`*Embedded region, interaction-surface`

*Embedded region, interaction-surface [, type=constant | type=Mohr-Coulomb]
                                      [, section=circular] [, ghost=yes | ghost=no]
                                      [, np=<int>] [, nb=<int>]

Embedded beam with implicit interaction surface (EB-I) for pile-type structures with realistic diameter and base resistance, following Truty (2023) and Granitzer et al. (2024). The guest (3D beam) elements retain their independent degrees of freedom (translations and rotations). Around the beam axis a virtual cylindrical surface of radius \(R = D/2\) is generated and segmented into rings of coupling points, at which interface springs act on the relative displacement between the beam cross-section motion, \(\mathbf{u} + \boldsymbol{\theta} \times \mathbf{r}_{cs}\), and the host displacement field. Optionally, a base disk couples the pile toe to the host with an isotropic spring and a one-sided compressive cap. See the Theory Manual.

In contrast to *Embedded region, interaction (line coupling, suitable for slender inclusions such as grouted anchors and soil nails), the interaction surface

transfers skin friction, lateral pressure and interaction moments at the correct radial offset \(R\) (pile diameter effects, group effects),
provides a stress-dependent (Mohr-Coulomb) skin capacity based on the local host normal stress \(\sigma_n = \mathbf{n} \cdot \boldsymbol{\sigma} \cdot \mathbf{n}\) recovered on the interaction surface (rather than a mean lateral stress on the axis), and
supports an explicit base resistance.

Keyword line parameters (all optional)

Parameter	Values	Default	Description
`type`	`constant`, `Mohr-Coulomb`	`constant`	Skin (tangential) capacity law
`section`	`circular`	`circular`	Cross-section of the interaction surface (rectangular sections are not yet available)
`ghost`	`yes`, `no`	`yes`	Automatic ghost (elastic) zone inside the virtual pile volume
`np`	integer \(\geq 3\)	8	Coupling points per circumferential ring (shaft)
`nb`	integer \(\geq 1\)	9	Coupling points on the base disk (1 central + `nb`-1 ring points at \(0.75R\))

Data lines

One line per region definition. With type=constant (default):

<guest elset>, <host elset>, <base nset | NONE>, D, k_t, k_n, k_b, t_ult, sigma_lim [, E_ghost, nu_ghost]

With type=Mohr-Coulomb:

<guest elset>, <host elset>, <base nset | NONE>, D, k_t, k_n, k_b, c, phi, sigma_lim [, E_ghost, nu_ghost]

Parameter	Unit	Description
guest elset	-	Element set with the guest elements; only 3D beam elements (`u2-beam-3D`, `u3-beam-3D`) are allowed, since the coupling requires the cross-section rotations
host elset	-	Element set defining the host region (3D continuum elements)
base nset	-	Node set with the pile toe node(s), one per pile; each must be an end node of a coupled guest element. `NONE` disables the base coupling
`D`	L	Diameter of the interaction surface (\(> 0\))
`k_t`	F/L³	Tangential interface stiffness per area, identical for the axial and circumferential direction (\(> 0\))
`k_n`	F/L³	Normal (radial) interface stiffness per area (\(\geq 0\))
`k_b`	F/L³	Base interface stiffness per area, isotropic (\(> 0\) if a base nset is given; ignored with `NONE`)
`t_ult`	F/L²	Constant skin capacity; \(\leq 0\): elastic shaft (no slider)
`c`, `phi`	F/L², deg	Interface cohesion (adhesion) and friction angle: \(t_{cap}(\tilde{x}) = \max(c - \sigma_n(\tilde{x}) \tan\varphi,\, 0)\) with tension cutoff
`sigma_lim`	F/L²	Compressive base capacity (one-sided cap on the axial base traction); \(\leq 0\): elastic base
`E_ghost`, `nu_ghost`	F/L², -	Stiffness of the ghost (elastic) zone; present on the data line if and only if the ghost zone is active

Remarks

3D only. The interaction surface requires *Model, dimension=3 and 3D beam guests. For 2D models and slender inclusions use *Embedded region, interaction.
Coupling points and skipped areas. At each of the 2 (linear) or 3 (quadratic) line integration stations of a guest element, np points are distributed over the circumference (offset by half a segment); their analytically assigned areas sum exactly to the shaft area \(\pi D L\). Points located outside the host region are skipped with their area dropped (surface analogue of the free anchor length); guest elements without any coupling point are skipped entirely (log message).
Mohr-Coulomb stress recovery. \(\sigma_n\) is recovered per coupling point from the converged host stresses (effective stresses for multi-phase host elements) of the neighbouring host integration points by inverse-distance weighting; integration points inside the virtual pile volume (+ half sphere below the toe) are excluded from the recovery. Since converged stresses are used, the capacity lags one increment behind and the consistent linearisation of the slider is preserved. Small load increments are advised when the host stress state changes strongly.
Base coupling. The base disk acts at the toe node with the lever arms of the disk points (it also transfers moments). Base tension is not limited (no gapping, see below); use sigma_lim only for compressive base resistance. Following Granitzer et al. (2024), the base capacity is a direct input rather than recovered from the host stresses (tip stress recovery is unreliable due to the singular stress concentration).
Ghost (elastic) zone. With ghost=YES (default), all host integration points inside the virtual pile volume are automatically switched to a linear elastic response (\(E_{ghost}\), \(\nu_{ghost}\), chosen similar to the soil stiffness) with frozen state variables. This prevents spurious plastification of the soil inside the pile. The override is available for the single-phase solid elements (element_U, element_Us families) and the saturated u-p elements (element_UP_saturated family); integration points of other element formulations inside the pile volume keep the real material response and a warning is issued. For u-p elements only the mechanical (effective stress) response is replaced; the fluid/storage terms are unaffected. Note that the ghost zone only has an effect if host integration points actually lie inside the virtual pile volume: for 8-point hexahedra of edge length \(h\) centred on the pile axis, the integration points nearest to the axis sit at \(0.408\,h\), i.e. \(D > 0.816\,h\) is required before any point is switched. The build reports the number of switched integration points in the log and issues a warning if the ghost zone is active but empty; with ghost=NO a log line confirms that the zone is disabled.
No gapping. The normal (radial) spring is linear elastic in compression and tension; interface separation is not modelled (consistent with Truty 2023 and Granitzer et al. 2024).
Mesh guidance. np=8 and host elements of size \(\approx 0.5D\) to \(2D\) near the pile are a robust choice (Granitzer et al. 2024). The coupling data are frozen at the reference configuration (geometrically linear coupling).
Explicit dynamics are not supported (error).

Example

*Embedded region, interaction-surface, type=Mohr-Coulomb
pile, soil, pile-toe, 0.9, 25000., 25000., 50000., 5., 27.5, 5000., 75000., 0.3

Pile with \(D = 0.9\) m, \(k_t = k_n = 25000\) kN/m³, base spring \(k_b = 50000\) kN/m³ with compressive capacity \(\sigma_{lim} = 5000\) kPa, Mohr-Coulomb skin capacity with \(c = 5\) kPa and \(\varphi = 27.5°\), ghost zone with \(E_{ghost} = 75000\) kPa, \(\nu_{ghost} = 0.3\).

References

Truty, A. (2023): Improved formulation of embedded beams with interaction surface. Studia Geotechnica et Mechanica, 45(2).
Granitzer, A.-N., Tschuchnigg, F., et al. (2024): Construction and verification of an improved embedded beam formulation with an explicit interaction surface. Computers and Geotechnics.