9.7 Slide-Line Element with Coulomb Friction

The slide-line element is defined by three nodes and two spring constants or "penalty parameters," k1 and k2, in the tangential and normal directions, respectively. The connection from node A to node B defines the "slide-line" direction, and node C is the contact node (see Fig. 9.7.1).

Figure 9.7.1 Slide Line with Coulomb Friction

The tangent vector is defined as:

t = AB / |AB|

The direction of the unit vector n normal to the slide-line direction is given by:

in 2D: by rotating the tangent vector 90 degrees conterclockwise: n = e₃ x t

in 3D: n = -(AB x AC) x t / |AB x AC|

where "x" denotes the cross product of two vectors. The projected distance of node C to node A onto the slide-line direction is denoted by a, and is given by

a = AB^.AC / |AB|² = AC ^. t / |AB| 0 a 1

where "." denotes the dot product of two vectors. The relative normal displacement, or gap, is computed as:

g_n = |AB x AC| / |AB|² = AC ^. n / |AB|

and the relative slip as:

g_t = a - a₀

where a₀ is the relative position at which node C first contacted the line AB. The normal and tangential stresses are computed as:

S_n = k2*g_n and S_t = k1*g_t

The normal stress must be compressive, i.e.,

S_n 0

and the tangential stress such that

|S_t| tan()*|S_n| (*)

where = friction angle. The Coulomb friction law is associated with a no-slip condition and a directional constraint that requires the friction force to always act opposite to the direction of the relative slip of node C with respect to nodes A and B. A return procedure is used to enforce inequality (*) when violated.

The local contact stiffness matrix K is given by:

where k = k1 and k2 for the tangential and normal directions, respectively, and where the rows and columns are arranged such that the first, second and third rows (columns) correspond to nodes A, B and C, respectively. The contact/release condition is defined as follows:

If ( 0 a 1 and S_n 0 ) ===> contact

otherwise, release.

When contact is noted, a contact element stiffness and out-of-balance force are added to the global equations, by rotating the local stiffness and force to the global axes.

SLIDE_COULOMB

Element_name = SLIDE_COULOMB

m, stiff(1, m), stiff(2, m), phi(m), c(m) < m = 1, numat >

< connectivity data >

< terminate with a blank record >.

9.7.1 Element Group Control Information

Must follow the element name (same data record), and define the control parameter as follows:

Note Variable Name Type Default Description

(1) Gapping list [on] Gapping code
on / off (only applicable to 2D cases)

Friction_load_time integer [0] Friction angle load time function
number

formulation list [penalty] Formulation

penalty

augmented_lagrangian

Notes /

(1) This allows the contact-release option to be deactivated if needed.

9.7.2 Geometric / Material Properties Data (Numat sets)

Note Variable Default Description

M [0] Geometric/material set number

STIFF(1,M) [0.0] Spring constant k1

STIFF(2,M) [0.0] Spring constant k2

PHI(M) [0.0] Friction angle (degrees)

C(M) [0.0] Cohesion

9.7.3 Element Nodal Connectivity Data

Consult Chapter 11 for details; for this element NEN = 3, and the nodes are entered in the following order; node A, node B, node C (see Figure 9.7.1).

Notes . .

Notes . .

Notes . .