5.2 Prandtl Model
211
The basic kinematic assumption of the Prandtl model is the additive decomposition
of the total strain into the elastic strain e (representing the elongation of the elastic
spring) and the plastic strain p (representing the elongation of the frictional slider),
i.e.
= e + p .
(5.57)
Note that the plastic strain p denotes the only element contained in the set of internal
variables α = { p } for the Prandtl model.
Ludwig Prandtl [b. 4.2.1875, Freising, Germany, d. 15.8.1953, Göttingen, Germany] was
Professor of Fluid Mechanics at the University Göttingen and the attached Kaiser Wilhelm Institute for Flow Research. He is the
founding father of the boundary layer theory
in fluid dynamics. Although mainly known for
his research in aerodynamics, he also worked
on problems of plasticity. His contribution to
dry friction from 1928 gave an atomistic explanation of the static friction force. The Prandtl
model thus describes solids displaying a yield
stress after an initial elastic phase.
5.2.1 Specific Prandtl Model: Formulation
The specific Prandtl model, displayed in Fig. 5.7, consists of a serial arrangement
of (1) a linear elastic spring with stiffness E and (2) a linear frictional slider with
threshold σ y .
For the specific Prandtl model the free energy density ψ is expressed as a quadratic
(and thus convex) function of − p (the elastic strain e )
ψ(, p ) =
1
2
E [ − p ]
2
.
(5.58)
Then the energetic stress σ
conjugated to the total strain and the energetic plastic
stress σ
p conjugated to the plastic strain p follow as
σ
(, p ) = ∂ ψ(, p ) = E [ − p ],
(5.59a)
σ
p (, p ) = ∂ p ψ(, p ) = −E [ − p ].
(5.59b)
211
The basic kinematic assumption of the Prandtl model is the additive decomposition
of the total strain into the elastic strain e (representing the elongation of the elastic
spring) and the plastic strain p (representing the elongation of the frictional slider),
i.e.
= e + p .
(5.57)
Note that the plastic strain p denotes the only element contained in the set of internal
variables α = { p } for the Prandtl model.
Ludwig Prandtl [b. 4.2.1875, Freising, Germany, d. 15.8.1953, Göttingen, Germany] was
Professor of Fluid Mechanics at the University Göttingen and the attached Kaiser Wilhelm Institute for Flow Research. He is the
founding father of the boundary layer theory
in fluid dynamics. Although mainly known for
his research in aerodynamics, he also worked
on problems of plasticity. His contribution to
dry friction from 1928 gave an atomistic explanation of the static friction force. The Prandtl
model thus describes solids displaying a yield
stress after an initial elastic phase.
5.2.1 Specific Prandtl Model: Formulation
The specific Prandtl model, displayed in Fig. 5.7, consists of a serial arrangement
of (1) a linear elastic spring with stiffness E and (2) a linear frictional slider with
threshold σ y .
For the specific Prandtl model the free energy density ψ is expressed as a quadratic
(and thus convex) function of − p (the elastic strain e )
ψ(, p ) =
1
2
E [ − p ]
2
.
(5.58)
Then the energetic stress σ
conjugated to the total strain and the energetic plastic
stress σ
p conjugated to the plastic strain p follow as
σ
(, p ) = ∂ ψ(, p ) = E [ − p ],
(5.59a)
σ
p (, p ) = ∂ p ψ(, p ) = −E [ − p ].
(5.59b)