Elasticity/Homogeneous and inhomogeneous displacements

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Homogeneous and inhomogeneous displacements[edit | edit source]

Homogeneous Displacement Field[edit | edit source]

A displacement field is called homogeneous if

where are independent of .

Pure Strain[edit | edit source]

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 | edit source]

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 | edit source]

For a uniform dilatation ,

and

If and , then (in matrix notation)

and

The volume change is given by .

Simple Shear[edit | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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