - 4\ \(cos( 4\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\ \[Theta])\))\))\)\ a\^2 + 8\ r\ \(cos(\[Theta] + 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\ \[Theta])\))\))\)\ a - r\ \(cos(5\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\ \[Theta])\))\))\)\ a + 8\ r\^2\ \(cos( 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\ \[Theta])\))\))\))\) + \((6\ r\^3 + 20\ a\^2\ r)\)\ \(cos( 2\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\ \[Theta])\))\))\) + \((5\ a\ r\^2 - 4\ a\^3)\)\ \(cos( 3\ \[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\ \[Theta])\))\))\))\) + 2\ \((10\ a\^4 + 12\ r\^2\ a\^2 + r\^4)\)\ \(cos(\[Theta] - 1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\[Theta])\))\ \))\))\)/\((2\ \@r\ \((4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + \ r\^2)\)\^\(7/4\))\);\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(TraditionalForm\`\(\[Sigma]\_xy = \(1\/\(\@r\ \((4\ a\^2 + 4\ r\ \ \(cos(\[Theta])\)\ a + r\^2)\)\^\(5/4\)\)\) \((P\ \((\@\(4\ a\^2 + 4\ r\ \ \(cos(\[Theta])\)\ a + r\^2\)\ \(cos( 2\ \[Theta] - 3\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r\^2 + 2\ a\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + r\^2\)\ \ \(cos(\[Theta] - 3\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\))\))\)\ r + \((\(-r\^3\) - 4\ a\ \(cos(\[Theta])\)\ r\^2 - 4\ a\^2\ r + 2\ a\^2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + \ r\^2\)\ \(cos(\(tan\^\(-1\)\)(2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \ \(sin(\[Theta])\)))\) - a\^2\ \@\(4\ a\^2 + 4\ r\ \(cos(\[Theta])\)\ a + \ r\^2\))\)\ \(cos(1\/2\ \(\(tan\^\(-1\)\)( 2\ a\ \(cos(\[Theta])\) + r\ \(cos(2\ \[Theta])\), 2\ \((a + r\ \(cos(\[Theta])\))\)\ \(sin(\[Theta])\))\ \))\))\)\ \(sin(\[Theta])\))\);\)\)}]] }], "Text"], Cell[TextData[{ "The following figures display these stress components curves for three \ selected directions \[Theta]. It's interesting to notice that, the provided \ solution satisfies to boundary conditions. In the immediate neighbourhood of \ the crack tip, stress components are infinites, moreover at infinity, ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_yy\/\[Sigma]\_\[Infinity]\)]], " tends to 1 when the other components vanish, to satisfy to the remote \ stress distribution at infinity. \nThe provided expressions are general, \ depending both on the crack dimension (", Cell[BoxData[ \(TraditionalForm\`a\)]], "), the loading magnitude (", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_\[Infinity]\)]], ") and can be evaluate at any point (", Cell[BoxData[ \(TraditionalForm\`r, \ \[Theta]\)]], ") of the considered elastic region. " }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ StyleBox["Figure ", "SB"], StyleBox[ CounterBox["NumberedFigure"], "SB"], StyleBox[". 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TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Through the concept of complex functions, it's also possible to determine \ the expression of the stress intensity factor ", Cell[BoxData[ \(TraditionalForm\`K\)]], " generally used in fracture mechanics. Indeed, this stress intensity \ factor is related to the complex function ", Cell[BoxData[ \(TraditionalForm\`\[Phi]\)]], ":" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ \(K = \(K\_I - iK\_II = 2 \( Limit\+\(z\[LongRightArrow] z\_0\)\) \(\@\(2\ \[Pi] \((z - z\_0)\)\)\) \[Phi] \((z)\)\)\)]]]], TextForm]], "NumberedEquation"], Cell[TextData[{ "From relation (12 a), it's easy to obtain for the particulate treated case \ (crack under Mode I loading), the well known result: ", Cell[BoxData[ \(TraditionalForm\`K = \(K\_I = \(\[Sigma]\_\[Infinity]\) \@\(\[Pi]\ \ a\)\)\)]], ". \n\nThis general routine can be easily applied to any specific boundary \ conditions as well at infinity as on the crack's contours in order to express \ stress field. Classical loadings such as Mode II and Mode III are also \ investigated and general analytical results obtained." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Subsection"], ". ", "Mode I for an infinite plate with two cracks " }], "Subsection"], Cell[TextData[{ StyleBox["Figure ", "SB"], StyleBox[ CounterBox["NumberedFigure"], "SB"], StyleBox[". 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\(b\ P\ \(E \((b\^2\/a\^2)\)\)\ \ a\^2 - b\ P\ \(K \((b\^2\/a\^2)\)\)\ a\^2 - b\^2\ P\ \(E \((a\^2\/b\^2)\)\)\ \ a + b\^2\ P\ \(K \((a\^2\/b\^2)\)\)\ a\)\/\(2\ \((a\ \(K \((a\^2\/b\^2)\)\) - \ b\ \(K \((b\^2\/a\^2)\)\))\)\)\)\/\@\(\((z\^2 - a\^2)\)\ \((z\^2 - b\^2)\)\) \ - P\/4;\)\n\), "\[IndentingNewLine]", \(\(\[CapitalOmega] = P\/4 + \(\(P\ z\^2\)\/2 - \(b\ P\ \(E \((b\^2\/a\^2)\)\)\ a\^2 \ - b\ P\ \(K \((b\^2\/a\^2)\)\)\ a\^2 - b\^2\ P\ \(E \((a\^2\/b\^2)\)\)\ a + b\ \^2\ P\ \(K \((a\^2\/b\^2)\)\)\ a\)\/\(2\ \((a\ \(K \((a\^2\/b\^2)\)\) - b\ \ \(K \((b\^2\/a\^2)\)\))\)\)\)\/\@\(\((z\^2 - a\^2)\)\ \((z\^2 - \ b\^2)\)\);\)\)}], FontFamily->"Helvetica"]]], TextForm]], "NumberedEquation"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`E(m)\)]], " gives the complete elliptic integral and ", Cell[BoxData[ \(TraditionalForm\`K(m)\)]], " the complete elliptic integral of the first kind. " }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Subsubsection"], ". Stress intensity factor" }], "Subsubsection"], Cell[TextData[{ "The stress intensity factor at the crack tip is defined by equation (14). \ For the particular case of plate with two cracks this factor can be evaluated \ at the iner (", Cell[BoxData[ \(TraditionalForm\`z\_0 = \(\[PlusMinus]a\)\)]], ") or outer crack tip (", Cell[BoxData[ \(TraditionalForm\`z\_0 = \(\[PlusMinus]b\)\)]], ")." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[{ \(TraditionalForm\`K\_Ia = \(\(-P\)\ \@\[Pi]\ \(K(a\^2\/b\^2)\)\ \ a\^3 + b\ P\ \@\[Pi]\ \(E(b\^2\/a\^2)\)\ a\^2 + b\^2\ P\ \@\[Pi]\ \(K(a\^2\/b\ \^2)\)\ a - a\ b\^2\ P\ \@\[Pi]\ \(E(a\^2\/b\^2)\)\)\/\(\@\(a\ b\^2 - a\^3\)\ \ \((a\ \(K(a\^2\/b\^2)\) - b\ \(K(b\^2\/a\^2)\))\)\)\n\), "\[IndentingNewLine]", \(TraditionalForm\`K\_Ib = \(\(-P\)\ \@\[Pi]\ \(K(b\^2\/a\^2)\)\ \ b\^3 + a\ P\ \@\[Pi]\ \(E(a\^2\/b\^2)\)\ b\^2 - a\^2\ P\ \@\[Pi]\ \(E(b\^2\/a\ \^2)\)\ b + a\^2\ P\ \@\[Pi]\ \(K(b\^2\/a\^2)\)\ b\)\/\(\@\(b\^3 - a\^2\ b\)\ \ \((a\ \(K(a\^2\/b\^2)\) - b\ \(K(b\^2\/a\^2)\))\)\)\)}], FontFamily->"Helvetica"]]], TextForm]], "NumberedEquation"], Cell[TextData[{ "For further analysis, we introduce the following notation: \n", Cell[BoxData[ \(TraditionalForm\`K\_I = F\ \ \ P \@\( \[Pi]\ \(b - a\)\/2\)\)]], " where ", Cell[BoxData[ \(TraditionalForm\`P \@\( \[Pi]\ \(b - a\)\/2\)\)]], " is the stress concentration factor for a single crack with ", Cell[BoxData[ \(TraditionalForm\`\((b - a)\)\)]], " length subjected to a mode I loading of magnitude ", Cell[BoxData[ \(TraditionalForm\`P\)]], ". The parameter ", Cell[BoxData[ \(TraditionalForm\`\(\(F \[GreaterEqual] 1\)\(,\)\)\)]], " therefore represents the stress amplification coefficient due to \ interactions between cracks ." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[{ \(TraditionalForm\`\(F\_a = \(\@\(\[Beta]\/\(\[Beta] + 1\)\)\ \((\ \[Beta]\ \(E(\((\[Beta] + 2)\)\^2\/\[Beta]\^2)\)\ \((\[Beta] + 2)\) - \((\ \[Beta] + 2)\)\^2\ \(E(\[Beta]\^2\/\((\[Beta] + 2)\)\^2)\) + 4\ \((\[Beta] + \ 1)\)\ \(K(\[Beta]\^2\/\((\[Beta] + 2)\)\^2)\))\)\)\/\(2\ \[Beta]\ \(K(\[Beta]\ \^2\/\((\[Beta] + 2)\)\^2)\) - 2\ \((\[Beta] + 2)\)\ \(K(\((\[Beta] + 2)\)\^2\ \/\[Beta]\^2)\)\);\)\n\), "\[IndentingNewLine]", \(TraditionalForm\`F\_b = \(\((\[Beta] + 2)\)\ \((\(-\(E(\((\[Beta] \ + 2)\)\^2\/\[Beta]\^2)\)\)\ \[Beta]\^2 + \((\[Beta] + 2)\)\ \ \(E(\[Beta]\^2\/\((\[Beta] + 2)\)\^2)\)\ \[Beta] - 4\ \((\[Beta] + 1)\)\ \(K(\ \((\[Beta] + 2)\)\^2\/\[Beta]\^2)\))\)\)\/\(2\ \@\(\[Beta]\^2 + 3\ \[Beta] + \ 2\)\ \((\[Beta]\ \(K(\[Beta]\^2\/\((\[Beta] + 2)\)\^2)\) - \((\[Beta] + 2)\)\ \ \(K(\((\[Beta] + 2)\)\^2\/\[Beta]\^2)\))\)\)\)}], FontFamily->"Helvetica"]]], TextForm]], "NumberedEquation"], Cell[TextData[{ "Where the ", Cell[BoxData[ FormBox[ RowBox[{"\[Beta]", "=", RowBox[{ StyleBox[\(\(distance\ between\ cracks\)\/\(crack\ length\)\), FontSize->12], "=", StyleBox[\(\(2\ a\)\/\(b - a\)\), FontSize->12]}]}], TraditionalForm]]], "." }], "Text"], Cell[TextData[{ StyleBox["Figure ", "SB"], StyleBox[ CounterBox["NumberedFigure"], "SB"], StyleBox[". 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It's also noticed that cracks interactions on the interior ends (", Cell[BoxData[ \(TraditionalForm\`F\_a\)]], ") are stronger than those in the external vicinity (", Cell[BoxData[ \(TraditionalForm\`F\_b\)]], ").\nWhen the distance ", Cell[BoxData[ \(TraditionalForm\`2\ a\)]], " between cracks is very weak relatively with their length ", Cell[BoxData[ \(TraditionalForm\`b - a\)]], " , interactions between the two cracks are important: higher than 1.2 and \ 1.5 respectively for inner and outer crack tips.\nIn the same way when the \ distance between the cracks becomes very large compared to the crack length, \ the interactions coefficients tend towards their minimal value 1.\nOne can \ conclude that for ", Cell[BoxData[ \(TraditionalForm\`\[Beta] \[GreaterEqual] 5\)]], ", interactions between the cracks become negligible.\n\nIn a previous \ investigation due to Erdogan and reported in [], the author suggested some \ expressions of these interactions coefficients at cracks tips. This \ semi-numerical approach is based on the following figure and provided the \ following results:" }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ StyleBox["Figure ", "SB"], StyleBox[ CounterBox["NumberedFigure"], "SB"], StyleBox[". 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Results provided by the present study \ match very well these initial semi-numerical data of Erdogan. Moreover, it's \ also possible to obtain full analytical expressions of stress fields in the \ considered elastic plate.\ \>", "Text", TextAlignment->Left, TextJustification->1] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Conclusion" }], "Section"], Cell[TextData[{ "This present paper reports a study on the determination of the analytical \ expressions of the stress fields and the stress intensity factor K in an \ elastic region containing cracks.\nExpressions of these parameters \ classically used in fracture mechanics are available but they appear to be \ crude approximations of our results which are applicable beyond the crack \ vicinity. Our expressions have thus the merit to be more general than those \ resulting from other analytical methods. \nThus the results obtained by Irwin \ by application of the method of Westergaard are approached expressions, only \ valid for ", Cell[BoxData[ \(TraditionalForm\`r \[LessLess] 1\)]], ". These results constitute a good approximation of the stress fields and \ displacements in the vicinity of the crack. \nFor the problem of the elastic \ region with two cracks, expressions of stress field are provided. Their use \ by computer algebra software such as Mathematica does not present a major \ difficulty. They can also be integrated in a optimized form into a numerical \ computer code.\nFor engineers concerned with design and technical calculation \ on the safety of structures and thus always willing to gain a more precise \ fracture criterion, results provided by the present study are a real \ progress. " }], "Text", TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". References" }], "Section"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "AIFANTIS (E. C.) and GERBERICH (W. W.) - ", StyleBox["A new form of exact solutions for mode I, II, III cracks problems \ and implications", FontSlant->"Italic"], ", Engineering Fracture Mechanics, Vol.10, 1978." }], "Reference", TextAlignment->Left, TextJustification->1], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "BARTHELEMY (Bernard) - ", StyleBox["Notions pratiques de m\[EAcute]canique de la rupture", FontSlant->"Italic"], " - Paris, Editions EYROLLES, 1980." }], "Reference"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "BUI (H.D.) - ", StyleBox["M\[EAcute]canique de la rupture ", FontSlant->"Italic"], "- Paris, Masson Editeurs, 1978." }], "Reference"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "GREEN (A.) and ZERNA (W.) - ", StyleBox["Theoritical elasticity", FontFamily->"Helvetica", FontSlant->"Italic"], ", Oxford University Press, Fair lawn, NJ, 1954." }], "Reference"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "KOLOSSOV (G. V.) - ", StyleBox["On an application of complex functions theory to a plane problem \ of the mathematical theory of elasticity", FontSlant->"Italic"], ", Communication, University of Dorpat, 1909." }], "Reference", TextAlignment->Left, TextJustification->1], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "MUSKHELISHVILI (N.I.) - ", StyleBox["Some basic problems of mathematical elasticity", FontSlant->"Italic"], "; translated by J.R.M.RADOK- Leyden: Noordhoff International Publishing, \ 1975." }], "Reference", TextAlignment->Left, TextJustification->1], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "SAADA (A.S.) - ", StyleBox["Elasticity theory and applications ", FontSlant->"Italic"], "- Krieger Publishing Company, 1973, ", Cell[BoxData[ \(2\^e\)]], " \[EAcute]dition." }], "Reference"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "SNEDDON (I.N.) - ", StyleBox["The distribution of stress in the neighborhood of crack in an \ elastic solid", FontSlant->"Italic"], ", Proc. R. Soc. London, A187, 1946." }], "Reference"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "SOKOLNIKOFF (I.S.) - ", StyleBox["Mathematical theory of elasticity", FontSlant->"Italic"], " - New-York, Krieger Publishing company, 1983." }], "Reference"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "SOLOMON (L.) - ", StyleBox["Elasticit\[EAcute] lin\[EAcute]aire", FontSlant->"Italic"], " - Paris:Masson et ", Cell[BoxData[ \(C\^ie\)]], " Editeurs, 1968." }], "Reference"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "MURAKAMI (Y.) - ", StyleBox["Stress Intensity factors handbook", FontSlant->"Italic"], " - Oxford:Pergamon press, 1987- 648 - Vol 1." }], "Reference"], Cell[TextData[{ Cell[TextData[{ "[", CounterBox["Reference"], "]\[ThickSpace]\[MediumSpace]" }], CellSize->{24, Inherited}, TextAlignment->Right], "WESTERGAARD (H. M.) - ", StyleBox["Bearing pressure and cracks", FontSlant->"Italic"], ", Trans. Am. Soc. Mech. Engrs. J. Appl. Mech. 61, A49, 1939." }], "Reference"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for X", ScreenRectangle->{{0, 1920}, {0, 1200}}, WindowToolbars->"EditBar", WindowSize->{831, 753}, WindowMargins->{{421, Automatic}, {Automatic, 151}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, Magnification->1, StyleDefinitions -> "IMS2006styles.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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