dependent. A class of plasticity theories where size effect is considered are usually
called strain gradient plasticity theories.
In strain gradient plasticity theories, it is assumed that classic plastic behavior is
due to slip of statistically stored dislocations and length-scale effects are due to slip
of geometrically necessary dislocations. As a result, the dislocation term in Taylor’s
flow stress equation is modified to add the geometrically necessary dislocation
density:
τ ¼ αGb
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ρ SSD þ ρ GND
p
ð5:118Þ
where α is geometrical factor that depends on the type and arrangement of the
interacting dislocations, G is the shear modulus, b is Burger’s vector, ρ SSD is the
Fig. 5.3 Thin layer solder joint attached to copper plates
Table 5.4 Material
parameters
Elastic (θ temperature in K)
Young’s modulus (GPa)
52.10–0.1059θ
Shear modulus (GPa)
19.44–0.0395θ
Isotropic hardening
R 00 (MPa)
37.47–0.0748θ
c (Dimensionless)
383.3
σ y (MPa)
60.069–0.140θ
Kinematic hardening
c 1 (MPa)
2040
c 2 (Dimensionless)
180
Creep strain rate
A (Dimensionless)
7.60E+09
D 0 (mm/s
2
)
48.8
b (mm)
3.18EÀ07
d (mm)
1.06EÀ02
n
1.67
p
3.34
Q (mJ/mol)
4.47E+07
Фcritical
0.1
5.6 Thermo-mechanical Analysis of Cosserat Continuum: Length-Scale Effects
235
called strain gradient plasticity theories.
In strain gradient plasticity theories, it is assumed that classic plastic behavior is
due to slip of statistically stored dislocations and length-scale effects are due to slip
of geometrically necessary dislocations. As a result, the dislocation term in Taylor’s
flow stress equation is modified to add the geometrically necessary dislocation
density:
τ ¼ αGb
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ρ SSD þ ρ GND
p
ð5:118Þ
where α is geometrical factor that depends on the type and arrangement of the
interacting dislocations, G is the shear modulus, b is Burger’s vector, ρ SSD is the
Fig. 5.3 Thin layer solder joint attached to copper plates
Table 5.4 Material
parameters
Elastic (θ temperature in K)
Young’s modulus (GPa)
52.10–0.1059θ
Shear modulus (GPa)
19.44–0.0395θ
Isotropic hardening
R 00 (MPa)
37.47–0.0748θ
c (Dimensionless)
383.3
σ y (MPa)
60.069–0.140θ
Kinematic hardening
c 1 (MPa)
2040
c 2 (Dimensionless)
180
Creep strain rate
A (Dimensionless)
7.60E+09
D 0 (mm/s
2
)
48.8
b (mm)
3.18EÀ07
d (mm)
1.06EÀ02
n
1.67
p
3.34
Q (mJ/mol)
4.47E+07
Фcritical
0.1
5.6 Thermo-mechanical Analysis of Cosserat Continuum: Length-Scale Effects
235