Elastic instability - AbsoluteAstronomy.com

Elastic instability is a form of instability occurring in elastic systems, such as buckling

Buckling

In science, buckling is a mathematical instability, leading to a failure mode.Theoretically, buckling is caused by a bifurcation in the solution to the equations of static equilibrium...

of beams and plates subject to large compressive loads.

Single degree of freedom-systems

Consider as a simple example a rigid beam of length L, hinged in one end and free in the other, and having an angular spring

Spring (device)

A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...

attached to the hinged end. The beam is loaded in the free end by a force F acting in the compressive axial direction of the beam, see the figure to the right.

Moment equilibrium condition

Assuming a clockwise angular deflection

, the clockwise moment

Moment (physics)

In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...

exerted by the force becomes

. The moment equilibrium

Mechanical equilibrium

A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....

equation is given by

where

is the spring constant of the angular spring (Nm/radian). Assuming

is small enough, implementing the taylor expansion of the sine

Sine

In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

function and keeping the two first terms yields

which has three solutions, the trivial

, and

which is imaginary

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

(i.e. not physical) for

and real

Complex number

otherwise. This implies that for small compressive forces, the only equilibrium state is given by

, while if the force exceeds the value

there is suddenly another mode of deformation possible.

Energy method

The same result can be obtained by considering energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

relations. The energy stored in the angular spring is

and the work done by the force is simply the force multiplied by the vertical displacement of the beam end, which is

. Thus,

The energy equilibrium condition

now yields

as before (besides from the trivial

Stability of the solutions

Any solution

is stable

Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

iff

IFF

IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

a small change in the deformation angle

results in a reaction moment trying to restore the original angle of deformation. The net clockwise moment acting on the beam is

An infinitesimal

Infinitesimal

Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

clockwise change of the deformation angle

results in a moment

which can be rewritten as

since

due to the moment equilibrium condition. Now, a solution

is stable iff a clockwise change

results in a negative change of moment

and vice versa. Thus, the condition for stability becomes

The solution

is stable only for

, which is expected. By expanding the cosine term in the equation, the approximate stability condition is obtained:

for

, which the two other solutions satisfy. Hence, these solutions are stable.

Multiple degrees of freedom-systems

By attaching another rigid beam to the original system by means of an angular spring a two degrees of freedom-system is obtained. Assume for simplicity that the beam lengths and angular springs are equal. The equilibrium conditions become

where

and

are the angles of the two beams. Linearizing by assuming these angles are small yields

The non-trivial solutions to the system is obtained by finding the roots of the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

of the system matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, i.e. for

Thus, for the two degrees of freedom-system there are two critical values for the applied force F. These correspond to two different modes of deformation which can be computed from the nullspace of the system matrix. Dividing the equations by

yields

For the lower critical force the ratio is positive and the two beams deflect in the same direction while for the higher force they form a "banana" shape. These two states of deformation represent the buckling