Elastic instability is a form of instability occurring in elastic systems, such as
buckling
In structural engineering, buckling is the sudden change in shape ( deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a ...
of beams and plates subject to large compressive loads.
There are a lot of ways to study this kind of instability. One of them is to use the method of
incremental deformations based on superposing a small perturbation on an equilibrium solution.
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 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 exerted by the force becomes
. The moment
equilibrium equation is given by
where
is the spring constant of the angular spring (Nm/radian). Assuming
is small enough, implementing the
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
of the
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
function and keeping the two first terms yields
which has three solutions, the trivial
, and
which is
imaginary (i.e. not physical) for
and
real
Real may refer to:
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* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
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
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
iff
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
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
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the system
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, 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
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of ...
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
In structural engineering, buckling is the sudden change in shape ( deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a ...
mode shape
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
s of the system.
See also
*
Buckling
In structural engineering, buckling is the sudden change in shape ( deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a ...
*
Cavitation (elastomers)
Cavitation is the unstable unhindered expansion of a microscopic void in a solid elastomer under the action of tensile hydrostatic stresses. This can occur whenever the hydrostatic tension exceeds 5/6 of Young's modulus
Young's modulus E, the ...
*
Drucker stability Drucker stability (also called the Drucker stability postulates) refers to a set of mathematical criteria that restrict the possible nonlinear stress-strain relations that can be satisfied by a solid material. The postulates are named after Daniel ...
Further reading
*''Theory of elastic stability'',
S. Timoshenko and J. Gere
Continuum mechanics
Structural analysis
Mechanics