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{"name": "Contact Mechanics: Ball on table", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Contact.png", "/srv/numbas/media/question-resources/Contact.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Contact Mechanics: Ball on table", "tags": [], "metadata": {"description": "A gold ball on a steel table - effect of its own weight.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "The size of the contact and the pressure between the contacting surfaces depends on:
\n\n- the materials, and here we assume them to be linear, elastic and isotropic;
\n- the geometry, and here we assume them to be either flat (infinite radius) or having a constant radius;
\n- the applied load, and here we assume the load is applied normal to the contact, i.e., there is no friction.
\n
\n![](\"resources/question-resources/Contact.png\"/)
\nMaterials: A single material property - the Elastic Contact Modulus ($E^*$) - combines the elastic properties of both contacting surfaces into an equivalent stiffness:
\n\\[{1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2}\\]
\nwhere $E_1$ and $E_2$ are the Young's elastic moduli of the two materials, and $\\nu_1$ and $\\nu_2$ are the corresponding Poisson's ratios.
\nGeometry: The two simplest forms of contact between curved surfaces are:
\n\n- 2D: cylindrical surfaces aligned along their length, or a cylinder on a flat surface, creating a rectangular contact area;
\n- 3D: spherical surfaces, or a sphere on a flat surface, creating a circular contact area.
\n
\nIn either case, an equivalent radius $R$ can be defined. If both surfaces are convex, e.g., two balls touching, then:
\n\\[{1 \\over R} = {1 \\over R_1} + {1 \\over R_2}\\]
\nwhere $R_1$ is the radius of Surface 1 and $R_2$ is the radius of Surface 2. If, however, one surface is concave, e.g., a ball in a cup, then:
\n\\[{1 \\over R} = {1 \\over R_1} - {1 \\over R_2}\\]
\nwhere $R_1$ is the radius of (convex) Surface 1 and $R_2$ is the radius of (concave) Surface 2. (The 'cup' radius must be larger than the 'ball' radius.)
\nApplied Load: This is a little tricky because the same notation, $P$, is used to mean different things:
\n\n- In 2D, $P$ is the applied load per unit length of the cylindrical contact. [Units: N/m]
\n- In 3D, $P$ is simply the applied load. [Units: N]
\n
\n2D Contact: The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:
\n\\[p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2}\\]
\n\\[a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2}\\]
\n3D Contact: The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:
\n\\[p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3}\\]
\n\\[a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3}\\]
", "advice": "\n- The weight of the sphere is the volume times the density times the acceleration from gravity, i.e.:
\n
\nweight $= \\rho g \\times {4 \\over 3} \\pi r^3 = 19320 \\times 9.81 \\times {4 \\over 3} \\times \\pi \\times \\left( {\\var{diameter} \\times 10^{-2} \\over 2} \\right)^3 = \\var{siground(weight,3)}$N.
\n\n- The Elastic Contact Modulus, $E^*$, is given by:
\n
\n${1 \\over E^*} = {1 - 0.4^2 \\over 79 \\times 10^9} + {1 - 0.3^2 \\over 209 \\times 10^9}$
\nwhich can be rearranged to give $E^* = \\var{siground(ECM,3)}$GPa.
\n\n- This is spherical (3D) contact, with the equivalent radius equal to the radius of the sphere. The peak contact pressure, $p_0$, is therefore given by:
\n
\n$p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3} = \\left({6 \\times \\var{siground(weight,3)} \\times \\left( \\var{siground(ECM,3)} \\times 10^9 \\right)^2 \\over \\pi^3 \\left( \\var{diameter} \\times 10^{-2} \\div 2 \\right)^2}\\right)^{1 \\over 3} = \\var{siground(p0,3)}$MPa.
\n\n- Similarly, the semi-contact width, $a$, is given by:
\n
\n$a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3} = \\left({3 \\times \\var{siground(weight,3)} \\times \\var{diameter} \\times 10^{-2} \\div 2 \\over 4 \\times \\var{siground(ECM,3)} \\times 10^9} \\right)^{1 \\over 3} = \\var{siground(scw,3)}$mm.
\n\n- The area of the contact patch is circular with radius equal to $a$, i.e.:
\n
\narea $= \\pi a^2 = \\pi \\left( \\var{siground(scw,3)} \\times 10^{-3} \\right)^2 = \\var{siground(area,3)}$mm$^2$.
", "rulesets": {}, "extensions": [], "variables": {"weight": {"name": "weight", "group": "Ungrouped variables", "definition": "volume*19320*9.81", "description": "Weight of gold ball. [Units: N]
", "templateType": "anything"}, "ECM": {"name": "ECM", "group": "Ungrouped variables", "definition": "1/((1-0.4^2)/79+(1-0.3^2)/209)", "description": "Elastic Contact Modulus. [Units: GPa]
", "templateType": "anything"}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "pi*scw^2", "description": "Contact area. [Units: mm$^2$]
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", "templateType": "anything"}, "scw": {"name": "scw", "group": "Ungrouped variables", "definition": "(3 * weight * R / (4 * ECM * 10^9))^(1/3) * 1000", "description": "Semi-contact width. [Units: mm]
", "templateType": "anything"}, "diameter": {"name": "diameter", "group": "Ungrouped variables", "definition": "random(6..40)", "description": "Diameter of gold ball. [Units: cm]
", "templateType": "anything"}, "volume": {"name": "volume", "group": "Ungrouped variables", "definition": "pi*R^3*4/3", "description": "Volume of gold ball. [Units: m3]
", "templateType": "anything"}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "diameter/200", "description": "Radius of gold ball. [Units: m]
", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["diameter", "R", "volume", "weight", "ECM", "p0", "scw", "area"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Reference values:
\n\n- Gold: Density, $\\rho = 19\\ 320$ kg/m$^3$; Young's modulus, $E = 79$ GPa; Poisson's ratio, $\\nu = 0.4$.
\n- Steel: Density, $\\rho = 8\\ 050$ kg/m$^3$; Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.
\n- Gravity: $g = 9.81$ m/s$^2$.
\n
\nA gold sphere of diameter $\\var{diameter}$ cm sits on a flat steel table.
\nCalculate:
\n\n- the weight of the ball: [[0]] [Units: N]
\n- the Elastic Contact Modulus: $E^* =$ [[1]] [Units: GPa]
\n- the peak contact pressure: $p_0 =$ [[2]] [Units: MPa]
\n- the semi-contact width: $a =$ [[3]] [Units: mm]
\n- the area of the contact patch: [[4]] [Units: mm$^2$]
\n
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