Chapter 28
Module of Additional Orthogonal Load
28.1 Problem Statement
In the studies of material behavior in the case of complex loading, a special place
is taken [6] by the problem of finding a link between the increments of stresses and
strain in the vicinity of the loading trajectory break (angular point). We have already
said (p. 286) that this type of problem based on the Batdorf and Budiansky model [1]
was first stated by Cicala [3]. This link is a kind of an applicability test for the
ratios of various variants of plasticity theory in the case of non-proportional loading.
Moreover, the knowledge of this link is necessary in the study of the flexure–torsion
form of stability loss of structural elements beyond the yield stress.
The simplest form of an angular point represents a so-called orthogonal loading
whose partial case is torsion with constant pre-applied elongating stress of a higher
stress yield. We are interested in the quantity of the instantaneous shear modulus G i
equal to the ratio of the tangential stress increment τ to the shear strain increment
γ at the moment after the loading trajectory break: G i = τ//γ .
28.2 Determining the Intensity of Additional Slips
Assume that an infinitely low tangential stress τ xz is applied to a plastic material
element elongated beyond the yield stress with the constant elongating stress
σ z ((σ z = 0). Let us find the component of the plastic strain γ xz caused by
the action of the component τ xz . To simplify the research, assume as follows in
formulas (25.2)–(25.3): ε = b = g = 0, i. e. let us represent the shear resistance as
S νλ ϕ nl = ψ + (aϕ νλ + cγ νλ ) − AA νλ − BB.
(28.1)
© 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_28
369
Module of Additional Orthogonal Load
28.1 Problem Statement
In the studies of material behavior in the case of complex loading, a special place
is taken [6] by the problem of finding a link between the increments of stresses and
strain in the vicinity of the loading trajectory break (angular point). We have already
said (p. 286) that this type of problem based on the Batdorf and Budiansky model [1]
was first stated by Cicala [3]. This link is a kind of an applicability test for the
ratios of various variants of plasticity theory in the case of non-proportional loading.
Moreover, the knowledge of this link is necessary in the study of the flexure–torsion
form of stability loss of structural elements beyond the yield stress.
The simplest form of an angular point represents a so-called orthogonal loading
whose partial case is torsion with constant pre-applied elongating stress of a higher
stress yield. We are interested in the quantity of the instantaneous shear modulus G i
equal to the ratio of the tangential stress increment τ to the shear strain increment
γ at the moment after the loading trajectory break: G i = τ//γ .
28.2 Determining the Intensity of Additional Slips
Assume that an infinitely low tangential stress τ xz is applied to a plastic material
element elongated beyond the yield stress with the constant elongating stress
σ z ((σ z = 0). Let us find the component of the plastic strain γ xz caused by
the action of the component τ xz . To simplify the research, assume as follows in
formulas (25.2)–(25.3): ε = b = g = 0, i. e. let us represent the shear resistance as
S νλ ϕ nl = ψ + (aϕ νλ + cγ νλ ) − AA νλ − BB.
(28.1)
© 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_28
369