In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient 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 non-linear or second-order 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,
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.
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 non-linear or second-order terms of the finite strain tensor can be neglected. Thus we have
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 second-order identity tensor, we have
Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have
Considering a two-dimensional 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
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 left-hand-side 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 quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions and V0 = a3, 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 over-determined 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 Saint-Venant, and are called the "Saint Venant compatibility equations".
The compatibility functions serve to assure a single-valued 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
In real engineering components, stress (and strain) are 3-D 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 3-direction) are constrained by nearby material and are small compared to the cross-sectional 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 non-zero is needed to maintain the constraint . This stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3-D problem to a much simpler 2-D problem.
The infinitesimal strain tensor is defined as
Therefore the displacement gradient can be expressed as
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 second-order tensor has three independent scalar components. These three components are used to define an axial vector, , as follows
where eijk 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, single-valued 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
From an important identity regarding the curl of a tensor we know that for a continuous, single-valued displacement field ,
Since we have