Study on Material Point Method with Different Influence Factors
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is stored at material points from step to step of analysis in MPM. The calculations are
computed on the surrounding background grids with input data obtained by interpolation
from the material points. The grid is restored to its original location at the end of each
time step to avoid the mesh distortion problems of finite element methods. Therefore,
MPM is suitable for simulating the process of impact, fracture and other phenomenon
which accompanies with large deformations. Meanwhile, compared to other mesh-less
methods, it is unnecessary to search a point’s neighborhood points in MPM as the governing equations are solved on a regular grid. Hence, MPM has a higher efficiency than
other mesh-less methods.
At present, MPM has been widely used in the engineering simulations. Many scholars have improved the algorithm according to their needs. Sulsky etc. [5, 6] simulated
the process of Taylor impact test and spherical steel fragments penetrating aluminum
target plates problem. Huang Peng [7], Ma Zhitao [8] simulated the problems of lowspeed impact penetration by MPM. Chen [9–11] simulated the dynamic failure of brittle
materials under impact loading and the failure of materials under local heating by MPM
either.
Temperature is one of the most important factors that affect the characteristics of a
material. During the simulation process mentioned above by MPM, most of the simulation process has only considered the effect of large deformations on the temperature
rise. However, during the high-speed impact problems, such as a Taylor impact test, the
adiabatic compression and shock wave dissipation effect caused by impact will cause
a drastic temperature change of a structure. The changed temperature is called shock
temperature, but the classic material point method does not take it into account. Meanwhile, a high pressure will increase the melting point of materials. Hence, the shock
temperature and melting point affected by high pressure will be considered in MPM in
this article.
2 The Material Point Method
2.1 Governing Equations
The accuracy of a numerical model depends on its ability to obey the three governing
equations: balance of mass, momentum and energy conservation.
Conservation of mass is ensured by the equation
Dρ
Dt
+ ρ∇ · ·
v = 0
( 1 )
Where
v = v( x, t) is the spatial velocity and ρ = ρ( x, t) is the current density.
∇ is the gradient operator and ∇· is the divergence of the vector field.
Conservation of momentum is ensured by the equation
ρ
D v
Dt
= ∇ · ·
σ + ρ
b
(2)
Where
σ = σ ( x, t) is the Cauchy stress tensor and
b = b( x, t) is the specific
body force.
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is stored at material points from step to step of analysis in MPM. The calculations are
computed on the surrounding background grids with input data obtained by interpolation
from the material points. The grid is restored to its original location at the end of each
time step to avoid the mesh distortion problems of finite element methods. Therefore,
MPM is suitable for simulating the process of impact, fracture and other phenomenon
which accompanies with large deformations. Meanwhile, compared to other mesh-less
methods, it is unnecessary to search a point’s neighborhood points in MPM as the governing equations are solved on a regular grid. Hence, MPM has a higher efficiency than
other mesh-less methods.
At present, MPM has been widely used in the engineering simulations. Many scholars have improved the algorithm according to their needs. Sulsky etc. [5, 6] simulated
the process of Taylor impact test and spherical steel fragments penetrating aluminum
target plates problem. Huang Peng [7], Ma Zhitao [8] simulated the problems of lowspeed impact penetration by MPM. Chen [9–11] simulated the dynamic failure of brittle
materials under impact loading and the failure of materials under local heating by MPM
either.
Temperature is one of the most important factors that affect the characteristics of a
material. During the simulation process mentioned above by MPM, most of the simulation process has only considered the effect of large deformations on the temperature
rise. However, during the high-speed impact problems, such as a Taylor impact test, the
adiabatic compression and shock wave dissipation effect caused by impact will cause
a drastic temperature change of a structure. The changed temperature is called shock
temperature, but the classic material point method does not take it into account. Meanwhile, a high pressure will increase the melting point of materials. Hence, the shock
temperature and melting point affected by high pressure will be considered in MPM in
this article.
2 The Material Point Method
2.1 Governing Equations
The accuracy of a numerical model depends on its ability to obey the three governing
equations: balance of mass, momentum and energy conservation.
Conservation of mass is ensured by the equation
Dρ
Dt
+ ρ∇ · ·
v = 0
( 1 )
Where
v = v( x, t) is the spatial velocity and ρ = ρ( x, t) is the current density.
∇ is the gradient operator and ∇· is the divergence of the vector field.
Conservation of momentum is ensured by the equation
ρ
D v
Dt
= ∇ · ·
σ + ρ
b
(2)
Where
σ = σ ( x, t) is the Cauchy stress tensor and
b = b( x, t) is the specific
body force.