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bond_style bpm/peri command

Syntax

bond_style bpm/peri keyword value attribute1 attribute2 ...

  • zero or more keyword/value pairs may be appended

  • keyword = store/local

    store/local values = fix_ID N attributes ...
   * fix_ID = ID of associated internal fix to store data
   * N = prepare data for output every this many timesteps
   * attributes = zero or more of the below attributes may be appended

     id1, id2 = IDs of two atoms in the bond
     time = the timestep the bond broke
     x, y, z = the center of mass position of the two atoms when the bond broke (distance units)
     x/ref, y/ref, z/ref = the initial center of mass position of the two atoms (distance units)

Examples

bond_style bpm/peri
bond_coeff 1 pmb 1.6863e22 0.0015001 0.0005 0.25
bond_coeff 1 lps 14.9e9 14.9e9 0.0015001 0.0005 0.25
bond_coeff 1 ves 14.9e9 14.9e9 0.5 0.001 0.0015001 0.0005 0.25
bond_coeff 1 eps 14.9e9 14.9e9 118.43 0.0015001 0.0005 0.25

Description

Added in version TBD.

The bpm/peri bond style implements the four peridynamic constitutive models of the PERI package — bond-based prototype microelastic brittle (pmb), state-based linear peridynamic solid (lps), state-based viscoelastic solid (ves), and state-based elastic-plastic solid (eps) — recast as a bond style within the BPM framework. A peridynamic “bond” is a BPM bond: it is created from the reference configuration, stores its reference length, and breaks individually on a stretch criterion. The model is selected by the first bond_coeff argument. This allows peridynamic materials to reuse BPM’s tooling (per-bond history in restart files, broken-bond output via store/local, hybrid composition) and the standard atom_style bond instead of a dedicated atom style.

Like the other BPM bond styles, the reference state is stored by each bond when it is first computed in the setup of a run, is preserved across run commands, and is written to binary restart files.

A short-range repulsive contact force between non-bonded near neighbors is provided by the companion pair_style bpm/peri, which must also be defined (see the Restrictions section).

Per-atom nodal volume

Every peridynamic node carries a nodal volume \(V\) (the discretized material volume it represents). Unlike the internal bookkeeping fields the bond style creates automatically, the nodal volume is input data and so it must be declared by the user with a fix property/atom named vfrac, using group all and with ghost communication enabled, before the bond style is defined:

fix vol all property/atom d_vfrac ghost yes
set group all d_vfrac 1.25e-10            # uniform nodal volume

For a body with non-uniform nodal volume the values may instead be read from a data file (see read_data and fix property/atom). The nodal mass is supplied independently, either as a uniform per-type mass (equal to density times nodal volume) or, for variable mass, by using atom_style hybrid sphere bond and setting the per-atom mass. See the peridynamics Howto for a complete worked example.

Constitutive models and coefficients

The model keyword is the first bond_coeff argument, followed by that model’s coefficients:

For pmb (bond-based prototype microelastic brittle):

  • c (energy/distance/volume^2 units)

  • horizon \(\delta\) (distance units)

  • s00 (unitless)

  • \(\alpha\) (unitless)

The bond force is \(F = c\, s\, V_j\) with stretch \(s = (r - r_0)/r_0\), where c is the micromodulus \(c = 18 K / (\pi \delta^4)\) for bulk modulus K.

For lps (state-based linear peridynamic solid):

  • K (force/area units), bulk modulus

  • G (force/area units), shear modulus

  • horizon \(\delta\) (distance units)

  • s00 (unitless)

  • \(\alpha\) (unitless)

For ves (state-based viscoelastic solid), as lps plus the viscoelastic parameters inserted after G:

  • K, G (force/area units)

  • \(\lambda\) (unitless), relaxation parameter in [0,1]

  • \(\tau\) (time units), relaxation time

  • horizon, s00, \(\alpha\)

For small \(\lambda\) the response approaches the linear-elastic lps model.

For eps (state-based elastic-plastic solid), as lps with the yield stress inserted after G:

  • K, G (force/area units)

  • yield stress (force/area units)

  • horizon, s00, \(\alpha\)

The lps, ves, and eps models are state-based: the force on a bond depends on the dilatation \(\theta\) and weighted volume of both endpoints, which the bond style accumulates and communicates internally each step. The canonical references are (Silling 2000), (Silling 2007), and (Parks). The ves and eps formulations are from (Mitchell 2011a) and (Mitchell 2011b); the underlying state-based viscoplasticity theory is (Foster 2010), and the original PDLAMMPS implementations of these two models by Rahman and Foster are described in (Rahman).

Note

The eps plastic deviatoric extension is updated by a radial return onto the yield surface that is consistent with the returned force state. This differs from the update in the legacy pair_style peri/eps, which is only conditionally stable under sustained plastic flow; the yield-surface norm here also uses the same deviatoric force state the model applies.

Bond breaking criterion

A peridynamic bond between particles i and j breaks irreversibly once its stretch \(s = (r - r_0)/r_0\) exceeds a critical stretch. Following (Parks) (eq. 9), the per-bond critical stretch is \(s_{00} - \alpha\, \max(s_{min,i}, s_{min,j})\), where \(s_{min}\) is the minimum (most compressive) stretch over a particle’s bonds on the previous step. The criterion is evaluated per bond using that bond’s own s00 and \(\alpha\), so type-dependent coefficients behave as intended (for example a weaker s00 at a material interface initiates a crack there). Broken bonds are removed from the bond topology by setting the bond type to 0.

The compute property/atom s0 and smin per-atom properties report the per-particle critical stretch and minimum stretch; for eps, lambda reports the accumulated plastic multiplier. Volume-weighted damage is available through compute bpm/peri/damage/atom.

For the state-based models (lps, ves, eps) the per-step dilatation \(\theta\) can optionally be exposed for visualization: declare a per-atom theta property before the bond style,

fix dil all property/atom d_theta ghost yes

and the bond style writes the dilatation of each owned node into it every step, readable with compute property/atom theta. (It must be declared up front, like vfrac, so a compute that references it can be defined from the start of the input.)

Restart and other info

This bond style writes the reference state of each bond and its per-type coefficients to binary restart files. Loading a restart file restores bonds and their reference state. The reference state is NOT written to data files.

If the store/local option is used, an internal fix records data for each breaking bond into a local vector or array accessible through a dump local command, as for the other BPM bond styles.

Restrictions

This bond style is part of the BPM package. It is only enabled if LAMMPS was built with that package. See the Build package page for more info.

This bond style requires the companion pair_style bpm/peri (or pair_style zero for a deliberate contact-free test), a per-atom vfrac property as described above, and a cubic lattice with equal spacing in x, y, and z (used by the partial-volume horizon correction).

As with the other BPM bond styles, newton must be set to bond off and the special bond weights must be

special_bonds lj 0 1 1 coul 1 1 1

See the peridynamics Howto for a complete input script and a guide to migrating PERI inputs to the BPM framework.

Default

none

(Parks) Parks, Lehoucq, Plimpton, Silling, Comp Phys Comm, 179(11), 777-783 (2008).

(Silling 2000) Silling, J Mech Phys Solids, 48, 175-209 (2000).

(Silling 2007) Silling, Epton, Weckner, Xu, Askari, J Elasticity, 88, 151-184 (2007).

(Mitchell 2011a) Mitchell. A non-local, ordinary-state-based viscoelasticity model for peridynamics. Sandia National Lab Report, 8064:1-28 (2011).

(Mitchell 2011b) Mitchell. A Nonlocal, Ordinary, State-Based Plasticity Model for Peridynamics. Sandia National Lab Report, 3166:1-34 (2011).

(Foster 2010) Foster, Silling, Chen, Viscoplasticity using peridynamics, Int J Numer Methods Eng, 81(10), 1242-1258 (2010).

(Rahman) Rahman, Foster, Implementation of linear viscoelasticity model in PDLAMMPS (2013) and Implementation of elastic-plastic model in PDLAMMPS, technical reports, University of Texas at San Antonio (included in the LAMMPS distribution as PDLammps_VES.pdf and PDLammps_EPS.pdf; see also the PDLAMMPS overview PDLammps_overview.pdf).