290
19 Other Variants of Plasticity Theories
2ω + β =
π
2
+ λ.
We have
q =
1
4
F (τ )(e 1 cos λ − e 2 sin λ)dλ,
(19.8)
where τ = Q cos λ and integration falls upon those values λ for which τ > τ s , e.g.
cos λ >
τ s
Q
.
If we designate cos λ 1 = τ s /Q, the integration area will be
−λ 1 λ λ 1 .
Assume that after proportional loading, additional loading δQ is done in an
arbitrary direction making the angle α with the unit axis e 2 (Fig. 19.4). By
designating the additional loading vector modulus as ε for shortness, we have as
follows for its components:
Q 1 = ε sin(α − β), ,Q 2 = ε cos(α − β).
To gain the tangential stress component, we obtain
τ = ε cos(α − β − ω) = ε sin(α − λ).
For the gain of the plastic strain vector, we have
q =
1
4
ε
F
(τ ) sin(α − λ)[e 1 cos λ − e 2 sin λ]dλ.
(19.9)
The simple formula obtained for finding the plastic strain vector gain bears the
same challenges as in the more general Batdorf–Budiansky model: in the case of low
additional loading, plastic strain is obtained for those elements only where τ > 0;
elastic unloading will occur in elements for which τ < 0. As per the graphical
picture of the process given in Fig. 19.4, the following variants are possible.
1. α −λ 1 . In this case α − λ < 0 ∀λ ∈ (−λ 1 ; λ 1 ), τ < 0 and elastic
unloading takes place. The straight lines α = ±λ 1 define the angle inside which
the additional loading vector causes elastic unloading in all elements. It is easy
to believe that these straight lines touch the initial circumference (see Fig. 19.4).
2. α λ 1 ; then, α − λ > 0 ∀λ ∈ (−λ 1 ; λ 1 ) and everywhere τ > 0—
additional loading takes place in all elements. The ratio (19.7) is obtained from
formula (19.5) where integration is replaced by varying.
19 Other Variants of Plasticity Theories
2ω + β =
π
2
+ λ.
We have
q =
1
4
F (τ )(e 1 cos λ − e 2 sin λ)dλ,
(19.8)
where τ = Q cos λ and integration falls upon those values λ for which τ > τ s , e.g.
cos λ >
τ s
Q
.
If we designate cos λ 1 = τ s /Q, the integration area will be
−λ 1 λ λ 1 .
Assume that after proportional loading, additional loading δQ is done in an
arbitrary direction making the angle α with the unit axis e 2 (Fig. 19.4). By
designating the additional loading vector modulus as ε for shortness, we have as
follows for its components:
Q 1 = ε sin(α − β), ,Q 2 = ε cos(α − β).
To gain the tangential stress component, we obtain
τ = ε cos(α − β − ω) = ε sin(α − λ).
For the gain of the plastic strain vector, we have
q =
1
4
ε
F
(τ ) sin(α − λ)[e 1 cos λ − e 2 sin λ]dλ.
(19.9)
The simple formula obtained for finding the plastic strain vector gain bears the
same challenges as in the more general Batdorf–Budiansky model: in the case of low
additional loading, plastic strain is obtained for those elements only where τ > 0;
elastic unloading will occur in elements for which τ < 0. As per the graphical
picture of the process given in Fig. 19.4, the following variants are possible.
1. α −λ 1 . In this case α − λ < 0 ∀λ ∈ (−λ 1 ; λ 1 ), τ < 0 and elastic
unloading takes place. The straight lines α = ±λ 1 define the angle inside which
the additional loading vector causes elastic unloading in all elements. It is easy
to believe that these straight lines touch the initial circumference (see Fig. 19.4).
2. α λ 1 ; then, α − λ > 0 ∀λ ∈ (−λ 1 ; λ 1 ) and everywhere τ > 0—
additional loading takes place in all elements. The ratio (19.7) is obtained from
formula (19.5) where integration is replaced by varying.