Study on Material Point Method with Different Influence Factors
75
4. The equation of momentum is solved at the grid nodes.
5. According to the results attained from step 4, the information is transferred back
to the material points and the state variables of material points, such as velocity,
position, are updated.
6. The strain and vortex increments of material points are solved and the density and
stress of material points are also updated.
7. The increases of temperatures caused by plastic deformations of material points are
computed and the temperatures of material points are updated.
8. The deformed mesh is abandoned and a new background mesh is generated.
As mentioned earlier, temperature is one of the most important factors that affect the
characteristics of a material and the adiabatic compression and shock wave dissipation
effect caused by impact will cause a drastic temperature change of a structure. But
the classic MPM algorithm has only considered the effect of large deformation on the
temperature rise. Therefore, we will consider the shock temperature in MPM as follows.
3 Improved MPM
Johnson-cook constitutive model is a visco-plastic model and it is suited to a circumstance with high strain rate and high temperature. The model is ensured by the
equation
σ y = (A + Bε
pn
)(1 + C ln ˙
ε
∗
)(1 − T
∗m
)
(5)
Where σ y is the von-Mises yield stress. A, B, n, C, m are five material constants
which are obtained from tests. ε p is the effective plastic strain. ˙
ε ∗ is the dimensionless
equivalent plastic strain-rate and T ∗m is the homologous temperature.
T
∗m
=
T − T r
T m − T r
∈ [0, 1]
(6)
Where T is the current temperature of the structure. T r is the room temperature and
T m is the melting temperature of the material.
According to Eqs. (5) and (6), σ y is approximately equal to A if the strain and the
strain rates are small enough and T = T r . σ y increases with the increase of the strain
or the strain rates. However, σ y is approximately equal to zero which lead a structure
to deform easily if T is approximately equal to T m . The phenomenon is called material
softening.
A Taylor impact test [12] is taken as an example to verify the material softening and
the importance of shock temperature in a high-speed impact simulation by MPM.
The material of the cylinder in the impact test is copper and it impact on a rigid
boundary with 190 m/s. The parameters of the cylinder are shown in Table 1.
75
4. The equation of momentum is solved at the grid nodes.
5. According to the results attained from step 4, the information is transferred back
to the material points and the state variables of material points, such as velocity,
position, are updated.
6. The strain and vortex increments of material points are solved and the density and
stress of material points are also updated.
7. The increases of temperatures caused by plastic deformations of material points are
computed and the temperatures of material points are updated.
8. The deformed mesh is abandoned and a new background mesh is generated.
As mentioned earlier, temperature is one of the most important factors that affect the
characteristics of a material and the adiabatic compression and shock wave dissipation
effect caused by impact will cause a drastic temperature change of a structure. But
the classic MPM algorithm has only considered the effect of large deformation on the
temperature rise. Therefore, we will consider the shock temperature in MPM as follows.
3 Improved MPM
Johnson-cook constitutive model is a visco-plastic model and it is suited to a circumstance with high strain rate and high temperature. The model is ensured by the
equation
σ y = (A + Bε
pn
)(1 + C ln ˙
ε
∗
)(1 − T
∗m
)
(5)
Where σ y is the von-Mises yield stress. A, B, n, C, m are five material constants
which are obtained from tests. ε p is the effective plastic strain. ˙
ε ∗ is the dimensionless
equivalent plastic strain-rate and T ∗m is the homologous temperature.
T
∗m
=
T − T r
T m − T r
∈ [0, 1]
(6)
Where T is the current temperature of the structure. T r is the room temperature and
T m is the melting temperature of the material.
According to Eqs. (5) and (6), σ y is approximately equal to A if the strain and the
strain rates are small enough and T = T r . σ y increases with the increase of the strain
or the strain rates. However, σ y is approximately equal to zero which lead a structure
to deform easily if T is approximately equal to T m . The phenomenon is called material
softening.
A Taylor impact test [12] is taken as an example to verify the material softening and
the importance of shock temperature in a high-speed impact simulation by MPM.
The material of the cylinder in the impact test is copper and it impact on a rigid
boundary with 190 m/s. The parameters of the cylinder are shown in Table 1.