Infinitesimal strain theory

Short description: Mathematical model for describing material deformation under stress

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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.

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]

## Contents

* 1 Infinitesimal strain tensor
* 1.1 Geometric derivation
* 1.2 Physical interpretation
* 1.3 Strain transformation rules
* 1.4 Strain invariants
* 1.5 Principal strains
* 1.6 Volumetric strain
* 1.7 Strain deviator tensor
* 1.8 Octahedral strains
* 1.9 Equivalent strain
* 2 Compatibility equations
* 3 Special cases
* 3.1 Plane strain
* 3.2 Antiplane strain
* 4 Relation to infinitesimal rotation tensor
* 4.1 The axial vector
* 4.2 Relation between the strain tensor and the rotation vector
* 4.3 Relation between rotation tensor and rotation vector
* 5 Strain tensor in non-Cartesian coordinates
* 5.1 Strain tensor in cylindrical coordinates
* 5.2 Strain tensor in spherical coordinates
* 6 See also
* 7 References
* 8 External links

## 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. ‖∇𝐮‖≪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

𝐄=12(∇𝐗𝐮+(∇𝐗𝐮)T+(∇𝐗𝐮)T∇𝐗𝐮)≈12(∇𝐗𝐮+(∇𝐗𝐮)T) or EKL=12(∂UK∂XL+∂UL∂XK+∂UM∂XK∂UM∂XL)≈12(∂UK∂XL+∂UL∂XK) and 𝐞=12(∇𝐱𝐮+(∇𝐱𝐮)T−∇𝐱𝐮(∇𝐱𝐮)T)≈12(∇𝐱𝐮+(∇𝐱𝐮)T) or ers=12(∂ur∂xs+∂us∂xr−∂uk∂xr∂uk∂xs)≈12(∂ur∂xs+∂us∂xr)

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 𝐄≈𝐞≈ε=12((∇𝐮)T+∇𝐮) or EKL≈ers≈εij=12(ui,j+uj,i) where εij are the components of the infinitesimal strain tensor ε, also called Cauchy's strain tensor, linear strain tensor, or small strain tensor.

εij=12(ui,j+uj,i)=[ε11ε12ε13ε21ε22ε23ε31ε32ε33]=[∂u1∂x112(∂u1∂x2+∂u2∂x1)12(∂u1∂x3+∂u3∂x1)12(∂u2∂x1+∂u1∂x2)∂u2∂x212(∂u2∂x3+∂u3∂x2)12(∂u3∂x1+∂u1∂x3)12(∂u3∂x2+∂u2∂x3)∂u3∂x3] or using different notation: [εxxεxyεxzεyxεyyεyzεzxεzyεzz]=[∂ux∂x12(∂ux∂y+∂uy∂x)12(∂ux∂z+∂uz∂x)12(∂uy∂x+∂ux∂y)∂uy∂y12(∂uy∂z+∂uz∂y)12(∂uz∂x+∂ux∂z)12(∂uz∂y+∂uy∂z)∂uz∂z]

Furthermore, since the deformation gradient can be expressed as F=∇𝐮+I where I is the second-order identity tensor, we have ε=12(FT+F)−I

Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have 𝐄(m)=12m(𝐔2m−I)=12m[(FTF)m−I]≈12m[{∇𝐮+(∇𝐮)T+I}m−I]≈ε𝐞(m)=12m(𝐕2m−I)=12m[(FFT)m−I]≈ε

### Geometric derivation

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

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

ab‾=(dx+∂ux∂xdx)2+(∂uy∂xdx)2=dx1+2∂ux∂x+(∂ux∂x)2+(∂uy∂x)2

For very small displacement gradients, i.e., ‖∇𝐮‖≪1, we have ab‾≈dx+∂ux∂xdx

The normal strain in the x-direction of the rectangular element is defined by εx=ab‾−AB‾AB‾ and knowing that AB‾=dx, we have εx=∂ux∂x

Similarly, the normal strain in the y-direction, and z-direction, becomes εy=∂uy∂y,εz=∂uz∂z

The engineering shear strain, or the change in angle between two originally orthogonal material lines, in this case line AC‾ and AB‾, is defined as γxy=α+β

From the geometry of Figure 1 we have tan⁡α=∂uy∂xdxdx+∂ux∂xdx=∂uy∂x1+∂ux∂x,tan⁡β=∂ux∂ydydy+∂uy∂ydy=∂ux∂y1+∂uy∂y

For small rotations, i.e., α and β are ≪1 we have tan⁡α≈α,tan⁡β≈β and, again, for small displacement gradients, we have α=∂uy∂x,β=∂ux∂y thus γxy=α+β=∂uy∂x+∂ux∂y By interchanging x and y and ux and uy, it can be shown that γxy=γyx.

Similarly, for the y-z and x-z planes, we have γyz=γzy=∂uy∂z+∂uz∂y,γzx=γxz=∂uz∂x+∂ux∂z

It can be seen that the tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, γ, as [εxxεxyεxzεyxεyyεyzεzxεzyεzz]=[εxxγxy/2γxz/2γyx/2εyyγyz/2γzx/2γzy/2εzz]

### Physical interpretation

From finite strain theory we have d𝐱2−d𝐗2=d𝐗⋅2𝐄⋅d𝐗or(dx)2−(dX)2=2EKLdXKdXL

For infinitesimal strains then we have d𝐱2−d𝐗2=d𝐗⋅2ε⋅d𝐗or(dx)2−(dX)2=2εKLdXKdXL

Dividing by (dX)2 we have dx−dXdXdx+dXdX=2εijdXidXdXjdX

For small deformations we assume that dx≈dX, thus the second term of the left hand side becomes: dx+dXdX≈2.

Then we have dx−dXdX=εijNiNj=𝐍⋅ε⋅𝐍 where Ni=dXidX, is the unit vector in the direction of d𝐗, 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 X1 direction, i.e., 𝐍=𝐈1, we have e(𝐈1)=𝐈1⋅ε⋅𝐈1=ε11.

Similarly, for 𝐍=𝐈2 and 𝐍=𝐈3 we can find the normal strains ε22 and ε33, 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 (𝐞1,𝐞2,𝐞3) we can write the tensor in terms of components with respect to those base vectors as ε=∑i=13∑j=13εij𝐞i⊗𝐞j In matrix form, ε__=[ε11ε12ε13ε12ε22ε23ε13ε23ε33] We can easily choose to use another orthonormal coordinate system (𝐞^1,𝐞^2,𝐞^3) instead. In that case the components of the tensor are different, say ε=∑i=13∑j=13ε^ij𝐞^i⊗𝐞^j⟹ε^__=[ε^11ε^12ε^13ε^12ε^22ε^23ε^13ε^23ε^33] The components of the strain in the two coordinate systems are related by ε^ij=ℓipℓjqεpq where the Einstein summation convention for repeated indices has been used and ℓij=𝐞^i⋅𝐞j. In matrix form ε^__=𝐋__ε__𝐋__T or [ε^11ε^12ε^13ε^21ε^22ε^23ε^31ε^32ε^33]=[ℓ11ℓ12ℓ13ℓ21ℓ22ℓ23ℓ31ℓ32ℓ33][ε11ε12ε13ε21ε22ε23ε31ε32ε33][ℓ11ℓ12ℓ13ℓ21ℓ22ℓ23ℓ31ℓ32ℓ33]T

### 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 I1=tr(ε)I2=12{[tr(ε)]2−tr(ε2)}I3=det⁡(ε) In terms of components I1=ε11+ε22+ε33I2=ε11ε22+ε22ε33+ε33ε11−ε122−ε232−ε312I3=ε11(ε22ε33−ε232)−ε12(ε21ε33−ε23ε31)+ε13(ε21ε32−ε22ε31)

### Principal strains

It can be shown that it is possible to find a coordinate system (𝐧1,𝐧2,𝐧3) in which the components of the strain tensor are ε__=[ε1000ε2000ε3]⟹ε=ε1𝐧1⊗𝐧1+ε2𝐧2⊗𝐧2+ε3𝐧3⊗𝐧3 The components of the strain tensor in the (𝐧1,𝐧2,𝐧3) coordinate system are called the principal strains and the directions 𝐧i 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 (ε__−εi𝐈__)𝐧i=𝟎_ This system of equations is equivalent to finding the vector 𝐧i 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: δ=ΔVV0=I1=ε11+ε22+ε33 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+ε11)×a⋅(1+ε22)×a⋅(1+ε33) and V0 = a3, thus ΔVV0=(1+ε11+ε22+ε33+ε11⋅ε22+ε11⋅ε33+ε22⋅ε33+ε11⋅ε22⋅ε33)⋅a3−a3a3 as we consider small deformations, 1≫εii≫εii⋅εjj≫ε11⋅ε22⋅ε33 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 εij, 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, εMδij, related to dilation or volume change; and
2. a deviatoric component called the strain deviator tensor, ε'ij, related to distortion.

εij=ε'ij+εMδij where εM is the mean strain given by εM=εkk3=ε11+ε22+ε333=13I1e

The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:  ε'ij=εij−εkk3δij[ε'11ε'12ε'13ε'21ε'22ε'23ε'31ε'32ε'33]=[ε11ε12ε13ε21ε22ε23ε31ε32ε33]−[εM000εM000εM]=[ε11−εMε12ε13ε21ε22−εMε23ε31ε32ε33−εM]

### Octahedral strains

Let (𝐧1,𝐧2,𝐧3) 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 γoct=23(ε1−ε2)2+(ε2−ε3)2+(ε3−ε1)2

The normal strain on an octahedral plane is given by εoct=13(ε1+ε2+ε3)

### 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 εeq=23εdev:εdev=23εijdevεijdev;εdev=ε−13tr(ε)I This quantity is work conjugate to the equivalent stress defined as σeq=32σdev:σdev

## Compatibility equations

Main article: Physics:Compatibility (mechanics)

For prescribed strain components εij the strain tensor equation ui,j+uj,i=2εij represents a system of six differential equations for the determination of three displacements components ui, 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 ui. 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 εij,km+εkm,ij−εik,jm−εjm,ik=0

In engineering notation,

* ∂2ϵx∂y2+∂2ϵy∂x2=2∂2ϵxy∂x∂y
* ∂2ϵy∂z2+∂2ϵz∂y2=2∂2ϵyz∂y∂z
* ∂2ϵx∂z2+∂2ϵz∂x2=2∂2ϵzx∂z∂x
* ∂2ϵx∂y∂z=∂∂x(−∂ϵyz∂x+∂ϵzx∂y+∂ϵxy∂z)
* ∂2ϵy∂z∂x=∂∂y(∂ϵyz∂x−∂ϵzx∂y+∂ϵxy∂z)
* ∂2ϵz∂x∂y=∂∂z(∂ϵyz∂x+∂ϵzx∂y−∂ϵxy∂z)

## Special cases

### Plane strain

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 ε33 and the shear strains ε13 and ε23 (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: ε__=[ε11ε120ε21ε220000] in which the double underline indicates a second order tensor. This strain state is called plane strain. The corresponding stress tensor is: σ__=[σ11σ120σ21σ22000σ33] in which the non-zero σ33 is needed to maintain the constraint ϵ33=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

Main article: Physics: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 ε__=[00ε1300ε23ε13ε230]

## Relation to infinitesimal rotation tensor

The infinitesimal strain tensor is defined as ε=12[∇𝐮+(∇𝐮)T] Therefore the displacement gradient can be expressed as ∇𝐮=ε+W where W:=12[∇𝐮−(∇𝐮)T] The quantity 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 W satisfy the condition |Wij|≪1. 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, 𝐰, as follows Wij=−ϵijkwk;wi=−12ϵijkWjk where ϵijk is the permutation symbol. In matrix form W__=[0−w3w2w30−w1−w2w10];𝐰_=[w1w2w3] The axial vector is also called the infinitesimal rotation vector. The rotation vector is related to the displacement gradient by the relation 𝐰=12∇×𝐮 In index notation wi=12ϵijkuk,j If ‖W‖≪1 and ε=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 ε, we have (see Tensor derivative (continuum mechanics)) ∇×ε=eijkεlj,i𝐞k⊗𝐞l=12eijk[ul,ji+uj,li]𝐞k⊗𝐞l Since a change in the order of differentiation does not change the result, ul,ji=ul,ij. Therefore eijkul,ji=(e12k+e21k)ul,12+(e13k+e31k)ul,13+(e23k+e32k)ul,32=0 Also 12eijkuj,li=(12eijkuj,i),l=(12ekijuj,i),l=wk,l Hence ∇×ε=wk,l𝐞k⊗𝐞l=∇𝐰

### 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, ∇×(∇𝐮)=0. Since ∇𝐮=ε+W we have ∇×W=−∇×ε=−∇𝐰.

## Strain tensor in non-Cartesian coordinates

### Strain tensor in cylindrical coordinates

In cylindrical polar coordinates (r,θ,z), the displacement vector can be written as 𝐮=ur𝐞r+uθ𝐞θ+uz𝐞z The components of the strain tensor in a cylindrical coordinate system are given by:[2] εrr=∂ur∂rεθθ=1r(∂uθ∂θ+ur)εzz=∂uz∂zεrθ=12(1r∂ur∂θ+∂uθ∂r−uθr)εθz=12(∂uθ∂z+1r∂uz∂θ)εzr=12(∂ur∂z+∂uz∂r)

### Strain tensor in spherical coordinates

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 (r,θ,ϕ), the displacement vector can be written as 𝐮=ur𝐞r+uθ𝐞θ+uϕ𝐞ϕ The components of the strain tensor in a spherical coordinate system are given by [2] εrr=∂ur∂rεθθ=1r(∂uθ∂θ+ur)εϕϕ=1rsin⁡θ(∂uϕ∂ϕ+ursin⁡θ+uθcos⁡θ)εrθ=12(1r∂ur∂θ+∂uθ∂r−uθr)εθϕ=12r(1sin⁡θ∂uθ∂ϕ+∂uϕ∂θ−uϕcot⁡θ)εϕr=12(1rsin⁡θ∂ur∂ϕ+∂uϕ∂r−uϕr)

## 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

## References

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.
2. ↑ 2.0 2.1 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.

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