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 edit

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 edit

For a uniform dilatation  ,

 

and

 

If   and  , then (in matrix notation)

 

and

 

The volume change is given by  .

Simple Shear edit

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 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 edit

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 edit

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