Introduction

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  • Kinematics
    • Displacements, strains and the relations between displacements and strains.
    • We view the theory of infinitesimal linear deformations as a first-order approximation of the theory of finite deformations.
    • Note that the linear theory can be derived independently of the finite theory and is completely self-consistent on its own.

Concepts and Definitions

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The following concepts and definitions are based on Gurtin (1972) and Truesdell and Noll (1992). These definitions are useful both for the linear and the nonlinear theory of elasticity.

Body

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We usually denote a body by the symbol  . A body is essentially a set of points in Euclidean space. For mathematical definition see Truesdell and Noll (1992)

Configuration

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A configuration of a body is denoted by the symbol  . A configuration of a body is just what the name suggests. Sometimes a configuration is also referred to as a placement.

Mathematically, we can think of a configuration as a smooth one-to-one mapping of a body into a region of three-dimensional Euclidean space.

Thus, we can have a reference configuration   and a current configuration  .

A one-to-one mapping is also called a homeomorphism.

Deformation

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A deformation is the relationship between two configurations and is usually denoted by  . Deformations include both volume and shape changes and rigid body motions.

For a continuous body, a deformation can be thought of as a smooth mapping from one configuration ( ) to another ( ). The inverse mapping should be possible.

This means that

 

For the inverse mapping to exist, we require that the Jacobian of the deformation is positive, i.e.,  .

Deformation Gradient

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The deformation gradient is usually denoted by   and is defined as

 

In index notation

 

For a deformation to be allowable, we must be able to invert  . That is why we require that  . Otherwise, the body may undergo deformations that are unphysical.

Displacement

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The displacement is usually denoted by the symbol  .

The displacement is defined as a vector from the location of a material point in one configuration to the location of the same material point in another configuration.

The definition is

 

In index notation

 

Displacement Gradient

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The gradient of the displacement is denoted by  .

The displacement gradient is given by

 

In index notation,

 

Strains

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Finite Strain Tensor

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The finite strain tensor ( ) is also called the Green-St. Venant Strain Tensor or the Lagrangian Strain Tensor.

This strain tensor is defined as

 

In index notation,

 

Infinitesimal Strain Tensor

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In the limit of small strains, the Lagrangian finite strain tensor reduces to the infinitesimal strain tensor ( ).

This strain tensor is defined as

 

In index notation,

 

Therefore we can see that the finite strain tensor and the infinitesimal strain tensor are related by

 

If  , then

 

For small strains,   and

 .

Infinitesimal Rotation Tensor

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For small deformation problems, in addition to small strains we can also have small rotations ( ). The infinitesimal rotation tensor is defined as

 

In index notation,

 

If   is a skew-symmetric tensor, then for any vector   we have

 

The vector   is called the axial vector of the skew-symmetric tensor.

In our case,   is the skew-symmetric infinitesimal rotation tensor. The corresponding axial vector is the rotation vector   defined as

 

where

 

Volume Change Due To Finite Deformation

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The change in volume ( ) during a finite deformation is given by

 

Volume Change Due To Infinitesimal Deformation

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The volume change during an infinitesimal deformation ( ) is given by

 

because

 

The quantity   is called the dilatation.

A volume change is isochoric (volume preserving) if

 .

Relation between axial vector and displacement

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Let   be a displacement field. The displacement gradient tensor is given by  . Let the skew symmetric part of the displacement gradient tensor (infinitesimal rotation tensor) be

 

Let   be the axial vector associated with the skew symmetric tensor  . Show that

 

Proof:

The axial vector   of a skew-symmetric tensor   satisfies the condition

 

for all vectors  . In index notation (with respect to a Cartesian basis), we have

 

Since  , we can write

 

or,

 

Therefore, the relation between the components of   and   is

 

Multiplying both sides by  , we get

 

Recall the identity

 

Therefore,

 

Using the above identity, we get

 

Rearranging,

 

Now, the components of the tensor   with respect to a Cartesian basis are given by

 

Therefore, we may write

 

Since the curl of a vector   can be written in index notation as

 

we have

 

where   indicates the  -th component of the vector inside the square brackets.

Hence,

 

Therefore,

 


Relation between axial vector and strain

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Let   be a displacement field. Let   be the strain field (infinitesimal) corresponding to the displacement field and let   be the corresponding infinitesimal rotation vector. Show that

 

Proof:

The infinitesimal strain tensor is given by

 

Therefore,

 

Recall that

 

Hence,

 

Also recall that

 

Therefore,

 


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