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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.
Contents
- Infinitesimal strain tensor
- Geometric derivation of the infinitesimal strain tensor
- Physical interpretation of the infinitesimal strain tensor
- Strain transformation rules
- Strain invariants
- Principal strains
- Volumetric strain
- Strain deviator tensor
- Octahedral strains
- Equivalent strain
- Compatibility equations
- Plane strain
- Antiplane strain
- Infinitesimal rotation tensor
- The axial vector
- Relation between the strain tensor and the rotation vector
- Relation between rotation tensor and rotation vector
- Strain tensor in cylindrical coordinates
- Strain tensor in spherical coordinates
- References
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.
Infinitesimal strain tensor
For infinitesimal deformations of a continuum body, in which the displacement (vector) and the displacement gradient (2nd order tensor) are small compared to unity, i.e.,
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
where
or using different notation:
Furthermore, since the deformation gradient can be expressed as
Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have
Geometric derivation of the infinitesimal strain tensor
Consider a two-dimensional deformation of an infinitesimal rectangular material element with dimensions
For very small displacement gradients, i.e.,
The normal strain in the
and knowing that
Similarly, the normal strain in the
The engineering shear strain, or the change in angle between two originally orthogonal material lines, in this case line
From the geometry of Figure 1 we have
For small rotations, i.e.
and, again, for small displacement gradients, we have
thus
By interchanging
Similarly, for the
It can be seen that the tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition,
Physical interpretation of the infinitesimal strain tensor
From finite strain theory we have
For infinitesimal strains then we have
Dividing by
For small deformations we assume that
Then we have
where
Similarly, for
Strain transformation rules
If we choose an orthonormal coordinate system (
In matrix form,
We can easily choose to use another orthonormal coordinate system (
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
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 (
The components of the strain tensor in the (
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
Volumetric strain
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
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.
Strain deviator tensor
The infinitesimal strain tensor
- a mean strain tensor or volumetric strain tensor or spherical strain tensor,
ε M δ i j , related to dilation or volume change; and - a deviatoric component called the strain deviator tensor,
ε i j ′ , related to distortion.
where
The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:
Octahedral strains
Let (
where
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
The compatibility functions serve to assure a single-valued continuous displacement function
In index notation, the compatibility equations are expressed as
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
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
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
The axial vector
A skew symmetric second-order tensor has three independent scalar components. These three components are used to define an axial vector,
where
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
Relation between the strain tensor and the rotation vector
Given a continuous, single-valued displacement field
Since a change in the order of differentiation does not change the result,
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
Since
Strain tensor in cylindrical coordinates
In cylindrical polar coordinates (
The components of the strain tensor in a cylindrical coordinate system are given by
Strain tensor in spherical coordinates
In spherical coordinates (
The components of the strain tensor in a spherical coordinate system are given by