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 C. Drucker. A material that does not satisfy these criteria is often found to be
unstable
In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability ...
in the sense that application of a load to a material point can lead to arbitrary
deformations at that material point unless an additional length or time scale is specified in the
constitutive relations.
The Drucker stability postulates are often invoked in nonlinear
finite element analysis
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of structural ...
. Materials that satisfy these criteria are generally well-suited for numerical analysis, while materials that fail to satisfy this criterion are likely to present difficulties (i.e. non-uniqueness or singularity) during the solution process.
Drucker's first stability criterion
Drucker's first stability criterion (first proposed by
Rodney Hill and also called Hill's stability criterion
) is a strong condition on the incremental internal energy of a material which states that the incremental internal energy can only increase. The criterion may be written as follows:
:
,
where dσ is the stress increment tensor associated with the strain increment tensor dε through the constitutive relation.
Drucker's stability postulate
Drucker's postulate is applicable to elastic-plastic materials and states that in a cycle of
plastic
Plastics are a wide range of synthetic polymers, synthetic or Semisynthesis, semisynthetic materials composed primarily of Polymer, polymers. Their defining characteristic, Plasticity (physics), plasticity, allows them to be Injection moulding ...
deformation the second degree plastic work is always positive. This postulate can be expressed in incremental form as
:
,
where dε
p is the incremental plastic strain tensor.
References
3. {{cite journal , last1=Drucker , first1=Daniel Charles , title=A definition of stable inelastic material , date=1957 , url=https://apps.dtic.mil/dtic/tr/fulltext/u2/143756.pdf, archive-url=https://web.archive.org/web/20190509075114/https://apps.dtic.mil/dtic/tr/fulltext/u2/143756.pdf, url-status=dead, archive-date=May 9, 2019
External links
Chapter 3, Constitutive Models -relations between stress and strain, Applied Mechanics of Solids, Allen Bower
Continuum mechanics
Mechanics