Wolfram Language

3D Printing

Detect Design Issues before 3D Printing

Use numerical differential equation-solving capabilities in the Wolfram Language to detect issues with your designs before undertaking the expense of 3D printing.

Perform stress analysis and model the deformation of a clamped crankshaft.

In[1]:=
Click for copyable input
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show complete Wolfram Language input
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(* load the finite element package and create a finite element mesh *) \ Needs["NDSolve`FEM`"] crankshaft = ToElementMesh[DiscretizeGraphics[crankshaft], "MeshOrder" -> 1]; (* Specify the 3D stress formulation *) stressOperator[ Y_, \[Nu]_] := {Inactive[ Div][({{0, 0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu])))}, {0, 0, 0}, {-(Y/(2 (1 + \[Nu]))), 0, 0}}.Inactive[Grad][ w[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0}, {-(Y/(2 (1 + \[Nu]))), 0, 0}, {0, 0, 0}}.Inactive[Grad][ v[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{-((Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 0}, {0, -(Y/(2 (1 + \[Nu]))), 0}, {0, 0, -(Y/(2 (1 + \[Nu])))}}.Inactive[Grad][ u[x, y, z], {x, y, z}]), {x, y, z}], Inactive[ Div][({{0, 0, 0}, {0, 0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu])))}, {0, -(Y/( 2 (1 + \[Nu]))), 0}}.Inactive[Grad][ w[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{0, -(Y/(2 (1 + \[Nu]))), 0}, {-((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 0}, {0, 0, 0}}.Inactive[Grad][u[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{-(Y/(2 (1 + \[Nu]))), 0, 0}, {0, -((Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu]))), 0}, {0, 0, -(Y/(2 (1 + \[Nu])))}}.Inactive[Grad][ v[x, y, z], {x, y, z}]), {x, y, z}], Inactive[ Div][({{0, 0, 0}, {0, 0, -(Y/(2 (1 + \[Nu])))}, {0, -(( Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0}}.Inactive[Grad][ v[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{0, 0, -(Y/(2 (1 + \[Nu])))}, {0, 0, 0}, {-((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 0}}.Inactive[Grad][u[x, y, z], {x, y, z}]), {x, y, z}] + Inactive[ Div][({{-(Y/(2 (1 + \[Nu]))), 0, 0}, {0, -(Y/(2 (1 + \[Nu]))), 0}, {0, 0, -(( Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu])))}}.Inactive[ Grad][w[x, y, z], {x, y, z}]), {x, y, z}]}; (* Specify constraints such that the boundary of the crankshaft is \ held fixed at these positions *) Subscript[\[CapitalGamma], D] = {DirichletCondition[{u[x, y, z] == 0., v[x, y, z] == 0., w[x, y, z] == 0.}, {y <= 0.1, y >= 890}]}; (* Specify a boundary load pushing in the negative z\[Hyphen]axis *) Subscript[\[CapitalGamma], L] = NeumannValue[-10000., ((217 <= y <= 256) && (40 <= z <= 103)) || ((584 <= y <= 622) && (40 <= z <= 103))]; (* visualization *) showStress[{uif_, vif_, wif_}] := Show[ ToBoundaryMesh[crankshaft][ "Wireframe"[ "MeshElementStyle" -> {Directive[EdgeForm[], FaceForm[Gray]]}]], ElementMeshDeformation[crankshaft, {uif, vif, wif}, "ScalingFactor" -> 100][ "Wireframe"[AspectRatio -> Automatic, "MeshElementStyle" -> Directive[EdgeForm[], FaceForm[Red]]]], ImageSize -> 300, ViewPoint -> {-2, -2, 0} ]

Find the material properties of steel.

In[3]:=
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WolframAlpha["young modulus of steel in Pa", "WolframResult"]
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WolframAlpha["Poisson ratio of steel", "WolframResult"]
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Solve and visualize the deformation.

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NDSolveValue[{stressOperator[2.05*10^11, 0.29] == {0, 0, Subscript[\[CapitalGamma], L]}, Subscript[\[CapitalGamma], D]}, {u, v, w}, {x, y, z} \[Element] crankshaft] // showStress
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