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Biomechanics_course_gw_08 stress transformation

# Biomechanics_course_gw_08 stress transformation - • A 2-D...

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Stress with different coordinate Stress: A scalar zero-order tensor: only magnitude A vector is a first-order tensor: magnitude and direction A stress is a second-order tensor: Magnitude, direction of force and direction of face Stress is mathematically a tensor (9 components), which is defined for each spatial point independent of a coordinate However, in practice, we always need a coordinate to calculate the stress In 3-D state of stress there are 9 components (6 independent, 3 normal, 3 shear components) may be nonzero In 2-D state of stress there are 4 components (3 independent, 2 normal) may be nonzero In 1-D state of stress, only one component (normal) is nonzero The stress can be calculated in some coordinate relatively easier Various coordinate systems can be related through transformation relation The various components of stress can be related through coordinate transformation Example

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Stress transformation

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Unformatted text preview: • A 2-D state of stress relative to one Cartesian coordinate system can be related to those of other Cartesian coordinate system sharing a common origin. • The only parameter that relates the two systems is the rotating angle. • 3-D state of stress has similar transformation. • Stress is defined at every point in continuum body. • Stress components are not unique, they depend of the coordinate system. • Once the components of a coordinate are know, the others in other coordinate can be calculated through transformation. • These transformation is valid for any solid or fluid material, as long as continuum assumption is satisfied. • A shearless state of stress corresponding to one coordinate is not shearless in other coordinate. • Derive equations (2.13) through (2.22) • Example Stress transformation Stress transformation Stress transformation Stress transformation...
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