27.7 Strain Creep and Stress Relaxation
363
The result (27.51) can be used when defining the model parameters.
Assume that after elongation at some rate at the moment t ∗ , full (elastic–plastic)
strain is fixed ε
p
z . Plastic strain ε z at t > t ∗ will change, since otherwise the equation
S νλ = τ νλ is violated. Plastic strain increment will change the shear resistance
and, therefore, a respective component of tangential stress. When the stress (σ z )
decreases, the elastic strain ((ε
y
z ) decrements under the law
ε
y
z =
1
E
σ
∗
z − σ z (t)
(t t ∗ ),
(27.52)
where σ ∗
z is the deforming stress at the moment t ∗ , and E is the Young modulus.
For the strain to remain constant, we must demand equations (in modulus) of plastic
strain increment ((ε z ) and changes in elastic strain (27.52), e.g.
ε z = ε z (t) − ε z (t ∗ ) =
1
E
[σ
∗
z − σ z (t)].
(27.53)
Hence, the plastic strain rate will be
˙
ε z (t) = −
1
E
d
dt
σ z (t) (t t ∗ ).
(27.54)
Since in this case of relaxation, after pre-elongation beyond the yield stress, the
shear resistance sign and the plastic strain rate sign do not change, the ratios (27.34)–
(27.36) are true . By substituting formulas (27.53)–(27.54) into the right part of
Eq. (27.36), we obtain
ε z (t ∗ ) +
σ ∗
z − σ z (t)
E
−
ε
E
d
dt
σ z (t) =
2
3
·
ψ[t] − BB
[t]
−
J [u(t)]
aδ[u(t)]
.
(27.55)
The latter equation together with formulas (27.34)–(27.36) represents a nonlinear differential equation of the first order relative to the function σ z (t), whereas
the initial conditions will be
σ z (t)
t=t ∗
= σ
∗
z .
(27.56)
The integration of equation (27.55) for the initial condition (27.56) is easily done
by numerical methods using a computer.
It should be noted that dependencies obtained in previous paragraphs for the case
of uniaxial homogeneous stressed state are true also for the case of pure shear (τ xy =
0), if we replace σ z /2 with τ xy and 3ε z /2, 3˙ ε z with γ xy and ˙
γ xy , respectively, and
instead of the function J (u), defined by formula (27.31), we use its representation
for pure shear given in [3].
363
The result (27.51) can be used when defining the model parameters.
Assume that after elongation at some rate at the moment t ∗ , full (elastic–plastic)
strain is fixed ε
p
z . Plastic strain ε z at t > t ∗ will change, since otherwise the equation
S νλ = τ νλ is violated. Plastic strain increment will change the shear resistance
and, therefore, a respective component of tangential stress. When the stress (σ z )
decreases, the elastic strain ((ε
y
z ) decrements under the law
ε
y
z =
1
E
σ
∗
z − σ z (t)
(t t ∗ ),
(27.52)
where σ ∗
z is the deforming stress at the moment t ∗ , and E is the Young modulus.
For the strain to remain constant, we must demand equations (in modulus) of plastic
strain increment ((ε z ) and changes in elastic strain (27.52), e.g.
ε z = ε z (t) − ε z (t ∗ ) =
1
E
[σ
∗
z − σ z (t)].
(27.53)
Hence, the plastic strain rate will be
˙
ε z (t) = −
1
E
d
dt
σ z (t) (t t ∗ ).
(27.54)
Since in this case of relaxation, after pre-elongation beyond the yield stress, the
shear resistance sign and the plastic strain rate sign do not change, the ratios (27.34)–
(27.36) are true . By substituting formulas (27.53)–(27.54) into the right part of
Eq. (27.36), we obtain
ε z (t ∗ ) +
σ ∗
z − σ z (t)
E
−
ε
E
d
dt
σ z (t) =
2
3
·
ψ[t] − BB
[t]
−
J [u(t)]
aδ[u(t)]
.
(27.55)
The latter equation together with formulas (27.34)–(27.36) represents a nonlinear differential equation of the first order relative to the function σ z (t), whereas
the initial conditions will be
σ z (t)
t=t ∗
= σ
∗
z .
(27.56)
The integration of equation (27.55) for the initial condition (27.56) is easily done
by numerical methods using a computer.
It should be noted that dependencies obtained in previous paragraphs for the case
of uniaxial homogeneous stressed state are true also for the case of pure shear (τ xy =
0), if we replace σ z /2 with τ xy and 3ε z /2, 3˙ ε z with γ xy and ˙
γ xy , respectively, and
instead of the function J (u), defined by formula (27.31), we use its representation
for pure shear given in [3].