Finite element approaches for wear discretization

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Ω_m(t)→Ω_t(t), X(t)˜ →x(t),

ϕ(X) =˜ x. (5.13)

The mapping ϕ in its general form is denoted as Arbitrary-Lagrangian-Eulerian (ALE) for-mulation, since the observer is neither fixed at a material point nor fixed at a spatial point. By degeneration of the material mapping φ to an identity mapping, the ALE approach reduces to a pureLagrangianrepresentation. On the contrary, when the spatial mapping ψ represents the identity mapping, the approach changes to anEulerianrepresentation. This concept of an ALE formulation is widely used, e.g. for fluid-structure interaction problems Klöppel et al. [138]

and Mayr et al. [166] and finite strain plasticity models Armero and Love [10]. More detailed explanations of ALE problems can for example be found in Belytschko et al. [16] and Huerta and Casadei [118].

5.3. Finite element approaches for wear discretization

In the following, it is distinguished between two fundamentally different finite element ap-proaches for discretizing the volume loss, i.e. the wear depth. Again, wear phenomena are con-sidered as a description for material loss driven by Archard’s law regardless of the underlying physical effects. First, the so-calledinternal state variable approachis introduced based on the investigations in Farah et al. [69] and Gitterle et al. [88]. It expresses the wear depth in a weak sense and in terms of the already existing quantities. Thus, no additional unknowns enter the discrete system of equations. The alternative method of wear discretization is based on the work

5.3. Finite element approaches for wear discretization in Farah et al. [74] and is namedprimary variable approach. Here, wear is completely discretized on the contact interface and enters the system of equations as additional set of unknowns.

5.3.1. Internal state variable approach

The internal state variable approach was developed in the context of 2D dual mortar methods in Gitterle [87] and was extended towards 3D applications in Farah et al. [69]. Thus, the follow-ing explanations are strongly based on the work in Farah et al. [69]. The internal state variable approach is based on expressing the wear depth in terms of already employed quantities, namely the normal contact pressurepn and the displacementsu⁽ⁱ⁾. Therefore, the basis for the finite el-ement discretization of wear with the internal state variable approach is the expression of the cumulative wear in (5.10), which reads

∆w = (k_w⁽¹⁾+k⁽²⁾_w )|pn| ||u_τ,rel||. (5.14) Here, the wear increment∆wrepresents the total wear depth per time step of the two involved slave and master surfaces. Now, the cumulative wear coefficient is introduced and reads

kw=k⁽¹⁾_w +k⁽²⁾_w . (5.15)

Keeping in mind the well-established slave-master principle of mortar contact discretizations introduced in Chapter 3, all following relations are stated on the slave side. Thus, the weighted wear increment∆ ˜w⁽¹⁾is defined on the slave side and is evaluated as integral expression of (5.14) over the slave side, via

∆ ˜w⁽¹⁾ =kw

Z

γc⁽¹⁾

pn||u_τ,rel|| dA. (5.16)

Herein, the normal contact pressurepn can be identified as normal part of the contact Lagrange multiplierλn. Thus, the discretized form of the weighted wear increment∆˜w⁽¹⁾_j for slave node-j reads

∆˜w_j⁽¹⁾ =kw

Z

γ⁽¹⁾_c,h

Φˆjλn||u_τ,rel|| dA , (5.17)

withΦˆ_j representing a test function for the wear equation. Its physical interpretation and defini-tion will be discussed in Secdefini-tion 5.4.2.1. Similar to the discrete weighted gap funcdefini-tion in (3.40), the discrete weighted wear increment represents a discrete distance quantity, which is integrated over the discrete slave surface. Therefore, the discrete weighted wear could be interpreted as volume loss per slave node.

In addition, it is important to point out that the employed slip increment in (5.17) is not related with the weighted relative tangential velocity in (3.43), but rather a point-wise evaluated quantity.

The weighted relative tangential velocity is created by a weighting introduced by the variation of Lagrange multipliers, which is represented by the shape functionΦj within the time derivative of the mortar matrices D and Min (3.43). Employing the frame indifferent relative tangential velocity from (5.17) is only appropriate when assuming a constant normal pressure over the local support of a node and would allow to exclude the normal part of the Lagrange multiplierλn

from the integral in (5.17). However, this simplification would decrease the accuracy of the traction interpolation, which directly affects the quality of the computational results. Therefore, the non-weighted slip increment is employed for the weighted wear and slight discrepancies will be accepted. Further information concerning the choice of the slip increment can be found in Gitterle [87].

5.3.2. Primary variable approach

The second possibility of discretizing the wear depth is the so-called primary variable approach, which was firstly presented in Cavalieri and Cardona [36] for a small amount of wear and has been extended to finite wear simulations in Farah et al. [74]. Here, Archard’s wear law is rewrit-ten by employing the method of weighted residuals

Z

γ⁽ⁱ⁾c

δw⁽ⁱ⁾(w⁽ⁱ⁾−k⁽ⁱ⁾_w |pn| ||uτ,rel||)dA= 0, (5.18) with the wear weighting functions δw⁽ⁱ⁾. In contrast to the internal state variable approach, the wear depth on slave and master side is explicitly discretized via

w_h⁽¹⁾|_Γ⁽¹⁾ wherew⁽¹⁾_k andw⁽²⁾_l represent discrete nodal wear variables on the slave and the master surface, respectively. As for the displacement interpolation, standard shape functions based on Lagrange polynomials N_k⁽¹⁾ and N_l⁽²⁾ are employed for wear representation. Moreover, the numbers of nodes carrying discrete wear unknowns are defined to be equal to the total numbers of slave and master nodes, i.e.n⁽¹⁾_w =n⁽¹⁾andn⁽²⁾_w =n⁽²⁾. In addition to the discrete wear depth unknowns, discrete weighting functions are introduced as

δw⁽¹⁾_h |_Γ⁽¹⁾_c,h = with the nodes carrying discrete wear weightings being chosen according to the wear interpo-lation: n⁽¹⁾_z = n⁽¹⁾ andn⁽²⁾_z = n⁽²⁾. The shape functions Θ⁽ⁱ⁾_w are defined in full analogy to the Lagrange multiplier interpolationΦ, which allows for employing either standard or dual shape functions, see Section 3.4.1. With the discrete wear unknowns and the weighting functions being defined, the calculation of the wear depth w⁽ⁱ⁾ can be stated based on the weak enforcement of Archard’s wear law in (5.18). Therefore, the normal contact pressurepnin (5.18) is identified as normal part of the contact Lagrange multiplierλnand the discretized wear depth and the discrete wear weightings are employed, which leads to the resulting discrete wear residuum in global form:

r⁽ⁱ⁾_w =E⁽ⁱ⁾w⁽ⁱ⁾−T⁽ⁱ⁾λ=0. (5.21)

5.3. Finite element approaches for wear discretization Here, the vector w⁽ⁱ⁾ contains all discrete wear unknowns. The new mortar matrices E⁽ⁱ⁾ are created by assembling the nodal values

E_jk⁽¹⁾ = It is obvious that E⁽¹⁾ can be interpreted as standard slave-sided mortar matrixD with reduced dimension. Thus, employing dual shape functions for the discrete wear weighting and standard shape functions for the wear interpolation yields a Petrov-Galerkin type formulation and con-sequently a diagonal matrix E⁽¹⁾ in (5.22). A detailed motivation for such a Petrov-Galerkin approach can be found in Popp et al. [214] in the context of the non-penetration constraint for contact. The beneficial effects ofE⁽¹⁾ becoming a diagonal matrix can be employed for an effi-cient calculation of the discrete wear unknowns in (5.21), via

w⁽¹⁾ =E^(1),−1T⁽¹⁾λ. (5.24)

Here, the mortar matrix E⁽¹⁾ is inverted to obtain a solution for the discrete wear unknowns.

The matrixE⁽²⁾in (5.23) arises by integrating the two shape functionsΘ⁽²⁾_j andN_l⁽²⁾defined on the master side over the slave surface. This is realized by applying the already introduced dis-crete mappingχhtwice. Integrating this matrix over the master surface would arguably be easier, because no discrete projection would be required, but to keep the established slave-master prin-ciple, and thus in some sense the consistency of the mortar integration scheme, all integration procedures are performed on the slave side. But, it is not guaranteed that the matrixE⁽²⁾in (5.23) is strictly diagonal, because the biorthogonality condition of the dual shape functionsΘ⁽²⁾_j is en-forced over the discrete master surface γ_c,h⁽²⁾, but the integral expression in (5.23) is evaluated over the discrete slave surfaceγ_c,h⁽¹⁾. A strictly diagonal matrixE⁽²⁾ could be created by enforcing the biorthogonality condition between the projected master side shape functions over the slave surface γ_c,h⁽¹⁾, or performing the integral evaluation over the master surface. However, from an engineering point of view, the resulting error in the problem formulation is negligible. The ma-tricesT⁽ⁱ⁾arise from integrating products of weighting and Lagrange multiplier shape functions with the norm of the non-weighted slip increment over the slave side, i.e.

T_jk⁽¹⁾ =k_w⁽¹⁾ These matrices complete the discrete wear equation.

Remark 5.1. The definition that the wear weighting functionsΘw,j act as dual shape functions and the interpolation of the discrete wear unknowns is realized by standard shape functions is somehow arbitrary. It is also possible to create a Petrov-Galerkin type formulation by defining the shape functions vice versa. This was also implemented and tested and no negative influence on the solution quality, the stability or robustness of the proposed algorithms was noticed.