Continuum mechanics  


In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacementgradient theory, deals with infinitesimal deformations of a continuum body. For an infinitesimal deformation the displacements and the displacement gradients are small compared to unity, i.e., and , allowing for the geometric linearization of the Lagrangian finite strain tensor , and the Eulerian finite strain tensor , i.e. the nonlinear or secondorder terms of the finite strain tensor can be neglected. The linearized Lagrangian and Eulerian strain tensors are approximately the same and can be approximated by the infinitesimal strain tensor or Cauchy's strain tensor, . Thus,
or
The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting elastic behaviour, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.
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For infinitesimal deformations of a continuum body, in which the displacements and the displacement gradients are small compared to unity, i.e., and , it is possible for the geometric linearization of the Lagrangian finite strain tensor , and the Eulerian finite strain tensor , i.e. the nonlinear or secondorder terms of the finite strain tensor can be neglected. Thus we have
or
and
or
This linearization implies that the Lagrangian description and the Eulerian description are approximately the same as there is little difference in the material and spatial coordinates of a given material point in the continuum. Therefore, the material displacement gradient components and the spatial displacement gradient components are approximately equal. Thus we have
where are the components of the infinitesimal strain tensor , also called Cauchy's strain tensor, linear strain tensor, or small strain tensor.
or using different notation:
Furthermore, since the deformation gradient can be expressed as where is the secondorder identity tensor, we have
Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have
Considering a twodimensional deformation of an infinitesimal rectangular material element with dimensions by (Figure 1), which after deformation, takes the form of a rhombus. From the geometry of Figure 1 we have
For very small displacement gradients, i.e., , we have
The normal strain in the direction of the rectangular element is defined by
and knowing that , we have
Similarly, the normal strain in the direction, and direction, becomes
The engineering shear strain, or the change in angle between two originally orthogonal material lines, in this case line and , is defined as
From the geometry of Figure 1 we have
For small rotations, i.e. and are we have
and, again, for small displacement gradients, we have
thus
By interchanging and and and , it can be shown that
Similarly, for the  and  planes, we have
It can be seen that the tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, , as
From finite strain theory we have
For infinitesimal strains then we have
Dividing by we have
For small deformations we assume that , thus the second term of the left hand side becomes: .
Then we have
where , is the unit vector in the direction of , and the lefthandside expression is the normal strain in the direction of . For the particular case of in the direction, i.e. , we have
Similarly, for and we can find the normal strains and , respectively. Therefore, the diagonal elements of the infinitesimal strain tensor are the normal strains in the coordinate directions.
The dilatation (the relative variation of the volume) is the trace of the tensor:
Actually, if we consider a cube with an edge length a, it is a quasicube after the deformation (the variations of the angles do not change the volume) with the dimensions and V_{0} = a^{3}, thus
as we consider small deformations,
therefore the formula.
Real variation of volume (top) and the approximated one (bottom): the green drawing shows the estimated volume and the orange drawing the neglected volume
In case of pure shear, we can see that there is no change of the volume.
The infinitesimal strain tensor , similarly to the stress tensor, can be expressed as the sum of two other tensors:
where is the mean stress given by
The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:
For prescribed strain components the strain tensor equation represents a system of six differential equations for the determination of three displacements components , giving an overdetermined system. Thus, a solution does not generally exist for an arbitrary choice of strain components. Therefore, some restrictions, named compatibility equations, are imposed upon the strain components. With the addition of the three compatibility equations the number of independent equations is reduced to three, matching the number of unknown displacement components. These constraints on the strain tensor were discovered by SaintVenant, and are called the "Saint Venant compatibility equations".
The compatibility functions serve to assure a singlevalued continuous displacement function . If the elastic medium is visualized as a set of infinitesimal cubes in the unstrained state, after the medium is strained, an arbitrary strain tensor may not yield a situation in which the distorted cubes still fit together without overlapping.
In index notation, the compatibility equations are expressed as
Engineering notation 

In real engineering components, stress (and strain) are 3D tensors but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. The strains associated with length, i.e. the normal strain and the shear strains and (if the length is the 3direction) are constrained by nearby material and are small compared to the crosssectional strains. The strain tensor can then be approximated by:
in which the double underline indicates a second order tensor. This strain state is called plane strain. The corresponding stress tensor is:
in which the nonzero is needed to maintain the constraint . This stress term can be temporarily removed from the analysis to leave only the inplane terms, effectively reducing the 3D problem to a much simpler 2D problem.
Antiplane strain is another special state of strain that can occur in a body, for instance in a region close to s screw dislocation. The strain tensor for antiplane strain is given by
The infinitesimal strain tensor is defined as
Therefore the displacement gradient can be expressed as
where
The quantity is the infinitesimal rotation tensor. This tensor is skew symmetric. For infinitesimal deformations the scalar components of satisfy the condition . Note that the displacement gradient is small only if both the strain tensor and the rotation tensor are infinitesimal.
A skew symmetric secondorder tensor has three independent scalar components. These three components are used to define an axial vector, , as follows
where e_{ijk} is the permutation symbol. In matrix form
The axial vector is also called the infinitesimal rotation vector. The rotation vector is related to the displacement gradient by the relation
In index notation
If and then the material undergoes an approximate rigid body rotation of magnitude around the vector .
Given a continuous, singlevalued displacement field and the corresponding infinitesimal strain tensor , we have (see Tensor derivative (continuum mechanics))
Since a change in the order of differentiation does not change the result, . Therefore
Also
Hence
From an important identity regarding the curl of a tensor we know that for a continuous, singlevalued displacement field ,
Since we have

