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.[1]

Infinitesimal strain tensor edit

For infinitesimal deformations of a continuum body, in which the displacement gradient tensor (2nd order tensor) is small compared to unity, i.e.  , it is possible to perform a geometric linearization of any one of the finite strain tensors used in finite strain theory, e.g. the Lagrangian finite strain tensor  , and the Eulerian finite strain tensor  . In such a linearization, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have

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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 tensor components and the spatial displacement gradient tensor 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


Geometric derivation edit

Figure 1. Two-dimensional geometric deformation of an infinitesimal material element.

Consider 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


Physical interpretation edit

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.

Strain transformation rules edit

If we choose an orthonormal coordinate system ( ) 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 ( ) 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  . In matrix form

Strain invariants edit

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 edit

It can be shown that it is possible to find a coordinate system ( ) in which the components of the strain tensor are

The components of the strain tensor in the ( ) coordinate system are called the principal strains and the directions   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   along which the strain tensor becomes a pure stretch with no shear component.

Volumetric strain edit

The volumetric strain, also called bulk strain, is the relative variation of the volume, as arising from dilation or compression; it 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   and V0 = a3, 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 edit

The infinitesimal strain tensor  , similarly to the Cauchy stress tensor, can be expressed as the sum of two other tensors:

  1. a mean strain tensor or volumetric strain tensor or spherical strain tensor,  , related to dilation or volume change; and
  2. a deviatoric component called the strain deviator tensor,  , related to distortion.

where   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 edit

Let ( ) 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   are the principal strains. [citation needed]

The normal strain on an octahedral plane is given by

[citation needed]

Equivalent strain edit

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 edit

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 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  . 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


In engineering notation,


Special cases edit

Plane strain edit

Plane strain state in a continuum.

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. 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   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.

Antiplane strain edit

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


Relation to infinitesimal rotation tensor edit

The infinitesimal strain tensor is defined as

Therefore the displacement gradient can be expressed as
The quantity   is the infinitesimal rotation tensor or infinitesimal angular displacement tensor (related to the infinitesimal rotation matrix). 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.

The axial vector edit

A skew symmetric second-order tensor has three independent scalar components. These three components are used to define an axial vector,  , as follows

where   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  .

Relation between the strain tensor and the rotation vector edit

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

Relation between rotation tensor and rotation vector edit

From an important identity regarding the curl of a tensor we know that for a continuous, single-valued displacement field  ,

Since   we have

Strain tensor in non-Cartesian coordinates edit

Strain tensor in cylindrical coordinates edit

In cylindrical polar coordinates ( ), the displacement vector can be written as

The components of the strain tensor in a cylindrical coordinate system are given by:[2]

Strain tensor in spherical coordinates edit

Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.

In spherical coordinates ( ), the displacement vector can be written as

The components of the strain tensor in a spherical coordinate system are given by [2]

See also edit

References edit

  1. ^ Boresi, Arthur P. (Arthur Peter), 1924- (2003). Advanced mechanics of materials. Schmidt, Richard J. (Richard Joseph), 1954- (6th ed.). New York: John Wiley & Sons. p. 62. ISBN 1601199228. OCLC 430194205.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  2. ^ a b Slaughter, William S. (2002). The Linearized Theory of Elasticity. New York: Springer Science+Business Media. doi:10.1007/978-1-4612-0093-2. ISBN 9781461266082.

External links edit