Fórmulas de conversão |
Materiais lineares homogêneos e isotrópicos tem suas propriedades elásticas determinadas unicamente por qualquer dois módulos dentre estes, e assim dados quaisquer dois, qualquer outro dos módulos elásticos pode ser determinado de acordo com estas fórmulas. |
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![{\displaystyle (K,\,E)}](//wikimedia.org/api/rest_v1/media/math/render/svg/996a015d617843aa00212a26839fbae210c834a4) |
![{\displaystyle (K,\,\lambda )}](//wikimedia.org/api/rest_v1/media/math/render/svg/423e5f4d1c8ee39ba6de7bc5e48df6837fff232a) |
![{\displaystyle (K,\,G)}](//wikimedia.org/api/rest_v1/media/math/render/svg/bae48568e7a818eb72621f7cd602d0effc7e3648) |
![{\displaystyle (K,\,\nu )}](//wikimedia.org/api/rest_v1/media/math/render/svg/b2f6df6fb9936ea5a0e15358e66802cafe9b747d) |
![{\displaystyle (E,\,G)}](//wikimedia.org/api/rest_v1/media/math/render/svg/ee52d776f2cb07c88fbac6567029158dff718e01) |
![{\displaystyle (E,\,\nu )}](//wikimedia.org/api/rest_v1/media/math/render/svg/7d858acfba62fc12da4ae6fa0759b9df87c51d96) |
![{\displaystyle (\lambda ,\,G)}](//wikimedia.org/api/rest_v1/media/math/render/svg/7bffd6d22cc67fd1a35a585b13d7b32a74ce6b1f) |
![{\displaystyle (\lambda ,\,\nu )}](//wikimedia.org/api/rest_v1/media/math/render/svg/cbafda646cd2c8251ec73dbe5efd428aee76544e) |
![{\displaystyle (G,\,\nu )}](//wikimedia.org/api/rest_v1/media/math/render/svg/03de031e73561f10f24638feaf69d90abe32aea5) |
![{\displaystyle (G,\,M)}](//wikimedia.org/api/rest_v1/media/math/render/svg/40060aa6964ead07257199a60095ec47599c9455) |
![{\displaystyle K=\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/d12e227970b986c1ff219badec7438f7383bfdb6) |
![{\displaystyle K}](//wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0) |
![{\displaystyle K}](//wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0) |
![{\displaystyle K}](//wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0) |
![{\displaystyle K}](//wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0) |
![{\displaystyle {\tfrac {EG}{3(3G-E)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/94fef5491daefd3b9667bfd32eb6e4b3f31a67aa) |
![{\displaystyle {\tfrac {E}{3(1-2\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/49073e20128a9ecf62b06e95a6e611808b4a1015) |
![{\displaystyle \lambda +{\tfrac {2G}{3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/bfcd78f0e042c88176c6ba5c0f8dcebf89fe223a) |
![{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f8668d7ae1d4ba0bea79bc24b22969de2e0020b1) |
![{\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e7f8f5046df3cc6a7152f9fa56845d5dbce78033) |
![{\displaystyle M-{\tfrac {4G}{3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/bf22c9741623f0209e7585845196a3fa1ed6282a) |
![{\displaystyle E=\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/b75f04c79769676b88c45d4fef023a01294b670a) |
![{\displaystyle E}](//wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b) |
![{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/3233ebb13df867ba5551fefa7e092733fea80c42) |
![{\displaystyle {\tfrac {9KG}{3K+G}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/5440c8c124cded6c55e978cbf920e76e4883a51b) |
![{\displaystyle 3K(1-2\nu )\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/dd567543f973005ad7031789e6219f93463e6d87) |
![{\displaystyle E}](//wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b) |
![{\displaystyle E}](//wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b) |
![{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d0994ccfa74411f35a8f0762379eddc3660bfec4) |
![{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/16ba478690ce90bc61e86d5b21ef89e8d97c0223) |
![{\displaystyle 2G(1+\nu )\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/8ecc39d8499aa0848351a6510b116bd2dfea06d8) |
![{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0b53b71a131246c16879e071ac283f0cf16bc255) |
![{\displaystyle \lambda =\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/fdf13c116950ce78ddbc7a016c3cd804e23ea86d) |
![{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0cad8348883df915b9b03fb2820ec6b096640f4f) |
![{\displaystyle \lambda }](//wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) |
![{\displaystyle K-{\tfrac {2G}{3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a5272b70cff4a6a17225acf399ea27a1948b4f59) |
![{\displaystyle {\tfrac {3K\nu }{1+\nu }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1194366850f7d5421ab50b6c2706e58498f7ea96) |
![{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/2b592ec0480b2ca3a99ea29d9cbd0b631d777d4d) |
![{\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/7aa443ecbbd2c92a028d26ee25863bbc1592b4b5) |
![{\displaystyle \lambda }](//wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) |
![{\displaystyle \lambda }](//wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a) |
![{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d011b18df127233873f4af03b50318509583006e) |
![{\displaystyle M-2G\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/4777994467bbd835a560ddf3d9b7a9bb88f6be15) |
![{\displaystyle G=\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/fa8a892590600356bf9957268977269f386821c6) |
![{\displaystyle {\tfrac {3KE}{9K-E}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/cdd679253fa436a6900d1cf949d909d3b09cf3c0) |
![{\displaystyle {\tfrac {3(K-\lambda )}{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e4d4ffacd8fb2de88a240a663a13067a7c3e921f) |
![{\displaystyle G}](//wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) |
![{\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/31c8588d1514e15c703af716b1d98c94ef2ad4c9) |
![{\displaystyle G}](//wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) |
![{\displaystyle {\tfrac {E}{2(1+\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ea91afdc4bb2364d6deb22ee5653d33451c8d7b2) |
![{\displaystyle G}](//wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) |
![{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c0ae2f9d9801984c3a421791f68ed6d206906ce4) |
![{\displaystyle G}](//wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) |
![{\displaystyle G}](//wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) |
![{\displaystyle \nu =\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/3b632d33a8bb2255e8a086b63953c599d3b6a718) |
![{\displaystyle {\tfrac {3K-E}{6K}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/52e22f238cec78b55c3b21bb03bd0f6bc122006b) |
![{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f1c70a68ad578ac446b83ae2af368a85554e2f4d) |
![{\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b11dec1ab89c26fc5ad5719fd2932e1dc68f4cb3) |
![{\displaystyle \nu }](//wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468) |
![{\displaystyle {\tfrac {E}{2G}}-1}](//wikimedia.org/api/rest_v1/media/math/render/svg/b9fb70b8d93a1714d230f745d4bb0a3aa81031a5) |
![{\displaystyle \nu }](//wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468) |
![{\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/569977b3b51798797b8ed7d287e828ef07cdb54a) |
![{\displaystyle \nu }](//wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468) |
![{\displaystyle \nu }](//wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468) |
![{\displaystyle {\tfrac {M-2G}{2M-2G}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d95ab5446ac8be62e025bbc4461d19ad03ddb0ba) |
![{\displaystyle M=\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/843dc939c547010c55bd7753f7c94e6996934ca4) |
![{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/9c924e463a20a0f63bd2f876283e3fb1a965db68) |
![{\displaystyle 3K-2\lambda \,}](//wikimedia.org/api/rest_v1/media/math/render/svg/dccc234a5195e4cc32a106e7280ce2e082a9da43) |
![{\displaystyle K+{\tfrac {4G}{3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0a71d02abd602def118ed9f9228f0f686a06448d) |
![{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c09e8d1337a766b61ba2b9948fa4af1503d2ede0) |
![{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/06fef6489910b32a86944f5fe2f5f0c1473e87c1) |
![{\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/7abf15539e84aef12ea7f6aed6ac61b939e0897a) |
![{\displaystyle \lambda +2G\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/f04bd3c70df3389cd672961e8ba5636aa8d5fb48) |
![{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a4b0e21d0522f123eb9043c4e4352bbfd0944365) |
![{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/036c5b70861f3145642c2d797f6cbf666ea26dbc) |
![{\displaystyle M}](//wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd) |
A matriz constitutiva (9 por 9, ou 6 por 6 na notação de Voigt) da lei de Hooke (em três dimensões) pode ser parametrizada com somente duas componentes independentes para materiais homogêneos isotrópicos. Qualquer par pode ser escolhido entre os módulos elásticos apresentados. Algumas das possíveis conversões são apresentadas na tabela. |
Bibliografia: G. Mavko, T. Mukerji, J. Dvorkin. The Rock Physics Handbook. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4 |