Strain for scientists and engineers

This page has two purposes:

• To introduce Cauchy's infinitesimal strain at the freshman college level
• To summarize available resources on Wikipedia:

Infinitesimal strain theory begins by defining Cauchy's strain tensor using a variety of notations. A notation not mentioned is very easy to use when writing by hand. It uses multiple underlines to identify the tensor's rank:

$${\displaystyle {\underline {\underline {\varepsilon }}}\,=\,{\tfrac {1}{2}}\left({\underline {\nabla }}\,{\underline {u}}({\underline {r}})+{\underline {\nabla }}\,{\underline {u}}^{T}({\underline {r}})\right)={\underline {\nabla }}\,{\underline {u}}_{\,S}(\,{\underline {r}}),}$$

where the number of underlines establishes the rank of the tensor, "T" denotes transpose, and "S" is used to established that the only the symmetric part of the tensor is included. The section #Geometric_derivation uses Figure 2 to help the reader visualize all this. This article also:

1. States transformation rules for ${\displaystyle {\underline {\underline {\varepsilon }}}}$ under 3-dimensional rotations, expressed in terms of the basis vectors.^[1]
2. Identifies three invariants under these rotations (including determinant and trace.)
3. Expresses ${\displaystyle {\underline {\underline {\varepsilon }}}}$ in cartesian and cylindrical coordinates.
4. Warns that infinitesimal strain theory is accurate only for very small strains, and is especially likely to yield incorrect results for thin plates and rods.

Figure 3 is not currently on Wikipedia. It illustrates the symmetric and antisymmetric components of ${\displaystyle {\underline {\nabla }}\,{\underline {u}}.}$ The symmetric part rotates the object so that the deformation's principle axes ("eigenvectors") are highlighted. The asymmetric part is a rotation that is "pure" only in the limit of small rotations (i.e., where ${\displaystyle \cos \theta \rightarrow 1.)}$

Deformation (physics) explains the distinction between a deformation and motion that is only a rigid body displacement as follows: The motion is not a deformation, but merely a rigid body displacement if all possible curves before the deformation maintain their original length after the motion has taken place. The article needs more references, but is noteworthy for its careful and detailed presentation of formulas, many of which clarify of what approximations are required for the transition from general deformations to Cauchy's more tractable infinitesimal (linearized) theory.

• The section #Affine deformation discusses deformations within the context of affine transformations.
• Cauchy's strain tensor is based on the partial derivatives of the displacement tensor. The article's discussion is based on w:Displacement (geometry) and w:Displacement field (mechanics).

• See also w:Piola–Kirchhoff stress tensors, w:Reciprocal lattice, w:Einstein notation, w:Covariance and contravariance of vectors, and w:Tensor#As_multidimensional_arrays

Finite strain theory is simultaneously essential, as well as a bit too confusing for beginners. This page resolves a difficulty that plagues a number of other Wikipedia articles: When a small portion of matter moves significantly as it is deformed, which coordinate system do you use?

The article utilizes covariant notation to almost effortlessly maintain the distinction. With this notation, both coordinate systems can be used at the same time, and the reader can quickly ascertain which coordinate system is being used. In its simplest form, we have something like this for vectors:

$${\displaystyle {\underline {V}}=\sum _{i}V_{i}{\hat {e}}^{i}=\sum _{i}V^{i}{\hat {e}}_{i}}$$

The reader needs to be informed of the nature of ${\displaystyle {\hat {e}}^{i}}$ and ${\displaystyle {\hat {e}}_{i}}$ only once. With tensors of higher rank, you have more than two choices:

$${\displaystyle {\underline {\underline {T}}}=\sum _{i,j}T_{ji}{\hat {e}}^{i}{\hat {e}}^{j}=\sum _{i,j}T_{i}^{j}{\hat {e}}_{j}{\hat {e}}^{i}=\sum _{i,j}T^{ji}{\hat {e}}_{i}{\hat {e}}_{j}}$$

This notation looks complicated, but is easy to use once you get familiar with it. This is because in most cases you can dispense with the underlines and unit vectors by utilizing summation notation:

$${\displaystyle V_{j}=T_{\,i}^{j}R^{i}}$$

Note how a lower summed subscript is always paired with a superscript. The reader automatically knows all the equivalent variations: ${\textstyle V_{j}=T_{ji}R^{i}=T_{j}^{\,i}R_{i}\,,}$ and so forth.

This notation is essential in General Relativity, and is extremely easy type (requiring only subscripts and superscripts.) A similar notation is also required in crystal structures where the three spatial basis vectors are not orthogonal, and where the natural basis vectors for displacements are orthogonal to the natural basis vectors for wavenumber.

Strain (mechanics) cryptically defines strain as as the spatial derivative of displacement: $${\displaystyle {\boldsymbol {\varepsilon }}\doteq {\cfrac {\partial }{\partial \mathbf {X} }}\left(\mathbf {x} -\mathbf {X} \right)={\boldsymbol {F}}'-{\boldsymbol {I}},}$$

where I is the identity tensor, where x = F(X), and F is defined with a link to w:Finite strain theory.

Other definitions of strain exist: Engineering (Cauchy) strain is, ${\displaystyle e(L-l)/L,}$ where ${\displaystyle l(L)}$ is the length before (after) deformation. The stretch ratio is ${\displaystyle 1+e}$, and the logarithmic strain is ${\displaystyle \ln(1+e).}$ Two other definitions of strain (Green and Euler/Alansi) are described at w:Finite strain theory.

Three types deformation theories are listed:

1. w:Finite strain theory deals with deformations in which both rotations and strains are arbitrarily large.
2. w:Infinitesimal strain theory is valid when both strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical.
3. Large-displacement or large-rotation theory assumes small strains but large rotations and/or displacements.

WARNING: I question the distinction between #2 and #3 on this list, because I don't see any reason why the rotations and displacements must be small in the linearized theory. Looking at figures 3 and 4, it seems likely that infinitesimal strain theory applies when the rotations are large. My intuition is is that it is important to distinguish between two types of rotation in infinitesimal strain theory. The large scale rotation and displacement in figure 4 is allowed and does not preclude the use of infinitesimal strain theory. On the other hand, in figure 3, the splitting of ${\displaystyle {\underline {\nabla }}\,{\underline {u}}}$ into a rotation and a deformation (i.e. symmetric and antisymmetric parts), is a different matter. It must be small, since the "rotation" is only a linear approximation to a true rotation, as can be seen in the approximations ${\displaystyle \cos \theta \approx 1}$ and ${\displaystyle \sin \theta \approx \theta .}$ The cosine approximation suggests that there is a small (second order) expansion in this so-called "rotation". That expansion releases or absorbs energy, and is ignored by the symmetrical strain tensor. Also, figure 5 shows how an arch can be constructed using infinitesimal rotations; the trick is to not excessively bend each square into a trapezoid, and allow the Lagrangian coordinate transformation to do the heavy lifting when it comes to large-scale rotations and deformations. I'm not saying that the list of three types of deformation theory is wrong, but it is misleading because types #2 and #3 involve almost the same mathematical structure.

The section #Strain_tensor reviews material found at w:Infinitesimal_strain_theory. The complexity of tensor strain in two or three dimensions are simplified in sections #Normal strain, #Shear strain, and #Volume strain.

Displacement field (mechanics) carefully defines the displacement field, beginning with writing the displacement vector. It also addresses the distinction between Lagrangian and Eulerian descriptions of the flow of matter. This article is difficult for beginners to follow. For that reason, what follows is a highly simplified attempt to explain what is behind the advanced mathematical language:

Figure 4 is designed to informally introduce introductory students to this Lagrangian and Eulerian perspectives: In contrast with most fluids, which can become chaotic, deformations of solid objects are simple (until something breaks!) Figure 4 illustrates how it often difficult to follow a fluid element, which suggests that a single stationary reference frame is best suited for the study of fluid dynamics. In contrast, it is not only easy to follow the path of atoms in a deformation, it is desirable because it permits us to focus on the interaction of these atoms with their nearest neighbors. It is these nearest neighbor interactions that define how solid matter behaves when deformations occur.

The bending of a two-dimensional "rod" in figure 4 also raises an interesting issue regarding how one visualizes strain (in both two and three dimensions). In two dimensions, a symmetric tensor has three independent terms. Two of them are "eigenvalues", or scalars that define how much stretching has occurred in two orthogonal directions. The third term orients the these two directions, and since they are orthogonal, a single angle is sufficient to establish their direction (or orientation of the "eigenvectors".)

The ellipse, rectangle, and rhombus all serve to represent the eigenvalues of a symmetrical two dimensional matrix. And all three have analogs in three dimensions. Figure 5 establishes that deformation into a trapezoid is not a tensor, but a tensor field: When bending a thin rod or sheet, elements near the outer radius are expanded, while those near the inner radius are compressed.

Deformation (engineering) is not a bad article, but the focus is on things like the elastic versus plastic regimes, fracture, strain hardening, and necking. These are important topics that have little to do with the linear algebra associated with infinitesimal stress and strain.

• browwn.edu/Departments/Engineering/Courses/En221 is interesting but not sufficiently clear.
• See also Talk:Strain_for_scientists_and_engineers#I'm_still_confused

1. ↑ The article has only two references, neither of which allows the casual reader to see the actual text. The rotation matrix expressed in terms of unit vectors is described at https://www1.udel.edu/biology/rosewc/kaap686/notes/matrices_rotations.pdf