Chapter 19
Other Variants of Plasticity Theories
19.1 Batdorf–Budiansky Slip Theory
By the mid-1920s, an idea was developed to account for the physical properties
of a real body when analyzing the mechanics of non-elastic strain. The first effort
to associate the properties of a poly-crystalline aggregation with the properties of
its component grains was undertaken in 1938 by Taylor [29]. The study was based
on the hypothesis that arbitrary strain not accompanied by changes in volume can
be represented as a result of shifts in five planes and directions. Taylor’s idea was
developed in the paper by Bishop and Hill [5] and later also used by Lin [18].
However, chronologically it is deemed that the completed theory was formulated
by Batdorf and Budiansky [2] . The proposed mechanical model of the plastic strain
of metals was called the slip theory by the authors. The authors used clearly defined
facts:
– a real metal represents an aggregate of disorderly oriented crystalline grains;
– the plastic strain of a single grain is caused by a shift in a specific crystallographic
plane and in a specific direction (highest density of atom packing).
A normal line to the slip plane with a single vector n and a single vector β
defining the slip direction in this plane forms the slip system. If the tangential stress
τ nβ in the plane n in the direction β exceeds the yield stress, the crystal undergoes
plastic strain of pure shift γ nβ . It is assumed that the value of this strain is a definite
function of stress τ nβ .
Real crystals have several systems of possible slips where the plane and direction
of the highest density of atomic packing are similar. For example, there are
12 systems in crystals with a cubic face-centered grating (Fig. 19.1). The figure
highlights one of the four planes of the highest density of atom arrangement. Atoms
belonging to the highlighted plane are marked by light highlighting. Arrows show
directions with the shortest distances between atoms.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_19
283
Other Variants of Plasticity Theories
19.1 Batdorf–Budiansky Slip Theory
By the mid-1920s, an idea was developed to account for the physical properties
of a real body when analyzing the mechanics of non-elastic strain. The first effort
to associate the properties of a poly-crystalline aggregation with the properties of
its component grains was undertaken in 1938 by Taylor [29]. The study was based
on the hypothesis that arbitrary strain not accompanied by changes in volume can
be represented as a result of shifts in five planes and directions. Taylor’s idea was
developed in the paper by Bishop and Hill [5] and later also used by Lin [18].
However, chronologically it is deemed that the completed theory was formulated
by Batdorf and Budiansky [2] . The proposed mechanical model of the plastic strain
of metals was called the slip theory by the authors. The authors used clearly defined
facts:
– a real metal represents an aggregate of disorderly oriented crystalline grains;
– the plastic strain of a single grain is caused by a shift in a specific crystallographic
plane and in a specific direction (highest density of atom packing).
A normal line to the slip plane with a single vector n and a single vector β
defining the slip direction in this plane forms the slip system. If the tangential stress
τ nβ in the plane n in the direction β exceeds the yield stress, the crystal undergoes
plastic strain of pure shift γ nβ . It is assumed that the value of this strain is a definite
function of stress τ nβ .
Real crystals have several systems of possible slips where the plane and direction
of the highest density of atomic packing are similar. For example, there are
12 systems in crystals with a cubic face-centered grating (Fig. 19.1). The figure
highlights one of the four planes of the highest density of atom arrangement. Atoms
belonging to the highlighted plane are marked by light highlighting. Arrows show
directions with the shortest distances between atoms.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_19
283