Adhesion.Interactions package

Submodules

Adhesion.Interactions.Base module

Defines the base class for contact description

class Adhesion.Interactions.Base.Dugdale(stress, length)

Bases: HardWall

Potential class representing a Dugdale cohesive zone model

evaluate(gap, tol=0.0)

property length

property stress

class Adhesion.Interactions.Base.HardWall

Bases: Interaction

base class for non-smooth contact mechanics

compute(gap, tol=0.0)

Parameters:

gap: array_like

array containing the point-wise gap values

tol: float, optional

tolerance for determining whether the gap is closed, default 0.

class Adhesion.Interactions.Base.Interaction

Bases: object

base class for all interactions, e.g. interatomic potentials

class Adhesion.Interactions.Base.SoftWall(communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: Interaction

base class for smooth contact mechanics

evaluate(gap, potential=True, gradient=False)

computes and returns the interaction energy and/or forces based on the as fuction of the gap Parameters: ———– gap: array like

the point-wise gap values

potential: bool

(default True) whether the energy should be evaluated

gradient: bool

(default False) whether the gradient should be evaluated

plot()

Adhesion.Interactions.Coulomb module

Electrostatic potential for Coulombic Interactions

Heß, M. and Forsbach, F., 2020. Macroscopic modeling of fingerpad friction under electroadhesion: Possibilities and limitations. Frontiers in Mechanical Engineering, 6, p.567386.

class Adhesion.Interactions.Coulomb.Coulomb(eps0, eps1, epsg, eps2, d1, d2, voltage, communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: Potential

Electrostatic potential for Coulombic interaction when a voltage is applied across a gap between two perfect insulators

@doi https://doi.org/10.3389/fmech.2020.567386

evaluate(gap, potential=True, gradient=False, curvature=False, mask=None)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

name = 'Coulomb'

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

property r_min

convenience function returning the location of the energy minimum

Adhesion.Interactions.Cutoff module

class Adhesion.Interactions.Cutoff.CutoffPotential(parent_potential, cutoff_radius)

Bases: DecoratedPotential

sets the potential to 0 above the cutoff radius and shifts it up to enforce continuity of the potential. This potential hence has a discontinuity in the force

evaluate(gap, potential=True, gradient=False, curvature=False, mask=None)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

property has_cutoff

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

property r_min

convenience function returning the location of the enery minimum

class Adhesion.Interactions.Cutoff.LinearCorePotential(parent_potential, r_ti=None, hardness=None, htol=1e-05)

Bases: DecoratedPotential

Replaces the singular repulsive part of potentials by a linear part. This makes potentials maximally robust for the use with very bad initial parameters. Consider using this instead of loosing time guessing initial states for optimization.

The repulsive part of the potential is linear, meaning that the pressure is constant. This thus corresponds to an ideal plastic model.

evaluate(gap, potential=True, gradient=False, curvature=False, mask=None)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

property r_min

convenience function returning the location of the enery minimum

class Adhesion.Interactions.Cutoff.ParabolicCutoffPotential(parent_potential, cutoff_radius)

Bases: DecoratedPotential

Implements a very simple smoothing of a potential, by complementing the functional form of the potential with a parabola that brings to zero the potential’s zeroth, first and second derivative at an imposed (and freely chosen) cut_off radius r_c

evaluate(gap, potential=True, gradient=False, curvature=False)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

property has_cutoff

property r_infl

convenience function returning the location of the inflection point

property r_min

convenience function returning the location of the energy minimum

Adhesion.Interactions.Exponential module

Exponential attraction.

class Adhesion.Interactions.Exponential.Exponential(gamma0, rho, communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: Potential

V(g) = -gamma0*e^(-g(r)/rho)

V(g) = -gamma0*e^(-g(r)/rho) V’(g) = (gamma0/rho)*e^(-g(r)/rho) V’’(g) = -(gamma0/rho^2)*e^(-g(r)/rho)

evaluate(gap, potential=True, gradient=False, curvature=False, mask=None)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

property max_tensile

convenience function returning the value of the maximum stress (i.e. the value of the gradient at the inflexion point pot.r_infl)

name = 'exp'

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

property r_min

convenience function returning the location of the energy minimum

class Adhesion.Interactions.Exponential.Morse(work_of_adhesion, interaction_range, communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: RepulsiveExponential

name = 'morse'

class Adhesion.Interactions.Exponential.RepulsiveExponential(gamma_rep, rho_rep, gamma_att, rho_att, communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: Potential

V(g) = -gamma_{rep}*e^(-r/rho_{rep}) - gamma_{att}*e^(-r/rho_{att})

evaluate(gap, potential=True, gradient=False, curvature=False, mask=(slice(None, None, None), slice(None, None, None)))

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

name = 'repulsive_exp'

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

property r_min

convenience function returning the location of the energy minimum

Adhesion.Interactions.Harmonic module

Harmonic potential for wall interaction

class Adhesion.Interactions.Harmonic.Harmonic(spring_constant, communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: Potential

Repulsive harmonic potential.

Harmonic potential: V (r) = 1/2 k r**2 for r < 0

evaluate(gap, potential=True, gradient=False, curvature=True)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

name = 'harmonic'

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

property r_min

convenience function returning the location of the energy minimum

Adhesion.Interactions.LJ93 module

9-3 Lennard-Jones potential for wall interactions

class Adhesion.Interactions.LJ93.LJ93(epsilon, sigma, communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: Potential

9-3 Lennard-Jones potential with optional cutoff radius.

9-3 Lennard-Jones potential: V_l (r) = ε[ 2/15 (σ/r)**9 - (σ/r)**3]

⎛ 3 9⎞ ⎜ σ 2⋅σ ⎟

V_l(r) = ε⋅⎜- ── + ─────⎟

⎜ 3 9⎟ ⎝ r 15⋅r ⎠

⎛ 3 9⎞ ⎜3⋅σ 6⋅σ ⎟

V_l’(r) = ε⋅⎜──── - ─────⎟

⎜ 4 10⎟ ⎝ r 5⋅r ⎠

⎛ 3 9⎞ ⎜ 12⋅σ 12⋅σ ⎟

V_l’’(r) = ε⋅⎜- ───── + ─────⎟

⎜ 5 11 ⎟ ⎝ r r ⎠

evaluate(gap, potential=True, gradient=False, curvature=False, mask=None)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

name = 'lj-93'

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

r_infl = σ

property r_min

convenience function returning the location of the energy minimum

6 ___ 5/6

r_min = ╲╱ 2 ⋅5 ⋅σ

────────────

5

Adhesion.Interactions.Potential module

Generic potential class, all potentials inherit it

class Adhesion.Interactions.Potential.DecoratedPotential(parent_potential)

Bases: Potential

pipeline()

class Adhesion.Interactions.Potential.Potential(communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: SoftWall

Describes the minimum interface to interaction potentials for Adhesion. These potentials are purely 1D, which allows for a few simplifications. For instance, the potential energy and forces can be computed at any point in the problem from just the one-dimensional gap (h(x,y)-z(x,y)) at that point

exception PotentialError

Bases: Exception

umbrella exception for potential-related issues

class SliceableNone

Bases: object

small helper class to remedy numpy’s lack of views on index-sliced array views. This construction avoid the computation of all interactions as with np.where, and copies

evaluate(gap, potential=True, gradient=False, curvature=False, mask=None)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

property has_cutoff

property max_tensile

convenience function returning the value of the maximum stress (i.e. the value of the gradient at the inflexion point pot.r_infl)

name = 'generic_potential'

pipeline()

abstract property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

abstract property r_min

convenience function returning the location of the energy minimum

classmethod register_function(name, function)

property v_min

convenience function returning the value of the energy minimum

Adhesion.Interactions.PowerLaw module

class Adhesion.Interactions.PowerLaw.PowerLaw(work_of_adhesion, cutoff_radius, exponent=3, communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: Potential

Polynomial interaction wiches value, first and second derivatives are 0 at the cutoff radius \(r_c\)

\[(r < r_c) \ (1 - r / r_c)^p\]

With the exponent \(p >= 3\)

evaluate(gap, potential=True, gradient=False, curvature=False, mask=None)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

property has_cutoff

property max_tensile

convenience function returning the value of the maximum stress (i.e. the value of the gradient at the inflexion point pot.r_infl)

name = 'PowerLaw'

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

property r_min

convenience function returning the location of the energy minimum

Adhesion.Interactions.SmoothPotential module

class Adhesion.Interactions.SmoothPotential.SmoothPotential(parent_potential, gamma=None, r_t=None)

Bases: DecoratedPotential

implements the splining of the potential tail using a fourth order polynomial than reaches zero within a finite cutoff radius

decoupled potential: V_lγ (r) = V_l(r) - V_l(r_t) + γ

spline: (Δr = r-r_t)

2 3 4

C₂⋅Δr C₃⋅Δr C₄⋅Δr

V_s(Δr) = C₀ - C₁⋅Δr - ────── - ────── - ──────

2 3 4

The formulation allows to choose the contact stiffness (the original repulsive zone) independently from the work of adhesion (the energy well). By default, the work of adhesion equals -min(V(r)) and the transition point r_t between the naive potential and the spline is at the minimumm, however they can be chosen freely.

The spline is chosen to guarantee continuity of the second derivative of the potential, leading to the following conditions: (1): V_s’ (Δr_t) = V_lγ’ (r_t) (2): V_s’’(Δr_t) = V_lγ’’(r_t) (3): V_s (Δr_c) = 0, where Δr_c = r_c-r_o (4): V_s’ (Δr_c) = 0 (5): V_s’’(Δr_c) = 0 (6): V_s (Δr_m) = -γ, where Δr_m = r_min-r_o The unknowns are C_i (i in {0,..4}) and r_c r_o is the origin for the spline, which can be chosen freely. The original choice was r_o = r_t, but it turned out to be a bad choice, leading to a system of nonlinear equation in which only two of the six unknowns can be eliminated.

With r_o = r_c, all coefficients C_i can be eliminated and a good initial guess for the remaining scalar nonlinear equation can be computed. see _eval_poly_and_cutoff

evaluate(gap, potential=True, gradient=False, curvature=False)

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

get_r_infl_spline()

compute the inflection Point of the spline. The inflexion Point of the smoothed potential could be below r_t and so staying the original one

property r_infl

convenience function returning the location of the potential’s inflection point Depending on where the transition between the parent_potential and the spline has been made this returns the inflection point of the spline or of the parent_potential

property r_min

convenience function returning the location of the enery minimum of the parent potential. (The spline begins at radii above r_min)

Adhesion.Interactions.VdW82 module

Implements a van der Waals-type interaction as described in http://dx.doi.org/10.1103/PhysRevLett.111.035502

Adhesion.Interactions.VdW82.Lj82(w, z0, **kwargs)

class Adhesion.Interactions.VdW82.VDW82(c_sr, hamaker, communicator=<NuMPI.MPIStub.Intracomm object>)

Bases: Potential

Van der Waals-type attractive potential with a fantasy repulsive model (like Lennard-Jones). The potential uses the formulation of Lessel et al. 2013 (http://dx.doi.org/10.1103/PhysRevLett.111.035502). However, the oxide layer is supposed do be thick

A C_sr

vdW(r) = - ─────── + ────

2 8

12⋅r ⋅π r

A 8⋅C_sr

vdW’(r) = ────── - ──────

3 9

6⋅r ⋅π r

A 72⋅C_sr

vdW”(r) = - ────── + ───────

4 10

2⋅r ⋅π r

evaluate(gap, potential=True, gradient=False, curvature=False, mask=(slice(None, None, None), slice(None, None, None)))

Evaluates the potential and its derivatives

Parameters:

gap: array_like of float

array of distances between the two surfaces

potential: bool, optional

if true, returns potential energy (default True)

gradient: bool, optional

if true, returns gradient (default False)

curvature: bool, optional

if true, returns second derivative (default False)

mask: ndarray of bool, optional

potential is only evaluated on gap[mask] this property is used by the child potential

name = 'v-d-Waals82'

property r_infl

convenience function returning the location of the potential’s inflection point (if applicable)

3 ____ ╱ C_sr⋅π

r_infl = ╲╱ 12 ⋅6 ╱ ──────

╲╱ A

property r_min

convenience function returning the location of the enery minimum

2/3 6 ___ ╱ C_sr⋅π

r_min = 2 ⋅╲╱ 3 ⋅6 ╱ ──────

╲╱ A

Module contents