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A Potential-Based Smoothed Particle Hydrodynamics Approach for the Dynamic Failure Assessment of Compact and Granular Materials

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Abstract

Computational simulation of solids has experienced a rapid development since the formulation of the finite element method. However a number of problems cannot be properly solved by using the finite element method because a severe mesh distortion in computations of Lagrangian scheme may arise for very large displacements, high speed impact, fragmentation, particulate solids, fluid-structure interaction, leading to lack of consistency between the numerical and the physical problem. The discretization of the problem domain with nodal points without any mesh connectivity would be useful to overcome this difficulty. Moreover the discrete nature of continuum matter—usually observed at the microscale—allows to adopt such a kind of discretization that is natural for granular materials and enables us to model very large deformations, handle damage—such as fracture, crushing, fragmentation, clustering—thanks to the variable interaction between particles.

In the context of meshless methods, smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation that has been widely applied to different engineering fields. In the present paper a unified computational potential-based particle method for the mechanical simulation of continuum and granular materials under dynamic condition, is proposed and framed in the SPH-like approaches. The particleparticle and particle-boundary interaction is modelled through force functionals related to the nature of the material being analyzed (solid, granular,...); large geometrical changes of the mechanical system, such as fracture, clustering, granular flow can be easily modelled. Some examples are finally proposed and discussed to underline the potentiality of the approach.

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Abbreviations

a(r infl):

function of the influence radius of a particl

A ij , A 0, ij :

cross section area of the truss assumed beytween particles i and j and its reference value, respectively

b i :

body force component in the i-th direction

Cʹ:

tangent elastic tensor of the material

c(s):

function accounting for the no-penetration condition

d * :

equivalent diameter of two particles in contact

d i = 2r i :

diameter of the generic particle i

Eʹ(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

F tot, i :

vector of the total force acting on the particle i

h :

smoothing length or support dimension

K(s):

stiffness of the particles bonding depending on their effective distance s

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 :

unit vector identifying the direction connecting the two particles centers

r :

distance between the centers of a couple of particles

\(\bar r\) :

average radius of a couple of particles

r infl :

radius of influence (or maximum interacting distance) of a particle r infl

r 0 :

distance between the centers of the particles at which F (r = r 0) → -∞

s, s e :

effective distance between particle surfaces and at the equilibrium state, respectively

sʹ = r - (d i /2 + d j/2):

distance between two particle 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 (|x - x i |):

Kernel or smoothing function for the smoothed particle hydrodynamics method

x 0 :

position vector identifying the equilibrium state

x i , ẋ i , ẍ i :

position, velocity and acceleration vector of the particle i, respectively

x, ẋ, ẍ :

position, velocity and acceleration vector, respectively

α:

coefficient defining the maximum copenetration depth

ε ij (x i ):

strain tensor components at the point identified by x i

γ:

thickness of a soft layer added to the boundary surface in order to smooth the contact forces

δ(| x - x i |):

Dirac delta function placed at x i

\(\delta = \alpha \bar r\) :

maximum copenetration amount between two particles

σ ij :

stress tensor components

Φ(x), Φtot(x):

generic strain energy potential and potential of the particle system, respectively

λd :

damping coefficient

η:

viscosity coefficient

χ(w):

smoothing function for the force particle-boundary contact evaluation

μd :

coefficient of dynamic friction between particles and boundaries

μ md :

coefficient of dynamic friction between particles

ν i :

Poissons ratio of the generic particle i

ρ:

mass density.

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  1. University of Parma, Parma, 43100, Italy

    R. Brighenti & N. Corbari

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  1. R. Brighenti
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Original Text © R. Brighenti, N. Corbari, 2015, published in Fizicheskaya Mezomekhanika, 2015, Vol. 18, No. 5, pp. 124-136.

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Brighenti, R., Corbari, N. A Potential-Based Smoothed Particle Hydrodynamics Approach for the Dynamic Failure Assessment of Compact and Granular Materials. Phys Mesomech 18, 402–415 (2015). https://doi.org/10.1134/S1029959915040128

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