Contacts

Numerical contact mechanics can roughly be divided into contact discretisation and constitutive contact description. While the discretisation detects, calculates and integrates contact contributions, the constitutive contact delivers the relation between contact distance and contact stress.

For the contact constraint enforcement, the penalty regularisation and the augmented Lagrange method are implemented in numgeo.

Details on the discretisation and constitutive interface formulation are provided in the following sections:

Below, the adopted notation can be found.

Contact mechanics notation

The basic features of the mechanical description of the contact problem are presented first, whereby these apply independently of the chosen contact discretisation method.

Classically, one surface of the pair is denoted as the slave (\(\sqcup^{(1)}\)) (i.e. the surface with the finer mesh) and the other as master (\(\sqcup^{(2)}\)) surface. The surface traction is defined by

\(\begin{aligned} \boldsymbol{t}=\boldsymbol{\sigma}\cdot\boldsymbol{n}^{(1)}, \end{aligned}\)

wherein \(\boldsymbol{n}^{(1)}\) is the normal vector of the slave surface. The normal vector of the master surface \(\boldsymbol{n}^{(2)}\) is given by

\(\begin{aligned} \boldsymbol{n}^{(2)}=-\boldsymbol{n}^{(1)}. \end{aligned}\)

The contact stress of the contact pair can be separated in its normal and tangential components, viz.

\(\begin{aligned} \boldsymbol{t}=\boldsymbol{t}_{N}+\boldsymbol{t}_{T}. \end{aligned}\)

The normal stress component \(t_{N}\) (negative for compression) and the normal stress vector \(\boldsymbol{t}_{N}\) are

\(\begin{aligned} t_{N}=-\boldsymbol{n}^{(2)}\cdot\boldsymbol{\sigma}\cdot\boldsymbol{n}^{(1)}~~\textrm{and}~~ \boldsymbol{t}_{N}=-\boldsymbol{n}^{(2)} t_{N}. \end{aligned}\)

The (total) normal stress can be decomposed according to Terzaghi's principle in

\(\begin{aligned} t_{N} =t'_{N} - p^w, \end{aligned}\)

where the effective normal stress \(t'_{N}\) and the (intrinsic) pore water pressure \(p^w\) (positive for compression) are introduced.

The tangential stress vector is defined by

\(\boldsymbol{t}_{T}=\boldsymbol{t}-\boldsymbol{t}_{N}= (\boldsymbol{ I} -\boldsymbol{n}^{(2)} \otimes \boldsymbol{n}^{(2)}) \cdot \boldsymbol{t} \label{deftangentialSpg}.\)

The minimum distance between slave and master nodes is evaluated by the euclidean norm

\(\begin{aligned} \boldsymbol{x}^{(2)}(\boldsymbol{x}^{(1)})&=\text{arg}(\min \|\boldsymbol{x}^{(1)}-\boldsymbol{x}^{(2)}\|) \label{Abstandsfunktion0} \end{aligned}\) and the gap \(\boldsymbol{g}\) calculated using

\(\begin{aligned} \boldsymbol{g}&=\boldsymbol{X}^{(1)}+\boldsymbol{u}^{(1)}-(\boldsymbol{X}^{(2)}+\boldsymbol{u}^{(2)})=\boldsymbol{g}_{0}+\boldsymbol{u}^{(1)}-\boldsymbol{u}^{(2)}\label{Abstandsfunktion1}, \end{aligned}\)

where \(\boldsymbol{g}_{0}\) is the gap in the reference configuration and \(\boldsymbol{u}\) the displacement. Similar to the contact stress, the contact gap has a normal

\(\begin{aligned} \boldsymbol{g}_{N}=\boldsymbol{n}^{(2)} \otimes \boldsymbol{n}^{(2)} \cdot \boldsymbol{g} \end{aligned}\)

and a tangential part

\(\begin{aligned} \boldsymbol{g}_{T}=(\boldsymbol{ I} -\boldsymbol{n}^{(2)} \otimes \boldsymbol{n}^{(2)}) \cdot \boldsymbol{g}. \end{aligned}\)

Conventionally, the contact pair has to satisfy the following conditions:

Only contact pressure is possible: \(t_{N} \leq 0\)
A penetration is not allowed: \(g_N \geq 0\)
If the surfaces are not in contact, the contact stress is zero. If the gap is zero, the stress is not equal to zero. These so-called complementary conditions are expressed by \(g_N t_{N} = 0\)

The three conditions are also known as the Karush-Kuhn-Tucker (KKT) conditions. The contact conditions are interpreted as constraints in mechanical terms. These constraints can represent a condition on the displacement of two contact points and prevent them from penetrating into each other. In water-saturated soil, a structure (e.g. a pile) can not separate from the water as long as no cavitation takes place allowing for positive values of \(t_{N}\) (and non-zero normal contact stress despite \(g_N > 0\)). This means that for positive values of \(g_N\), the effective stress component \(t'_{N}\) is zero and the pore water pressure \(p^w\) takes negative values. Thus, \(t_{N} =-p^w\) holds in this case.

Basic FE notation

The derivation of the global coordinates with respect to the local coordinates is

\[\begin{align} \boldsymbol{x}_{,\xi} (\boldsymbol{\xi})=\sum_I \frac{dN_I(\boldsymbol{\xi})}{d\boldsymbol{\xi}}\boldsymbol{x}_I, \end{align}\label{eq_element_jacobian}\]

which is also known as the element Jacobian \(\boldsymbol{J}\). The normal vector \(\boldsymbol{n} (\xi)\) at the edge of an element is given for 2D by

\[\begin{align} \boldsymbol{n}(\xi) =\frac{\boldsymbol{x}_{,\xi}(\xi) \times \boldsymbol{\tau}_3}{\|\boldsymbol{x}_{,\xi}(\xi)\|}, \end{align}\]

where \(\boldsymbol{\tau}_3=[0,0,1]^T\) is used and \(\boldsymbol{x}_{,\xi}(\xi)\) is the derivative of the global coordinate with respect to the local coordinate \(\xi\) evaluated at position \(\xi\) (only relevant for interpolation with order two or higher). \(\times\) marks the cross product. For three-dimensional analyses, the normal vector is defined by

\[\begin{align} \boldsymbol{n}(\xi, \eta) =\frac{\boldsymbol{x}_{,\xi}(\xi, \eta) \times \boldsymbol{x}_{,\eta}(\xi, \eta) }{\|\boldsymbol{x}_{,\xi}(\xi, \eta) \times \boldsymbol{x}_{,\eta}(\xi, \eta)\|}. \end{align}\label{eq_normal_vector3D_fem}\]

The tangential vectors orientated in the local coordinate system are determined by

\[\begin{align} \boldsymbol{\tau}_{1}(\xi) =\frac{\boldsymbol{x}_{,\xi}(\xi)}{\|\boldsymbol{x}_{,\xi}(\xi)\|} \end{align}\label{eq_tangVector_fem2D}\]

for 2D and by

\[\begin{align} \boldsymbol{\tau}_{1}(\xi, \eta) =\frac{\boldsymbol{x}_{, \xi}(\xi,\eta)}{\|\boldsymbol{x}_{, \xi}(\xi,\eta)\|} ~\textrm{and}~ \boldsymbol{\tau}_{2}(\xi, \eta) =\frac{\boldsymbol{x}_{, \eta}(\xi,\eta)}{\|\boldsymbol{x}_{, \eta}(\xi,\eta)\|} \end{align}\label{eq_tangVector_fem3D} \]

for the 3D case.