{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T20:24:33+00:00","modifiedTime":"2024-03-20T13:45:41+00:00","timestamp":"2024-03-20T15:01:17+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Careers","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34256"},"slug":"careers","categoryId":34256},{"name":"Trades, Tech, & Engineering Careers","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34271"},"slug":"trades-tech-engineering-careers","categoryId":34271}],"title":"Using a Generalized Hooke's Law for Stress and Strain","strippedTitle":"using a generalized hooke's law for stress and strain","slug":"using-a-generalized-hookes-law-for-stress-and-strain","canonicalUrl":"","seo":{"metaDescription":"Clearly find out why Hooke’s law is important, what the formula is, and how to apply it to uniaxial stresses and three-dimensional objects.","noIndex":0,"noFollow":0},"content":"In mechanics of materials, <i>Hooke</i><i>'</i><i>s law</i> is the relationship that connects stresses to strains. Although Hooke's original law was developed for uniaxial stresses, you can use a generalized version of Hooke's law to connect stress and strain in three-dimensional objects, as well. Eventually, Hooke's law helps you relate stresses (which are based on loads) to strains (which are based on deformations).\r\n\r\nFor a three-dimensional state of stress, the normal strain in a given direction (such as <i>x</i>) is a function of the stresses in all three orthogonal directions (usually the Cartesian <i>x</i>-<i>, </i><i>y-,</i><i> </i>and <i>z-</i>directions), as shown by this equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/273815.image0.jpg\" alt=\"image0.jpg\" width=\"201\" height=\"29\" />\r\n\r\nwhere <i>E</i> is the modulus of elasticity and ν is Poisson's ratio for the material. For a uniaxial stress, two of the stresses in the equation are zero. For a biaxial stress condition, one of the stresses in this equation is zero.\r\n\r\nThe generalized relationship for Hooke's law for shear in the XY plane can be given as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/273816.image1.jpg\" alt=\"image1.jpg\" width=\"155\" height=\"29\" />","description":"In mechanics of materials, <i>Hooke</i><i>'</i><i>s law</i> is the relationship that connects stresses to strains. Although Hooke's original law was developed for uniaxial stresses, you can use a generalized version of Hooke's law to connect stress and strain in three-dimensional objects, as well. Eventually, Hooke's law helps you relate stresses (which are based on loads) to strains (which are based on deformations).\r\n\r\nFor a three-dimensional state of stress, the normal strain in a given direction (such as <i>x</i>) is a function of the stresses in all three orthogonal directions (usually the Cartesian <i>x</i>-<i>, </i><i>y-,</i><i> </i>and <i>z-</i>directions), as shown by this equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/273815.image0.jpg\" alt=\"image0.jpg\" width=\"201\" height=\"29\" />\r\n\r\nwhere <i>E</i> is the modulus of elasticity and ν is Poisson's ratio for the material. For a uniaxial stress, two of the stresses in the equation are zero. For a biaxial stress condition, one of the stresses in this equation is zero.\r\n\r\nThe generalized relationship for Hooke's law for shear in the XY plane can be given as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/273816.image1.jpg\" alt=\"image1.jpg\" width=\"155\" height=\"29\" />","blurb":"","authors":[{"authorId":10166,"name":"James H. Allen III","slug":"james-h-allen","description":"James H. Allen III, PE, PhD, is an assistant professor of civil engineering and a registered professional engineer. 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Using a Generalized Hooke's Law for Stress and Strain

By: James H. Allen III and III and

Updated: 03-20-2024

From The Book: Mechanics of Materials For Dummies

Mechanics of Materials For Dummies

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In mechanics of materials, Hooke's law is the relationship that connects stresses to strains. Although Hooke's original law was developed for uniaxial stresses, you can use a generalized version of Hooke's law to connect stress and strain in three-dimensional objects, as well. Eventually, Hooke's law helps you relate stresses (which are based on loads) to strains (which are based on deformations).

For a three-dimensional state of stress, the normal strain in a given direction (such as x) is a function of the stresses in all three orthogonal directions (usually the Cartesian x-, y-, and z-directions), as shown by this equation:

where E is the modulus of elasticity and ν is Poisson's ratio for the material. For a uniaxial stress, two of the stresses in the equation are zero. For a biaxial stress condition, one of the stresses in this equation is zero.

The generalized relationship for Hooke's law for shear in the XY plane can be given as

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Mechanics of Materials For Dummies ,

About the book author:

James H. Allen III, PE, PhD is an assistant professor of civil engineering and a registered professional engineer. His specialty areas include structural engineering, numerical analysis and error control, and steel design. He is also the author of Statics For Dummies.

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