Infinitesimal strain theory explained

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

For infinitesimal deformations of a continuum body, in which the displacement gradient tensor (2nd order tensor) is small compared to unity, i.e.

\|\nabla u\| \ll 1 , 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 E , and the Eulerian finite strain tensor e . In such a linearization, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have

$$\backslash mathbf\; E\; =\; \backslash frac\; \backslash left(\backslash nabla\_\backslash mathbf\; u\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\backslash nabla\_\backslash mathbf\; u\backslash right)\backslash approx\; \backslash frac\backslash left(\backslash nabla\_\backslash mathbf\; u\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\backslash right)$$or$$E\_=\; \backslash frac\; \backslash left(\backslash frac\; +\backslash frac+\; \backslash frac\; \backslash frac\backslash right)\backslash approx\; \backslash frac\backslash left(\backslash frac+\backslash frac\backslash right)$$and$$\backslash mathbf\; e\; =\backslash frac\; \backslash left(\backslash nabla\_\backslash mathbf\; u\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\; -\; \backslash nabla\_\backslash mathbf\; u(\backslash nabla\_\backslash mathbf\; u)^T\backslash right)\backslash approx\; \backslash frac\backslash left(\backslash nabla\_\backslash mathbf\; u\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\backslash right)$$or$$e\_=\backslash frac\; \backslash left(\backslash frac\; +\backslash frac\; -\backslash frac\; \backslash frac\backslash right)\backslash approx\; \backslash frac\backslash left(\backslash frac\; +\backslash frac\backslash right)$$

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$$\backslash mathbf\; E\; \backslash approx\; \backslash mathbf\; e\; \backslash approx\; \backslash boldsymbol\; \backslash varepsilon\; =\; \backslash frac\backslash left((\backslash nabla\backslash mathbf\; u)^T\; +\; \backslash nabla\backslash mathbf\; u\backslash right)$$or$$E\_\backslash approx\; e\_\backslash approx\backslash varepsilon\_\; =\; \backslash frac\; \backslash left(u\_+u\_\backslash right)$$where

\varepsilon _{ij} are the components of the infinitesimal strain tensor \boldsymbol \varepsilon , 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

\boldsymbol{F} = \boldsymbol{\nabla} u + \boldsymbol{I} where \boldsymbol{I} is the second-order identity tensor, we have$$\backslash boldsymbol\backslash varepsilon\; =\; \backslash frac\; \backslash left(\backslash boldsymbol^T+\backslash boldsymbol\backslash right)-\backslash boldsymbol$$

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

dx by dy (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.,

\|\nabla u\| \ll 1 , we have$$\backslash overline\; \backslash approx\; dx\; +\; \backslash frac\; dx$$

The normal strain in the

x -direction of the rectangular element is defined by$$\backslash varepsilon\_x\; =\; \backslash frac$$and knowing that \overline {AB} = dx , we have$$\backslash varepsilon\_x\; =\; \backslash frac$$

Similarly, the normal strain in the and becomes$$\backslash varepsilon\_y\; =\; \backslash frac\; \backslash quad,\; \backslash qquad\; \backslash varepsilon\_z\; =\; \backslash frac$$

The engineering shear strain, or the change in angle between two originally orthogonal material lines, in this case line

\overline {AC} and \overline {AB} , is defined as$$\backslash gamma\_=\; \backslash alpha\; +\; \backslash beta$$

From the geometry of Figure 1 we have$$\backslash tan\; \backslash alpha\; =\; \backslash frac\; =\; \backslash frac\; \backslash quad,\; \backslash qquad\; \backslash tan\; \backslash beta=\backslash frac=\backslash frac$$

For small rotations, i.e.,

\alpha and \beta are \ll 1 we have$$\backslash tan\; \backslash alpha\; \backslash approx\; \backslash alpha\; \backslash quad,\; \backslash qquad\; \backslash tan\; \backslash beta\; \backslash approx\; \backslash beta$$and, again, for small displacement gradients, we have$$\backslash alpha=\backslash frac\; \backslash quad,\; \backslash qquad\; \backslash beta=\backslash frac$$thus$$\backslash gamma\_=\; \backslash alpha\; +\; \backslash beta\; =\; \backslash frac\; +\; \backslash frac$$By interchanging x and y and u _{x} and u _{y} , it can be shown that \gamma _{xy} = \gamma _{yx} .

Similarly, for the

y -z and x -z planes, we have$$\backslash gamma\_\; =\; \backslash gamma\_\; =\; \backslash frac\; +\; \backslash frac\; \backslash quad,\; \backslash qquad\; \backslash gamma\_\; =\; \backslash gamma\_\; =\; \backslash frac\; +\; \backslash frac$$

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

From finite strain theory we have$$d\backslash mathbf^2\; -\; d\backslash mathbf^2\; =\; d\backslash mathbf\; X\; \backslash cdot\; 2\backslash mathbf\; E\; \backslash cdot\; d\backslash mathbf\; X\; \backslash quad\backslash text\backslash quad\; (dx)^2\; -\; (dX)^2\; =\; 2E\_\backslash ,dX\_K\backslash ,dX\_L$$

For infinitesimal strains then we have$$d\backslash mathbf^2\; -\; d\backslash mathbf^2\; =\; d\backslash mathbf\; X\; \backslash cdot\; 2\backslash mathbf\; \backslash cdot\; d\backslash mathbf\; X\; \backslash quad\backslash text\backslash quad\; (dx)^2\; -\; (dX)^2\; =\; 2\backslash varepsilon\_\backslash ,dX\_K\backslash ,dX\_L$$

Dividing by

(dX) ^{2} we have$$\backslash frac\backslash frac=2\backslash varepsilon\_\backslash frac\backslash frac$$

For small deformations we assume that

dx ≈ dX , thus the second term of the left hand side becomes: (dx + dX)/(dX) ≈ 2 .

Then we have$$\backslash frac\; =\; \backslash varepsilon\_N\_iN\_j\; =\; \backslash mathbf\; N\; \backslash cdot\; \backslash boldsymbol\; \backslash varepsilon\; \backslash cdot\; \backslash mathbf\; N$$where

N _{i = (dX _{i})/(dX)} , is the unit vector in the direction of d X , and the left-hand-side expression is the normal strain e _{(N)} in the direction of N . For the particular case of N in the X _{1} direction, i.e., N = I _{1} , we have$$e\_=\backslash mathbf\; I\_1\; \backslash cdot\; \backslash boldsymbol\; \backslash varepsilon\; \backslash cdot\; \backslash mathbf\; I\_1\; =\; \backslash varepsilon\_.$$

Similarly, for

N= I _{2} and N= I _{3} we can find the normal strains \varepsilon _{22} and \varepsilon _{33} , 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 (

e _{1,e} _{2,e} _{3} ) we can write the tensor in terms of components with respect to those base vectors as$$\backslash boldsymbol\; =\; \backslash sum\_^3\; \backslash sum\_^3\; \backslash varepsilon\_\; \backslash mathbf\_i\backslash otimes\backslash mathbf\_j$$In matrix form,We can easily choose to use another orthonormal coordinate system (\hat{ e }_1,\hat_2,\hat_3) instead. In that case the components of the tensor are different, sayThe components of the strain in the two coordinate systems are related by$$\backslash hat\_\; =\; \backslash ell\_~\backslash ell\_~\backslash varepsilon\_$$where the Einstein summation convention for repeated indices has been used and \ell _{ij} = \hat{ e }_i\cdot_j. In matrix form$$\backslash underline\; =\; \backslash underline\; ~\backslash underline~\; \backslash underline^T$$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 areIn terms of components

Principal strains

It can be shown that it is possible to find a coordinate system (

n _{1,n} _{2,n} _{3} ) in which the components of the strain tensor areThe components of the strain tensor in the (n _{1,n} _{2,n} _{3} ) coordinate system are called the principal strains and the directions n _{i} 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$$(\backslash underline\; -\; \backslash varepsilon\_i~\backslash underline)~\backslash mathbf\_i\; =\; \backslash underline$$This system of equations is equivalent to finding the vector

n _{i} along which the strain tensor becomes a pure stretch with no shear component.

Volumetric strain

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:$$\backslash delta=\backslash frac\; =\; I\_1\; =\; \backslash varepsilon\_\; +\; \backslash varepsilon\_\; +\; \backslash varepsilon\_$$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

a ⋅ ( 1 + \varepsilon _{11} ) x a ⋅ ( 1 + \varepsilon _{22} ) x a ⋅ ( 1 + \varepsilon _{33} ) and V₀ = a³, thus$$\backslash frac\; =\; \backslash frac$$as we consider small deformations,$$1\; \backslash gg\; \backslash varepsilon\_\; \backslash gg\; \backslash varepsilon\_\; \backslash cdot\; \backslash varepsilon\_\; \backslash gg\; \backslash varepsilon\_\; \backslash cdot\; \backslash varepsilon\_\; \backslash cdot\; \backslash varepsilon\_$$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

\varepsilon _{ij} , 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,

\varepsilon _{M\delta} _{ij} , related to dilation or volume change; and

1. a deviatoric component called the strain deviator tensor,

\varepsilon' _{ij} , related to distortion.

$$\backslash varepsilon\_=\; \backslash varepsilon\text{'}\_\; +\; \backslash varepsilon\_M\backslash delta\_$$where

\varepsilon _{M} is the mean strain given by$$\backslash varepsilon\_M\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash tfracI^e\_1$$

The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:

Octahedral strains

Let (

n _{1,} n _{2,} n _{3} ) 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$$\backslash gamma\_\; =\; \backslash tfrac\backslash sqrt$$where \varepsilon _{1,} \varepsilon _{2,} \varepsilon _{3} are the principal strains.

The normal strain on an octahedral plane is given by$$\backslash varepsilon\_\; =\; \backslash tfrac(\backslash varepsilon\_1\; +\; \backslash varepsilon\_2\; +\; \backslash varepsilon\_3)$$

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$$\backslash varepsilon\_\; =\; \backslash sqrt\; =\; \backslash sqrt\; ~;~~\; \backslash boldsymbol^\; =\; \backslash boldsymbol\; -\; \backslash tfrac\backslash mathrm(\backslash boldsymbol)~\backslash boldsymbol$$This quantity is work conjugate to the equivalent stress defined as$$\backslash sigma\_\; =\; \backslash sqrt$$

Compatibility equations

See main article: Compatibility (mechanics). For prescribed strain components

\varepsilon _{ij} the strain tensor equation u _{i,j} + u _{j,i} = 2 \varepsilon _{ij} represents a system of six differential equations for the determination of three displacements components u _{i} , 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

u _{i} . 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$$\backslash varepsilon\_+\backslash varepsilon\_-\backslash varepsilon\_-\backslash varepsilon\_=0$$

In engineering notation,

(\partial ^{2} \epsilon _{x})/(\partial y ^{2}) + (\partial ^{2} \epsilon _{y})/(\partial x ^{2}) = 2 (\partial ^{2} \epsilon _{xy})/(\partial x \partial y) (\partial ^{2} \epsilon _{y})/(\partial z ^{2}) + (\partial ^{2} \epsilon _{z})/(\partial y ^{2}) = 2 (\partial ^{2} \epsilon _{yz})/(\partial y \partial z) (\partial ^{2} \epsilon _{x})/(\partial z ^{2}) + (\partial ^{2} \epsilon _{z})/(\partial x ^{2}) = 2 (\partial ^{2} \epsilon _{zx})/(\partial z \partial x) (\partial ^{2} \epsilon _{x})/(\partial y \partial z) = (\partial)/(\partial x) \left ( - (\partial \epsilon _{yz})/(\partial x) + (\partial \epsilon _{zx})/(\partial y) + (\partial \epsilon _{xy})/(\partial z) \right) (\partial ^{2} \epsilon _{y})/(\partial z \partial x) = (\partial)/(\partial y) \left ( (\partial \epsilon _{yz})/(\partial x) - (\partial \epsilon _{zx})/(\partial y) + (\partial \epsilon _{xy})/(\partial z) \right) (\partial ^{2} \epsilon _{z})/(\partial x \partial y) = (\partial)/(\partial z) \left ( (\partial \epsilon _{yz})/(\partial x) + (\partial \epsilon _{zx})/(\partial y) - (\partial \epsilon _{xy})/(\partial z) \right)

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

\varepsilon _{33} and the shear strains \varepsilon _{13} and \varepsilon _{23} (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 \sigma _{33} is needed to maintain the constraint \epsilon _{33} = 0 . 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

See main article: Antiplane shear. 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

The infinitesimal strain tensor is defined as$$\backslash boldsymbol\; =\; \backslash frac\; [\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\}\; +\; (\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\})^T]$$Therefore the displacement gradient can be expressed as$$\backslash boldsymbol\backslash mathbf\; =\; \backslash boldsymbol\; +\; \backslash boldsymbol$$where$$\backslash boldsymbol\; :=\; \backslash frac\; [\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\}\; -\; (\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\})^T]$$The quantity

\boldsymbol{W} 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 \boldsymbol{W} satisfy the condition |W _{ij} | \ll 1 . Note that the displacement gradient is small only if 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,

w , as follows$$W\_\; =\; -\backslash epsilon\_~w\_k\; ~;~~\; w\_i\; =\; -\backslash tfrac~\backslash epsilon\_~W\_$$where \epsilon _{ijk} is the permutation symbol. In matrix formThe axial vector is also called the infinitesimal rotation vector. The rotation vector is related to the displacement gradient by the relation$$\backslash mathbf\; =\; \backslash tfrac~\; \backslash boldsymbol\; \backslash times\; \backslash mathbf$$In index notation$$w\_i\; =\; \backslash tfrac~\backslash epsilon\_~u\_$$If \lVert\boldsymbol{W}\rVert \ll 1 and \boldsymbol{\varepsilon} = \boldsymbol{ 0 } then the material undergoes an approximate rigid body rotation of magnitude | w | around the vector w .

Relation between the strain tensor and the rotation vector

Given a continuous, single-valued displacement field

u and the corresponding infinitesimal strain tensor \boldsymbol{\varepsilon} , we have (see Tensor derivative (continuum mechanics))$$\backslash boldsymbol\backslash times\backslash boldsymbol\; =\; e\_~\backslash varepsilon\_~\backslash mathbf\_k\backslash otimes\backslash mathbf\_l\; =\; \backslash tfrac~e\_~[u\_\{l,ji\}\; +\; u\_\{j,li\}]~\backslash mathbf\_k\backslash otimes\backslash mathbf\_l$$Since a change in the order of differentiation does not change the result, u _{l,ji} = u _{l,ij} . Therefore$$e\_\; u\_\; =\; (e\_+e\_)\; u\_\; +\; (e\_+e\_)\; u\_\; +\; (e\_\; +\; e\_)\; u\_\; =\; 0$$Also$$\backslash tfrac~e\_~u\_\; =\; \backslash left(\backslash tfrac~e\_~u\_\backslash right)\_\; =\; \backslash left(\backslash tfrac\; ~\; e\_~u\_\backslash right)\_\; =\; w\_$$Hence$$\backslash boldsymbol\; \backslash times\; \backslash boldsymbol\; =\; w\_~\backslash mathbf\_k\backslash otimes\backslash mathbf\_l\; =\; \backslash boldsymbol\backslash mathbf$$

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

u ,$$\backslash boldsymbol\backslash times(\backslash boldsymbol\backslash mathbf)\; =\; \backslash boldsymbol.$$Since \boldsymbol{\nabla} u = \boldsymbol{\varepsilon} + \boldsymbol{W} we have$$\backslash boldsymbol\backslash times\backslash boldsymbol\; =\; -\backslash boldsymbol\backslash times\backslash boldsymbol\; =\; -\; \backslash boldsymbol\; \backslash mathbf.$$

Strain tensor in non-Cartesian coordinates

Strain tensor in cylindrical coordinates

In cylindrical polar coordinates (

r, \theta, z ), the displacement vector can be written as$$\backslash mathbf\; =\; u\_r~\backslash mathbf\_r\; +\; u\_\backslash theta~\backslash mathbf\_\backslash theta\; +\; u\_z~\backslash mathbf\_z$$The components of the strain tensor in a cylindrical coordinate system are given by:^[2]

Strain tensor in spherical coordinates

In spherical coordinates (

r, \theta, \phi ), the displacement vector can be written as$$\backslash mathbf\; =\; u\_r~\backslash mathbf\_r\; +\; u\_\backslash theta~\backslash mathbf\_\backslash theta\; +\; u\_\backslash phi~\backslash mathbf\_\backslash phi$$The components of the strain tensor in a spherical coordinate system are given by ^[2]

See also

• Deformation (mechanics)
• Compatibility (mechanics)
• Stress tensor
• Strain gauge
• Elasticity tensor
• Stress–strain curve
• Hooke's law
• Poisson's ratio
• Finite strain theory
• Strain rate
• Plane stress
• Digital image correlation

Notes and References

1. Book: Boresi, Arthur P. (Arthur Peter), 1924– . Advanced mechanics of materials . 2003 . John Wiley & Sons . Schmidt, Richard J. (Richard Joseph), 1954– . 1601199228 . 6th . New York . 62 . 430194205.
2. Book: Slaughter . William S. . The Linearized Theory of Elasticity . 2002 . Springer Science+Business Media . New York . 9781461266082 . 10.1007/978-1-4612-0093-2.