Problem statement

Starting point for the derivation of a contact formulation being valid for various different contact scenarios is the standard initial boundary value problem (IBVP) for finite deformation elastody-namics as introduced in Section 2.1.4. For the sake of completeness, it is restated here for the two considered bodiesB⁽¹⁾ andB⁽²⁾:

DivP⁽ⁱ⁾+ ˆb⁽ⁱ⁾₀ =ρ⁽ⁱ⁾₀ u¨⁽ⁱ⁾ inΩ⁽ⁱ⁾₀ ×[0, T], (4.1) u⁽ⁱ⁾ = ˆu⁽ⁱ⁾ onΓ⁽ⁱ⁾_u ×[0, T], (4.2) P⁽ⁱ⁾·N⁽ⁱ⁾ = ˆt⁽ⁱ⁾₀ onΓ⁽ⁱ⁾_σ ×[0, T], (4.3) u⁽ⁱ⁾(X⁽ⁱ⁾,0) = ˆu⁽ⁱ⁾₀ (X⁽ⁱ⁾) inΩ⁽ⁱ⁾₀ , (4.4)

˙

u⁽ⁱ⁾(X⁽ⁱ⁾,0) = ˆ˙u⁽ⁱ⁾₀ (X⁽ⁱ⁾) inΩ⁽ⁱ⁾₀ . (4.5) As explained for standard surface-to-surface contact in Section 3.1, these bodies share one po-tential contact interface, which is defined by the popo-tential contact boundaries in the reference configurationΓ⁽ⁱ⁾_c . Again, the overall boundary of each body is divided into three disjoint sets as

∂Ω⁽ⁱ⁾₀ = Γ⁽ⁱ⁾_u ∪Γ⁽ⁱ⁾_σ ∪Γ⁽ⁱ⁾_c , (4.6) Γ⁽ⁱ⁾_u ∩Γ⁽ⁱ⁾_σ = Γ⁽ⁱ⁾_u ∩Γ⁽ⁱ⁾_c = Γ⁽ⁱ⁾_σ ∩Γ⁽ⁱ⁾_c =∅, (4.7) with the Dirichlet boundariesΓ⁽ⁱ⁾_u and the Neumann boundariesΓ⁽ⁱ⁾_σ . Since the special focus of this chapter is to investigate contact situations involving non-smooth geometries, the potential contact boundariesΓ⁽ⁱ⁾_c are further divided into three disjoint subsets, viz.

Γ⁽ⁱ⁾_c = Γ⁽ⁱ⁾_◦ ∪Γ^(i)p ∪Γ⁽ⁱ⁾_⋆ , (4.8) Γ⁽ⁱ⁾_◦ ∩Γ⁽ⁱ⁾p = Γ⁽ⁱ⁾_◦ ∩Γ⁽ⁱ⁾_⋆ = Γ⁽ⁱ⁾p ∩Γ⁽ⁱ⁾_⋆ =∅, (4.9) where Γ⁽ⁱ⁾_◦ are the potential contact boundaries of surfaces, Γ^(i)p represent edges and Γ⁽ⁱ⁾⋆ are the sets of all vertices within the contact boundaries, see also Figure 4.2. Similar to the other boundaries, their spatial counterparts are denoted asγ_◦⁽ⁱ⁾,γp⁽ⁱ⁾andγ⋆⁽ⁱ⁾. Due to the assumed finite deformation regime, the geometrical contact entities, namely surfaces, edges and vertices can deform significantly, meaning that an initial vertex could be flattened to become part of a new surface, or a surface could be deformed in a way to create a new edge. However, in this thesis it is assumed that the spatial points are assigned to the set definition of its reference boundary and consequently extreme deformations, such as a complete flattening of an edge, are not allowed.

In order to define suitable contact conditions the possibly arising contact scenarios have to be specified. For this purpose, it is assumed that the contact entity with the lower geometrical di-mension acts as slave part and the corresponding contact entity of equal or higher geometrical order is defined to be the master boundary. Concretely, the first class of possible contact scenar-ios is characterized by the active contacting area reducing to a point, see Figure 4.3. Namely, these scenarios are vertex-to-vertex, vertex-to-edge, vertex-to-surface and non-parallel edge-to-edge settings. These contact situations are denoted as point contact in the following. The next class is defined by the contacting area being a 1D line, which could arise due to edge-to-surface

4.2. Problem statement

Figure 4.2: Kinematics and basic notation for the description of contact of vertices, edges and surfaces for two elastic bodies. Note, only the grey surface represent the potential contact regions and thus all notations are restricted to these zones. The figure is taken from Farah et al. [71].

Figure 4.3: Classification of contact scenarios leading to point contact. From left to right: vertex-to-vertex, vertex-to-edge, vertex-to-surface and non-parallel edge-to-edge. Contact point is highlighted in red. The figure is taken from Farah et al. [71].

Figure 4.4: Classification of contact scenarios leading to line contact: edge-to-surface (left) and parallel edge-to-edge (right). Contact line is highlighted in red. The figure is taken from Farah et al. [71].

Figure 4.5: Classical surface contact scenario with contact area highlighted in red. The figure is taken from Farah et al. [71].

4.2. Problem statement and parallel edge-to-edge contact, see Figure 4.4. The last setting is the classical surface contact scenario, which is well-investigated in the context of computational contact mechanics and is schematically visualized in Figure 4.5. This classification represents a hierarchy of contact situ-ations, where the contact boundary on which the contact constraints are going to be formulated is the involved slave entity with the lowest dimension. The only exception to this scheme is the non-parallel edge-to-edge contact. Here, the geometrical slave entity is an edge but the contact scenario reduces to a point contact. Therefore, these special contact points are denoted as x_× and the set of all slave contact points resulting from crossing edges isγ_×⁽¹⁾. For the non-parallel edge-to-edge setting the discrete enforcement of the contact constraints will be treated in a spe-cial way later on. However, in order to keep a convenient notation, it is postulated that the set of edge-to-edge crossing points and the set of all vertices are united to the set of potential point contactsγ_•⁽¹⁾:

γ_•⁽¹⁾ =γ_⋆⁽¹⁾∪γ_×⁽¹⁾. (4.10)

Furthermore, it is assumed that the potential line contact setting is defined onγ^p(1) and the poten-tial surface contact scenario is defined on γ_◦⁽¹⁾. The corresponding contact force quantity which acts on the slave contact boundary is now different for point, line and surface contact. The point

Figure 4.6: Contact force quantities for different contact scenarios: point contact (left), line con-tact (middle) and surface concon-tact (right). The figure is taken from Farah et al. [71].

contact formulation is subjected to the force vector f_c⁽¹⁾, which is a concentrated load on the contact point. For the line contact setting, a line load vectorl⁽¹⁾_c is introduced, and consequently a surface traction vector t⁽¹⁾_c is employed for the surface contact. Due to the balance of linear momentum, the corresponding master load vectors are identical except for the opposite sign, i.e.

f_c⁽¹⁾ =−f_c⁽²⁾, l⁽¹⁾_c =−l⁽²⁾_c , t⁽¹⁾_c =−t⁽²⁾_c . (4.11) Similar to the gap function and the relative tangential velocity in Chapter 3, the contact load vectorsf_c⁽¹⁾,l_c⁽¹⁾andt⁽¹⁾_c on the slave surface can be split into their normal and tangential com-ponents, yielding

f_c⁽¹⁾ =fnn+f_τ, l_c⁽¹⁾ =lnn+l_τ, t⁽¹⁾_c =tnn+t_τ. (4.12)

Thus, the contact constraints in normal direction are given in form of the well-known Hertz-Signorini-Moreau conditions for the point, line and surface contact scenarios:

gn ≥0 onγ_c⁽¹⁾×[0, T], (4.13)

fn≤0, fngn= 0 onγ_•⁽¹⁾×[0, T], (4.14) ln ≤0, lngn= 0 onγp⁽¹⁾×[0, T], (4.15) tn ≤0, tngn = 0 onγ_◦⁽¹⁾×[0, T]. (4.16) In addition, frictional sliding is formulated under the assumption that Coulomb’s law is also valid for point contact and line contact, see Pandolfi et al. [194]. The frictional sliding constraints for point contact read

Υ_• :=kf_τk −F|f_n| ≤0,

vτ,rel+β_•fτ =0, β_• ≥0, Υ_•β_• = 0 onγ_•⁽¹⁾×[0, T]. (4.17)

The corresponding constraints for line contact are given as Υ^p:=kl_τk −F|l_n| ≤0,

vτ,rel+β^plτ =0, β^p≥0, Υ^pβ^p = 0 onγ^p(1)×[0, T]. (4.18)

Finally, the tangential part of surface contact is defined by Υ_◦ :=ktτk −F|tn| ≤0,

vτ,rel+β_◦tτ =0, β_◦ ≥0, Υ_◦β_◦ = 0 onγ_◦⁽¹⁾×[0, T]. (4.19)

In (4.17)-(4.19) the friction coefficient F ≥ 0is assumed to be equal for all contact scenarios for the sake of simplicity. Again, k · k denotes the L²-norm in R³ and the parameters β_i are complementarity parameters that are necessary to describe the separation of the stick and slip branch. These parameters will vanish by reformulating the sets of tangential constraints within so-called nonlinear complementarity functions, see Section 3.5.