Drucker stability (also called the Drucker stability postulates) refers to a set of mathematical criteria that restrict the possible nonlinear
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
-
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
relations that can be satisfied by a solid material. The postulates are named after
Daniel C. Drucker
Daniel Charles Drucker (June 3, 1918 – September 1, 2001) was American civil and mechanical engineer and academic, who served as president of the Society for Experimental Stress Analysis (now Society for Experimental Mechanics) in 1960–1961, ...
. A material that does not satisfy these criteria is often found to be
unstable
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
in the sense that application of a load to a material point can lead to arbitrary
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
s at that material point unless an additional length or time scale is specified in the
constitutive relations
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
.
The Drucker stability postulates are often invoked in nonlinear
finite element analysis
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
. 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
Rodney Hill FRS (11 June 1921 – 2 February 2011) was an applied mathematician and a former Professor of Mechanics of Solids at Gonville and Caius College, Cambridge.
Career
In 1953 he was appointed Professor of Applied Mathematics at the ...
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 or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
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