Introduction Edit

  • 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 Edit

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 Edit

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 Edit

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 Edit

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 Edit

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 Edit

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 Edit

The gradient of the displacement is denoted by  .

The displacement gradient is given by


In index notation,


Strains Edit

Finite Strain Tensor Edit

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 Edit

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 Edit

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




Volume Change Due To Finite Deformation Edit

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


Volume Change Due To Infinitesimal Deformation Edit

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




The quantity   is called the dilatation.

A volume change is isochoric (volume preserving) if


Relation between axial vector and displacement Edit

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



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




Therefore, the relation between the components of   and   is


Multiplying both sides by  , we get


Recall the identity




Using the above identity, we get




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.





Relation between axial vector and strain Edit

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



The infinitesimal strain tensor is given by




Recall that




Also recall that




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