Abstract
Computational approaches of mechanical systems based on the continuum hypothesis, are sometimes inaccurate and not reliable. As an example, problems involving severe mesh distortion, geometric discontinuities or characterized by an assembly of discrete parts, are not easily solvable within the Lagrangian continuum framework, such as by using the classical finite element method. Computational approaches based on the description of the domain without the need of a mesh connectivity, would be useful to overcome this drawbacks. On the other hand a discrete approach is a particularly suitable tool for modeling materials at the microscale where its particulate nature becomes evident. The Lagrangian-based meshless formulation—known as smoothed particle hydrodynamics (SPH)—has been widely applied to different engineering fields. In the present research a general force potential-based particle method falling within the SPH framework for the mechanical simulation of granular and continuum materials under dynamic condition, is developed. The particle–particle and particle-boundary interaction is modeled through force functionals, tuned according to the nature of the material being analyzed (solid, granular, …). The proposed potential-based formulation allows the description of the forces existing between the discrete elements of generic materials. Thanks to the capability to deal with short and long distance actions (namely mechanical and/or electrostatic), general force–deformation laws, etc. it allows a straightforward mechanical simulation of fine particles assemblies such as powders. The theoretical basis of the computational approach are presented and some examples involving powder motion and a continuum mechanical problem are illustrated and discussed.
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Abbreviations
- a(r infl):
-
Function of the influence radius of a particle
- A ij , A 0,ij :
-
Cross-section area of the truss assumed between particles i and j and its reference value, respectively
- b i :
-
Component of the body force in the i-th direction
- c(s):
-
Function accounting for the no-penetration condition
- \( {\mathbf{C}}^{\prime } \) :
-
Current elastic tensor of the material
- \( d^{*} \) :
-
Equivalent diameter of two particles in contact
- d i :
-
Diameter of the generic particle i
- \( E^{\prime } (s),E_{0} \) :
-
Elastic modulus of a linear element that represents the material connecting the two particles i, j and corresponding reference value for s ≫ 0
- E i :
-
Elastic modulus of the generic particle i
- \( \bar{E} \) :
-
Equivalent Young modulus of the elastic contact between two particles
- E tot (x) = Π(x):
-
Total energy of the particle system
- F ij (s):
-
Generic force acting between a couple of particles at the effective distance s
- F i , F d , F b , F e :
-
Internal force, damping force, boundary and external force vectors, respectively
- h :
-
Smoothing length or support dimension
- H(w):
-
Heaviside function
- K(s):
-
Stiffness of the particles bonding depending on their effective distance s
- \( \kappa_{e} = \frac{1}{{4\pi \varepsilon_{0} }} \) :
-
Coulomb’s constant
- K n :
-
Normal stiffness of the particle-boundary contact
- m i , M :
-
Mass of the particle i and mass matrix, respectively
- n, t :
-
Unit vectors normal and parallel to the tangential plane in the contact point between a generic particle and the boundary surface, respectively
- P i :
-
Force vector acting on particle i
- q, qi :
-
Unit vector (and corresponding components) identifying the direction connecting the centers of two particles
- q i :
-
Magnitude of the charges of the i particle (expressed in Coulombs)
- r :
-
Distance between the centers of a couple of particles
- r infl :
-
Radius of influence (or maximum interacting distance) of a particle
- r 0 :
-
Distance between the centers of the particles at which \( F(r_{{}} = r_{0} ) \to - \infty \)
- s, s e :
-
Effective distance between particles’ surfaces and at the equilibrium state, respectively
- \( \begin{aligned} s^{\prime } & = \\ & = r - \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} (d_{i} + d_{j} ) \\ \end{aligned} \) :
-
Distance between two particles’ surfaces
- t, Δt :
-
Time variable and time integration step amplitude, respectively
- T n , T t :
-
Particle-boundary contact surface forces normal and parallel to the tangential plane at the contact point, respectively
- T t , ij :
-
Tangential force between the colliding particles i, j
- V p :
-
Volume of the particle p
- w :
-
Displaced distance of one particle into another or into the contact surface
- \( W(\left| {{\mathbf{x}} - {\mathbf{x}}_{i} } \right|) \) :
-
Kernel or smoothing function for the SPH method
- x 0 :
-
Position vector identifying the equilibrium state
- \( {\mathbf{x}}_{i} \), \( {\dot{\mathbf{x}}}_{i} \), \( {\ddot{\mathbf{x}}}_{i} \) :
-
Position, velocity and acceleration vector of the particle i, respectively
- \( {\mathbf{x}} \), \( {\dot{\mathbf{x}}} \), \( {\ddot{\mathbf{x}}} \) :
-
Generic position, velocity and acceleration vector, respectively
- α :
-
Coefficient defining the maximum co-penetration depth
- γ :
-
Thickness of a soft layer added to the boundary surface in order to smooth the contact forces
- \( \delta (\left| {{\mathbf{x}} - {\mathbf{x}}_{i} } \right|) \) :
-
Dirac delta function placed at x i
- \( \delta = \alpha \cdot \bar{r} \) :
-
Maximum co-penetration amount between two particles
- ε ij :
-
Strain tensor components
- \( \varepsilon_{0} = \frac{{10^{ - 9} }}{36\pi }{\text{ C}}^{ 2} {\text{N}}^{ - 1} {\text{m}}^{ - 2} \) :
-
Permittivity of the free space
- Φ(x), Φ tot (x):
-
Generic strain energy potential and potential of the particle system, respectively
- λ d :
-
Damping coefficient
- η :
-
Viscosity coefficient
- χ(w):
-
Smoothing function for the force particle-boundary contact evaluation
- μ d :
-
Coefficient of dynamic friction between particles and boundaries
- μ md :
-
Coefficient of dynamic friction between particles
- ρ :
-
Mass density
- σ ij :
-
Stress tensor components
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The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR).
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Brighenti, R., Corbari, N. A general discrete element approach for particulate materials. Int J Mech Mater Des 13, 267–286 (2017). https://doi.org/10.1007/s10999-015-9332-z
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DOI: https://doi.org/10.1007/s10999-015-9332-z