elasticity in polar coordinates

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Elasticity in Polar Coordinates Structural Analysis: Polar Coordinates © 2021 Mayuresh Patil. Licensed under a Creative Commons Attribution 4.0 license https://creativecommons.org/licenses/by-nc-sa/4.0/ [email protected]

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Page 1: Elasticity in Polar Coordinates

Elasticity in Polar Coordinates

Structural Analysis: Polar Coordinates© 2021 Mayuresh Patil. Licensed under a Creative Commons Attribution 4.0 license https://creativecommons.org/licenses/by-nc-sa/4.0/ [email protected]

Page 2: Elasticity in Polar Coordinates

Polar Coordinates: Coordinate Transformation

tan ✓ =y

xr2 = x2 + y2

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• Nonlinear transformation• Not the same as rotation of a Cartesian reference frame• Transformation depends on the coordinates (x,y) or (r,θ)

x = r cos ✓

y = r sin ✓<latexit sha1_base64="C+olspmgGyGidcDvA0o3ItjyOHo=">AAACE3icbZDLSgMxFIYz9VbrbdSlm2BRxEWZEUE3QkEElxXsBTpDyaRpG5pJhuSMWErfwY2v4saFIm7duPNtTNsRtPWHwM93zuHk/FEiuAHP+3JyC4tLyyv51cLa+sbmlru9UzMq1ZRVqRJKNyJimOCSVYGDYI1EMxJHgtWj/uW4Xr9j2nAlb2GQsDAmXck7nBKwqOUe3+NDfIE1DqgyOIAeA4KDoDD4wYbLDLfcolfyJsLzxs9MEWWqtNzPoK1oGjMJVBBjmr6XQDgkGjgVbFQIUsMSQvuky5rWShIzEw4nN43wgSVt3FHaPgl4Qn9PDElszCCObGdMoGdma2P4X62ZQuc8HHKZpMAknS7qpAKDwuOAcJtrRkEMrCFUc/tXTHtEEwo2xoINwZ89ed7UTkq+9TenxfJVFkce7aF9dIR8dIbK6BpVUBVR9ICe0At6dR6dZ+fNeZ+25pxsZhf9kfPxDSbmmzo=</latexit><latexit sha1_base64="C+olspmgGyGidcDvA0o3ItjyOHo=">AAACE3icbZDLSgMxFIYz9VbrbdSlm2BRxEWZEUE3QkEElxXsBTpDyaRpG5pJhuSMWErfwY2v4saFIm7duPNtTNsRtPWHwM93zuHk/FEiuAHP+3JyC4tLyyv51cLa+sbmlru9UzMq1ZRVqRJKNyJimOCSVYGDYI1EMxJHgtWj/uW4Xr9j2nAlb2GQsDAmXck7nBKwqOUe3+NDfIE1DqgyOIAeA4KDoDD4wYbLDLfcolfyJsLzxs9MEWWqtNzPoK1oGjMJVBBjmr6XQDgkGjgVbFQIUsMSQvuky5rWShIzEw4nN43wgSVt3FHaPgl4Qn9PDElszCCObGdMoGdma2P4X62ZQuc8HHKZpMAknS7qpAKDwuOAcJtrRkEMrCFUc/tXTHtEEwo2xoINwZ89ed7UTkq+9TenxfJVFkce7aF9dIR8dIbK6BpVUBVR9ICe0At6dR6dZ+fNeZ+25pxsZhf9kfPxDSbmmzo=</latexit><latexit sha1_base64="C+olspmgGyGidcDvA0o3ItjyOHo=">AAACE3icbZDLSgMxFIYz9VbrbdSlm2BRxEWZEUE3QkEElxXsBTpDyaRpG5pJhuSMWErfwY2v4saFIm7duPNtTNsRtPWHwM93zuHk/FEiuAHP+3JyC4tLyyv51cLa+sbmlru9UzMq1ZRVqRJKNyJimOCSVYGDYI1EMxJHgtWj/uW4Xr9j2nAlb2GQsDAmXck7nBKwqOUe3+NDfIE1DqgyOIAeA4KDoDD4wYbLDLfcolfyJsLzxs9MEWWqtNzPoK1oGjMJVBBjmr6XQDgkGjgVbFQIUsMSQvuky5rWShIzEw4nN43wgSVt3FHaPgl4Qn9PDElszCCObGdMoGdma2P4X62ZQuc8HHKZpMAknS7qpAKDwuOAcJtrRkEMrCFUc/tXTHtEEwo2xoINwZ89ed7UTkq+9TenxfJVFkce7aF9dIR8dIbK6BpVUBVR9ICe0At6dR6dZ+fNeZ+25pxsZhf9kfPxDSbmmzo=</latexit><latexit sha1_base64="C+olspmgGyGidcDvA0o3ItjyOHo=">AAACE3icbZDLSgMxFIYz9VbrbdSlm2BRxEWZEUE3QkEElxXsBTpDyaRpG5pJhuSMWErfwY2v4saFIm7duPNtTNsRtPWHwM93zuHk/FEiuAHP+3JyC4tLyyv51cLa+sbmlru9UzMq1ZRVqRJKNyJimOCSVYGDYI1EMxJHgtWj/uW4Xr9j2nAlb2GQsDAmXck7nBKwqOUe3+NDfIE1DqgyOIAeA4KDoDD4wYbLDLfcolfyJsLzxs9MEWWqtNzPoK1oGjMJVBBjmr6XQDgkGjgVbFQIUsMSQvuky5rWShIzEw4nN43wgSVt3FHaPgl4Qn9PDElszCCObGdMoGdma2P4X62ZQuc8HHKZpMAknS7qpAKDwuOAcJtrRkEMrCFUc/tXTHtEEwo2xoINwZ89ed7UTkq+9TenxfJVFkce7aF9dIR8dIbK6BpVUBVR9ICe0At6dR6dZ+fNeZ+25pxsZhf9kfPxDSbmmzo=</latexit>

(x, y)

(r, ✓)<latexit sha1_base64="v/Q94IkfRS3v3dvB2xps1PmFZoM=">AAAB/HicbZDLSsNAFIYnXmu8Rbt0M1iEFkpJRNBl0Y3LCvYCTSiT6aQdOrkwcyKGUF/FjQtF3Pog7nwbp20W2vrDwMd/zuGc+f1EcAW2/W2srW9sbm2Xdszdvf2DQ+vouKPiVFLWprGIZc8nigkesTZwEKyXSEZCX7CuP7mZ1bsPTCoeR/eQJcwLySjiAacEtDWwytXHelbDrmtWZd2FMQNSG1gVu2HPhVfBKaCCCrUG1pc7jGkasgioIEr1HTsBLycSOBVsarqpYgmhEzJifY0RCZny8vnxU3ymnSEOYqlfBHju/p7ISahUFvq6MyQwVsu1mflfrZ9CcOXlPEpSYBFdLApSgSHGsyTwkEtGQWQaCJVc34rpmEhCQedl6hCc5S+vQue84Wi+u6g0r4s4SugEnaIqctAlaqJb1EJtRFGGntErejOejBfj3fhYtK4ZxUwZ/ZHx+QMSk5Ma</latexit><latexit sha1_base64="v/Q94IkfRS3v3dvB2xps1PmFZoM=">AAAB/HicbZDLSsNAFIYnXmu8Rbt0M1iEFkpJRNBl0Y3LCvYCTSiT6aQdOrkwcyKGUF/FjQtF3Pog7nwbp20W2vrDwMd/zuGc+f1EcAW2/W2srW9sbm2Xdszdvf2DQ+vouKPiVFLWprGIZc8nigkesTZwEKyXSEZCX7CuP7mZ1bsPTCoeR/eQJcwLySjiAacEtDWwytXHelbDrmtWZd2FMQNSG1gVu2HPhVfBKaCCCrUG1pc7jGkasgioIEr1HTsBLycSOBVsarqpYgmhEzJifY0RCZny8vnxU3ymnSEOYqlfBHju/p7ISahUFvq6MyQwVsu1mflfrZ9CcOXlPEpSYBFdLApSgSHGsyTwkEtGQWQaCJVc34rpmEhCQedl6hCc5S+vQue84Wi+u6g0r4s4SugEnaIqctAlaqJb1EJtRFGGntErejOejBfj3fhYtK4ZxUwZ/ZHx+QMSk5Ma</latexit><latexit sha1_base64="v/Q94IkfRS3v3dvB2xps1PmFZoM=">AAAB/HicbZDLSsNAFIYnXmu8Rbt0M1iEFkpJRNBl0Y3LCvYCTSiT6aQdOrkwcyKGUF/FjQtF3Pog7nwbp20W2vrDwMd/zuGc+f1EcAW2/W2srW9sbm2Xdszdvf2DQ+vouKPiVFLWprGIZc8nigkesTZwEKyXSEZCX7CuP7mZ1bsPTCoeR/eQJcwLySjiAacEtDWwytXHelbDrmtWZd2FMQNSG1gVu2HPhVfBKaCCCrUG1pc7jGkasgioIEr1HTsBLycSOBVsarqpYgmhEzJifY0RCZny8vnxU3ymnSEOYqlfBHju/p7ISahUFvq6MyQwVsu1mflfrZ9CcOXlPEpSYBFdLApSgSHGsyTwkEtGQWQaCJVc34rpmEhCQedl6hCc5S+vQue84Wi+u6g0r4s4SugEnaIqctAlaqJb1EJtRFGGntErejOejBfj3fhYtK4ZxUwZ/ZHx+QMSk5Ma</latexit><latexit sha1_base64="v/Q94IkfRS3v3dvB2xps1PmFZoM=">AAAB/HicbZDLSsNAFIYnXmu8Rbt0M1iEFkpJRNBl0Y3LCvYCTSiT6aQdOrkwcyKGUF/FjQtF3Pog7nwbp20W2vrDwMd/zuGc+f1EcAW2/W2srW9sbm2Xdszdvf2DQ+vouKPiVFLWprGIZc8nigkesTZwEKyXSEZCX7CuP7mZ1bsPTCoeR/eQJcwLySjiAacEtDWwytXHelbDrmtWZd2FMQNSG1gVu2HPhVfBKaCCCrUG1pc7jGkasgioIEr1HTsBLycSOBVsarqpYgmhEzJifY0RCZny8vnxU3ymnSEOYqlfBHju/p7ISahUFvq6MyQwVsu1mflfrZ9CcOXlPEpSYBFdLApSgSHGsyTwkEtGQWQaCJVc34rpmEhCQedl6hCc5S+vQue84Wi+u6g0r4s4SugEnaIqctAlaqJb1EJtRFGGntErejOejBfj3fhYtK4ZxUwZ/ZHx+QMSk5Ma</latexit>

Page 3: Elasticity in Polar Coordinates

Polar Coordinates: Vector Transformation

• Cartesian coordinate system unit vectors: • Polar coordinate system unit vectors:

• At each individual point this is similar to a rotation of a Cartesian coordinate system

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er e✓<latexit sha1_base64="3IJbKXqaihPLil8ikGjKAPJdXAo=">AAACB3icbZDJSgNBEIZ74h63UY+CNAbBk8yIoMegF48KZoFMCDWdimnSs9hdI4QhNy++ihcPinj1Fbz5NnYWQRN/aPj6ryq66w9TJQ153pdTmJtfWFxaXimurq1vbLpb21WTZFpgRSQq0fUQDCoZY4UkKaynGiEKFdbC3sWwXrtHbWQS31A/xWYEt7HsSAFkrZa7F3SBchy0NA/uMmjzn3tAXSRouSXvyBuJz4I/gRKb6KrlfgbtRGQRxiQUGNPwvZSaOWiSQuGgGGQGUxA9uMWGxRgiNM18tMeAH1inzTuJticmPnJ/T+QQGdOPQtsZAXXNdG1o/ldrZNQ5a+YyTjPCWIwf6mSKU8KHofC21ChI9S2A0NL+lYsuaBBkoyvaEPzplWehenzkW74+KZXPJ3Ess122zw6Zz05ZmV2yK1Zhgj2wJ/bCXp1H59l5c97HrQVnMrPD/sj5+AZ1P5ms</latexit><latexit sha1_base64="3IJbKXqaihPLil8ikGjKAPJdXAo=">AAACB3icbZDJSgNBEIZ74h63UY+CNAbBk8yIoMegF48KZoFMCDWdimnSs9hdI4QhNy++ihcPinj1Fbz5NnYWQRN/aPj6ryq66w9TJQ153pdTmJtfWFxaXimurq1vbLpb21WTZFpgRSQq0fUQDCoZY4UkKaynGiEKFdbC3sWwXrtHbWQS31A/xWYEt7HsSAFkrZa7F3SBchy0NA/uMmjzn3tAXSRouSXvyBuJz4I/gRKb6KrlfgbtRGQRxiQUGNPwvZSaOWiSQuGgGGQGUxA9uMWGxRgiNM18tMeAH1inzTuJticmPnJ/T+QQGdOPQtsZAXXNdG1o/ldrZNQ5a+YyTjPCWIwf6mSKU8KHofC21ChI9S2A0NL+lYsuaBBkoyvaEPzplWehenzkW74+KZXPJ3Ess122zw6Zz05ZmV2yK1Zhgj2wJ/bCXp1H59l5c97HrQVnMrPD/sj5+AZ1P5ms</latexit><latexit sha1_base64="3IJbKXqaihPLil8ikGjKAPJdXAo=">AAACB3icbZDJSgNBEIZ74h63UY+CNAbBk8yIoMegF48KZoFMCDWdimnSs9hdI4QhNy++ihcPinj1Fbz5NnYWQRN/aPj6ryq66w9TJQ153pdTmJtfWFxaXimurq1vbLpb21WTZFpgRSQq0fUQDCoZY4UkKaynGiEKFdbC3sWwXrtHbWQS31A/xWYEt7HsSAFkrZa7F3SBchy0NA/uMmjzn3tAXSRouSXvyBuJz4I/gRKb6KrlfgbtRGQRxiQUGNPwvZSaOWiSQuGgGGQGUxA9uMWGxRgiNM18tMeAH1inzTuJticmPnJ/T+QQGdOPQtsZAXXNdG1o/ldrZNQ5a+YyTjPCWIwf6mSKU8KHofC21ChI9S2A0NL+lYsuaBBkoyvaEPzplWehenzkW74+KZXPJ3Ess122zw6Zz05ZmV2yK1Zhgj2wJ/bCXp1H59l5c97HrQVnMrPD/sj5+AZ1P5ms</latexit><latexit sha1_base64="3IJbKXqaihPLil8ikGjKAPJdXAo=">AAACB3icbZDJSgNBEIZ74h63UY+CNAbBk8yIoMegF48KZoFMCDWdimnSs9hdI4QhNy++ihcPinj1Fbz5NnYWQRN/aPj6ryq66w9TJQ153pdTmJtfWFxaXimurq1vbLpb21WTZFpgRSQq0fUQDCoZY4UkKaynGiEKFdbC3sWwXrtHbWQS31A/xWYEt7HsSAFkrZa7F3SBchy0NA/uMmjzn3tAXSRouSXvyBuJz4I/gRKb6KrlfgbtRGQRxiQUGNPwvZSaOWiSQuGgGGQGUxA9uMWGxRgiNM18tMeAH1inzTuJticmPnJ/T+QQGdOPQtsZAXXNdG1o/ldrZNQ5a+YyTjPCWIwf6mSKU8KHofC21ChI9S2A0NL+lYsuaBBkoyvaEPzplWehenzkW74+KZXPJ3Ess122zw6Zz05ZmV2yK1Zhgj2wJ/bCXp1H59l5c97HrQVnMrPD/sj5+AZ1P5ms</latexit>

f = fxi+ fy j

= fr er + f✓ e✓<latexit sha1_base64="tMysyb0H3yn/IGnuDA77Zi4Z/uc=">AAACPHicbZC7SwNBEMb3fBtfUUubxaAIgtyJoI0giGCpaFTIhTC3mTOrew9258Rw3B9m4x9hZ2VjoYittZtcEF9T/eabb9idL0iVNOS6j87Q8Mjo2PjEZGVqemZ2rjq/cGaSTAusi0Ql+iIAg0rGWCdJCi9SjRAFCs+D6/3e/PwGtZFJfErdFJsRXMYylALISq3qiR+A5iFf5bs8bN1yvwOUy4Kv265bdlcF9/1KadClhIWlnsWnDhJ8iWXbqtbcDbdf/C94A6ixQR21qg9+OxFZhDEJBcY0PDelZg6apFBYVPzMYAriGi6xYTGGCE0z7x9f8BWrtHmY2CuSmHhf/b6RQ2RMNwqsMwLqmN+znvjfrJFRuNPMZZxmhLEoHwozxSnhvSR5W2oUpLoWQGhp/8pFBzQIsnlXbAje75P/wtnmhmf5eKu2dzCIY4ItsWW2xjy2zfbYITtidSbYHXtiL+zVuXeenTfnvbQOOYOdRfajnI9PhsSsDQ==</latexit><latexit sha1_base64="tMysyb0H3yn/IGnuDA77Zi4Z/uc=">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</latexit><latexit sha1_base64="tMysyb0H3yn/IGnuDA77Zi4Z/uc=">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</latexit><latexit sha1_base64="tMysyb0H3yn/IGnuDA77Zi4Z/uc=">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</latexit>

er = cos ✓ i+ sin ✓ j

e✓ = � sin ✓ i+ cos ✓ j<latexit sha1_base64="LJEaEkVm3HtKGjtQJrZca2BsESo=">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</latexit><latexit sha1_base64="LJEaEkVm3HtKGjtQJrZca2BsESo=">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</latexit><latexit sha1_base64="LJEaEkVm3HtKGjtQJrZca2BsESo=">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</latexit><latexit sha1_base64="LJEaEkVm3HtKGjtQJrZca2BsESo=">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</latexit>

fr = fx cos ✓ + fy sin ✓

f✓ = �fx sin ✓ + fy cos ✓<latexit sha1_base64="4nafCsZMu+uEwhZa22H+GmvAiEg=">AAACTHicbVDLSgMxFM3Ud31VXbq5WBRBLDMi6EYQRHCpYG2hU4ZMmmmDmcyQ3BFL6Qe6ceHOr3DjQhHBTDtIfVwInJxHHidMpTDous9OaWp6ZnZufqG8uLS8slpZW78xSaYZr7NEJroZUsOlULyOAiVvpprTOJS8Ed6e5XrjjmsjEnWN/ZS3Y9pVIhKMoqWCCosCDTtwAlFwDz5LDPjY40gB9izVB98I9U35fjkKik2e2YdxbNJTxCZOKgeVqltzRwN/gVeAKinmMqg8+Z2EZTFXyCQ1puW5KbYHVKNgkg/LfmZ4Stkt7fKWhYrG3LQHozKGsG2ZDkSJtkshjNjJxIDGxvTj0Dpjij3zW8vJ/7RWhtFxeyBUmiFXbHxRlEnABPJmoSM0Zyj7FlCmhX0rsB7VlKHtPy/B+/3lv+DmoOZZfHVYPT0v6pgnm2SL7BKPHJFTckEuSZ0w8kBeyBt5dx6dV+fD+RxbS06R2SA/pjT7Bcbbrnw=</latexit><latexit sha1_base64="4nafCsZMu+uEwhZa22H+GmvAiEg=">AAACTHicbVDLSgMxFM3Ud31VXbq5WBRBLDMi6EYQRHCpYG2hU4ZMmmmDmcyQ3BFL6Qe6ceHOr3DjQhHBTDtIfVwInJxHHidMpTDous9OaWp6ZnZufqG8uLS8slpZW78xSaYZr7NEJroZUsOlULyOAiVvpprTOJS8Ed6e5XrjjmsjEnWN/ZS3Y9pVIhKMoqWCCosCDTtwAlFwDz5LDPjY40gB9izVB98I9U35fjkKik2e2YdxbNJTxCZOKgeVqltzRwN/gVeAKinmMqg8+Z2EZTFXyCQ1puW5KbYHVKNgkg/LfmZ4Stkt7fKWhYrG3LQHozKGsG2ZDkSJtkshjNjJxIDGxvTj0Dpjij3zW8vJ/7RWhtFxeyBUmiFXbHxRlEnABPJmoSM0Zyj7FlCmhX0rsB7VlKHtPy/B+/3lv+DmoOZZfHVYPT0v6pgnm2SL7BKPHJFTckEuSZ0w8kBeyBt5dx6dV+fD+RxbS06R2SA/pjT7Bcbbrnw=</latexit><latexit sha1_base64="4nafCsZMu+uEwhZa22H+GmvAiEg=">AAACTHicbVDLSgMxFM3Ud31VXbq5WBRBLDMi6EYQRHCpYG2hU4ZMmmmDmcyQ3BFL6Qe6ceHOr3DjQhHBTDtIfVwInJxHHidMpTDous9OaWp6ZnZufqG8uLS8slpZW78xSaYZr7NEJroZUsOlULyOAiVvpprTOJS8Ed6e5XrjjmsjEnWN/ZS3Y9pVIhKMoqWCCosCDTtwAlFwDz5LDPjY40gB9izVB98I9U35fjkKik2e2YdxbNJTxCZOKgeVqltzRwN/gVeAKinmMqg8+Z2EZTFXyCQ1puW5KbYHVKNgkg/LfmZ4Stkt7fKWhYrG3LQHozKGsG2ZDkSJtkshjNjJxIDGxvTj0Dpjij3zW8vJ/7RWhtFxeyBUmiFXbHxRlEnABPJmoSM0Zyj7FlCmhX0rsB7VlKHtPy/B+/3lv+DmoOZZfHVYPT0v6pgnm2SL7BKPHJFTckEuSZ0w8kBeyBt5dx6dV+fD+RxbS06R2SA/pjT7Bcbbrnw=</latexit><latexit sha1_base64="4nafCsZMu+uEwhZa22H+GmvAiEg=">AAACTHicbVDLSgMxFM3Ud31VXbq5WBRBLDMi6EYQRHCpYG2hU4ZMmmmDmcyQ3BFL6Qe6ceHOr3DjQhHBTDtIfVwInJxHHidMpTDous9OaWp6ZnZufqG8uLS8slpZW78xSaYZr7NEJroZUsOlULyOAiVvpprTOJS8Ed6e5XrjjmsjEnWN/ZS3Y9pVIhKMoqWCCosCDTtwAlFwDz5LDPjY40gB9izVB98I9U35fjkKik2e2YdxbNJTxCZOKgeVqltzRwN/gVeAKinmMqg8+Z2EZTFXyCQ1puW5KbYHVKNgkg/LfmZ4Stkt7fKWhYrG3LQHozKGsG2ZDkSJtkshjNjJxIDGxvTj0Dpjij3zW8vJ/7RWhtFxeyBUmiFXbHxRlEnABPJmoSM0Zyj7FlCmhX0rsB7VlKHtPy/B+/3lv+DmoOZZfHVYPT0v6pgnm2SL7BKPHJFTckEuSZ0w8kBeyBt5dx6dV+fD+RxbS06R2SA/pjT7Bcbbrnw=</latexit>

fx = fr cos ✓ � f✓ sin ✓

fy = fr sin ✓ + f✓ cos ✓<latexit sha1_base64="FgkTIxx2PsudPwUCOgJMLUnC+XA=">AAACT3icbZFLSwMxFIUz9dXWV9Wlm2BRBLHMiKAboSCCywr2AZ1hyKSZNjSTGZI7Yin9h250599w40IR0+kgtXoh8HHOvXmcBIngGmz71SosLa+srhVL5fWNza3tys5uS8epoqxJYxGrTkA0E1yyJnAQrJMoRqJAsHYwvJ767QemNI/lPYwS5kWkL3nIKQEj+ZUw9B/xEb7Coa+wS2ONXRgwIBifGilnV3P5o7tuOfRH2cxsaN48mRua26zsV6p2zc4K/wUnhyrKq+FXXtxeTNOISaCCaN117AS8MVHAqWCTsptqlhA6JH3WNShJxLQ3zvKY4EOj9HAYK7Mk4EydnxiTSOtRFJjOiMBAL3pT8T+vm0J46Y25TFJgks4OClOBIcbTcHGPK0ZBjAwQqri5K6YDoggF8wXTEJzFJ/+F1lnNMXx3Xq3f5HEU0T46QMfIQReojm5RAzURRU/oDX2gT+vZere+CnlrwcphD/2qQukbEgSvQw==</latexit><latexit sha1_base64="FgkTIxx2PsudPwUCOgJMLUnC+XA=">AAACT3icbZFLSwMxFIUz9dXWV9Wlm2BRBLHMiKAboSCCywr2AZ1hyKSZNjSTGZI7Yin9h250599w40IR0+kgtXoh8HHOvXmcBIngGmz71SosLa+srhVL5fWNza3tys5uS8epoqxJYxGrTkA0E1yyJnAQrJMoRqJAsHYwvJ767QemNI/lPYwS5kWkL3nIKQEj+ZUw9B/xEb7Coa+wS2ONXRgwIBifGilnV3P5o7tuOfRH2cxsaN48mRua26zsV6p2zc4K/wUnhyrKq+FXXtxeTNOISaCCaN117AS8MVHAqWCTsptqlhA6JH3WNShJxLQ3zvKY4EOj9HAYK7Mk4EydnxiTSOtRFJjOiMBAL3pT8T+vm0J46Y25TFJgks4OClOBIcbTcHGPK0ZBjAwQqri5K6YDoggF8wXTEJzFJ/+F1lnNMXx3Xq3f5HEU0T46QMfIQReojm5RAzURRU/oDX2gT+vZere+CnlrwcphD/2qQukbEgSvQw==</latexit><latexit sha1_base64="FgkTIxx2PsudPwUCOgJMLUnC+XA=">AAACT3icbZFLSwMxFIUz9dXWV9Wlm2BRBLHMiKAboSCCywr2AZ1hyKSZNjSTGZI7Yin9h250599w40IR0+kgtXoh8HHOvXmcBIngGmz71SosLa+srhVL5fWNza3tys5uS8epoqxJYxGrTkA0E1yyJnAQrJMoRqJAsHYwvJ767QemNI/lPYwS5kWkL3nIKQEj+ZUw9B/xEb7Coa+wS2ONXRgwIBifGilnV3P5o7tuOfRH2cxsaN48mRua26zsV6p2zc4K/wUnhyrKq+FXXtxeTNOISaCCaN117AS8MVHAqWCTsptqlhA6JH3WNShJxLQ3zvKY4EOj9HAYK7Mk4EydnxiTSOtRFJjOiMBAL3pT8T+vm0J46Y25TFJgks4OClOBIcbTcHGPK0ZBjAwQqri5K6YDoggF8wXTEJzFJ/+F1lnNMXx3Xq3f5HEU0T46QMfIQReojm5RAzURRU/oDX2gT+vZere+CnlrwcphD/2qQukbEgSvQw==</latexit><latexit sha1_base64="FgkTIxx2PsudPwUCOgJMLUnC+XA=">AAACT3icbZFLSwMxFIUz9dXWV9Wlm2BRBLHMiKAboSCCywr2AZ1hyKSZNjSTGZI7Yin9h250599w40IR0+kgtXoh8HHOvXmcBIngGmz71SosLa+srhVL5fWNza3tys5uS8epoqxJYxGrTkA0E1yyJnAQrJMoRqJAsHYwvJ767QemNI/lPYwS5kWkL3nIKQEj+ZUw9B/xEb7Coa+wS2ONXRgwIBifGilnV3P5o7tuOfRH2cxsaN48mRua26zsV6p2zc4K/wUnhyrKq+FXXtxeTNOISaCCaN117AS8MVHAqWCTsptqlhA6JH3WNShJxLQ3zvKY4EOj9HAYK7Mk4EydnxiTSOtRFJjOiMBAL3pT8T+vm0J46Y25TFJgks4OClOBIcbTcHGPK0ZBjAwQqri5K6YDoggF8wXTEJzFJ/+F1lnNMXx3Xq3f5HEU0T46QMfIQReojm5RAzURRU/oDX2gT+vZere+CnlrwcphD/2qQukbEgSvQw==</latexit>

Page 4: Elasticity in Polar Coordinates

Polar Coordinates: Vector Field Transformation

• Curved (as opposed to Cartesian) spaces – different transformation at different location• Differentiation in curved spaces is more involved

• Variation of components of a vector field may be due to variation of the reference frame• No variation of the components of a vector field may still lead to non-zero derivatives

Page 5: Elasticity in Polar Coordinates

Displacements and Strains• Displacement: Cartesian & Polar

• Strains written in terms of differentiation of the components of the displacement vector

✏xx =@u(x, y)

@x

✏yy =@v(x, y)

@y

�xy = 2✏xy =@u(x, y)

@y+

@v(x, y)

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✏rr =?

✏✓✓ =?

�r✓ = 2✏r✓ =?<latexit sha1_base64="GMths2tm56QXXT5eWx69llRqjh8=">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</latexit><latexit sha1_base64="GMths2tm56QXXT5eWx69llRqjh8=">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</latexit><latexit sha1_base64="GMths2tm56QXXT5eWx69llRqjh8=">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</latexit><latexit sha1_base64="GMths2tm56QXXT5eWx69llRqjh8=">AAACUXicbVFNSyNBEK3Mrh+brJp1j16aDcqeZEYEvYiCCHt0wWggE0JNp5I0ds8M3TVCGPIXPawn/4eXPeyynQ/WmFjQ9OO9etXdr5NcK8dh+FwJPnxcW9/Y/FStfd7a3ql/2b11WWElNWWmM9tK0JFWKTVZsaZWbglNoukuub+c6HcPZJ3K0hse5dQxOEhVX0lkT3Xrw5hyp7SHpbVjcSDOxLmI4+orHfOQGMVsW+wYoDHobf+lM3EkFsa9tXTrjfAwnJZYBdEcNGBe1936r7iXycJQylKjc+0ozLlTomUlNY2rceEoR3mPA2p7mKIh1ymniYzFvmd6op9Zv1IWU3bRUaJxbmQS32mQh25Zm5Dvae2C+6edUqV5wZTK2UH9QgvOxCRe0VOWJOuRByit8ncVcogWJftPqPoQouUnr4Lbo8PI45/HjYureRybsAff4DtEcAIX8AOuoQkSHuEF/sDfylPldwBBMGsNKnPPV3hTQe0fKQGxBw==</latexit>

u(x, y)

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ur(r, ✓)

u✓(r, ✓)<latexit sha1_base64="xtqhHdKQ30eBmTXq+ilWqjweMcE=">AAACDXicbZDLSsNAFIYn9VbjLerSzWAVKkhJRNBlQQSXFewFmhAm00k7dHJh5kQooS/gxldx40IRt+7d+TZO24Da+sPAN/85h5nzB6ngCmz7yygtLa+srpXXzY3Nre0da3evpZJMUtakiUhkJyCKCR6zJnAQrJNKRqJAsHYwvJrU2/dMKp7EdzBKmReRfsxDTgloy7eOMl9W5akLAwbkBLsuNjN/dvuxfati1+yp8CI4BVRQoYZvfbq9hGYRi4EKolTXsVPwciKBU8HGppsplhI6JH3W1RiTiCkvn24zxsfa6eEwkfrEgKfu74mcREqNokB3RgQGar42Mf+rdTMIL72cx2kGLKazh8JMYEjwJBrc45JRECMNhEqu/4rpgEhCQQdo6hCc+ZUXoXVWczTfnlfq10UcZXSADlEVOegC1dENaqAmougBPaEX9Go8Gs/Gm/E+ay0Zxcw++iPj4xs9PJpm</latexit><latexit sha1_base64="xtqhHdKQ30eBmTXq+ilWqjweMcE=">AAACDXicbZDLSsNAFIYn9VbjLerSzWAVKkhJRNBlQQSXFewFmhAm00k7dHJh5kQooS/gxldx40IRt+7d+TZO24Da+sPAN/85h5nzB6ngCmz7yygtLa+srpXXzY3Nre0da3evpZJMUtakiUhkJyCKCR6zJnAQrJNKRqJAsHYwvJrU2/dMKp7EdzBKmReRfsxDTgloy7eOMl9W5akLAwbkBLsuNjN/dvuxfati1+yp8CI4BVRQoYZvfbq9hGYRi4EKolTXsVPwciKBU8HGppsplhI6JH3W1RiTiCkvn24zxsfa6eEwkfrEgKfu74mcREqNokB3RgQGar42Mf+rdTMIL72cx2kGLKazh8JMYEjwJBrc45JRECMNhEqu/4rpgEhCQQdo6hCc+ZUXoXVWczTfnlfq10UcZXSADlEVOegC1dENaqAmougBPaEX9Go8Gs/Gm/E+ay0Zxcw++iPj4xs9PJpm</latexit><latexit sha1_base64="xtqhHdKQ30eBmTXq+ilWqjweMcE=">AAACDXicbZDLSsNAFIYn9VbjLerSzWAVKkhJRNBlQQSXFewFmhAm00k7dHJh5kQooS/gxldx40IRt+7d+TZO24Da+sPAN/85h5nzB6ngCmz7yygtLa+srpXXzY3Nre0da3evpZJMUtakiUhkJyCKCR6zJnAQrJNKRqJAsHYwvJrU2/dMKp7EdzBKmReRfsxDTgloy7eOMl9W5akLAwbkBLsuNjN/dvuxfati1+yp8CI4BVRQoYZvfbq9hGYRi4EKolTXsVPwciKBU8HGppsplhI6JH3W1RiTiCkvn24zxsfa6eEwkfrEgKfu74mcREqNokB3RgQGar42Mf+rdTMIL72cx2kGLKazh8JMYEjwJBrc45JRECMNhEqu/4rpgEhCQQdo6hCc+ZUXoXVWczTfnlfq10UcZXSADlEVOegC1dENaqAmougBPaEX9Go8Gs/Gm/E+ay0Zxcw++iPj4xs9PJpm</latexit><latexit sha1_base64="xtqhHdKQ30eBmTXq+ilWqjweMcE=">AAACDXicbZDLSsNAFIYn9VbjLerSzWAVKkhJRNBlQQSXFewFmhAm00k7dHJh5kQooS/gxldx40IRt+7d+TZO24Da+sPAN/85h5nzB6ngCmz7yygtLa+srpXXzY3Nre0da3evpZJMUtakiUhkJyCKCR6zJnAQrJNKRqJAsHYwvJrU2/dMKp7EdzBKmReRfsxDTgloy7eOMl9W5akLAwbkBLsuNjN/dvuxfati1+yp8CI4BVRQoYZvfbq9hGYRi4EKolTXsVPwciKBU8HGppsplhI6JH3W1RiTiCkvn24zxsfa6eEwkfrEgKfu74mcREqNokB3RgQGar42Mf+rdTMIL72cx2kGLKazh8JMYEjwJBrc45JRECMNhEqu/4rpgEhCQQdo6hCc+ZUXoXVWczTfnlfq10UcZXSADlEVOegC1dENaqAmougBPaEX9Go8Gs/Gm/E+ay0Zxcw++iPj4xs9PJpm</latexit>

Normal strain in r-directionNormal strain in θ-directionShear strain

<latexit sha1_base64="CmZkoY1V/e4MJBNiFJ+QZZoG/94=">AAACLXicbZDJSgNBEIZ73I1b1KOXxiAIgTAjil6EoB48KpgFMiHUdCqmtWehuyYQgi/kxVcRwYMiXn0NO5kR3AoKvvqriu76g0RJQ6774kxNz8zOzS8sFpaWV1bXiusbdROnWmBNxCrWzQAMKhlhjSQpbCYaIQwUNoLb03G/MUBtZBxd0TDBdgjXkexJAWSlTvHMD0DzlB/b9PtAXPIyH2R4M1Y7OivQQtmWPvWR4EvLqk6x5FbcSfC/4OVQYnlcdIpPfjcWaYgRCQXGtDw3ofYINEmh8K7gpwYTELdwjS2LEYRo2qPJtXd8xypd3ou1zYj4RP2+MYLQmGEY2MkQqG9+98bif71WSr2j9khGSUoYieyhXqo4xXxsHe9KjYLU0AIILe1fueiDBkHW4II1wft98l+o71W8g4p7uV+qnuR2LLAtts12mccOWZWdswtWY4Lds0f2wl6dB+fZeXPes9EpJ9/ZZD/C+fgEmq6lZw==</latexit>

u = ui+ vj = ur er + u✓ e✓

Page 6: Elasticity in Polar Coordinates

Derivation: Strain-Displacement Equations in Polar Coordinates

• Lets write in terms of the displacements field in the polar coordinates• We know how to differentiate in the Cartesian coordinate frame• So transform displacements from polar to Cartesian coordinates

• Use the strain displacement equations

u = ur cos ✓ � u✓ sin ✓

v = ur sin ✓ + u✓ cos ✓<latexit sha1_base64="LG9Y34Cbfzb2iq8PvoTpw+zbWe8=">AAACS3icbZDNS8MwGMbT6XTOr6pHL8GhCOJoRdCLMBDB4wT3AesYaZZtYWlakreDMfb/efHizX/CiwdFPJh2RebmC4Efz5MnH48fCa7BcV6t3Mpqfm29sFHc3Nre2bX39us6jBVlNRqKUDV9opngktWAg2DNSDES+II1/OFt4jdGTGkeykcYR6wdkL7kPU4JGKlj+zE+wTc47ijs0VBjDwYMCMbnRsrY01z+6p5XHKWJWWTeOpuLzB1V7Nglp+ykg5fBzaCEsql27BevG9I4YBKoIFq3XCeC9oQo4FSwadGLNYsIHZI+axmUJGC6PUm7mOJjo3RxL1RmScCpOp+YkEDrceCbnQGBgV70EvE/rxVD77o94TKKgUk6u6gXCwwhTorFXa4YBTE2QKji5q2YDogiFEz9SQnu4peXoX5Rdg0/XJYqd1kdBXSIjtApctEVqqB7VEU1RNETekMf6NN6tt6tL+t7tjVnZZkD9Gdy+R8RP67G</latexit><latexit sha1_base64="LG9Y34Cbfzb2iq8PvoTpw+zbWe8=">AAACS3icbZDNS8MwGMbT6XTOr6pHL8GhCOJoRdCLMBDB4wT3AesYaZZtYWlakreDMfb/efHizX/CiwdFPJh2RebmC4Efz5MnH48fCa7BcV6t3Mpqfm29sFHc3Nre2bX39us6jBVlNRqKUDV9opngktWAg2DNSDES+II1/OFt4jdGTGkeykcYR6wdkL7kPU4JGKlj+zE+wTc47ijs0VBjDwYMCMbnRsrY01z+6p5XHKWJWWTeOpuLzB1V7Nglp+ykg5fBzaCEsql27BevG9I4YBKoIFq3XCeC9oQo4FSwadGLNYsIHZI+axmUJGC6PUm7mOJjo3RxL1RmScCpOp+YkEDrceCbnQGBgV70EvE/rxVD77o94TKKgUk6u6gXCwwhTorFXa4YBTE2QKji5q2YDogiFEz9SQnu4peXoX5Rdg0/XJYqd1kdBXSIjtApctEVqqB7VEU1RNETekMf6NN6tt6tL+t7tjVnZZkD9Gdy+R8RP67G</latexit><latexit sha1_base64="LG9Y34Cbfzb2iq8PvoTpw+zbWe8=">AAACS3icbZDNS8MwGMbT6XTOr6pHL8GhCOJoRdCLMBDB4wT3AesYaZZtYWlakreDMfb/efHizX/CiwdFPJh2RebmC4Efz5MnH48fCa7BcV6t3Mpqfm29sFHc3Nre2bX39us6jBVlNRqKUDV9opngktWAg2DNSDES+II1/OFt4jdGTGkeykcYR6wdkL7kPU4JGKlj+zE+wTc47ijs0VBjDwYMCMbnRsrY01z+6p5XHKWJWWTeOpuLzB1V7Nglp+ykg5fBzaCEsql27BevG9I4YBKoIFq3XCeC9oQo4FSwadGLNYsIHZI+axmUJGC6PUm7mOJjo3RxL1RmScCpOp+YkEDrceCbnQGBgV70EvE/rxVD77o94TKKgUk6u6gXCwwhTorFXa4YBTE2QKji5q2YDogiFEz9SQnu4peXoX5Rdg0/XJYqd1kdBXSIjtApctEVqqB7VEU1RNETekMf6NN6tt6tL+t7tjVnZZkD9Gdy+R8RP67G</latexit><latexit sha1_base64="LG9Y34Cbfzb2iq8PvoTpw+zbWe8=">AAACS3icbZDNS8MwGMbT6XTOr6pHL8GhCOJoRdCLMBDB4wT3AesYaZZtYWlakreDMfb/efHizX/CiwdFPJh2RebmC4Efz5MnH48fCa7BcV6t3Mpqfm29sFHc3Nre2bX39us6jBVlNRqKUDV9opngktWAg2DNSDES+II1/OFt4jdGTGkeykcYR6wdkL7kPU4JGKlj+zE+wTc47ijs0VBjDwYMCMbnRsrY01z+6p5XHKWJWWTeOpuLzB1V7Nglp+ykg5fBzaCEsql27BevG9I4YBKoIFq3XCeC9oQo4FSwadGLNYsIHZI+axmUJGC6PUm7mOJjo3RxL1RmScCpOp+YkEDrceCbnQGBgV70EvE/rxVD77o94TKKgUk6u6gXCwwhTorFXa4YBTE2QKji5q2YDogiFEz9SQnu4peXoX5Rdg0/XJYqd1kdBXSIjtApctEVqqB7VEU1RNETekMf6NN6tt6tL+t7tjVnZZkD9Gdy+R8RP67G</latexit>

✏xx<latexit sha1_base64="eYvBlC4raievpFEOTKGrUQFpoXI=">AAAB+HicbZBNS8NAEIYn9avWj0Y9elksgqeSiKDHohePFWwrtKFstpN26eaD3Y20hv4SLx4U8epP8ea/cdPmoK0vLDy8M8PMvn4iuNKO822V1tY3NrfK25Wd3b39qn1w2FZxKhm2WCxi+eBThYJH2NJcC3xIJNLQF9jxxzd5vfOIUvE4utfTBL2QDiMecEa1sfp2tYeJ4sJgNpnMKpW+XXPqzlxkFdwCalCo2be/eoOYpSFGmgmqVNd1Eu1lVGrOBM4qvVRhQtmYDrFrMKIhKi+bHz4jp8YZkCCW5kWazN3fExkNlZqGvukMqR6p5Vpu/lfrpjq48jIeJanGiC0WBakgOiZ5CmTAJTItpgYok9zcStiISsq0ySoPwV3+8iq0z+uu4buLWuO6iKMMx3ACZ+DCJTTgFprQAgYpPMMrvFlP1ov1bn0sWktWMXMEf2R9/gB5zpLy</latexit><latexit sha1_base64="eYvBlC4raievpFEOTKGrUQFpoXI=">AAAB+HicbZBNS8NAEIYn9avWj0Y9elksgqeSiKDHohePFWwrtKFstpN26eaD3Y20hv4SLx4U8epP8ea/cdPmoK0vLDy8M8PMvn4iuNKO822V1tY3NrfK25Wd3b39qn1w2FZxKhm2WCxi+eBThYJH2NJcC3xIJNLQF9jxxzd5vfOIUvE4utfTBL2QDiMecEa1sfp2tYeJ4sJgNpnMKpW+XXPqzlxkFdwCalCo2be/eoOYpSFGmgmqVNd1Eu1lVGrOBM4qvVRhQtmYDrFrMKIhKi+bHz4jp8YZkCCW5kWazN3fExkNlZqGvukMqR6p5Vpu/lfrpjq48jIeJanGiC0WBakgOiZ5CmTAJTItpgYok9zcStiISsq0ySoPwV3+8iq0z+uu4buLWuO6iKMMx3ACZ+DCJTTgFprQAgYpPMMrvFlP1ov1bn0sWktWMXMEf2R9/gB5zpLy</latexit><latexit sha1_base64="eYvBlC4raievpFEOTKGrUQFpoXI=">AAAB+HicbZBNS8NAEIYn9avWj0Y9elksgqeSiKDHohePFWwrtKFstpN26eaD3Y20hv4SLx4U8epP8ea/cdPmoK0vLDy8M8PMvn4iuNKO822V1tY3NrfK25Wd3b39qn1w2FZxKhm2WCxi+eBThYJH2NJcC3xIJNLQF9jxxzd5vfOIUvE4utfTBL2QDiMecEa1sfp2tYeJ4sJgNpnMKpW+XXPqzlxkFdwCalCo2be/eoOYpSFGmgmqVNd1Eu1lVGrOBM4qvVRhQtmYDrFrMKIhKi+bHz4jp8YZkCCW5kWazN3fExkNlZqGvukMqR6p5Vpu/lfrpjq48jIeJanGiC0WBakgOiZ5CmTAJTItpgYok9zcStiISsq0ySoPwV3+8iq0z+uu4buLWuO6iKMMx3ACZ+DCJTTgFprQAgYpPMMrvFlP1ov1bn0sWktWMXMEf2R9/gB5zpLy</latexit><latexit sha1_base64="eYvBlC4raievpFEOTKGrUQFpoXI=">AAAB+HicbZBNS8NAEIYn9avWj0Y9elksgqeSiKDHohePFWwrtKFstpN26eaD3Y20hv4SLx4U8epP8ea/cdPmoK0vLDy8M8PMvn4iuNKO822V1tY3NrfK25Wd3b39qn1w2FZxKhm2WCxi+eBThYJH2NJcC3xIJNLQF9jxxzd5vfOIUvE4utfTBL2QDiMecEa1sfp2tYeJ4sJgNpnMKpW+XXPqzlxkFdwCalCo2be/eoOYpSFGmgmqVNd1Eu1lVGrOBM4qvVRhQtmYDrFrMKIhKi+bHz4jp8YZkCCW5kWazN3fExkNlZqGvukMqR6p5Vpu/lfrpjq48jIeJanGiC0WBakgOiZ5CmTAJTItpgYok9zcStiISsq0ySoPwV3+8iq0z+uu4buLWuO6iKMMx3ACZ+DCJTTgFprQAgYpPMMrvFlP1ov1bn0sWktWMXMEf2R9/gB5zpLy</latexit>

<latexit sha1_base64="CmZkoY1V/e4MJBNiFJ+QZZoG/94=">AAACLXicbZDJSgNBEIZ73I1b1KOXxiAIgTAjil6EoB48KpgFMiHUdCqmtWehuyYQgi/kxVcRwYMiXn0NO5kR3AoKvvqriu76g0RJQ6774kxNz8zOzS8sFpaWV1bXiusbdROnWmBNxCrWzQAMKhlhjSQpbCYaIQwUNoLb03G/MUBtZBxd0TDBdgjXkexJAWSlTvHMD0DzlB/b9PtAXPIyH2R4M1Y7OivQQtmWPvWR4EvLqk6x5FbcSfC/4OVQYnlcdIpPfjcWaYgRCQXGtDw3ofYINEmh8K7gpwYTELdwjS2LEYRo2qPJtXd8xypd3ou1zYj4RP2+MYLQmGEY2MkQqG9+98bif71WSr2j9khGSUoYieyhXqo4xXxsHe9KjYLU0AIILe1fueiDBkHW4II1wft98l+o71W8g4p7uV+qnuR2LLAtts12mccOWZWdswtWY4Lds0f2wl6dB+fZeXPes9EpJ9/ZZD/C+fgEmq6lZw==</latexit>

u = ui+ vj = ur er + u✓ e✓

Page 7: Elasticity in Polar Coordinates

• Use chain rule since displacements are functions of (r,𝜃)

• Compare to the strain transformation equation

• Strain-displacement equations in polar coordinates✏xx = ✏rr cos

2 ✓ � 2✏r✓ sin ✓ cos ✓ + ✏✓✓ sin2 ✓

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✏rr =@ur

@r

✏✓✓ =1

r

@u✓

@✓+

ur

r

2✏r✓ =1

r

@ur

@✓+

@u✓

@r� u✓

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Page 8: Elasticity in Polar Coordinates

Understanding Strain Displacement Equations

• Why is there showing up?

• Why is there a in ?

• Why is there a in ?

1

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ur

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�u✓

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Page 9: Elasticity in Polar Coordinates

Stresses

• Stresses in polar coordinates: �rr �✓✓ �r✓<latexit sha1_base64="dqTvo/XJO5Rx+is8gff0P1R9Kqw=">AAACLHicbZDLSgMxFIYzXmu9VV26CRbBVZkRQZeFIrisYC/QKcOZTNqGZi4mZ4Qy9IHc+CqCuLCIW5/DtB1FWw+EfPz/OSTn9xMpNNr2xFpZXVvf2CxsFbd3dvf2SweHTR2nivEGi2Ws2j5oLkXEGyhQ8naiOIS+5C1/WJv6rQeutIijOxwlvBtCPxI9wQCN5JVqrhb9ELxMqTF171MI6Lfi4oAj0Pk1psUFW/04XqlsV+xZ0WVwciiTvOpe6cUNYpaGPEImQeuOYyfYzUChYJKPi26qeQJsCH3eMRhByHU3my07pqdGCWgvVuZESGfq74kMQq1HoW86Q8CBXvSm4n9eJ8XeVTcTUZIij9j8oV4qKcZ0mhwNhOIM5cgAMCXMXykbgAKGJt+iCcFZXHkZmucVx/DtRbl6ncdRIMfkhJwRh1ySKrkhddIgjDySZ/JGJtaT9Wq9Wx/z1hUrnzkif8r6/AKNfqhc</latexit><latexit sha1_base64="dqTvo/XJO5Rx+is8gff0P1R9Kqw=">AAACLHicbZDLSgMxFIYzXmu9VV26CRbBVZkRQZeFIrisYC/QKcOZTNqGZi4mZ4Qy9IHc+CqCuLCIW5/DtB1FWw+EfPz/OSTn9xMpNNr2xFpZXVvf2CxsFbd3dvf2SweHTR2nivEGi2Ws2j5oLkXEGyhQ8naiOIS+5C1/WJv6rQeutIijOxwlvBtCPxI9wQCN5JVqrhb9ELxMqTF171MI6Lfi4oAj0Pk1psUFW/04XqlsV+xZ0WVwciiTvOpe6cUNYpaGPEImQeuOYyfYzUChYJKPi26qeQJsCH3eMRhByHU3my07pqdGCWgvVuZESGfq74kMQq1HoW86Q8CBXvSm4n9eJ8XeVTcTUZIij9j8oV4qKcZ0mhwNhOIM5cgAMCXMXykbgAKGJt+iCcFZXHkZmucVx/DtRbl6ncdRIMfkhJwRh1ySKrkhddIgjDySZ/JGJtaT9Wq9Wx/z1hUrnzkif8r6/AKNfqhc</latexit><latexit sha1_base64="dqTvo/XJO5Rx+is8gff0P1R9Kqw=">AAACLHicbZDLSgMxFIYzXmu9VV26CRbBVZkRQZeFIrisYC/QKcOZTNqGZi4mZ4Qy9IHc+CqCuLCIW5/DtB1FWw+EfPz/OSTn9xMpNNr2xFpZXVvf2CxsFbd3dvf2SweHTR2nivEGi2Ws2j5oLkXEGyhQ8naiOIS+5C1/WJv6rQeutIijOxwlvBtCPxI9wQCN5JVqrhb9ELxMqTF171MI6Lfi4oAj0Pk1psUFW/04XqlsV+xZ0WVwciiTvOpe6cUNYpaGPEImQeuOYyfYzUChYJKPi26qeQJsCH3eMRhByHU3my07pqdGCWgvVuZESGfq74kMQq1HoW86Q8CBXvSm4n9eJ8XeVTcTUZIij9j8oV4qKcZ0mhwNhOIM5cgAMCXMXykbgAKGJt+iCcFZXHkZmucVx/DtRbl6ncdRIMfkhJwRh1ySKrkhddIgjDySZ/JGJtaT9Wq9Wx/z1hUrnzkif8r6/AKNfqhc</latexit><latexit sha1_base64="dqTvo/XJO5Rx+is8gff0P1R9Kqw=">AAACLHicbZDLSgMxFIYzXmu9VV26CRbBVZkRQZeFIrisYC/QKcOZTNqGZi4mZ4Qy9IHc+CqCuLCIW5/DtB1FWw+EfPz/OSTn9xMpNNr2xFpZXVvf2CxsFbd3dvf2SweHTR2nivEGi2Ws2j5oLkXEGyhQ8naiOIS+5C1/WJv6rQeutIijOxwlvBtCPxI9wQCN5JVqrhb9ELxMqTF171MI6Lfi4oAj0Pk1psUFW/04XqlsV+xZ0WVwciiTvOpe6cUNYpaGPEImQeuOYyfYzUChYJKPi26qeQJsCH3eMRhByHU3my07pqdGCWgvVuZESGfq74kMQq1HoW86Q8CBXvSm4n9eJ8XeVTcTUZIij9j8oV4qKcZ0mhwNhOIM5cgAMCXMXykbgAKGJt+iCcFZXHkZmucVx/DtRbl6ncdRIMfkhJwRh1ySKrkhddIgjDySZ/JGJtaT9Wq9Wx/z1hUrnzkif8r6/AKNfqhc</latexit>

Page 10: Elasticity in Polar Coordinates

Equations of Equilibrium• Variation of stresses over a differential element

• Sum all the forces in the r-direction• Because of the curved box, some stresses will have components in the

unexpected directions

Page 11: Elasticity in Polar Coordinates

• Divide by volume and take a limit as

• Similarly take summation of forces in the 𝜃-direction• Equations of equilibrium in the polar coordinates

@�rr

@r+

1

r

@�r✓

@✓+

1

r(�rr � �✓✓) + br = 0

@�r✓

@r+

1

r

@�✓✓

@✓+

1

r2�r✓ + b✓ = 0

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Page 12: Elasticity in Polar Coordinates

2D Elasticity equation in polar coordinates

• 8 equations and 8 unknowns• Strain-displacement

• Constitutive

• Equilibrium

✏rr =@ur

@r

✏✓✓ =1

r

@u✓

@✓+

ur

r

2✏r✓ =1

r

@ur

@✓+

@u✓

@r� u✓

r<latexit sha1_base64="QvGUCLyK2ntwFe5HqR4WKWCeY0s=">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</latexit><latexit sha1_base64="QvGUCLyK2ntwFe5HqR4WKWCeY0s=">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</latexit><latexit sha1_base64="QvGUCLyK2ntwFe5HqR4WKWCeY0s=">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</latexit><latexit sha1_base64="QvGUCLyK2ntwFe5HqR4WKWCeY0s=">AAADEXicfVJNa9tAEF2paZMq/bDbYy9LTNpCqZFMIbkEQnrpMYE4CVjCrNajePHqg91RwAj9hVz6V3rpoaH02ltu+TdZyXLsOG0GhN7OvDdvNNowk0Kj695Y9pO1p8/WN547my9evnrdar850WmuOPR5KlN1FjINUiTQR4ESzjIFLA4lnIaTr1X99AKUFmlyjNMMgpidJyISnKFJDdvWBx8yLaTBhVIlfU/3qB8pxgs/YwoFkzQfqnJxMhzfd+hC5eMYkNHZa7mBVxYVebVZQ1yk5spPDbf2U2Xl0lvyUf+1eHzguequ/2OzmG6f6d0c87IqHWfY6rhdtw76EHgN6JAmDoeta3+U8jyGBLlkWg88N8OgqIy4hNLxcw0Z4xN2DgMDExaDDor6j5Z022RGNEqVeRKkdXZZUbBY62kcGmbMcKxXa1XyX7VBjtFuUIgkyxESPjOKckkxpdX1oCOhgKOcGsC4EmZWysfMrAPNJaqW4K1+8kNw0ut6Bh996ewfNOvYIO/IFvlIPLJD9sk3ckj6hFuX1g/rl3Vlf7d/2r/tPzOqbTWat+Re2H9vAc2z+8w=</latexit>

@�rr

@r+

1

r

@�r✓

@✓+

1

r(�rr � �✓✓) + br = 0

@�r✓

@r+

1

r

@�✓✓

@✓+

1

r2�r✓ + b✓ = 0

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8<

:

✏rr✏✓✓2✏r✓

9=

; =

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41/E �⌫/E 0�⌫/E 1/E 0

0 0 1/G

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Linear PDEs with varying coefficients

Linear PDEs with varying coefficients

Linear algebraic equations

Page 13: Elasticity in Polar Coordinates

Airy Stress Function

• Laplacian operator in polar coordinates

• Biharmonic equation in polar coordinates

• Stress in terms on Airy stress function in polar coordinates

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