Infinitesimal strain theory

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.

For infinitesimal deformations of a continuum body, in which the displacement (vector) and the displacement gradient (2nd order tensor) are small compared to unity, i.e., ${\displaystyle \|\mathbf {u} \|\ll 1\,\!}$ and ${\displaystyle \|\nabla \mathbf {u} \|\ll 1\,\!}$, it is possible to perform a geometric linearization of any one of the (infinitely many possible) strain tensors used in finite strain theory, e.g. the Lagrangian strain tensor ${\displaystyle \mathbf {E} \,\!}$, and the Eulerian strain tensor ${\displaystyle \mathbf {e} \,\!}$. In such a linearization, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have

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