Infinitesimal strain theory

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation.
With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the finite strain theory where the opposite assumption is made.
The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the stress analysis of structures built from relatively stiff elastic materials like concrete and steel, since a common goal in the design of such structures is to minimize their deformation under typical loads. However, this approximation demands caution in the case of thin flexible bodies, such as rods, plates, and shells which are susceptible to significant rotations, thus making the results unreliable.
Infinitesimal strain tensor.
For "infinitesimal deformations" of a continuum body, in which the displacement gradient (2nd order tensor) is small compared to unity, i.e. formula_1, 
it is possible to perform a "geometric linearization" of any one of the (infinitely many possible) strain tensors used in finite strain theory, e.g. the Lagrangian strain tensor formula_2, and the Eulerian strain tensor formula_3. In such a linearization, the non-linear or second-order terms of the finite strain tensor are 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
or
formula_9
where formula_10 are the components of the "infinitesimal strain tensor" formula_11, 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 formula_14 where formula_15 is the second-order identity tensor, we have
Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have
Geometric derivation.
Consider a two-dimensional deformation of an infinitesimal rectangular material element with dimensions formula_18 by formula_19 (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., formula_1, we have
The normal strain in the formula_23-direction of the rectangular element is defined by
and knowing that formula_25, we have
Similarly, the normal strain in the formula_27-direction, and formula_28-direction, becomes
The engineering shear strain, or the change in angle between two originally orthogonal material lines, in this case line formula_30 and formula_31, is defined as
From the geometry of Figure 1 we have
For small rotations, i.e. formula_34 and formula_35 are formula_36 we have
and, again, for small displacement gradients, we have
thus
By interchanging formula_23 and formula_27 and formula_42 and formula_43, it can be shown that formula_44
Similarly, for the formula_27-formula_28 and formula_23-formula_28 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, formula_50, as
formula_51
Physical interpretation.
From finite strain theory we have
For infinitesimal strains then we have
Dividing by formula_54 we have
For small deformations we assume that formula_56, thus the second term of the left hand side becomes: formula_57.
Then we have
where formula_59, is the unit vector in the direction of formula_60, and the left-hand-side expression is the normal strain formula_61 in the direction of formula_62. For the particular case of formula_62 in the formula_64 direction, i.e. formula_65, we have
Similarly, for formula_67 and formula_68 we can find the normal strains formula_69 and formula_70, respectively. Therefore, the diagonal elements of the infinitesimal strain tensor are the normal strains in the coordinate directions.
Strain transformation rules.
If we choose an orthonormal coordinate system (formula_71) we can write the tensor in terms of components with respect to those base vectors as
In matrix form,
We can easily choose to use another orthonormal coordinate system (formula_74) instead. In that case the components of the tensor are different, say
The components of the strain in the two coordinate systems are related by
where the Einstein summation convention for repeated indices has been used and formula_77. In matrix form
or
Strain invariants.
Certain operations on the strain tensor give the same result without regard to which orthonormal coordinate system is used to represent the components of strain. The results of these operations are called strain invariants. The most commonly used strain invariants are
In terms of components
Principal strains.
It can be shown that it is possible to find a coordinate system (formula_82) in which the components of the strain tensor are
The components of the strain tensor in the (formula_82) coordinate system are called the principal strains and the directions formula_85 are called the directions of principal strain. Since there are no shear strain components in this coordinate system, the principal strains represent the maximum and minimum stretches of an elemental volume.
If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an eigenvalue decomposition determined by solving the system of equations
This system of equations is equivalent to finding the vector formula_85 along which the strain tensor becomes a pure stretch with no shear component.
Volumetric strain.
The "dilatation" (the relative variation of the volume) is the first strain invariant or 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 formula_89 and "V"0 = "a"3, thus
as we consider small deformations,
therefore the formula.
In case of pure shear, we can see that there is no change of the volume.
Strain deviator tensor.
The infinitesimal strain tensor formula_10, similarly to the Cauchy stress tensor, can be expressed as the sum of two other tensors: 
where formula_96 is the mean strain given by
The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:
Octahedral strains.
Let (formula_99) be the directions of the three principal strains. An octahedral plane is one whose normal makes equal angles with the three principal directions. The engineering shear strain on an octahedral plane is called the octahedral shear strain and is given by
where formula_101 are the principal strains. 
The normal strain on an octahedral plane is given by
Equivalent strain.
A scalar quantity called the equivalent strain, or the von Mises equivalent strain, is often used to describe the state of strain in solids. Several definitions of equivalent strain can be found in the literature. A definition that is commonly used in the literature on plasticity is
This quantity is work conjugate to the equivalent stress defined as
Compatibility equations.
For prescribed strain components formula_10 the strain tensor equation formula_106 represents a system of six differential equations for the determination of three displacements components formula_107, 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 are 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 formula_107. If the elastic medium is visualised 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
Special cases.
Plane strain.
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 formula_70 and the shear strains formula_111 and formula_112 (if the length is the 3-direction) are constrained by nearby material and are small compared to the "cross-sectional strains". Plane strain is then an acceptable approximation. The strain tensor for plane strain is written as:
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 formula_115 is needed to maintain the constraint formula_116. 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.
Antiplane strain.
Antiplane strain is another special state of strain that can occur in a body, for instance in a region close to a screw dislocation. The strain tensor for antiplane strain is given by
Infinitesimal rotation tensor.
The infinitesimal strain tensor is defined as
Therefore the displacement gradient can be expressed as
where
The quantity formula_121 is the infinitesimal rotation tensor. This tensor is skew symmetric. For infinitesimal deformations the scalar components of formula_121 satisfy the condition formula_123. Note that the displacement gradient is small only if both the strain tensor and the rotation tensor are infinitesimal.
The axial vector.
A skew symmetric second-order tensor has three independent scalar components. These three components are used to define an axial vector, formula_124, as follows
where formula_126 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 formula_130 and formula_131 then the material undergoes an approximate rigid body rotation of magnitude formula_132 around the vector formula_124.
Relation between the strain tensor and the rotation vector.
Given a continuous, single-valued displacement field formula_134 and the corresponding infinitesimal strain tensor formula_135, we have (see Tensor derivative (continuum mechanics))
Since a change in the order of differentiation does not change the result, formula_137. Therefore
Also
Hence
Relation between rotation tensor and rotation vector.
From an important identity regarding the curl of a tensor we know that for a continuous, single-valued displacement field formula_134,
Since formula_143 we have
formula_144
Strain tensor in cylindrical coordinates.
In cylindrical polar coordinates (formula_145), the displacement vector can be written as
The components of the strain tensor in a cylindrical coordinate system are given by:
formula_147
Strain tensor in spherical coordinates.
In spherical coordinates (formula_148), the displacement vector can be written as
The components of the strain tensor in a spherical coordinate system are given by 