Elasticity/Homogeneous and inhomogeneous displacements

Homogeneous and inhomogeneous displacements

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Homogeneous Displacement Field

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A displacement field   is called homogeneous if

 

where   are independent of  .

Pure Strain

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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
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For 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
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For a uniform dilatation  ,

 

and

 

If   and  , then (in matrix notation)

 

and

 

The volume change is given by  .

Simple Shear
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For 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

  1. If   is a homogeneous displacement field, then  , where   is a rigid displacement and   is a pure strain from an arbitrary point  .
  2. Every pure strain   can be decomposed into the the sum of three simple extensions in mutually perpendicular directions,  .
  3. Every pure strain   can be decomposed into a uniform dilatation and an isochoric pure strain,   where  ,  , and  .
  4. Every simple shear   of amount   with respect to the direction pair ( ) can be decomposed into the sum of two simple extensions of the amount   in the directions  .
  5. Every simple shear is isochoric. Every isochoric pure strain is the sum of simple shears.

Inhomogeneous Displacement Field

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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

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Let   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

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Let   be a displacement field on B that is   continuous and let   on  . Then,