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{"name": "Crack growth with Paris equation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Crack growth with Paris equation", "tags": [], "metadata": {"description": "<p>Integrate Paris Law equation to estimate life to failure.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Crack growth can be modelled using the Paris equation:</p>\n<p>${da \\over dN}=C(\\Delta K)^m$</p>\n<p>where $m$ is typically about 3 and $C$ is of the order $10^{-11}$. The stress intensity factor range, $\\Delta K$, is the difference in stress intensity factor $K$ during a single stress cycle. If the maximum and minimum stresses are $\\sigma_{\\text {max}}$ and $\\sigma_{\\text {min}}$ respectively, then:</p>\n<p>$\\Delta K = K_{\\text{max}} - K_{\\text {min}} = Y(\\sigma_{\\text {max}} - \\sigma_{\\text {min}})\\sqrt{\\pi a}$</p>\n<p>where $a$ is the crack length and $Y$ is a geometry factor, typically about 1.12.</p>\n<p>Given the initial crack length, $a_i$, and the final crack length, $a_f$, it is possible to estimate the time required for the crack to grow, $N$, by integrating the Paris equation beteen these points, i.e.:</p>\n<p>$\\int_{a_i}^{a_f}a^{-m/2}da = C \\left(Y(\\sigma_{\\text {max}} - \\sigma_{\\text {min}})\\sqrt{\\pi}\\right)^m N$.</p>\n<p><strong>Important</strong>: The units of $\\Delta K$ in the Paris equations are ${\\text {MPa}}\\sqrt{\\text m}$.</p>", "advice": "<p>The Paris Law equation is:</p>\n<p>${da \\over dN}=10^\\var{C}(\\Delta K)^\\var{m} =10^\\var{C}(&nbsp;Y(\\sigma_{\\text {max}} - \\sigma_{\\text {min}})\\sqrt{\\pi a} )^\\var{m} = 10^\\var{C}(\\var{Y}(\\var{sigmax} -\\var{sigmin})\\sqrt{\\pi a} )^\\var{m}$.</p>\n<p>Integrating from the initial crack length $\\var{ai}$mm&nbsp;to the final critical crack length $\\var{af}$mm:</p>\n<p>$\\int_{\\var{ai} \\times 10^{-3}}^{\\var{af} \\times 10^{-3}}a^{-\\var{m}/2}da =10^\\var{C}(\\var{Y}(\\var{sigmax} -\\var{sigmin})\\sqrt{\\pi} )^\\var{m} N$</p>\n<p>$\\left[ {a^{1-\\var{m}/2} \\over 1-\\var{m}/2} \\right]_{\\var{ai} \\times 10^{-3}}^{\\var{af} \\times 10^{-3}} =10^\\var{C}(\\var{siground(Y*(sigmax-sigmin)*sqrt(pi),3)} )^\\var{m} N$</p>\n<p>${(\\var{ai} \\times 10^{-3})^{1-\\var{m}/2} -&nbsp;(\\var{af} \\times 10^{-3})^{1-\\var{m}/2} \\over \\var{m}/2 - 1} = \\var{siground(10^C*(Y*(sigmax-sigmin)*sqrt(pi))^m,3)} N$</p>\n<p>$N = {\\var{siground((ai/1000)^(1-m/2),3)} - \\var{siground((af/1000)^(1-m/2),3)} \\over \\var{siground(m/2-1,3)} \\times \\var{siground(10^C*(Y*(sigmax-sigmin)*sqrt(pi))^m,3)}} = \\var{siground(N,3)}$ cycles</p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"sigmin": {"name": "sigmin", "group": "Ungrouped variables", "definition": "random(0..100)", "description": "<p>Minimum tensile stress during stress cycle.</p>", "templateType": "anything"}, "sigmax": {"name": "sigmax", "group": "Ungrouped variables", "definition": "random(150..300)", "description": "<p>Maximum tensile stress&nbsp;during stress cycle.</p>", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "<p>Paris Law exponent.</p>", "templateType": "anything"}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "random(-12..-10)", "description": "<p>Paris Law power of constant.</p>", "templateType": "anything"}, "Y": {"name": "Y", "group": "Ungrouped variables", "definition": "1.0+random(0..10)/20", "description": "<p>Geometry factor.</p>", "templateType": "anything"}, "ai": {"name": "ai", "group": "Ungrouped variables", "definition": "0.0+random(1..200)/100", "description": "<p>Initial crack size.</p>", "templateType": "anything"}, "af": {"name": "af", "group": "Ungrouped variables", "definition": "random(10..20)", "description": "<p>Final crack size.</p>", "templateType": "anything"}, "N": {"name": "N", "group": "Ungrouped variables", "definition": "((af/1000)^(1-m/2)-(ai/1000)^(1-m/2))/(1-m/2)/((10^C)*(Y*(sigmax-sigmin)*sqrt(pi))^m)", "description": "<p>Number of cycles to failure.</p>\n<p></p>", "templateType": "anything"}, "Nmin": {"name": "Nmin", "group": "Ungrouped variables", "definition": "floor(0.99*N)", "description": "<p>Minimum accepted value of N.</p>", "templateType": "anything"}, "Nmax": {"name": "Nmax", "group": "Ungrouped variables", "definition": "ceil(1.01*N)", "description": "<p>Maximum accepted value of N.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["sigmin", "sigmax", "m", "C", "Y", "ai", "af", "N", "Nmin", "Nmax"], "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": "<p>A structural beam is subject to a cycle of stress. Within each stress cycle, the minimum stress is $\\var{sigmin}$MPa and the maximum stress is $\\var{sigmax}$MPa. A crack of length $\\var{ai}$mm is observed. For this geometry of crack, the shape factor is $Y=\\var{Y}$, and for the material of the beam the Paris Law is:</p>\n<p>${da \\over dN}=10^\\var{C}(\\Delta K)^\\var{m}$.</p>\n<p>Before the crack reaches a length of $\\var{af}$mm, the beam will need to be replaced, or else there will be a risk of fast fracture and catastrophic failure.</p>\n<p>The estimated number of cycles to failure is: [[0]].</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "N", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Nmin", "maxValue": "Nmax", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}]}]}], "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}]}