Elastica theory
Encyclopedia
The elastica theory is a theory of mechanics of solid materials
developed by Leonhard Euler
that allows for very large scale elastic deflections of structures. Euler (1744) and Jakob Bernoulli developed the theory for elastic lines (yielding the solution known as the elastica curve) and studied buckling. Certain situations can be solved exactly by elliptic functions
. Later elastica theory was generalized by F. and E. Cosserat into a geometric theory with intrinsic directions at each point (1907).
Elastica theory is an example of bifurcation theory
. For most boundary conditions several solutions exist simultaneously.
When small deflections of a structure are to be analyzed, elastica theory is not required and an approximate solution may be found using the simpler linear elasticity theory
or (for 1-dimensional components) beam theory.
Solid mechanics
Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation...
developed by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
that allows for very large scale elastic deflections of structures. Euler (1744) and Jakob Bernoulli developed the theory for elastic lines (yielding the solution known as the elastica curve) and studied buckling. Certain situations can be solved exactly by elliptic functions
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
. Later elastica theory was generalized by F. and E. Cosserat into a geometric theory with intrinsic directions at each point (1907).
Elastica theory is an example of bifurcation theory
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
. For most boundary conditions several solutions exist simultaneously.
When small deflections of a structure are to be analyzed, elastica theory is not required and an approximate solution may be found using the simpler linear elasticity theory
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
or (for 1-dimensional components) beam theory.