Material Failure and Distortional Strain Energy

19
Material Failure and Distortional Strain Energy Structural Analysis: Failure Theories © 2021 Mayuresh Patil. Licensed under a Creative Commons Attribution 4.0 license https://creativecommons.org/licenses/by-nc-sa/4.0/ [email protected]

Transcript of Material Failure and Distortional Strain Energy

Page 1: Material Failure and Distortional Strain Energy

Material Failure and Distortional Strain Energy

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

https://creativecommons.org/licenses/by-nc-sa/4.0/
Page 2: Material Failure and Distortional Strain Energy

Yield Stress (Uniaxial Loading)

• Yield Stress: Uniaxial stress at material yield• After yield, the material undergoes plastic deformation

• Plastic deformation vs Elastic deformation• Elastic deformation is reversible (not dependent on load path)• Plastic deformation leads to residual strain even after the load is removed

Page 3: Material Failure and Distortional Strain Energy

Multi-axial loading

• Maximum shear stress (or Tresca) criteria• Maximum distortion strain energy (or von Mises or Octahedral shear

stress) criteria

Page 4: Material Failure and Distortional Strain Energy

Maximum Shear Stress (review)

• 3D Mohr circle

• 2D Mohr circle – still look at the 3D stresses

Normal stress

Shear stress

Normal stress

Shear stress

Normal stress

Shear stress

Normal stress

Shear stress

⌧max =�1 � �3

2<latexit sha1_base64="k+5mq4PoKh6NnS9jWHjeTybb2hw=">AAACGXicbZBNS8NAEIY3flu/qh69LBbBiyVRQS+K4MWjgm2FpoTJdlMXd5OwO5GWkL/hxb/ixYMiHvXkv3Fbc9DWFxYe3plhdt4wlcKg6345U9Mzs3PzC4uVpeWV1bXq+kbTJJlmvMESmeibEAyXIuYNFCj5Tao5qFDyVnh3Pqy37rk2IomvcZDyjoJeLCLBAK0VVF0fIQtyH3kfqYJ+QU+oH2lguW9ET0Hg0T1a4kGR7xdBtebW3ZHoJHgl1Eipy6D64XcTlikeI5NgTNtzU+zkoFEwyYuKnxmeAruDHm9bjEFx08lHlxV0xzpdGiXavhjpyP09kYMyZqBC26kAb814bWj+V2tnGB13chGnGfKY/SyKMkkxocOYaFdozlAOLADTwv6VsluwuaANs2JD8MZPnoTmft1z697VYe3stIxjgWyRbbJLPHJEzsgFuSQNwsgDeSIv5NV5dJ6dN+f9p3XKKWc2yR85n9+s36AL</latexit><latexit sha1_base64="k+5mq4PoKh6NnS9jWHjeTybb2hw=">AAACGXicbZBNS8NAEIY3flu/qh69LBbBiyVRQS+K4MWjgm2FpoTJdlMXd5OwO5GWkL/hxb/ixYMiHvXkv3Fbc9DWFxYe3plhdt4wlcKg6345U9Mzs3PzC4uVpeWV1bXq+kbTJJlmvMESmeibEAyXIuYNFCj5Tao5qFDyVnh3Pqy37rk2IomvcZDyjoJeLCLBAK0VVF0fIQtyH3kfqYJ+QU+oH2lguW9ET0Hg0T1a4kGR7xdBtebW3ZHoJHgl1Eipy6D64XcTlikeI5NgTNtzU+zkoFEwyYuKnxmeAruDHm9bjEFx08lHlxV0xzpdGiXavhjpyP09kYMyZqBC26kAb814bWj+V2tnGB13chGnGfKY/SyKMkkxocOYaFdozlAOLADTwv6VsluwuaANs2JD8MZPnoTmft1z697VYe3stIxjgWyRbbJLPHJEzsgFuSQNwsgDeSIv5NV5dJ6dN+f9p3XKKWc2yR85n9+s36AL</latexit><latexit sha1_base64="k+5mq4PoKh6NnS9jWHjeTybb2hw=">AAACGXicbZBNS8NAEIY3flu/qh69LBbBiyVRQS+K4MWjgm2FpoTJdlMXd5OwO5GWkL/hxb/ixYMiHvXkv3Fbc9DWFxYe3plhdt4wlcKg6345U9Mzs3PzC4uVpeWV1bXq+kbTJJlmvMESmeibEAyXIuYNFCj5Tao5qFDyVnh3Pqy37rk2IomvcZDyjoJeLCLBAK0VVF0fIQtyH3kfqYJ+QU+oH2lguW9ET0Hg0T1a4kGR7xdBtebW3ZHoJHgl1Eipy6D64XcTlikeI5NgTNtzU+zkoFEwyYuKnxmeAruDHm9bjEFx08lHlxV0xzpdGiXavhjpyP09kYMyZqBC26kAb814bWj+V2tnGB13chGnGfKY/SyKMkkxocOYaFdozlAOLADTwv6VsluwuaANs2JD8MZPnoTmft1z697VYe3stIxjgWyRbbJLPHJEzsgFuSQNwsgDeSIv5NV5dJ6dN+f9p3XKKWc2yR85n9+s36AL</latexit><latexit sha1_base64="k+5mq4PoKh6NnS9jWHjeTybb2hw=">AAACGXicbZBNS8NAEIY3flu/qh69LBbBiyVRQS+K4MWjgm2FpoTJdlMXd5OwO5GWkL/hxb/ixYMiHvXkv3Fbc9DWFxYe3plhdt4wlcKg6345U9Mzs3PzC4uVpeWV1bXq+kbTJJlmvMESmeibEAyXIuYNFCj5Tao5qFDyVnh3Pqy37rk2IomvcZDyjoJeLCLBAK0VVF0fIQtyH3kfqYJ+QU+oH2lguW9ET0Hg0T1a4kGR7xdBtebW3ZHoJHgl1Eipy6D64XcTlikeI5NgTNtzU+zkoFEwyYuKnxmeAruDHm9bjEFx08lHlxV0xzpdGiXavhjpyP09kYMyZqBC26kAb814bWj+V2tnGB13chGnGfKY/SyKMkkxocOYaFdozlAOLADTwv6VsluwuaANs2JD8MZPnoTmft1z697VYe3stIxjgWyRbbJLPHJEzsgFuSQNwsgDeSIv5NV5dJ6dN+f9p3XKKWc2yR85n9+s36AL</latexit>

Page 5: Material Failure and Distortional Strain Energy

Factor of Safety• Uniaxial tension tests are commonly used to calculate the (uniaxial)

yield stress at failure 𝜎y• Maximum shear stress at yield can be related to 𝜎y

• Factor of Safety:

• Margin of Safety:

⌧y =�y � 0

2=

1

2�y

<latexit sha1_base64="N01bh9dSqUSh4seMJaXSarzl+mU=">AAACH3icbVDLSsNAFJ3UV62vqEs3g0VwY0mqaDdCwY3LCvYBTSiT6aQdOpmEmYkQQv7Ejb/ixoUi4q5/4zSNoq0HLpw5517m3uNFjEplWVOjtLK6tr5R3qxsbe/s7pn7Bx0ZxgKTNg5ZKHoekoRRTtqKKkZ6kSAo8BjpepObmd99IELSkN+rJCJugEac+hQjpaWBeekoFA8SeA0dXyCcOpKOAqSFM2hlaT37Mez89W0PzKpVs3LAZWIXpAoKtAbmpzMMcRwQrjBDUvZtK1JuioSimJGs4sSSRAhP0Ij0NeUoINJN8/syeKKVIfRDoYsrmKu/J1IUSJkEnu4MkBrLRW8m/uf1Y+U33JTyKFaE4/lHfsygCuEsLDikgmDFEk0QFlTvCvEY6TiUjrSiQ7AXT14mnXrNPq/V7y6qzUYRRxkcgWNwCmxwBZrgFrRAG2DwCJ7BK3gznowX4934mLeWjGLmEPyBMf0CVL2h5Q==</latexit>

�1 = �y �2 = 0 �3 = 0<latexit sha1_base64="fNIkSpPG7/yfbDy2xQJgFbiufM4=">AAACI3icbZBNS8MwGMdTX+d8q3r0EhyCp9FugkMQBl48TnAvsJWSpukWlrQ1SYVS9l28+FW8eFCGFw9+F7Ougtt8IPDP7/88JM/fixmVyrK+jLX1jc2t7dJOeXdv/+DQPDruyCgRmLRxxCLR85AkjIakrahipBcLgrjHSNcb38787hMRkkbhg0pj4nA0DGlAMVIaueb1QNIhR64Nb2AhUzh4TJD/e61px1pE9Ry5ZsWqWnnBVWEXogKKarnmdOBHOOEkVJghKfu2FSsnQ0JRzMikPEgkiREeoyHpaxkiTqST5TtO4LkmPgwioU+oYE7/TmSIS5lyT3dypEZy2ZvB/7x+ooKGk9EwThQJ8fyhIGFQRXAWGPSpIFixVAuEBdV/hXiEBMJKx1rWIdjLK6+KTq1q16u1+8tKs1HEUQKn4AxcABtcgSa4Ay3QBhg8g1fwDj6MF+PNmBqf89Y1o5g5AQtlfP8AlrKhrQ==</latexit>

FS =⌧y

⌧max=

�y

2⌧max

<latexit sha1_base64="PWUuk5eWXzGYuk+Te39yq4HJWz0=">AAACLnicbZDLSgMxFIYz3q23qks3wSK4KjOlohtBEMVlRVuFTiln0kwbmswMyRmxDPNEbnwVXQgq4tbHML0s6uVAyM//nUNy/iCRwqDrvjozs3PzC4tLy4WV1bX1jeLmVsPEqWa8zmIZ69sADJci4nUUKPltojmoQPKboH865Dd3XBsRR9c4SHhLQTcSoWCA1moXz3zk95idX+X0mPqhBpb5CGl7kI/vTMF9PsWM6CoY0gqd4u1iyS27o6J/hTcRJTKpWrv47HdilioeIZNgTNNzE2xloFEwyfOCnxqeAOtDlzetjEBx08pG6+Z0zzodGsbangjpyJ2eyEAZM1CB7VSAPfObDc3/WDPF8KiViShJkUds/FCYSooxHWZHO0JzhnJgBTAt7F8p64HNBW3CBRuC93vlv6JRKXvV8sFltXRyNIljieyQXbJPPHJITsgFqZE6YeSBPJE38u48Oi/Oh/M5bp1xJjPb5Ec5X98qN6py</latexit>

MS = FS� 1<latexit sha1_base64="vpyi99BjraPHRfUr4Uv4AalAMmU=">AAACAnicbZDLSgMxFIYz9VbrbdSVuAkWwY1lpgp2IxQEcSNUtBdoh5JJ0zY0kxmSM2IZihtfxY0LRdz6FO58G9N2Ftr6Q+DjP+dwcn4/ElyD43xbmYXFpeWV7GpubX1jc8ve3qnpMFaUVWkoQtXwiWaCS1YFDoI1IsVI4AtW9wcX43r9ninNQ3kHw4h5AelJ3uWUgLHa9l4L2AMk17cjfI6nfGn42MVtO+8UnInwPLgp5FGqStv+anVCGgdMAhVE66brROAlRAGngo1yrViziNAB6bGmQUkCpr1kcsIIHxqng7uhMk8Cnri/JxISaD0MfNMZEOjr2drY/K/WjKFb8hIuoxiYpNNF3VhgCPE4D9zhilEQQwOEKm7+immfKELBpJYzIbizJ89DrVhwTwrFm9N8uZTGkUX76AAdIRedoTK6QhVURRQ9omf0it6sJ+vFerc+pq0ZK53ZRX9kff4AexqWJQ==</latexit>

Page 6: Material Failure and Distortional Strain Energy

Energy: Spring example• Linear spring

• Work done• Force applied very slowly

• Potential energy

k

F

u

F = ku

u =

✓1

k

◆F

U =1

2ku2

W =

Z u

0fdu =

Z u

0kudu =

1

2ku2

F

u

k: stiffness

1/k: flexibility

Page 7: Material Failure and Distortional Strain Energy

Strain Energy• The work done to deform a structure is stored as strain energy in the

solid• At any given point in the structure, the strain energy per unit volume

is called the strain energy density and given by

U =1

2(�xx✏xx + �yy✏yy + �zz✏zz + 2 �yz✏yz + 2 �xz✏xz + 2 �xy✏xy)

<latexit sha1_base64="PQhzLjKlmLjDlX+wcVOFAaI0uo8=">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</latexit><latexit sha1_base64="PQhzLjKlmLjDlX+wcVOFAaI0uo8=">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</latexit><latexit sha1_base64="PQhzLjKlmLjDlX+wcVOFAaI0uo8=">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</latexit><latexit sha1_base64="PQhzLjKlmLjDlX+wcVOFAaI0uo8=">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</latexit>

<latexit sha1_base64="R0YMtPuKo68zZQ9mKz9UEKxBITs=">AAAB/nicbVBNS8NAFHzxs9avqHjyslgETyURxV6EghePFUxbaELZbDbt0s0m7G6EEgr+FS8eFPHq7/Dmv3Hb5qCtAwvDzHu82QkzzpR2nG9rZXVtfWOzslXd3tnd27cPDtsqzSWhHkl5KrshVpQzQT3NNKfdTFKchJx2wtHt1O88UqlYKh70OKNBggeCxYxgbaS+feyhG+QzoVHhE8yRN0FRG/XtmlN3ZkDLxC1JDUq0+vaXH6UkT6jQhGOleq6T6aDAUjPC6aTq54pmmIzwgPYMFTihKihm8SfozCgRilNpngkyU39vFDhRapyEZjLBeqgWvan4n9fLddwICiayXFNB5ofinCOdomkXKGKSEs3HhmAimcmKyBBLTLRprGpKcBe/vEzaF3X3qu7cX9aajbKOCpzAKZyDC9fQhDtogQcECniGV3iznqwX6936mI+uWOXOEfyB9fkDuEeUAw==</latexit>

U =

ZUdV

Page 8: Material Failure and Distortional Strain Energy

Strain Decomposition

• Strain decomposition into volumetric and distortional (shape)

• Volumetric strain is given by

2

4✏xx ✏xy ✏xz✏xy ✏yy ✏yz✏xz ✏yz ✏zz

3

5 =

2

4✏m 0 00 ✏m 00 0 ✏m

3

5+

2

4✏xx � ✏m ✏xy ✏xz

✏xy ✏yy � ✏m ✏yz✏xz ✏yz ✏zz � ✏m

3

5

<latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">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</latexit><latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">AAAD3XicpVNNa9swGFbsfbTeR9P12ItY2BiMBbsUlkuhsEuPLTRpaRyCrLxJRWTZSHKJYwK77LBRet3/2q2/o3+gcmJoonS99AXB87zP+6UXKUo5U9r3b2uO++Llq9cbm96bt+/eb9W3P3RUkkkKbZrwRJ5HRAFnAtqaaQ7nqQQSRxzOovGPUj+7AqlYIk51nkIvJiPBhowSbVz97dpdGMGIiSKKiZZsMsNeCKli3IjFxNDPeInnFp/OcBh6/w/Ibb6WMF0PWObT6cxEi8HDeAf4iYHjMtufH9PGXylVaZXf1qwu+OtTbcq9fLNLP3NNq/WeuzSrnHW7fr3hN/254XUQVKCBKjvu1/+Fg4RmMQhNOVGqG/ip7hVEakY5mPKZgpTQMRlB10BBYlC9Yv46Z/iT8QzwMJHmCI3n3uWMgsRK5XFkIs2Al8rWSudjWjfTw1avYCLNNAi6aDTMONYJLp86HjAJVPPcAEIlM7Niekkkodp8CM8sIbCvvA46e83A4JP9xmGrWscG2kUf0RcUoO/oEB2hY9RG1Llwfjq/nT9u3/3lXrs3i1CnVuXsoBVz/94Dn8dFDg==</latexit><latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">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</latexit><latexit sha1_base64="ck8pdC+ekZH4nUmSP+ZG7r8lEyk=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odOn4MoA7ncAFXEMIN3MEDdKALAhJ4hXdv4r15H6uuat66tDP4I+/zBzjGijg=</latexit><latexit sha1_base64="Gk86LCOeYQ8f6sZXggfm6DIiRas=">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</latexit><latexit sha1_base64="Gk86LCOeYQ8f6sZXggfm6DIiRas=">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</latexit><latexit sha1_base64="RjACmTg4AICFMo1zDUt13cYcNtg=">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</latexit><latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">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</latexit><latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">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</latexit><latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">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</latexit><latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">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</latexit><latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">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</latexit><latexit sha1_base64="P/bnKlwr+m4inoKONRnq480JhsI=">AAAD3XicpVNNa9swGFbsfbTeR9P12ItY2BiMBbsUlkuhsEuPLTRpaRyCrLxJRWTZSHKJYwK77LBRet3/2q2/o3+gcmJoonS99AXB87zP+6UXKUo5U9r3b2uO++Llq9cbm96bt+/eb9W3P3RUkkkKbZrwRJ5HRAFnAtqaaQ7nqQQSRxzOovGPUj+7AqlYIk51nkIvJiPBhowSbVz97dpdGMGIiSKKiZZsMsNeCKli3IjFxNDPeInnFp/OcBh6/w/Ibb6WMF0PWObT6cxEi8HDeAf4iYHjMtufH9PGXylVaZXf1qwu+OtTbcq9fLNLP3NNq/WeuzSrnHW7fr3hN/254XUQVKCBKjvu1/+Fg4RmMQhNOVGqG/ip7hVEakY5mPKZgpTQMRlB10BBYlC9Yv46Z/iT8QzwMJHmCI3n3uWMgsRK5XFkIs2Al8rWSudjWjfTw1avYCLNNAi6aDTMONYJLp86HjAJVPPcAEIlM7Niekkkodp8CM8sIbCvvA46e83A4JP9xmGrWscG2kUf0RcUoO/oEB2hY9RG1Llwfjq/nT9u3/3lXrs3i1CnVuXsoBVz/94Dn8dFDg==</latexit>

✏m =✏xx + ✏yy + ✏zz

3<latexit sha1_base64="7ZYr91ZPAD+qZDY0CWdevv1CGOA=">AAACLnicbZBLSwMxEMez9VXra9Wjl2ARBKHsqmAvQkEEjxXsA9qlZNNsG5p9kGSl2yWfyItfRQ+Cinj1Y5i2C62tA4Hf/GeGyfzdiFEhLevdyK2srq1v5DcLW9s7u3vm/kFdhDHHpIZDFvKmiwRhNCA1SSUjzYgT5LuMNNzBzbjeeCRc0DB4kElEHB/1AupRjKSWOuZtm0SCMo2pr+A1bHsc4XQmDocKnsFZniRqPh2NlEovVMcsWiVrEnAZ7AyKIItqx3xtd0Mc+ySQmCEhWrYVSSdFXFLMiCq0Y0EihAeoR1oaA+QT4aSTcxU80UoXeiHXL5Bwos5PpMgXIvFd3ekj2ReLtbH4X60VS6/spDSIYkkCPF3kxQzKEI69g13KCZYs0YAwp/qvEPeRNkxqhwvaBHvx5GWon5dszfeXxUo5syMPjsAxOAU2uAIVcAeqoAYweAIv4AN8Gs/Gm/FlfE9bc0Y2cwj+hPHzC3KYqpg=</latexit><latexit sha1_base64="7ZYr91ZPAD+qZDY0CWdevv1CGOA=">AAACLnicbZBLSwMxEMez9VXra9Wjl2ARBKHsqmAvQkEEjxXsA9qlZNNsG5p9kGSl2yWfyItfRQ+Cinj1Y5i2C62tA4Hf/GeGyfzdiFEhLevdyK2srq1v5DcLW9s7u3vm/kFdhDHHpIZDFvKmiwRhNCA1SSUjzYgT5LuMNNzBzbjeeCRc0DB4kElEHB/1AupRjKSWOuZtm0SCMo2pr+A1bHsc4XQmDocKnsFZniRqPh2NlEovVMcsWiVrEnAZ7AyKIItqx3xtd0Mc+ySQmCEhWrYVSSdFXFLMiCq0Y0EihAeoR1oaA+QT4aSTcxU80UoXeiHXL5Bwos5PpMgXIvFd3ekj2ReLtbH4X60VS6/spDSIYkkCPF3kxQzKEI69g13KCZYs0YAwp/qvEPeRNkxqhwvaBHvx5GWon5dszfeXxUo5syMPjsAxOAU2uAIVcAeqoAYweAIv4AN8Gs/Gm/FlfE9bc0Y2cwj+hPHzC3KYqpg=</latexit><latexit sha1_base64="7ZYr91ZPAD+qZDY0CWdevv1CGOA=">AAACLnicbZBLSwMxEMez9VXra9Wjl2ARBKHsqmAvQkEEjxXsA9qlZNNsG5p9kGSl2yWfyItfRQ+Cinj1Y5i2C62tA4Hf/GeGyfzdiFEhLevdyK2srq1v5DcLW9s7u3vm/kFdhDHHpIZDFvKmiwRhNCA1SSUjzYgT5LuMNNzBzbjeeCRc0DB4kElEHB/1AupRjKSWOuZtm0SCMo2pr+A1bHsc4XQmDocKnsFZniRqPh2NlEovVMcsWiVrEnAZ7AyKIItqx3xtd0Mc+ySQmCEhWrYVSSdFXFLMiCq0Y0EihAeoR1oaA+QT4aSTcxU80UoXeiHXL5Bwos5PpMgXIvFd3ekj2ReLtbH4X60VS6/spDSIYkkCPF3kxQzKEI69g13KCZYs0YAwp/qvEPeRNkxqhwvaBHvx5GWon5dszfeXxUo5syMPjsAxOAU2uAIVcAeqoAYweAIv4AN8Gs/Gm/FlfE9bc0Y2cwj+hPHzC3KYqpg=</latexit><latexit sha1_base64="7ZYr91ZPAD+qZDY0CWdevv1CGOA=">AAACLnicbZBLSwMxEMez9VXra9Wjl2ARBKHsqmAvQkEEjxXsA9qlZNNsG5p9kGSl2yWfyItfRQ+Cinj1Y5i2C62tA4Hf/GeGyfzdiFEhLevdyK2srq1v5DcLW9s7u3vm/kFdhDHHpIZDFvKmiwRhNCA1SSUjzYgT5LuMNNzBzbjeeCRc0DB4kElEHB/1AupRjKSWOuZtm0SCMo2pr+A1bHsc4XQmDocKnsFZniRqPh2NlEovVMcsWiVrEnAZ7AyKIItqx3xtd0Mc+ySQmCEhWrYVSSdFXFLMiCq0Y0EihAeoR1oaA+QT4aSTcxU80UoXeiHXL5Bwos5PpMgXIvFd3ekj2ReLtbH4X60VS6/spDSIYkkCPF3kxQzKEI69g13KCZYs0YAwp/qvEPeRNkxqhwvaBHvx5GWon5dszfeXxUo5syMPjsAxOAU2uAIVcAeqoAYweAIv4AN8Gs/Gm/FlfE9bc0Y2cwj+hPHzC3KYqpg=</latexit>

Page 9: Material Failure and Distortional Strain Energy

Stress Decomposition

• Stress decomposition into hydrostatic and distortional

• Hydrostatic stress is given by

2

4�xx �xy �xz

�xy �yy �yz

�xz �yz �zz

3

5 =

2

4�m 0 00 �m 00 0 �m

3

5+

2

4�xx � �m �xy �xz

�xy �yy � �m �yz

�xz �yz �zz � �m

3

5

<latexit sha1_base64="iUu5XgwJuiYmuvcyEVfKTlu+buc=">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</latexit><latexit sha1_base64="iUu5XgwJuiYmuvcyEVfKTlu+buc=">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</latexit><latexit sha1_base64="iUu5XgwJuiYmuvcyEVfKTlu+buc=">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</latexit><latexit sha1_base64="iUu5XgwJuiYmuvcyEVfKTlu+buc=">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</latexit>

�m =�xx + �yy + �zz

3<latexit sha1_base64="v76MqR2TyQKaG8JwPtNxTGTChBU=">AAACJnicbVDLSsNAFJ3UV62vqEs3g0UQhJKoYDeFghuXFewD2hIm00k7dCYJMxNpGvI1bvwVNy4qIu78FKdtitp6YODcc+7lzj1uyKhUlvVp5NbWNza38tuFnd29/QPz8Kghg0hgUscBC0TLRZIw6pO6ooqRVigI4i4jTXd4O/Wbj0RIGvgPKg5Jl6O+Tz2KkdKSY1Y6kvY5chKewgrseALhZCGNRim8gIsqjtOfYjxO0+QqdcyiVbJmgKvEzkgRZKg55qTTC3DEia8wQ1K2bStU3QQJRTEjaaETSRIiPER90tbUR5zIbjI7M4VnWulBLxD6+QrO1N8TCeJSxtzVnRypgVz2puJ/XjtSXrmbUD+MFPHxfJEXMagCOM0M9qggWLFYE4QF1X+FeIB0VEonW9Ah2Msnr5LGZcnW/P66WC1nceTBCTgF58AGN6AK7kAN1AEGT+AFTMCb8Wy8Gu/Gx7w1Z2Qzx+APjK9v6P6mpA==</latexit><latexit sha1_base64="v76MqR2TyQKaG8JwPtNxTGTChBU=">AAACJnicbVDLSsNAFJ3UV62vqEs3g0UQhJKoYDeFghuXFewD2hIm00k7dCYJMxNpGvI1bvwVNy4qIu78FKdtitp6YODcc+7lzj1uyKhUlvVp5NbWNza38tuFnd29/QPz8Kghg0hgUscBC0TLRZIw6pO6ooqRVigI4i4jTXd4O/Wbj0RIGvgPKg5Jl6O+Tz2KkdKSY1Y6kvY5chKewgrseALhZCGNRim8gIsqjtOfYjxO0+QqdcyiVbJmgKvEzkgRZKg55qTTC3DEia8wQ1K2bStU3QQJRTEjaaETSRIiPER90tbUR5zIbjI7M4VnWulBLxD6+QrO1N8TCeJSxtzVnRypgVz2puJ/XjtSXrmbUD+MFPHxfJEXMagCOM0M9qggWLFYE4QF1X+FeIB0VEonW9Ah2Msnr5LGZcnW/P66WC1nceTBCTgF58AGN6AK7kAN1AEGT+AFTMCb8Wy8Gu/Gx7w1Z2Qzx+APjK9v6P6mpA==</latexit><latexit sha1_base64="v76MqR2TyQKaG8JwPtNxTGTChBU=">AAACJnicbVDLSsNAFJ3UV62vqEs3g0UQhJKoYDeFghuXFewD2hIm00k7dCYJMxNpGvI1bvwVNy4qIu78FKdtitp6YODcc+7lzj1uyKhUlvVp5NbWNza38tuFnd29/QPz8Kghg0hgUscBC0TLRZIw6pO6ooqRVigI4i4jTXd4O/Wbj0RIGvgPKg5Jl6O+Tz2KkdKSY1Y6kvY5chKewgrseALhZCGNRim8gIsqjtOfYjxO0+QqdcyiVbJmgKvEzkgRZKg55qTTC3DEia8wQ1K2bStU3QQJRTEjaaETSRIiPER90tbUR5zIbjI7M4VnWulBLxD6+QrO1N8TCeJSxtzVnRypgVz2puJ/XjtSXrmbUD+MFPHxfJEXMagCOM0M9qggWLFYE4QF1X+FeIB0VEonW9Ah2Msnr5LGZcnW/P66WC1nceTBCTgF58AGN6AK7kAN1AEGT+AFTMCb8Wy8Gu/Gx7w1Z2Qzx+APjK9v6P6mpA==</latexit><latexit sha1_base64="v76MqR2TyQKaG8JwPtNxTGTChBU=">AAACJnicbVDLSsNAFJ3UV62vqEs3g0UQhJKoYDeFghuXFewD2hIm00k7dCYJMxNpGvI1bvwVNy4qIu78FKdtitp6YODcc+7lzj1uyKhUlvVp5NbWNza38tuFnd29/QPz8Kghg0hgUscBC0TLRZIw6pO6ooqRVigI4i4jTXd4O/Wbj0RIGvgPKg5Jl6O+Tz2KkdKSY1Y6kvY5chKewgrseALhZCGNRim8gIsqjtOfYjxO0+QqdcyiVbJmgKvEzkgRZKg55qTTC3DEia8wQ1K2bStU3QQJRTEjaaETSRIiPER90tbUR5zIbjI7M4VnWulBLxD6+QrO1N8TCeJSxtzVnRypgVz2puJ/XjtSXrmbUD+MFPHxfJEXMagCOM0M9qggWLFYE4QF1X+FeIB0VEonW9Ah2Msnr5LGZcnW/P66WC1nceTBCTgF58AGN6AK7kAN1AEGT+AFTMCb8Wy8Gu/Gx7w1Z2Qzx+APjK9v6P6mpA==</latexit>

Page 10: Material Failure and Distortional Strain Energy

Strain Energy Density• Strain energy density is given by:

which can be written as

U =1

2(�xx✏xx + �yy✏yy + �zz✏zz + 2 �yz✏yz + 2 �xz✏xz + 2 �xy✏xy)

<latexit sha1_base64="PQhzLjKlmLjDlX+wcVOFAaI0uo8=">AAACtnicbZHfa9swEMdlt127rF2z7nEvoqGQUih2WmgYFAp72WMHS5MSZ0FWzo6ILBvpXOIa/4l92Vv/myo/WPNjB4Lv9z46TroLMykMet6r4+7s7n3YP/hY+3R49Pm4/uXkwaS55tDhqUx1L2QGpFDQQYESepkGloQSuuHkx4x3n0AbkarfWGQwSFisRCQ4Q5sa1l/KgDNJOxW9pUGkGS/9qmxVgYQIm4ERccLosJxOqwAyI2SqFu7iHyqKVWTdO3p+XkXWXbSC7++Fa7TYoNM1Ot2ka01nTot4jOfDesO79OZBt4W/FA2yjPth/W8wSnmegEIumTF938twUDKNgkuoakFuIGN8wmLoW6lYAmZQzsde0TObGdEo1fYopPPsakXJEmOKJLQ3E4Zjs8lmyf+xfo5Re1AKleUIii8aRbmkmNLZDulIaOAoCysY18K+lfIxs8tDu+maHYK/+eVt8dC69K3+dd24ay/HcUC+kVPSJD65IXfkJ7knHcKdK+fRCR3utt0/Lrjx4qrrLGu+krVwszf8ntrs</latexit><latexit sha1_base64="PQhzLjKlmLjDlX+wcVOFAaI0uo8=">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</latexit><latexit sha1_base64="PQhzLjKlmLjDlX+wcVOFAaI0uo8=">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</latexit><latexit sha1_base64="PQhzLjKlmLjDlX+wcVOFAaI0uo8=">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</latexit>

U =1

2

0

BBBBBB@

�xx

�yy

�zz

�yz

�xz

�xy

1

CCCCCCA

T 0

BBBBBB@

✏xx✏yy✏zz2✏yz2✏xz2✏xy

1

CCCCCCA

<latexit sha1_base64="gYod5aOZ7/ZqB7FHRJ8i6Dmsn7M=">AAADBnicbVJBT9swFHYytrGMjTKO28FaNYlTlRTE4DAJaZcdmUQBqekqx30pFo4T2Q5qsHLahb/CZYehadf9Bm78mzltiCjhSZa+73vP/p7tF2WcKe37d477bOX5i5err7zXa2/ernc23h2rNJcUBjTlqTyNiALOBAw00xxOMwkkiTicROdfq/zJBUjFUnGkiwxGCZkKFjNKtJXGG84HE1LC8aDEX3AYS0JNUJp+icMIpkyYLCFaslmJvVCxaULGZmZJGDa0KJbo5eVydpnOHtGiOhfEpLH5cWSFljNkinHbbePdCPfujVD79/GDmrY0e0Jq9YI9b9zp+j1/HrgNghp0UR2H485tOElpnoDQlBOlhoGf6ZEhUjPKofTCXEFG6DmZwtBCQRJQIzP/xhJ/ssoEx6m0S2g8Vx/uMCRRqkgiW2lbPFOPc5X4VG6Y63hvZJjIcg2CLozinGOd4mom8IRJoJoXFhAqme0V0zNiZ0HbyVk8wn4Vu82V2+C43wu2e9vfd7oHe/VzrKL36CPaQgH6jA7QN3SIBog6P51r57dz4165v9w/7t9FqevUezbRUrj//gMfsvdB</latexit>

Page 11: Material Failure and Distortional Strain Energy

• The last two terms are zero, thus we get

U =1

2

0

BBBBBB@

�xx

�yy

�zz

�yz

�xz

�xy

1

CCCCCCA

T 0

BBBBBB@

✏xx✏yy✏zz2✏yz2✏xz2✏xy

1

CCCCCCA=

1

2

2

6666664

0

BBBBBB@

�m

�m

�m

000

1

CCCCCCA

T

+

0

BBBBBB@

�xx � �m

�yy � �m

�zz � �m

�yz

�xz

�xy

1

CCCCCCA

T3

7777775

2

6666664

0

BBBBBB@

✏m✏m✏m000

1

CCCCCCA+

0

BBBBBB@

✏xx � ✏m✏yy � ✏m✏zz � ✏m2✏yz2✏xz2✏xy

1

CCCCCCA

3

7777775

=1

2

0

BBBBBB@

�m

�m

�m

000

1

CCCCCCA

T 0

BBBBBB@

✏m✏m✏m000

1

CCCCCCA+

1

2

0

BBBBBB@

�xx � �m

�yy � �m

�zz � �m

�yz

�xz

�xy

1

CCCCCCA

T 0

BBBBBB@

✏xx � ✏m✏yy � ✏m✏zz � ✏m2✏yz2✏xz2✏xy

1

CCCCCCA+

1

2

0

BBBBBB@

�m

�m

�m

000

1

CCCCCCA

T 0

BBBBBB@

✏xx � ✏m✏yy � ✏m✏zz � ✏m2✏yz2✏xz2✏xy

1

CCCCCCA+

1

2

0

BBBBBB@

�xx � �m

�yy � �m

�zz � �m

�yz

�xz

�xy

1

CCCCCCA

T 0

BBBBBB@

✏m✏m✏m000

1

CCCCCCA

<latexit sha1_base64="+Rrz3xd8Mu0ArYX6DLSBrwxifoY=">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</latexit>

Strain Energy Density Decomposition

U =1

2

0

BBBBBB@

�m

�m

�m

000

1

CCCCCCA

T 0

BBBBBB@

✏m✏m✏m000

1

CCCCCCA+

1

2

0

BBBBBB@

�xx � �m

�yy � �m

�zz � �m

�yz

�xz

�xy

1

CCCCCCA

T 0

BBBBBB@

✏xx � ✏m✏yy � ✏m✏zz � ✏m2✏yz2✏xz2✏xy

1

CCCCCCA= Uv + Ud

<latexit sha1_base64="Gr0Mhmqll2udFEq9BpDeJZ3S3Gg=">AAAEM3icjVPPb9MwFPYSYCP8WAdHLhYVEhKiSjrExgFpEhe4DWndJjUlchyns2Y7ke1MTa38T1z4RzggIQ4gxJX/AbfNSrq2gycleu/77OfPn57jnFGlff/rhuPeuHlrc+u2d+fuvfvbrZ0HxyorJCY9nLFMnsZIEUYF6WmqGTnNJUE8ZuQkPn8z4U8uiFQ0E0e6zMmAo6GgKcVIWyjacd6ZECMGexV8DcNUImyCynQrGMZkSIXJOdKSjirohYoOOYoMt1x4beU3fl5IRDLv8uEIWmSpM8kVZVbNZbd/1Ov7Qw8++79bjGzxHP7V3bhEWa6lxuP1u8YLToyulGW1woprnKj1rXWiFtngF+ha6CLdbQC13CY0WgEtybZTcjkw0YU1e14kXtRq+x1/GnA5CeqkDeo4jFqfwyTDBSdCY4aU6gd+rgcGSU0xI5UXForkCJ+jIenbVCBO1MBMZ76CTyySwDST9hMaTtHmDoO4UiWP7Uor/Uxd5SbgKq5f6HR/YKjIC00Enh2UFgzqDE4eEEyoJFiz0iYIS2q1QnyG7Mhp+8xmJryaxMv5lZeT424n2O3svn/RPtiv7dgCj8Bj8BQEYA8cgLfgEPQAdj46X5zvzg/3k/vN/en+mi11Nuo9D8FCuL//AIVBY54=</latexit>

Page 12: Material Failure and Distortional Strain Energy

Distortion Strain Energy Density Criteria• Distortion strain energy is a more accurate indicator of internal

deformation that may cause failure in ductile material• Uniaxial tension tests give the yield stress at failure 𝜎y• Distortional strain energy density at yield is

• Factor of Safety:

Udy =1

2

0

BBBBBB@

�y � �y

30� �y

30� �y

3000

1

CCCCCCA

T 0

BBBBBB@

�y

E � 1�2⌫3

�y

E�⌫ �y

E � 1�2⌫3

�y

E�⌫ �y

E � 1�2⌫3

�y

E000

1

CCCCCCA=

1 + ⌫

3E�2y

<latexit sha1_base64="59iVvlHPuxsQVi6UO+g3pphNpgg=">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</latexit><latexit sha1_base64="59iVvlHPuxsQVi6UO+g3pphNpgg=">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</latexit><latexit sha1_base64="59iVvlHPuxsQVi6UO+g3pphNpgg=">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</latexit><latexit sha1_base64="59iVvlHPuxsQVi6UO+g3pphNpgg=">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</latexit>

FS2 =Udy

Ud<latexit sha1_base64="fhJAgN7rYX5Ynmcj/pEQIk0rEso=">AAACGHicbZDLSgMxFIYzXmu9jbp0EyyCqzpTBLsRCoK4rGgv0BlLJs20oZkLyRmxDPMYbnwVNy4Ucdudb2PazsK2/hD48p9zSM7vxYIrsKwfY2V1bX1js7BV3N7Z3ds3Dw6bKkokZQ0aiUi2PaKY4CFrAAfB2rFkJPAEa3nD60m99cSk4lH4AKOYuQHph9znlIC2uua5A+wZ0pv77LGCr7DjS0LTNHUoEbiRdXtZd5TNXc2SVbamwstg51BCuepdc+z0IpoELAQqiFId24rBTYkETgXLik6iWEzokPRZR2NIAqbcdLpYhk+108N+JPUJAU/dvxMpCZQaBZ7uDAgM1GJtYv5X6yTgV92Uh3ECLKSzh/xEYIjwJCXc45JRECMNhEqu/4rpgOhsQGdZ1CHYiysvQ7NStjXfXZRq1TyOAjpGJ+gM2egS1dAtqqMGougFvaEP9Gm8Gu/Gl/E9a10x8pkjNCdj/AsW56Bm</latexit><latexit sha1_base64="fhJAgN7rYX5Ynmcj/pEQIk0rEso=">AAACGHicbZDLSgMxFIYzXmu9jbp0EyyCqzpTBLsRCoK4rGgv0BlLJs20oZkLyRmxDPMYbnwVNy4Ucdudb2PazsK2/hD48p9zSM7vxYIrsKwfY2V1bX1js7BV3N7Z3ds3Dw6bKkokZQ0aiUi2PaKY4CFrAAfB2rFkJPAEa3nD60m99cSk4lH4AKOYuQHph9znlIC2uua5A+wZ0pv77LGCr7DjS0LTNHUoEbiRdXtZd5TNXc2SVbamwstg51BCuepdc+z0IpoELAQqiFId24rBTYkETgXLik6iWEzokPRZR2NIAqbcdLpYhk+108N+JPUJAU/dvxMpCZQaBZ7uDAgM1GJtYv5X6yTgV92Uh3ECLKSzh/xEYIjwJCXc45JRECMNhEqu/4rpgOhsQGdZ1CHYiysvQ7NStjXfXZRq1TyOAjpGJ+gM2egS1dAtqqMGougFvaEP9Gm8Gu/Gl/E9a10x8pkjNCdj/AsW56Bm</latexit><latexit sha1_base64="fhJAgN7rYX5Ynmcj/pEQIk0rEso=">AAACGHicbZDLSgMxFIYzXmu9jbp0EyyCqzpTBLsRCoK4rGgv0BlLJs20oZkLyRmxDPMYbnwVNy4Ucdudb2PazsK2/hD48p9zSM7vxYIrsKwfY2V1bX1js7BV3N7Z3ds3Dw6bKkokZQ0aiUi2PaKY4CFrAAfB2rFkJPAEa3nD60m99cSk4lH4AKOYuQHph9znlIC2uua5A+wZ0pv77LGCr7DjS0LTNHUoEbiRdXtZd5TNXc2SVbamwstg51BCuepdc+z0IpoELAQqiFId24rBTYkETgXLik6iWEzokPRZR2NIAqbcdLpYhk+108N+JPUJAU/dvxMpCZQaBZ7uDAgM1GJtYv5X6yTgV92Uh3ECLKSzh/xEYIjwJCXc45JRECMNhEqu/4rpgOhsQGdZ1CHYiysvQ7NStjXfXZRq1TyOAjpGJ+gM2egS1dAtqqMGougFvaEP9Gm8Gu/Gl/E9a10x8pkjNCdj/AsW56Bm</latexit><latexit sha1_base64="fhJAgN7rYX5Ynmcj/pEQIk0rEso=">AAACGHicbZDLSgMxFIYzXmu9jbp0EyyCqzpTBLsRCoK4rGgv0BlLJs20oZkLyRmxDPMYbnwVNy4Ucdudb2PazsK2/hD48p9zSM7vxYIrsKwfY2V1bX1js7BV3N7Z3ds3Dw6bKkokZQ0aiUi2PaKY4CFrAAfB2rFkJPAEa3nD60m99cSk4lH4AKOYuQHph9znlIC2uua5A+wZ0pv77LGCr7DjS0LTNHUoEbiRdXtZd5TNXc2SVbamwstg51BCuepdc+z0IpoELAQqiFId24rBTYkETgXLik6iWEzokPRZR2NIAqbcdLpYhk+108N+JPUJAU/dvxMpCZQaBZ7uDAgM1GJtYv5X6yTgV92Uh3ECLKSzh/xEYIjwJCXc45JRECMNhEqu/4rpgOhsQGdZ1CHYiysvQ7NStjXfXZRq1TyOAjpGJ+gM2egS1dAtqqMGougFvaEP9Gm8Gu/Gl/E9a10x8pkjNCdj/AsW56Bm</latexit>

Page 13: Material Failure and Distortional Strain Energy

Review: Cauchy Formula for traction on a surface • Given the state of stress at a point we can calculate the traction on

any cut using the components of the tensor and unit vector in a common basis (axis)

• The normal and any shear component is given by

• The magnitude of shear stress on the surface is given by

T(n)

=

0

B@T (n)x

T (n)y

T (n)z

1

CA =

2

4�xx �xy �xz

�xy �yy �yz

�xz �yz �zz

3

5

0

@nx

ny

nz

1

A

<latexit sha1_base64="j1N5jljYiKe2SqQq9D0WZJaZRzM=">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</latexit><latexit sha1_base64="j1N5jljYiKe2SqQq9D0WZJaZRzM=">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</latexit><latexit sha1_base64="j1N5jljYiKe2SqQq9D0WZJaZRzM=">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</latexit><latexit sha1_base64="j1N5jljYiKe2SqQq9D0WZJaZRzM=">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</latexit>

�nn = T(n) · n̂

�nt = T(n) · t̂

<latexit sha1_base64="YZOTqGTxbG+4e4hKOuetRwh7Eio=">AAACTnicjVFJSwMxGM3Urdat6tFLsCj1UmZEsBeh4MVjhW7QqSWTpm1oJhmSb4QyzC/0It78GV48KKLpgmjrwQ8Cj7dkeQkiwQ247rOTWVldW9/Ibua2tnd29/L7Bw2jYk1ZnSqhdCsghgkuWR04CNaKNCNhIFgzGF1P9OY904YrWYNxxDohGUje55SApbp55hs+CEk3kTLFp/gK+8raJ7sltfQuKcqzFPu0pwD7QwKJNfl+7jsD/8lYUzdfcEvudPAy8OaggOZT7eaf/J6iccgkUEGMaXtuBJ2EaOBUsDTnx4ZFhI7IgLUtlCRkppNM60jxiWV6uK+0XRLwlP2ZSEhozDgMrDMkMDSL2oT8S2vH0C93Ei6jGJiks4P6scCg8KRb3OOaURBjCwjV3N4V0yHRhIL9gZwtwVt88jJonJc8i28vCpXyvI4sOkLHqIg8dIkq6AZVUR1R9IBe0Bt6dx6dV+fD+ZxZM848c4h+TSb7Be1/tDQ=</latexit><latexit sha1_base64="YZOTqGTxbG+4e4hKOuetRwh7Eio=">AAACTnicjVFJSwMxGM3Urdat6tFLsCj1UmZEsBeh4MVjhW7QqSWTpm1oJhmSb4QyzC/0It78GV48KKLpgmjrwQ8Cj7dkeQkiwQ247rOTWVldW9/Ibua2tnd29/L7Bw2jYk1ZnSqhdCsghgkuWR04CNaKNCNhIFgzGF1P9OY904YrWYNxxDohGUje55SApbp55hs+CEk3kTLFp/gK+8raJ7sltfQuKcqzFPu0pwD7QwKJNfl+7jsD/8lYUzdfcEvudPAy8OaggOZT7eaf/J6iccgkUEGMaXtuBJ2EaOBUsDTnx4ZFhI7IgLUtlCRkppNM60jxiWV6uK+0XRLwlP2ZSEhozDgMrDMkMDSL2oT8S2vH0C93Ei6jGJiks4P6scCg8KRb3OOaURBjCwjV3N4V0yHRhIL9gZwtwVt88jJonJc8i28vCpXyvI4sOkLHqIg8dIkq6AZVUR1R9IBe0Bt6dx6dV+fD+ZxZM848c4h+TSb7Be1/tDQ=</latexit><latexit sha1_base64="YZOTqGTxbG+4e4hKOuetRwh7Eio=">AAACTnicjVFJSwMxGM3Urdat6tFLsCj1UmZEsBeh4MVjhW7QqSWTpm1oJhmSb4QyzC/0It78GV48KKLpgmjrwQ8Cj7dkeQkiwQ247rOTWVldW9/Ibua2tnd29/L7Bw2jYk1ZnSqhdCsghgkuWR04CNaKNCNhIFgzGF1P9OY904YrWYNxxDohGUje55SApbp55hs+CEk3kTLFp/gK+8raJ7sltfQuKcqzFPu0pwD7QwKJNfl+7jsD/8lYUzdfcEvudPAy8OaggOZT7eaf/J6iccgkUEGMaXtuBJ2EaOBUsDTnx4ZFhI7IgLUtlCRkppNM60jxiWV6uK+0XRLwlP2ZSEhozDgMrDMkMDSL2oT8S2vH0C93Ei6jGJiks4P6scCg8KRb3OOaURBjCwjV3N4V0yHRhIL9gZwtwVt88jJonJc8i28vCpXyvI4sOkLHqIg8dIkq6AZVUR1R9IBe0Bt6dx6dV+fD+ZxZM848c4h+TSb7Be1/tDQ=</latexit><latexit sha1_base64="YZOTqGTxbG+4e4hKOuetRwh7Eio=">AAACTnicjVFJSwMxGM3Urdat6tFLsCj1UmZEsBeh4MVjhW7QqSWTpm1oJhmSb4QyzC/0It78GV48KKLpgmjrwQ8Cj7dkeQkiwQ247rOTWVldW9/Ibua2tnd29/L7Bw2jYk1ZnSqhdCsghgkuWR04CNaKNCNhIFgzGF1P9OY904YrWYNxxDohGUje55SApbp55hs+CEk3kTLFp/gK+8raJ7sltfQuKcqzFPu0pwD7QwKJNfl+7jsD/8lYUzdfcEvudPAy8OaggOZT7eaf/J6iccgkUEGMaXtuBJ2EaOBUsDTnx4ZFhI7IgLUtlCRkppNM60jxiWV6uK+0XRLwlP2ZSEhozDgMrDMkMDSL2oT8S2vH0C93Ei6jGJiks4P6scCg8KRb3OOaURBjCwjV3N4V0yHRhIL9gZwtwVt88jJonJc8i28vCpXyvI4sOkLHqIg8dIkq6AZVUR1R9IBe0Bt6dx6dV+fD+ZxZM848c4h+TSb7Be1/tDQ=</latexit>

⌧ = |T (n) � �nnn̂|<latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="ck8pdC+ekZH4nUmSP+ZG7r8lEyk=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odOn4MoA7ncAFXEMIN3MEDdKALAhJ4hXdv4r15H6uuat66tDP4I+/zBzjGijg=</latexit><latexit sha1_base64="lOEz7Zhg1waJevmwbl3IrZEHFGk=">AAACEnicbZC9SgNBFIXv+m+MGm1tBkWIhbIrgjaCYGOpYIyQjeHuZJIMzs4uM3eFsO6L2PgqNhaKWNj5Ns7GFBq9MMzHOXd+7olSJS35/qc3NT0zOze/sFhZqi6vrNbWqlc2yQwXDZ6oxFxHaIWSWjRIkhLXqREYR0o0o9vT0m/eCWNloi9pmIp2jH0te5IjOalTOwgJM3bM7lmYuL7ymvyyuMnreqdguyy0sh9jJ9e6YOEAKXf7faVT2/L3/FGxvxCMYQvGdd6pfYTdhGex0MQVWtsK/JTaORqSXImiEmZWpMhvsS9aDjXGwrbz0XQF23ZKl/US45YmNlJ/nsgxtnYYR64zRhrYSa8U//NaGfWO2rnUaUZC8++HeplilLAyKtaVRnBSQwfIjXR/ZXyABjm5QMsQgsmR/8LV/l7g+MKHBdiATahDAIdwAmdwDg3g8ABP8AKv3qP37L19xzXljXNbh1/lvX8BVpqf3Q==</latexit><latexit sha1_base64="lOEz7Zhg1waJevmwbl3IrZEHFGk=">AAACEnicbZC9SgNBFIXv+m+MGm1tBkWIhbIrgjaCYGOpYIyQjeHuZJIMzs4uM3eFsO6L2PgqNhaKWNj5Ns7GFBq9MMzHOXd+7olSJS35/qc3NT0zOze/sFhZqi6vrNbWqlc2yQwXDZ6oxFxHaIWSWjRIkhLXqREYR0o0o9vT0m/eCWNloi9pmIp2jH0te5IjOalTOwgJM3bM7lmYuL7ymvyyuMnreqdguyy0sh9jJ9e6YOEAKXf7faVT2/L3/FGxvxCMYQvGdd6pfYTdhGex0MQVWtsK/JTaORqSXImiEmZWpMhvsS9aDjXGwrbz0XQF23ZKl/US45YmNlJ/nsgxtnYYR64zRhrYSa8U//NaGfWO2rnUaUZC8++HeplilLAyKtaVRnBSQwfIjXR/ZXyABjm5QMsQgsmR/8LV/l7g+MKHBdiATahDAIdwAmdwDg3g8ABP8AKv3qP37L19xzXljXNbh1/lvX8BVpqf3Q==</latexit><latexit sha1_base64="K6aiLfYmpkPQlgCFmQB18UbVLGc=">AAACHXicbVDLSgMxFM34rPU16tJNsAi6sMyIYDeC4MZlhbYKnVrupGkbmskMyR2hjPMjbvwVNy4UceFG/BsztQtfB0IO59x7k3vCRAqDnvfhzMzOzS8slpbKyyura+vuxmbLxKlmvMliGeurEAyXQvEmCpT8KtEcolDyy3B0VviXN1wbEasGjhPeiWCgRF8wQCt13aMAIaUn9JYGsa0rxmSN/DrbU/s5PaCBEYMIuplSOQ2GgJm9b8tdt+JVvQnoX+JPSYVMUe+6b0EvZmnEFTIJxrR9L8FOBhoFkzwvB6nhCbARDHjbUgURN51ssl1Od63So/1Y26OQTtTvHRlExoyj0FZGgEPz2yvE/7x2iv1aJxMqSZEr9vVQP5UUY1pERXtCc4ZybAkwLexfKRuCBoY20CIE//fKf0nrsOpbfuFVTmvTOEpkm+yQPeKTY3JKzkmdNAkjd+SBPJFn5955dF6c16/SGWfas0V+wHn/BD4KoVc=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit>

Page 14: Material Failure and Distortional Strain Energy

Review: Principal stresses from 3D state of stress• Eigenvalue Analysis:

• Three eigenvalues or principal stresses: 𝜎1, 𝜎2, 𝜎3• Typically 𝜎1 > 𝜎2 > 𝜎3

• The eigenvectors are orthogonal – form an axis system• Stress in the original and principal axis system (its just a transformation!)

2

4�xx �xy �xz

�xy �yy �yz

�xz �yz �zz

3

5 !

2

4�1 0 00 �2 00 0 �3

3

5

<latexit sha1_base64="QftwJfIQwGrNwVCwQZtAz3qux2o=">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</latexit><latexit sha1_base64="QftwJfIQwGrNwVCwQZtAz3qux2o=">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</latexit><latexit sha1_base64="QftwJfIQwGrNwVCwQZtAz3qux2o=">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</latexit><latexit sha1_base64="QftwJfIQwGrNwVCwQZtAz3qux2o=">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</latexit>

[�]{n} = �p{n} {n}T {n} = 1

<latexit sha1_base64="ewZHdEUAIIQAx11onQDNnYvW2pk=">AAACLnicbVDLSgMxFM34rPVVdenmYhFclRlRdCMIIris0Bd0xiGTSWswk5kmGaEM/SI3/oouBBVx62eYtqPU1gOBk3POJbknSDhT2rZfrbn5hcWl5cJKcXVtfWOztLXdUHEqCa2TmMeyFWBFORO0rpnmtJVIiqOA02ZwdzH0m/dUKhaLmu4n1ItwV7AOI1gbyS9dQhtcxboRBg/cDAS4AziDXPOTXw3cXi/F4c/9pjaRdvxS2a7YI8AscXJSRjmqfunZDWOSRlRowrFSbcdOtJdhqRnhdFB0U0UTTO5wl7YNFTiiystG6w5g3yghdGJpjtAwUicnMhwp1Y8Ck4ywvlXT3lD8z2ununPqZUwkqaaCjB/qpBx0DMPuIGSSEs37hmAimfkrkFssMdGm4aIpwZleeZY0DivOUeX4+qh8fprXUUC7aA8dIAedoHN0haqojgh6QE/oDb1bj9aL9WF9jqNzVj6zg/7A+voG6r+kyQ==</latexit>

Page 15: Material Failure and Distortional Strain Energy

Octahedral Planes

• Use Cauchy formula to calculate the normal and shear stress on eight symmetrically placed planes around the principal axis system• Octahedral planes illustration:

Page 16: Material Failure and Distortional Strain Energy

Octahedral normal stress and magnitude of shear stress

• Stress tensor in the principal plane

• Normal vector to octahedral plane

• Traction on the octahedral plane

• Normal stress on the octahedral plane

• Magnitude of shear stress on the octahedral plane

n̂ =

✓1p3,1p3,1p3

◆T

<latexit sha1_base64="Wxqv5qMl4uy0H38JAjmBOAt54p0=">AAACPnicjZA7SwNBFIVn4yvGV9TSZjAIESTsRkEbIWBjGSEvyMYwO5lNhszOrjN3hbDsL7PxN9hZ2lgoYmvp5FFoYuGBgY9z7+XOPV4kuAbbfrYyS8srq2vZ9dzG5tb2Tn53r6HDWFFWp6EIVcsjmgkuWR04CNaKFCOBJ1jTG16N6817pjQPZQ1GEesEpC+5zykBY3XzdXdAIJEpvsSuYD4UsesrQhMnTVx9pyA5TdOTf1kYu4r3B3B8W8PdfMEu2RPhRXBmUEAzVbv5J7cX0jhgEqggWrcdO4JOQhRwKliac2PNIkKHpM/aBiUJmO4kk/NTfGScHvZDZZ4EPHF/TiQk0HoUeKYzIDDQ87Wx+VetHYN/0Um4jGJgkk4X+bHAEOJxlrjHFaMgRgYIVdz8FdMBMcGASTxnQnDmT16ERrnkGL45K1TKsziy6AAdoiJy0DmqoGtURXVE0QN6QW/o3Xq0Xq0P63PamrFmM/vol6yvb89WsAM=</latexit><latexit sha1_base64="Wxqv5qMl4uy0H38JAjmBOAt54p0=">AAACPnicjZA7SwNBFIVn4yvGV9TSZjAIESTsRkEbIWBjGSEvyMYwO5lNhszOrjN3hbDsL7PxN9hZ2lgoYmvp5FFoYuGBgY9z7+XOPV4kuAbbfrYyS8srq2vZ9dzG5tb2Tn53r6HDWFFWp6EIVcsjmgkuWR04CNaKFCOBJ1jTG16N6817pjQPZQ1GEesEpC+5zykBY3XzdXdAIJEpvsSuYD4UsesrQhMnTVx9pyA5TdOTf1kYu4r3B3B8W8PdfMEu2RPhRXBmUEAzVbv5J7cX0jhgEqggWrcdO4JOQhRwKliac2PNIkKHpM/aBiUJmO4kk/NTfGScHvZDZZ4EPHF/TiQk0HoUeKYzIDDQ87Wx+VetHYN/0Um4jGJgkk4X+bHAEOJxlrjHFaMgRgYIVdz8FdMBMcGASTxnQnDmT16ERrnkGL45K1TKsziy6AAdoiJy0DmqoGtURXVE0QN6QW/o3Xq0Xq0P63PamrFmM/vol6yvb89WsAM=</latexit><latexit sha1_base64="Wxqv5qMl4uy0H38JAjmBOAt54p0=">AAACPnicjZA7SwNBFIVn4yvGV9TSZjAIESTsRkEbIWBjGSEvyMYwO5lNhszOrjN3hbDsL7PxN9hZ2lgoYmvp5FFoYuGBgY9z7+XOPV4kuAbbfrYyS8srq2vZ9dzG5tb2Tn53r6HDWFFWp6EIVcsjmgkuWR04CNaKFCOBJ1jTG16N6817pjQPZQ1GEesEpC+5zykBY3XzdXdAIJEpvsSuYD4UsesrQhMnTVx9pyA5TdOTf1kYu4r3B3B8W8PdfMEu2RPhRXBmUEAzVbv5J7cX0jhgEqggWrcdO4JOQhRwKliac2PNIkKHpM/aBiUJmO4kk/NTfGScHvZDZZ4EPHF/TiQk0HoUeKYzIDDQ87Wx+VetHYN/0Um4jGJgkk4X+bHAEOJxlrjHFaMgRgYIVdz8FdMBMcGASTxnQnDmT16ERrnkGL45K1TKsziy6AAdoiJy0DmqoGtURXVE0QN6QW/o3Xq0Xq0P63PamrFmM/vol6yvb89WsAM=</latexit><latexit sha1_base64="Wxqv5qMl4uy0H38JAjmBOAt54p0=">AAACPnicjZA7SwNBFIVn4yvGV9TSZjAIESTsRkEbIWBjGSEvyMYwO5lNhszOrjN3hbDsL7PxN9hZ2lgoYmvp5FFoYuGBgY9z7+XOPV4kuAbbfrYyS8srq2vZ9dzG5tb2Tn53r6HDWFFWp6EIVcsjmgkuWR04CNaKFCOBJ1jTG16N6817pjQPZQ1GEesEpC+5zykBY3XzdXdAIJEpvsSuYD4UsesrQhMnTVx9pyA5TdOTf1kYu4r3B3B8W8PdfMEu2RPhRXBmUEAzVbv5J7cX0jhgEqggWrcdO4JOQhRwKliac2PNIkKHpM/aBiUJmO4kk/NTfGScHvZDZZ4EPHF/TiQk0HoUeKYzIDDQ87Wx+VetHYN/0Um4jGJgkk4X+bHAEOJxlrjHFaMgRgYIVdz8FdMBMcGASTxnQnDmT16ERrnkGL45K1TKsziy6AAdoiJy0DmqoGtURXVE0QN6QW/o3Xq0Xq0P63PamrFmM/vol6yvb89WsAM=</latexit>

T(n)

=

✓�1p3,�2p3,�3p3

◆T

<latexit sha1_base64="PFPc0dpq4FaYnhavgykV5xbU4os=">AAACXnicdZFLSwMxFIUz46tWa6tuBDfBIrQgMtMKuhEKblxW6As6tWTSTBvMZMbkjlCG+ZPuxI0/xfSx6EMvBA7nu5ebnPix4Boc58uyd3b39g9yh/mj48JJsXR61tFRoihr00hEqucTzQSXrA0cBOvFipHQF6zrvz3NePeDKc0j2YJpzAYhGUsecErAWMNS4kUGz6bTVvaaVmQ1w4/YEyyACvYCRWjqaT4OydDNjHpXkNaz7GaN1P4l9RWCsaf4eALV1xYelsrOrTMvvC3cpSijZTWHpU9vFNEkZBKoIFr3XSeGQUoUcCpYlvcSzWJC38iY9Y2UJGR6kM7jyfC1cUY4iJQ5EvDcXZ1ISaj1NPRNZ0hgojfZzPyL9RMIHgYpl3ECTNLFoiARGCI8yxqPuGIUxNQIQhU3d8V0Qkw+YH4kb0JwN5+8LTq1W9fol7tyo7aMI4cu0RWqIBfdowZ6Rk3URhR9W5aVt46sH3vfLtjFRattLWfO0VrZF795QbbT</latexit><latexit sha1_base64="PFPc0dpq4FaYnhavgykV5xbU4os=">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</latexit><latexit sha1_base64="PFPc0dpq4FaYnhavgykV5xbU4os=">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</latexit><latexit sha1_base64="PFPc0dpq4FaYnhavgykV5xbU4os=">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</latexit>

2

4�1 0 00 �2 00 0 �3

3

5

<latexit sha1_base64="Xg1vnGsqHvwaPEJp20tyzuxJaAM=">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</latexit><latexit sha1_base64="Xg1vnGsqHvwaPEJp20tyzuxJaAM=">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</latexit><latexit sha1_base64="Xg1vnGsqHvwaPEJp20tyzuxJaAM=">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</latexit><latexit sha1_base64="Xg1vnGsqHvwaPEJp20tyzuxJaAM=">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</latexit>

�oct = �nn =�1 + �2 + �3

3= �m

<latexit sha1_base64="qeXdiZixQabvC9aFHs7b+R567fM=">AAACPnicbZBNSwMxEIaz9avWr6pHL8EiCELZrYK9CAUvHivYD2jLkk2zbWiSXZKsUJb9ZV78Dd48evGgiFePptut1daBgWfemSGZ1wsZVdq2n63cyura+kZ+s7C1vbO7V9w/aKogkpg0cMAC2faQIowK0tBUM9IOJUHcY6Tlja4n/dY9kYoG4k6PQ9LjaCCoTzHSRnKLja6iA47cOMA6gVdwVgqRVr5EOJ5pTgLPfgYqyZzPk8TkfJu7xZJdttOAy+BkUAJZ1N3iU7cf4IgToTFDSnUcO9S9GElNMSNJoRspEiI8QgPSMSgQJ6oXp+cn8MQofegH0qTQMFV/b8SIKzXmnpnkSA/VYm8i/tfrRNqv9mIqwkgTgacP+RGDOoATL2GfSoI1GxtAWFLzV4iHyFimjeMFY4KzePIyNCtlx/DtRalWzezIgyNwDE6BAy5BDdyAOmgADB7AC3gD79aj9Wp9WJ/T0ZyV7RyCP2F9fQMeoq8A</latexit><latexit sha1_base64="qeXdiZixQabvC9aFHs7b+R567fM=">AAACPnicbZBNSwMxEIaz9avWr6pHL8EiCELZrYK9CAUvHivYD2jLkk2zbWiSXZKsUJb9ZV78Dd48evGgiFePptut1daBgWfemSGZ1wsZVdq2n63cyura+kZ+s7C1vbO7V9w/aKogkpg0cMAC2faQIowK0tBUM9IOJUHcY6Tlja4n/dY9kYoG4k6PQ9LjaCCoTzHSRnKLja6iA47cOMA6gVdwVgqRVr5EOJ5pTgLPfgYqyZzPk8TkfJu7xZJdttOAy+BkUAJZ1N3iU7cf4IgToTFDSnUcO9S9GElNMSNJoRspEiI8QgPSMSgQJ6oXp+cn8MQofegH0qTQMFV/b8SIKzXmnpnkSA/VYm8i/tfrRNqv9mIqwkgTgacP+RGDOoATL2GfSoI1GxtAWFLzV4iHyFimjeMFY4KzePIyNCtlx/DtRalWzezIgyNwDE6BAy5BDdyAOmgADB7AC3gD79aj9Wp9WJ/T0ZyV7RyCP2F9fQMeoq8A</latexit><latexit sha1_base64="qeXdiZixQabvC9aFHs7b+R567fM=">AAACPnicbZBNSwMxEIaz9avWr6pHL8EiCELZrYK9CAUvHivYD2jLkk2zbWiSXZKsUJb9ZV78Dd48evGgiFePptut1daBgWfemSGZ1wsZVdq2n63cyura+kZ+s7C1vbO7V9w/aKogkpg0cMAC2faQIowK0tBUM9IOJUHcY6Tlja4n/dY9kYoG4k6PQ9LjaCCoTzHSRnKLja6iA47cOMA6gVdwVgqRVr5EOJ5pTgLPfgYqyZzPk8TkfJu7xZJdttOAy+BkUAJZ1N3iU7cf4IgToTFDSnUcO9S9GElNMSNJoRspEiI8QgPSMSgQJ6oXp+cn8MQofegH0qTQMFV/b8SIKzXmnpnkSA/VYm8i/tfrRNqv9mIqwkgTgacP+RGDOoATL2GfSoI1GxtAWFLzV4iHyFimjeMFY4KzePIyNCtlx/DtRalWzezIgyNwDE6BAy5BDdyAOmgADB7AC3gD79aj9Wp9WJ/T0ZyV7RyCP2F9fQMeoq8A</latexit><latexit sha1_base64="qeXdiZixQabvC9aFHs7b+R567fM=">AAACPnicbZBNSwMxEIaz9avWr6pHL8EiCELZrYK9CAUvHivYD2jLkk2zbWiSXZKsUJb9ZV78Dd48evGgiFePptut1daBgWfemSGZ1wsZVdq2n63cyura+kZ+s7C1vbO7V9w/aKogkpg0cMAC2faQIowK0tBUM9IOJUHcY6Tlja4n/dY9kYoG4k6PQ9LjaCCoTzHSRnKLja6iA47cOMA6gVdwVgqRVr5EOJ5pTgLPfgYqyZzPk8TkfJu7xZJdttOAy+BkUAJZ1N3iU7cf4IgToTFDSnUcO9S9GElNMSNJoRspEiI8QgPSMSgQJ6oXp+cn8MQofegH0qTQMFV/b8SIKzXmnpnkSA/VYm8i/tfrRNqv9mIqwkgTgacP+RGDOoATL2GfSoI1GxtAWFLzV4iHyFimjeMFY4KzePIyNCtlx/DtRalWzezIgyNwDE6BAy5BDdyAOmgADB7AC3gD79aj9Wp9WJ/T0ZyV7RyCP2F9fQMeoq8A</latexit>

⌧2oct =1

9

h(�1 � �2)

2 + (�2 � �3)2 + (�1 � �3)

2i

<latexit sha1_base64="xDv+Syl4/wV6zTlbcFFUzBxU6Dk=">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</latexit><latexit sha1_base64="xDv+Syl4/wV6zTlbcFFUzBxU6Dk=">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</latexit><latexit sha1_base64="xDv+Syl4/wV6zTlbcFFUzBxU6Dk=">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</latexit><latexit sha1_base64="xDv+Syl4/wV6zTlbcFFUzBxU6Dk=">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</latexit>

�nn = T(n) · n̂

�nt = T(n) · t̂

<latexit sha1_base64="YZOTqGTxbG+4e4hKOuetRwh7Eio=">AAACTnicjVFJSwMxGM3Urdat6tFLsCj1UmZEsBeh4MVjhW7QqSWTpm1oJhmSb4QyzC/0It78GV48KKLpgmjrwQ8Cj7dkeQkiwQ247rOTWVldW9/Ibua2tnd29/L7Bw2jYk1ZnSqhdCsghgkuWR04CNaKNCNhIFgzGF1P9OY904YrWYNxxDohGUje55SApbp55hs+CEk3kTLFp/gK+8raJ7sltfQuKcqzFPu0pwD7QwKJNfl+7jsD/8lYUzdfcEvudPAy8OaggOZT7eaf/J6iccgkUEGMaXtuBJ2EaOBUsDTnx4ZFhI7IgLUtlCRkppNM60jxiWV6uK+0XRLwlP2ZSEhozDgMrDMkMDSL2oT8S2vH0C93Ei6jGJiks4P6scCg8KRb3OOaURBjCwjV3N4V0yHRhIL9gZwtwVt88jJonJc8i28vCpXyvI4sOkLHqIg8dIkq6AZVUR1R9IBe0Bt6dx6dV+fD+ZxZM848c4h+TSb7Be1/tDQ=</latexit><latexit sha1_base64="YZOTqGTxbG+4e4hKOuetRwh7Eio=">AAACTnicjVFJSwMxGM3Urdat6tFLsCj1UmZEsBeh4MVjhW7QqSWTpm1oJhmSb4QyzC/0It78GV48KKLpgmjrwQ8Cj7dkeQkiwQ247rOTWVldW9/Ibua2tnd29/L7Bw2jYk1ZnSqhdCsghgkuWR04CNaKNCNhIFgzGF1P9OY904YrWYNxxDohGUje55SApbp55hs+CEk3kTLFp/gK+8raJ7sltfQuKcqzFPu0pwD7QwKJNfl+7jsD/8lYUzdfcEvudPAy8OaggOZT7eaf/J6iccgkUEGMaXtuBJ2EaOBUsDTnx4ZFhI7IgLUtlCRkppNM60jxiWV6uK+0XRLwlP2ZSEhozDgMrDMkMDSL2oT8S2vH0C93Ei6jGJiks4P6scCg8KRb3OOaURBjCwjV3N4V0yHRhIL9gZwtwVt88jJonJc8i28vCpXyvI4sOkLHqIg8dIkq6AZVUR1R9IBe0Bt6dx6dV+fD+ZxZM848c4h+TSb7Be1/tDQ=</latexit><latexit sha1_base64="YZOTqGTxbG+4e4hKOuetRwh7Eio=">AAACTnicjVFJSwMxGM3Urdat6tFLsCj1UmZEsBeh4MVjhW7QqSWTpm1oJhmSb4QyzC/0It78GV48KKLpgmjrwQ8Cj7dkeQkiwQ247rOTWVldW9/Ibua2tnd29/L7Bw2jYk1ZnSqhdCsghgkuWR04CNaKNCNhIFgzGF1P9OY904YrWYNxxDohGUje55SApbp55hs+CEk3kTLFp/gK+8raJ7sltfQuKcqzFPu0pwD7QwKJNfl+7jsD/8lYUzdfcEvudPAy8OaggOZT7eaf/J6iccgkUEGMaXtuBJ2EaOBUsDTnx4ZFhI7IgLUtlCRkppNM60jxiWV6uK+0XRLwlP2ZSEhozDgMrDMkMDSL2oT8S2vH0C93Ei6jGJiks4P6scCg8KRb3OOaURBjCwjV3N4V0yHRhIL9gZwtwVt88jJonJc8i28vCpXyvI4sOkLHqIg8dIkq6AZVUR1R9IBe0Bt6dx6dV+fD+ZxZM848c4h+TSb7Be1/tDQ=</latexit><latexit sha1_base64="YZOTqGTxbG+4e4hKOuetRwh7Eio=">AAACTnicjVFJSwMxGM3Urdat6tFLsCj1UmZEsBeh4MVjhW7QqSWTpm1oJhmSb4QyzC/0It78GV48KKLpgmjrwQ8Cj7dkeQkiwQ247rOTWVldW9/Ibua2tnd29/L7Bw2jYk1ZnSqhdCsghgkuWR04CNaKNCNhIFgzGF1P9OY904YrWYNxxDohGUje55SApbp55hs+CEk3kTLFp/gK+8raJ7sltfQuKcqzFPu0pwD7QwKJNfl+7jsD/8lYUzdfcEvudPAy8OaggOZT7eaf/J6iccgkUEGMaXtuBJ2EaOBUsDTnx4ZFhI7IgLUtlCRkppNM60jxiWV6uK+0XRLwlP2ZSEhozDgMrDMkMDSL2oT8S2vH0C93Ei6jGJiks4P6scCg8KRb3OOaURBjCwjV3N4V0yHRhIL9gZwtwVt88jJonJc8i28vCpXyvI4sOkLHqIg8dIkq6AZVUR1R9IBe0Bt6dx6dV+fD+ZxZM848c4h+TSb7Be1/tDQ=</latexit>

⌧ = |T (n) � �nnn̂|<latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="ck8pdC+ekZH4nUmSP+ZG7r8lEyk=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odOn4MoA7ncAFXEMIN3MEDdKALAhJ4hXdv4r15H6uuat66tDP4I+/zBzjGijg=</latexit><latexit sha1_base64="lOEz7Zhg1waJevmwbl3IrZEHFGk=">AAACEnicbZC9SgNBFIXv+m+MGm1tBkWIhbIrgjaCYGOpYIyQjeHuZJIMzs4uM3eFsO6L2PgqNhaKWNj5Ns7GFBq9MMzHOXd+7olSJS35/qc3NT0zOze/sFhZqi6vrNbWqlc2yQwXDZ6oxFxHaIWSWjRIkhLXqREYR0o0o9vT0m/eCWNloi9pmIp2jH0te5IjOalTOwgJM3bM7lmYuL7ymvyyuMnreqdguyy0sh9jJ9e6YOEAKXf7faVT2/L3/FGxvxCMYQvGdd6pfYTdhGex0MQVWtsK/JTaORqSXImiEmZWpMhvsS9aDjXGwrbz0XQF23ZKl/US45YmNlJ/nsgxtnYYR64zRhrYSa8U//NaGfWO2rnUaUZC8++HeplilLAyKtaVRnBSQwfIjXR/ZXyABjm5QMsQgsmR/8LV/l7g+MKHBdiATahDAIdwAmdwDg3g8ABP8AKv3qP37L19xzXljXNbh1/lvX8BVpqf3Q==</latexit><latexit sha1_base64="lOEz7Zhg1waJevmwbl3IrZEHFGk=">AAACEnicbZC9SgNBFIXv+m+MGm1tBkWIhbIrgjaCYGOpYIyQjeHuZJIMzs4uM3eFsO6L2PgqNhaKWNj5Ns7GFBq9MMzHOXd+7olSJS35/qc3NT0zOze/sFhZqi6vrNbWqlc2yQwXDZ6oxFxHaIWSWjRIkhLXqREYR0o0o9vT0m/eCWNloi9pmIp2jH0te5IjOalTOwgJM3bM7lmYuL7ymvyyuMnreqdguyy0sh9jJ9e6YOEAKXf7faVT2/L3/FGxvxCMYQvGdd6pfYTdhGex0MQVWtsK/JTaORqSXImiEmZWpMhvsS9aDjXGwrbz0XQF23ZKl/US45YmNlJ/nsgxtnYYR64zRhrYSa8U//NaGfWO2rnUaUZC8++HeplilLAyKtaVRnBSQwfIjXR/ZXyABjm5QMsQgsmR/8LV/l7g+MKHBdiATahDAIdwAmdwDg3g8ABP8AKv3qP37L19xzXljXNbh1/lvX8BVpqf3Q==</latexit><latexit sha1_base64="K6aiLfYmpkPQlgCFmQB18UbVLGc=">AAACHXicbVDLSgMxFM34rPU16tJNsAi6sMyIYDeC4MZlhbYKnVrupGkbmskMyR2hjPMjbvwVNy4UceFG/BsztQtfB0IO59x7k3vCRAqDnvfhzMzOzS8slpbKyyura+vuxmbLxKlmvMliGeurEAyXQvEmCpT8KtEcolDyy3B0VviXN1wbEasGjhPeiWCgRF8wQCt13aMAIaUn9JYGsa0rxmSN/DrbU/s5PaCBEYMIuplSOQ2GgJm9b8tdt+JVvQnoX+JPSYVMUe+6b0EvZmnEFTIJxrR9L8FOBhoFkzwvB6nhCbARDHjbUgURN51ssl1Od63So/1Y26OQTtTvHRlExoyj0FZGgEPz2yvE/7x2iv1aJxMqSZEr9vVQP5UUY1pERXtCc4ZybAkwLexfKRuCBoY20CIE//fKf0nrsOpbfuFVTmvTOEpkm+yQPeKTY3JKzkmdNAkjd+SBPJFn5955dF6c16/SGWfas0V+wHn/BD4KoVc=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit><latexit sha1_base64="RVYMjEsI/jJSNiwZl4YPrRpnv3k=">AAACHXicbVDLSgMxFM34tr6qLt0Ei6ALZUYKdiMIblwq2Ad0armTpm0wkxmSO0KZzo+48VfcuFDEhRvxb8y0XWjrgZDDOffe5J4glsKg6347c/MLi0vLK6uFtfWNza3i9k7NRIlmvMoiGelGAIZLoXgVBUreiDWHMJC8Htxf5n79gWsjInWLg5i3Qugp0RUM0ErtYtlHSOg5HVI/snX5mPQ2u0sP1VFGj6lvRC+EdqpURv0+YGrvYaFdLLkn7gh0lngTUiITXLeLn34nYknIFTIJxjQ9N8ZWChoFkzwr+InhMbB76PGmpQpCblrpaLuMHlilQ7uRtkchHam/O1IIjRmEga0MAftm2svF/7xmgt1KKxUqTpArNn6om0iKEc2joh2hOUM5sASYFvavlPVBA0MbaB6CN73yLKmdnniW35RLF5VJHCtkj+yTQ+KRM3JBrsg1qRJGHskzeSVvzpPz4rw7H+PSOWfSs0v+wPn6AT9KoVs=</latexit>

Page 17: Material Failure and Distortional Strain Energy

Octahedral Shear Stress Criteria• Uniaxial tension tests give the yield stress at failure 𝜎y• Octahedral shear stress at yield is

• Factor of Safety:

• The distortional strain energy density is related to the octahedral shear stress

FS =⌧octy⌧oct

<latexit sha1_base64="zHT8vb8akYvIPf6fBEQ6O8ze2PI=">AAACFnicbVDLSsNAFJ34rPUVdelmsAhuLEkV7EYoCOKyon1AE8JkOmmHTh7M3Igl5Cvc+CtuXCjiVtz5N04fC9t64MKZc+5l7j1+IrgCy/oxlpZXVtfWCxvFza3tnV1zb7+p4lRS1qCxiGXbJ4oJHrEGcBCsnUhGQl+wlj+4GvmtByYVj6N7GCbMDUkv4gGnBLTkmacOsEfIru9yfImdQBKaZQ6Q1MtiCnnuDfOZp1myytYYeJHYU1JCU9Q989vpxjQNWQRUEKU6tpWAmxEJnAqWF51UsYTQAemxjqYRCZlys/FZOT7WShcHsdQVAR6rfycyEio1DH3dGRLoq3lvJP7ndVIIqm7GoyQFFtHJR0EqMMR4lBHucskoiKEmhEqud8W0T3Q2oJMs6hDs+ZMXSbNSts/KldvzUq06jaOADtEROkE2ukA1dIPqqIEoekIv6A29G8/Gq/FhfE5al4zpzAGagfH1C3xCoNQ=</latexit>

Ud =3

4G⌧2oct

<latexit sha1_base64="jpBRC/7rHkshxfZT8ZPP8fifoQU=">AAACEHicbVDLSsNAFJ34rPVVdelmsIiuStIW7EYouNBlBfuAJobJdNIOnTyYuRFKyCe48VfcuFDErUt3/o3TNgttPXDhcM693HuPFwuuwDS/jZXVtfWNzcJWcXtnd2+/dHDYUVEiKWvTSESy5xHFBA9ZGzgI1oslI4EnWNcbX0397gOTikfhHUxi5gRkGHKfUwJacktnqU2JwO3MHeBLbPuS0LSWpXV8nWEbSOKmEYXsvuqWymbFnAEvEysnZZSj5Za+7EFEk4CFQAVRqm+ZMTgpkcCpYFnRThSLCR2TIetrGpKAKSedPZThU60MsB9JXSHgmfp7IiWBUpPA050BgZFa9Kbif14/Ab/hpDyME2AhnS/yE4EhwtN08IBLRkFMNCFUcn0rpiOiQwGdYVGHYC2+vEw61YpVq1Rv6+VmI4+jgI7RCTpHFrpATXSDWqiNKHpEz+gVvRlPxovxbnzMW1eMfOYI/YHx+QM3SJwN</latexit>

⌧oct2y =

1

9

h(�y � 0)2 + (0� 0)2 + (�y � 0)2

i=

2

9�2y

⌧octy =

p2

3�y

<latexit sha1_base64="g8WP0ju8o/3ZrILiC19QyyAgevI=">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</latexit>

Page 18: Material Failure and Distortional Strain Energy

von Mises Stress• Expression for von Mises Stress

• von Mises Stress at yield is

• Factor of SafetyFS =

�y

�vm<latexit sha1_base64="WD9G8/Ey0uAji2MRThgQAIWNdqo=">AAACE3icbZDLSgMxFIYz9VbrrerSTbAI4qLMVMFuhIIgLivaC3RKyaSZNjSZGZIzxTLMO7jxVdy4UMStG3e+jekF0dYfAh//OYeT83uR4Bps+8vKLC2vrK5l13Mbm1vbO/ndvboOY0VZjYYiVE2PaCZ4wGrAQbBmpBiRnmANb3A5rjeGTGkeBncwilhbkl7AfU4JGKuTP3GB3UNydZviC+z6itDE1bwnSScZpekPD2WadvIFu2hPhBfBmUEBzVTt5D/dbkhjyQKggmjdcuwI2glRwKlgac6NNYsIHZAeaxkMiGS6nUxuSvGRcbrYD5V5AeCJ+3siIVLrkfRMpyTQ1/O1sflfrRWDX24nPIhiYAGdLvJjgSHE44BwlytGQYwMEKq4+SumfWKCARNjzoTgzJ+8CPVS0Tktlm7OCpXyLI4sOkCH6Bg56BxV0DWqohqi6AE9oRf0aj1az9ab9T5tzVizmX30R9bHN6Rhn0Q=</latexit>

�vm =

s(�xx � �yy)2 + (�yy � �zz)2 + (�xx � �zz)2 + 6

��2xy + �2

yz + �2xz

�

2

=

r(�1 � �2)2 + (�2 � �3)2 + (�1 � �3)2

2<latexit sha1_base64="iwIQHmW1QgCqmO8AKv/5+qr7Sfc=">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</latexit>

�vmy =

r(�y � 0)2 + (0� 0)2 + (�y � 0)2

2= �y

<latexit sha1_base64="B7OHepITFHNCRlYCxk9LgekX29I=">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</latexit>

Page 19: Material Failure and Distortional Strain Energy

Failure Criteria: Maximum distortion strain energy or von Mises or Octahedral shear stress• All the following failure criteria are equivalent – lead to the same FS• Maximum distortion strain energy• Octahedral shear stress• von Mises


High-strain, high-strain-rate flow and failure in PTFE/Al/W ...maeresearch.ucsd.edu/nesterenko/PUB_PDFS/A1/119, A I, 2008.pdf · High-strain, high-strain-rate flow and failure in

High-strain, high-strain-rate flow and failure in PTFE/Al/W ...maeresearch.ucsd.edu/nesterenko/PUB_PDFS/A1/119, A I, 2008.pdf · High-strain, high-strain-rate flow and failure in

The effect of film thickness on the failure strain of ...

The effect of film thickness on the failure strain of ...

Evaluation of strain path independent material models for ...publications.lib.chalmers.se/records/fulltext/238675/238675.pdf · material models for failure prediction ... strain path

Evaluation of strain path independent material models for ...publications.lib.chalmers.se/records/fulltext/238675/238675.pdf · material models for failure prediction ... strain path

Fatigue Failure of Notched Specimen A Strain-Life Approach · Fatigue Failure of Notched Specimen—A Strain-Life Approach Bikash Joadder ... 316 Stainless steel (SS316) is selected

Fatigue Failure of Notched Specimen A Strain-Life Approach · Fatigue Failure of Notched Specimen—A Strain-Life Approach Bikash Joadder ... 316 Stainless steel (SS316) is selected

''T: -7AIN STRAIN AGING AND DELAYED FAILURE IN · ''t: -7ain strain aging and "delayed failure in 0 ahigh-strength steels technical documentary report no. asd-tdr-62-968 november

''T: -7AIN STRAIN AGING AND DELAYED FAILURE IN · ''t: -7ain strain aging and "delayed failure in 0 ahigh-strength steels technical documentary report no. asd-tdr-62-968 november

DESIGN AIDS AND EXAMPLES FOR DISTORTIONAL BUCKLING

DESIGN AIDS AND EXAMPLES FOR DISTORTIONAL BUCKLING

EFFECT OF STRAIN RATE ON PLASTIC FLOW AND FAILURE IN POLYCRYSTALLINE TUNGSTEN

EFFECT OF STRAIN RATE ON PLASTIC FLOW AND FAILURE IN POLYCRYSTALLINE TUNGSTEN

I Ma, John · stress linearly to zero at a total strain of about 10 times the strain at failure. The strain at failure in standard concretes is typically 10-4, which suggests that

I Ma, John · stress linearly to zero at a total strain of about 10 times the strain at failure. The strain at failure in standard concretes is typically 10-4, which suggests that

A Finite Element Ductile Failure Simulation Method Usig Stress Modified Fracture Strain Model

A Finite Element Ductile Failure Simulation Method Usig Stress Modified Fracture Strain Model

Distortional Failure of Cold-Formed Steel Beams under Uniform … · 2016. 11. 2. · Distortional Failure of Cold-Formed Steel Beams under Uniform Bending: Behavior, Strength and

Distortional Failure of Cold-Formed Steel Beams under Uniform … · 2016. 11. 2. · Distortional Failure of Cold-Formed Steel Beams under Uniform Bending: Behavior, Strength and

GY403 Structural Geology - southalabama.edu · Strain Ellipsoid A three-dimensional ellipsoid that describes the magnitude of dilational and distortional strain PAssume a perfect

GY403 Structural Geology - southalabama.edu · Strain Ellipsoid A three-dimensional ellipsoid that describes the magnitude of dilational and distortional strain PAssume a perfect

EFFECT OF STRAIN RATE ON PLASTIC FLOW AND FAILURE IN ...

EFFECT OF STRAIN RATE ON PLASTIC FLOW AND FAILURE IN ...

On The Nearby-Tip Strain Investigation and Failure ...

On The Nearby-Tip Strain Investigation and Failure ...

Full-field Strain Methods for Investigating Failure Mechanisms ......1 Full-field Strain Methods for Investigating Failure Mechanisms in Triaxial Braided Composites Justin D. Littell

Full-field Strain Methods for Investigating Failure Mechanisms ......1 Full-field Strain Methods for Investigating Failure Mechanisms in Triaxial Braided Composites Justin D. Littell

Time Dependent Failure of Open Stopes at Target Mine...Stress-Strain Index, the extent of failure can be determined. The relation between mean stress 𝜎 and volumetric strain 𝜀𝑣𝑜

Time Dependent Failure of Open Stopes at Target Mine...Stress-Strain Index, the extent of failure can be determined. The relation between mean stress 𝜎 and volumetric strain 𝜀𝑣𝑜

Distortional buckling behavior and design method of cold ...

Distortional buckling behavior and design method of cold ...

An Improved Failure Envelop for Concrete under High Strain ... · An Improved Failure Envelop for Concrete under High Strain Rate ... An Improved Failure Envelop for concrete to ...

An Improved Failure Envelop for Concrete under High Strain ... · An Improved Failure Envelop for Concrete under High Strain Rate ... An Improved Failure Envelop for concrete to ...

Theories of Failure - WordPress.com · 1 Theories of Failure • Failure: Every material has certain strength, expressed in terms of stress or strain, beyond which it fractures or

Theories of Failure - WordPress.com · 1 Theories of Failure • Failure: Every material has certain strength, expressed in terms of stress or strain, beyond which it fractures or

A Bi-axial Failure Envelop for Concrete under High Strain ...€¦ · the high strain rate bi-axial behavior of concrete, it is difficult to reflect the coupled effect of strain rate

A Bi-axial Failure Envelop for Concrete under High Strain ...€¦ · the high strain rate bi-axial behavior of concrete, it is difficult to reflect the coupled effect of strain rate

LOCAL AND DISTORTIONAL BUCKLING OF PERFORATED …

LOCAL AND DISTORTIONAL BUCKLING OF PERFORATED …