Chapter 5
Plasticity
I have discovered that it is enough when a single note is
beautifully played.
— Arvo Pärt, b. 1935 —
Plasticity is the paradigm for non-reversible, rate-independent material behavior.
Experimental evidence for various classes of materials, in particular for ductile metals, suggests to decompose the total strain into an elastic, stress producing part and
a plastic, irreversible part. Irreversibility is rooted in sub-scale mechanisms in the
material, e.g. dislocation motion in ductile metallic materials with crystalline subscale structure. Conceptually, the plastic part of the strain remains after removing
the external load, however, in general together with reversible residual strains. These
only vanish when the material is allowed to locally relax into a non-strained state
(which usually requires to sacrifice global compatibility). The onset and evolution of
irreversibility are defined by a yield condition and a flow rule, which are fundamental
concepts at the core of plasticity. They are intimately related to the non-smoothness
of the convex dissipation potential and its dual.
The elementary rheological model to capture plastic, i.e. non-reversible, rateindependent material behavior is the frictional slider. The rheological model for a
perfect rigid-plastic solid consisting of a frictional slider only is denoted the St.
Venant model. Parallel and serial arrangements of a frictional slider with an elastic
spring render the St. Venant hardening model for a hardening rigid-plastic solid (not
considered here) and the Prandtl model for a perfect elasto-plastic solid, respectively.
Further, the serial arrangement of an elastic spring and the St. Venant hardening model
is established as the Prandtl model for a hardening elasto-plastic solid.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. Steinmann and K. Runesson, The Catalogue of Computational Material Models,
https://doi.org/10.1007/978-3-030-63684-5_5
193
Plasticity
I have discovered that it is enough when a single note is
beautifully played.
— Arvo Pärt, b. 1935 —
Plasticity is the paradigm for non-reversible, rate-independent material behavior.
Experimental evidence for various classes of materials, in particular for ductile metals, suggests to decompose the total strain into an elastic, stress producing part and
a plastic, irreversible part. Irreversibility is rooted in sub-scale mechanisms in the
material, e.g. dislocation motion in ductile metallic materials with crystalline subscale structure. Conceptually, the plastic part of the strain remains after removing
the external load, however, in general together with reversible residual strains. These
only vanish when the material is allowed to locally relax into a non-strained state
(which usually requires to sacrifice global compatibility). The onset and evolution of
irreversibility are defined by a yield condition and a flow rule, which are fundamental
concepts at the core of plasticity. They are intimately related to the non-smoothness
of the convex dissipation potential and its dual.
The elementary rheological model to capture plastic, i.e. non-reversible, rateindependent material behavior is the frictional slider. The rheological model for a
perfect rigid-plastic solid consisting of a frictional slider only is denoted the St.
Venant model. Parallel and serial arrangements of a frictional slider with an elastic
spring render the St. Venant hardening model for a hardening rigid-plastic solid (not
considered here) and the Prandtl model for a perfect elasto-plastic solid, respectively.
Further, the serial arrangement of an elastic spring and the St. Venant hardening model
is established as the Prandtl model for a hardening elasto-plastic solid.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. Steinmann and K. Runesson, The Catalogue of Computational Material Models,
https://doi.org/10.1007/978-3-030-63684-5_5
193