Strain (materials science) - AbsoluteAstronomy.com

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

, the infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformation
Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...

s of a continuum body

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

. For an infinitesimal deformation the displacements

and the displacement gradients

are small compared to unity, i.e.,

and

, allowing for the geometric linearisation 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 linearised 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

Elasticity (physics)

In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

behaviour, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.

Infinitesimal strain tensor

For infinitesimal deformations of a continuum body

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

, in which the displacements and the displacement gradients are small compared to unity, i.e.,

and

, it is possible to perform a geometric linearisation of the Lagrangian finite strain tensor

, and the Eulerian finite strain tensor

. In such a linearisation, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have

and

This linearisation 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

Geometric derivation of the infinitesimal strain tensor

Considering a two-dimensional deformation of an infinitesimal rectangular material element with dimensions

(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

, 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 of the infinitesimal strain tensor

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

If we choose an orthonormal coordinate system

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

(

) 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

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 (

) 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

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.

Strain deviator tensor

The infinitesimal strain tensor

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

a mean strain tensor or volumetric strain tensor or spherical strain tensor, , related to dilation or volume change; and
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

Let (

) be the directions of the three principal strains. An octahedral plane is 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.

The normal strain on an octahedral plane is given by

Equivalent strain

A scalar quantity called the equivalent strain, or the von Mises

Von Mises

von Mises may refer to:* Ludwig von Mises Institute* von Mises distribution, named after Richard von Mises* von Mises yield criterion, named after Richard von Mises- People :* Dr...

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

Plasticity

Plasticity may refer to:Science* Plasticity , in physics and engineering, plasticity is the propensity of a material to undergo permanent deformation under load...

This quantity is work conjugate to the equivalent stress defined as

Compatibility equations

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

Adhémar Jean Claude Barré de Saint-Venant

Adhémar Jean Claude Barré de Saint-Venant was a mechanician and mathematician who contributed to early stress analysis and also developed the one-dimensional unsteady open channel flow shallow water equations or Saint-Venant equations that are a fundamental set of equations used in modern...

, 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

Engineering notation

Plane strain

In real engineering components, stress

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

(and strain) are 3-D tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s 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

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

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

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.

The axial vector

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

and

then the material undergoes an approximate rigid body rotation of magnitude

around the vector

Relation between the strain tensor and the rotation vector

Given a continuous, single-valued displacement field

and the corresponding infinitesimal strain tensor

, we have (see Tensor derivative (continuum mechanics)

Tensor derivative (continuum mechanics)

The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical...

)