Contact Force: Hertz–MD (Mindlin and Deresiewicz) Model
Select the Hertz–MD from the Contact Force Model list in the physics interface Force section to compute the contact forces using the Hertz–MD model. The equations for the forces in Contact Force: Linear Elastic Model used the linear equations where the stiffness coefficients, kn and kt, and damping coefficients, cn and ct are known and constant. For the case where these coefficients are not known, a nonlinear method is used where the coefficients depend on the material properties and normal overlap between grains δn (Ref. 1).
Normal Force
The equation of normal force remains same as Equation 3-9; however, the spring and damping coefficients are not constant and depend on the material properties of grains i and j in contact:
(3-12)
(3-13)
where
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kn is the normal spring stiffness
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cn is the normal damping
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en is the coefficient of restitution in normal direction
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E is the Young’s modulus
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ν is the Poisson’s ratio
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m and R are the mass and radius of grain respectively
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Eeq is the equivalent Young’s modulus for grains in contact
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Req is the equivalent radius for grains in contact
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meff is the effective mass for grains in contact
Tangential Force
The equation of normal force remains same as Equation 3-10; however, the spring and damping coefficients are defined as
(3-14)
where
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kt is the tangential spring stiffness
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ct is the tangential damping
•
et is the coefficient of restitution in tangential direction
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G is the shear modulus
•
Geq is the equivalent shear modulus for grains in contact
Contact Force Between Grain and Wall
The contact force between grain and wall is the same as the contact force between grains. The expressions that are different are discussed in this section. For the contact between grain i and wall w,
(3-15)
(3-16)
(3-17)