PDF Index| SDT-nlsim
Contents
Functions
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This functionality is distributed as part of the SDT contact. For general theory see section 3.1.2.
The current chandle implementation (SDT ≥ 7.4) uses mex files and uses the convention of positive pressures for positive overclosure (opposite of gap −g). Observations of overclosures are thus of the usual form −g=unl = [c]{u}+{unl0}. For sliding, the convention of using tangential observation and commands with the same sign convention give contact forces with a minus sign.
For chandle runtime formatting, the implementation considers Gauss points with 3 gap strain components g=uN, uT1, uT2 and possibly xxx (unllab={'gn';'gt1';'gt2';'utt'}).
For 3 strain velocities vN, w1, w2 the formulation is given in (3.25)
| (3.34) |
Laws are selected using a URN combining normal (see sdtm.enum('nlCtc') ) and tangent laws (see sdtm.enum('nlSli') ) .
The nlutil('@Ctc') subfunction implements conversion of standard combinations of parameters to tabular laws
Ctc=nlutil('@Ctc'); Ctc('urn','n1s13{Kc1e12,Lambda2}')
Older versions use .m file implementation, gaps and not overclosures and the offsets, g=[c]{u}+{cqoff} is stored in field .cqoff. Laws are selected using the ContactLaw value listed below.
The simplest contact representation is to apply a linear reaction force to contact points with negative gaps. The resulting law is bilinear, expressing contact pressure at a contact integration point i as
| (3.35) |
This law has a single parameter kc (in Pa / m, or N / m3).
The contact stiffness for tangent contact states can thus have two values depending on the gap at contact integration point i
| (3.36) |
kc, of input name Kc, can be set to Val in Fu as a string of format 'Kc Val' or as a scalar of value Val.
The exponential contact law applies a non linear reaction force for all gaps. It is expressed at a contact integration point i as
| (3.37) |
The pressure applied is strictly positive for all gaps, but its fast decay for low gap values easily generates negligible values. Exponential increase of the pressure as function of the gap provides reasonable behavior in terms of penetrations for large gap variations over a surface. Convergence can become difficult to obtain for very fast varying contact conditions since penetration overestimation can lead to numerically infinite forces.
This law has two parameters, pz in Pa being the pressure at zero gap (curve offsetting as function of the gap), and λ in m−1 being the scaling factor (curvature of the exponential).
The contact stiffness for tangent contact states at contact integration point i is thus
| (3.38) |
The tangent contact stiffness varies with the gap at each points, which allows fine observation of system sensitivities to contact states in the modal domain.
As the definition of λ can be difficult, the exponential law can be alternatively set with parameter pe instead of λ. This parameter stands of expected pressure at structural stiffness, and represents a maximal working target pressure for the model at which one will expect the exponential law to generate a stiffness density higher than a stiff linear coupling. The estimated stiffness is the same than for the bilinear law with Kc=-1, and λ is resolved as λ=Kc/pe. Parameter lambda is then set in the law after initialization.
Using an exponential law can lead to very high stiffness densities, that for larger gaps may lead to very high values that would compromise numerical conditioning. It is thus possible to define a stiffness bound that will saturate the exponential growth after a critical gap value. One then defines parameter KcMax as the maximum stiffness density allowed to the exponential formulation. An associated gap gmax = −log(KcMax/pz λ)/λ defines the threshold after which the exponential law is bounded so that for gni>gmax
| (3.39) |
pz, of input name pz, λ of input name lambda, and Kcmax can be set to ValP, ValL and ValKM in Fu as a string of format 'Kc ValP lambda ValL' or as a line vector [ValP ValL ValKM]. The string format only accepts defining parameter pe of input name pe instead of lambda. Note that pe and lambda are mutually exclusive parameters, with the priority set on lambda.
Tabulated contact laws are user defined or built using nlutil('@Ctc'). Using chandle both linear an piecewise cubic interpolation are supported. The curves are referenced using a curve identifier (see sdtweb curve for more details). The table is a gap pressure relationship, with field X being the gap and field Y being the pressure.
For linear interpolation, a contact point i the output pressure is interpolated between the two gap values bounding the exact gap, noting G the set of provided gap values in the table
| (3.40) |
If gi is outside the table bounds, extrapolation is performed following the slopes of the first or last table segment, such that if gi is lower than the lower bound g1i and g2i are the first two values of the table, and if gi is higher than the higher bound g1i and g2i are the last two values of the table.
The contact stiffness for tangent contact states at contact integration point i is taken as the slope of the segment in which gi is found
| (3.41) |
In case of extrapolation, the slopes of the first or last segments are taken.
The curve must be stacked in the model with a proper .ID field. The curve of ID ID can be assigned to Fu as a string of format 'CurveID ID' or as a a scalar ID.
xxx give reference to cube example
Power laws can be used in the same way as exponential laws, for which it can be easier to fit parameters in some applications.
This law has a list of parameters kpm defining a stiffness density associated to power m as
| (3.42) |
The associated contact stiffness is
| (3.43) |
kpm, of input name kpm, can be set to ValM as a string format kpm ValM or as a 2 colum matrix [m ValM;...].
The Coulomb law is here expressed dynamically as function of the sliding direction (unidirectional for 1, bidirectional for 11), the friction constraint is a two direction vector at contact point i, expressed as function of the sliding velocity w (see (3.25)) as
| (3.44) |
This law takes in input the friction coefficient µ (input name Mu), and optionally the tangent sticking stiffness parameter κ (input name kappa).
.Fv='Mu ValM kappa ValK' was used prior to chandle. .Fu='n3s13{Kt1e12,Mu.1,kappa.2}' is the new format.
The Coulomb law is here expressed dynamically as function of the sliding direction, the friction constraint is a two direction vector at contact point i, expressed as function of the sliding velocity w as
| (3.45) |
The slope is then to the friction force between zero sliding velocity and ||wi||2 = µ p(xi) / kt. The saturation threshold thus depends on the level of contact pressure at each contact point.
This law takes in input the friction coefficient µ (input name Mu), the regularization slope kt (input name CtLin), and optionally the tangent sticking stiffness parameter κ (input name kappa).
µ, of input name Mu, ct of input name CtLin, and κ of input name kappa, can be set to ValM, ValCT and ValK in Fv as a string of format 'Mu ValM CtLin ValCT kappa ValK' or as a line vector [ValM ValCT ValK].
The Coulomb law is here expressed dynamically as function of the sliding direction, the friction constraint is a two direction vector at contact point i, expressed as function of the sliding velocity w as
| (3.46) |
The arctangent shape is here exploited to approximate the friction law, the advantage being that this function is infinitely continuously derivable thus alleviating potential threshold effects.
This law takes in input the friction coefficient µ, the regularization slope ct, and optionally the tangent sticking stiffness parameter κ.
µ, of input name Mu, ct of input name CtLin, and κ of input name kappa, can be set to ValM, ValCT and ValK in Fv as a string of format 'Mu ValM KtLin ValCT kappa ValK' or as a line vector [ValM ValCT ValK].
The Coulomb law is here expressed dynamically as function of the sliding direction, the friction constraint is a two direction vector at contact point i, expressed as function of the sliding velocity w as
| (3.47) |
The slope is then to the friction force between zero sliding velocity and ||wi||2 = µ / kt. The saturation threshold is thus independent on the level of contact pressure at each contact point.
This law takes in input the friction coefficient µ, the regularization slope ct, and optionally the tangent sticking stiffness parameter κ.
µ, of input name Mu, kt of input name CtLin, and κ of input name kappa, can be set to ValM, ValCT and ValK in Fv as a string of format 'Mu ValM CtLin ValCT kappa ValK' or as a line vector [ValM ValCT ValK].
