Infinitesimal strain theory

The infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with "infinitesimal deformations" of a continuum body. For an infinitesimal deformation the displacements and the displacement gradients are small compared to unity, i.e., $|mathbf\; u|\; ll\; 1$ and $|\; abla\; mathbf\; u|\; ll\; 1$, allowing for the "geometric linearization" of the Lagrangian finite strain tensor $mathbf\; E$, and the Eulerian finite strain tensor $mathbf\; e$, i.e. the non-linear or second-order terms of the finite strain tensor can be neglected. The linearized Lagrangian and Eulerian strain tensors are approximately the same and can be approximated by the infinitesimal strain tensor or Cauchy's strain tensor, $varepsilon$. Thus,

:$mathbf\; E\; approx\; mathbf\; e\; approx\; varepsilon\; =\; frac\{1\}\{2\}left(mathbf\; u\; abla^T\; +\; mathbf\; u\; abla\; ight)$or:$E\_\{KL\}approx\; e\_\{rs\}approxvarepsilon\_\{ij\}=frac\{1\}\{2\}left(u\_\{i,j\}+u\_\{j,i\}\; ight)$

The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting elastic 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, in which the displacements and the displacement gradients are small compared to unity, i.e., $|mathbf\; u|\; ll\; 1$ and $|\; abla\; mathbf\; u|\; ll\; 1$, it is possible for the "geometric linearization" of the Lagrangian finite strain tensor $mathbf\; E$, and the Eulerian finite strain tensor $mathbf\; e$, i.e. the non-linear or second-order terms of the finite strain tensor can be neglected. Thus we have

:$mathbf\; E\; =frac\{1\}\{2\}left(mathbf\; u\; abla\_\{mathbf\; X\}^T\; +\; mathbf\; u\; abla\_\{mathbf\; X\}\; +\; mathbf\; u\; abla\_\{mathbf\; X\}^T\; mathbf\; u\; abla\_\{mathbf\; X\}\; ight)approx\; frac\{1\}\{2\}left(mathbf\; u\; abla\_\{mathbf\; X\}^T\; +\; mathbf\; u\; abla\_\{mathbf\; X\}\; ight)$or:$E\_\{KL\}=frac\{1\}\{2\}left(frac\{partial\; U\_K\}\{partial\; X\_L\}+frac\{partial\; U\_L\}\{partial\; X\_K\}+frac\{partial\; U\_M\}\{partial\; X\_K\}frac\{partial\; U\_M\}\{partial\; X\_L\}\; ight)approx\; frac\{1\}\{2\}left(frac\{partial\; U\_K\}\{partial\; X\_L\}+frac\{partial\; U\_L\}\{partial\; X\_K\}\; ight)$and:$mathbf\; e\; =frac\{1\}\{2\}left(mathbf\; u\; abla\_\{mathbf\; x\}^T\; +\; mathbf\; u\; abla\_\{mathbf\; x\}\; -\; mathbf\; u\; abla\_\{mathbf\; x\}^T\; mathbf\; u\; abla\_\{mathbf\; x\}\; ight)approx\; frac\{1\}\{2\}left(mathbf\; u\; abla\_\{mathbf\; x\}^T\; +\; mathbf\; u\; abla\_\{mathbf\; x\}\; ight)$or:$e\_\{rs\}=frac\{1\}\{2\}left(frac\{partial\; u\_r\}\{partial\; x\_s\}\; +frac\{partial\; u\_s\}\{partial\; x\_r\}-frac\{partial\; u\_k\}\{partial\; x\_r\}frac\{partial\; u\_k\}\{partial\; x\_s\}\; ight)approx\; frac\{1\}\{2\}left(frac\{partial\; u\_r\}\{partial\; x\_s\}\; +frac\{partial\; u\_s\}\{partial\; x\_r\}\; ight)$

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

:$mathbf\; E\; approx\; mathbf\; e\; approx\; varepsilon\; =\; frac\{1\}\{2\}left(mathbf\; u\; abla^T\; +\; mathbf\; u\; abla\; ight)\; qquad\; ext\{or\}\; qquad\; E\_\{KL\}approx\; e\_\{rs\}approxvarepsilon\_\{ij\}=frac\{1\}\{2\}left(u\_\{i,j\}+u\_\{j,i\}\; ight)$

where $varepsilon\_\{ij\}$ are the components of the "infinitesimal strain tensor", $varepsilon$, also called "Cauchy's strain tensor", "linear strain tensor", or "small strain tensor".

:

left [egin{matrix}varepsilon_{11} & varepsilon_{12} & varepsilon_{13} \ varepsilon_{21} & varepsilon_{22} & varepsilon_{23} \ varepsilon_{31} & varepsilon_{32} & varepsilon_{33} \ end{matrix} ight] &=left [egin{matrix} frac{partial u_1}{partial x_1} & frac{1}{2} left(frac{partial u_1}{partial x_2}+frac{partial u_2}{partial x_1} ight) & frac{1}{2} left(frac{partial u_1}{partial x_3}+frac{partial u_3}{partial x_1} ight) \ frac{1}{2} left(frac{partial u_2}{partial x_1}+frac{partial u_1}{partial x_2} ight) & frac{partial u_2}{partial x_2} & frac{1}{2} left(frac{partial u_2}{partial x_3}+frac{partial u_3}{partial x_2} ight) \ frac{1}{2} left(frac{partial u_3}{partial x_1}+frac{partial u_1}{partial x_3} ight) & frac{1}{2} left(frac{partial u_3}{partial x_2}+frac{partial u_2}{partial x_3} ight) & frac{partial x_3}{partial x_3} \ end{matrix} ight] \

left [egin{matrix}varepsilon_{xx} & varepsilon_{xy} & varepsilon_{xz} \ varepsilon_{yx} & varepsilon_{yy} & varepsilon_{yz} \ varepsilon_{zx} & varepsilon_{zy} & varepsilon_{zz} \ end{matrix} ight] &=left [egin{matrix} frac{partial u_x}{partial x} & frac{1}{2} left(frac{partial u_x}{partial y}+frac{partial u_y}{partial x} ight) & frac{1}{2} left(frac{partial u_x}{partial z}+frac{partial u_z}{partial x} ight) \ frac{1}{2} left(frac{partial u_y}{partial x}+frac{partial u_x}{partial y} ight) & frac{partial u_y}{partial y} & frac{1}{2} left(frac{partial u_y}{partial z}+frac{partial u_z}{partial y} ight) \ frac{1}{2} left(frac{partial u_z}{partial x}+frac{partial u_x}{partial z} ight) & frac{1}{2} left(frac{partial u_z}{partial y}+frac{partial u_y}{partial z} ight) & frac{partial u_z}{partial z} \ end{matrix} ight] end{align} ,

Furthermore,

:$varepsilon=frac\{1\}\{2\}left(mathbf\; F+mathbf\; F^T\; ight)-mathbf\; I$

Considering the general expression for the Lagrangian finite strain tensor and the Eulerian finite strain tensor we have

:$mathbf\; E\_\{(m)\}=frac\{1\}\{2m\}(mathbf\; U^\{2m\}-I)approx\; varepsilon$

:$mathbf\; e\_\{(m)\}=frac\{1\}\{2m\}(mathbf\; V^\{2m\}-I)approx\; varepsilon$

Geometric derivation of the infinitesimal strain tensor

Considering 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 form. From the geometry of Figure 1 we have

:

For very small displacement gradients, i.e., $|\; abla\; mathbf\; u|\; ll\; 1$, we have

:$overline\; \{ab\}\; approx\; dx\; +frac\{partial\; u\_x\}\{partial\; x\}dx$

The normal strain in the $x$-direction of the rectangular element is defined by

:$varepsilon\_x\; =\; frac\{overline\; \{ab\}-overline\; \{AB\{overline\; \{AB$and knowing that $overline\; \{AB\}=\; dx$, we have

:$varepsilon\_x\; =\; frac\{partial\; u\_x\}\{partial\; x\}$

Similarly, the normal strain in the $y$-direction, and $z$-direction, becomes:$varepsilon\_y\; =\; frac\{partial\; u\_y\}\{partial\; y\}\; quad\; ,\; qquad\; varepsilon\_z\; =\; frac\{partial\; u\_z\}\{partial\; z\}$

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

:

From the geometry of Figure 1 we have

:

For small rotations, i.e. $alpha$ and are $ll\; 1$ we have

:

and, again, for small displacement gradients, we have

:

thus

: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

:$gamma\_\{yz\}=gamma\_\{zy\}\; =\; frac\{partial\; u\_y\}\{partial\; z\}\; +\; frac\{partial\; u\_z\}\{partial\; y\}\; quad\; ,\; qquad\; gamma\_\{zx\}=gamma\_\{xz\}=\; frac\{partial\; u\_z\}\{partial\; x\}\; +\; frac\{partial\; u\_x\}\{partial\; z\}$

It can be seen that the tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, $gamma$, as

Physical interpretation of the infinitesimal strain tensor

Principal strains

Volumetric strain

The "dilatation" (the relative variation of the volume) ? = ?"V"/"V"₀, is the trace of the tensor::$delta=frac\{Delta\; V\}\{V\_0\}\; =\; varepsilon\_\{11\}\; +\; varepsilon\_\{22\}\; +\; varepsilon\_\{33\}$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 $a\; cdot\; (1\; +\; varepsilon\_\{11\})\; imes\; a\; cdot\; (1\; +\; varepsilon\_\{22\})\; imes\; a\; cdot\; (1\; +\; varepsilon\_\{33\})$ and "V"₀ = "a"³, thus:$frac\{Delta\; V\}\{V\_0\}\; =\; frac\{left\; (\; 1\; +\; varepsilon\_\{11\}\; +\; varepsilon\_\{22\}\; +\; varepsilon\_\{33\}\; +\; varepsilon\_\{11\}\; cdot\; varepsilon\_\{22\}\; +\; varepsilon\_\{11\}\; cdot\; varepsilon\_\{33\}+\; varepsilon\_\{22\}\; cdot\; varepsilon\_\{33\}\; +\; varepsilon\_\{11\}\; cdot\; varepsilon\_\{22\}\; cdot\; varepsilon\_\{33\}\; ight\; )\; cdot\; a^3\; -\; a^3\}\{a^3\}$as we consider small deformations,:$1\; gg\; varepsilon\_\{ii\}\; gg\; varepsilon\_\{ii\}\; cdot\; varepsilon\_\{jj\}\; gg\; varepsilon\_\{11\}\; cdot\; varepsilon\_\{22\}\; cdot\; varepsilon\_\{33\}$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.

train deviator tensor

Octahedral strains

Compatibility equations

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 is 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 visualized 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:$varepsilon\_\{ij,km\}+varepsilon\_\{km,ij\}-varepsilon\_\{ik,jm\}-varepsilon\_\{jm,ik\}=0$

:

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 $epsilon\_\{33\}$ and the shear strains $epsilon\_\{13\}$ and $epsilon\_\{23\}$ (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. 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.

ee also

*Deformation (mechanics)
*Stress
*Strain gauge
*Stress-strain curve
*Hooke's law
*Poisson's ratio
*Finite strain theory
*Strain Rate
*plane stress
*Digital image correlation

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