Elasticity/Homogeneous and inhomogeneous displacements
Homogeneous and inhomogeneous displacements edit
Homogeneous Displacement Field edit
A displacement field is called homogeneous if
where are independent of .
Pure Strain edit
If and , then is called a pure strain from , i.e.,
Examples of pure strain If is a given point, , and is an orthonormal basis, then Simple Extension editFor a simple extension in the direction of the unit vector and If and , then (in matrix notation) and The volume change is given by . Uniform Dilatation editFor a uniform dilatation , and If and , then (in matrix notation) and The volume change is given by . Simple Shear editFor a simple shear with respect to the perpendicular unit vectors and , and If , , , and , then (in matrix notation) The volume change is given by . |
Properties of homogeneous displacement fields
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Inhomogeneous Displacement Field edit
Any displacement field that does not satisfy the condition of homogeneity is inhomogenous. Most deformations in engineering materials lead to inhomogeneous displacements.
Properties of inhomogeneous displacement fields Average strain editLet be a displacement field, be the corresponding strain field. Let and be continuous on B. Then, the mean strain depends only on the boundary values of . where is the unit normal to the infinitesimal surface area . Korn's Inequality editLet be a displacement field on B that is continuous and let on . Then, |