Solution method

The consideration of wear effects and finite shape changes are again realized in a partitioned scheme as introduced in Section 5.5.2. Here, a Lagrangian step is performed followed by a shape evolution step. In contrast to Section 5.5.2, no wear effects are considered during the Lagrangian

step, which means that no modification of the gap function is performed. Thus, the Lagrangian step consists of a pure thermo-elastic problem which is solved in a fully monolithic manner.

Herein, the structural field is considered as quasi-static problem and thus inertia effects are ne-glected. For the thermal field, dynamic effects are considered and time integration is done by the well-known One-Step-θ scheme. The contact interaction is discretized with the mortar method and the thermal heat fluxes are identified as additional Lagrange multipliers. However, further details on the spatial and temporal discretization of the coupled thermal-mechanical problem are beyond the scope of this thesis and the interested reader is referred to Danowski [51], Danowski et al. [52], Gitterle [87], Hüeber and Wohlmuth [116] and Seitz et al. [245] for detailed infor-mation on this topic. Therefore, the Lagrangian step can be considered in a black-box manner.

After a completely converged Lagrangian step, the wear increment for the considered time step is post-processed according to Section 5.3.1 and afterwards utilized in the shape evolution step.

The shape evolution procedure for the structural problem, i.e. for the material configuration, is identical to the explanations in Section 5.5.4. In addition, the temperatures have to be adapted according to the mesh movement. For this purpose, the discrete temperatureθ^m_j,h at a considered relocated nodej is consistently interpolated, via

θ^m_j,h=

nn

X

b=1

N_b(ξ˜_j)θb,h , (5.96)

where the parameter space coordinateξ˜_j of the updated node within the non-updated element is calculated as described in (5.72).

The complete algorithm for the thermo-structure-contact-wear interaction with finite shape changes follows to:

Algorithm5.3. Explicit partitioned scheme

1. Solve the nonlinear thermo-mechanical contact problem including friction for fixed ma-terial displacements according to Seitz et al. [245] as Lagrangian step. Herein, no effects due to the actual wear increment are considered

2. Post-Process the wear increment for the completed Lagrangian step according to Sec-tion 5.3.1.

3. Solve the nonlinear shape evolution step as a pseudo-elasticity problem (ALE problem) for the calculated amount of wear. Then update the material and the spatial configuration as well as the temperatures with regard to the calculated wear by employing the advection map procedure.

5.6.3. Numerical examples

In the following, two numerical examples are investigated to verify the developed thermo-structure-contact-wear interaction algorithm and to demonstrate its applicability to finite de-formations and finite shape changes.

5.6. Thermo-structure-contact-wear interaction with finite shape changes

Figure 5.27: Geometry with finite element mesh and boundary conditions for the oscillating punch example. The right part visualizes the given displacements for one cycle.

5.6.3.1. Validation – oscillating punch

The first example is taken from Ireman et al. [125] and is employed to compare the accuracy of the developed thermo-structure-contact-wear interaction framework with already existing al-gorithms. The example consists of two thermo-elastic bodies, namely a punch and a founda-tion. The dimensions of the punch are 20mm× 4mm and the dimensions of the foundation are 40mm× 20mm. In contrast to the original setting, a 3D simulation is performed. Thus, both bodies have a thickness of d = 0.5mm. The punch is loaded with a prescribed trac-tion pˆ = 200_mm^N in vertical direction, see Figure 5.27. Additionally, a horizontal oscillation uˆ is prescribed at the top of the punch, see again Figure 5.27. For the numerical tests, 100 cycles are carried out. The bottom of the foundation is completely supported in vertical direction and the midpoint is also fixed in horizontal direction. In order to achieve a problem setting, which corresponds to the plane strain assumption in Ireman et al. [125], symmetry conditions are en-forced at the front and back of the bodies. The spatial discretization is visualized in 5.27 with one element layer in thickness direction. The finite element mesh employed in Ireman et al. [125]

was initially matching at the contact interface and the mesh created in this thesis is non-matching in order to demonstrate the applicability of the mortar method to non-matching discretizations.

The contact formulation is manipulated by an additional initial gapgi(x) = 0.0005x², where x is the coordinate along the contact surface with its origin at the midpoint of the contact sur-face. This initial gap is included in the non-penetration constraints in the same way as the wear depth was included in (5.33). Consequently, the two bodies slightly overlap. The mate-rial properties are chosen corresponding to steel. Thus, Young’s modulus is E = 210000_mm^N2, Poisson’s ratio isν = 0.3, the initial density isρ0 = 7800_m^kg3, the thermal expansion coefficient is αθ = 12·10^{−6 1}_K, the specific heat capacity is CV = 460_kg·K^J and the thermal conduction iskθ = 46_m·K^W . For the contact interaction, the friction coefficient is defined as F = 0.3and no damage effects are assumed. Additionally, the thermal heat transfer parameters are assumed to be equal for the two bodies and set toα¯⁽ⁱ⁾_c = 10⁻³_N^W_·K. Finally, the wear coefficient is defined askw = 10^−5mm_N². The time integration factor for the One-Step-θ scheme for thermal effects is chosen asθt = 0.5and the structural problem is considered to be quasi-static. Moreover, a time

20

Figure 5.28: Results for oscillating punch compared to Ireman et al. [125] after 100 cycles: tem-perature difference with respect to reference temtem-perature for simulations with wear effects and without wear effects (left) and wear depth when thermal effects are in-cluded (right).

step size of ∆t = 3.125·10⁻⁴s is chosen. The punch is defined to be the slave body and the foundation is the master body, respectively.

In order to judge the accuracy of the results, the basic differences in the employed finite element frameworks must be pointed out. The algorithm used in Ireman et al. [125] is based on the assumption of small displacements, small deviations from the reference temperature and a small wear depth. The contact discretization was realized in a node-to-node manner and thus the employed finite element mesh must initially match at the interface. Again, the algorithm developed in this thesis is created in order to be valid in a finite deformation and finite wear regime and the contact interaction is based on the mortar method. Consequently, it should also be able to perform in a small deformation regime with small expected wear depths.

The results compared to Ireman et al. [125] are visualized in the Figure 5.28. In the left part of this figure, the differences of the interface temperature compared to the reference temperature are plotted for simulations with included and excluded wear effects. When wear is not consid-ered, the temperatures in the middle part of the contact interface match qualitatively well. At the outer parts of the contact zone relatively high discrepancies occur. This could originate from the additional initial gapgi(x)that becomes large at the outer parts of the contact zone. Therefore, the node-to-node contact assumption in Ireman et al. [125] is strongly violated for the outer con-tact regions and the solution quality drastically decreases. The presented mortar-based approach is constructed in order to deal with non-matching meshes and thus the solution quality is not drastically influenced by the mesh-to-mesh constellation. When wear is included, the overall temperature distribution of the contact interface matches very well with the result from litera-ture. The additional loss of material, which reaches its peak at the center of the contact interface, balances the initial gap overlap and by an ongoing simulation the matching node assumption in Ireman et al. [125] becomes more and more valid. In the right part of Figure 5.28, the wear depth is plotted over the contact interface for included thermal effects. The maximum wear depth

5.6. Thermo-structure-contact-wear interaction with finite shape changes

Figure 5.29: Geometry with finite element mesh and boundary conditions for block on rotating disc example.

at the center of the contact interface is excellently captured by the algorithm developed in this thesis. Also the wear depth distribution nicely matches the result from literature.

In summary, the presented algorithm yields excellent solutions and the results from Ireman et al. [125] can be reproduced.

5.6.3.2. Finite shape changes – block on rotating disc

The final wear example is introduced in order to demonstrate the applicability of the proposed thermo-structure-contact-wear interaction algorithm to finite deformations and finite shape chan-ges. The example consists of a block which is pressed onto a rotating hollow cylinder. The geometry and the employed finite element mesh is shown in Figure 5.29. In addition to the information in 5.29, the thickness of the block is 5mm and the thickness of the disc is defined as 6mm. For both bodies, first-order hex8 elements are employed. The material properties for both bodies are chosen according to steel. Thus, the material parameters are equal to the previous example. The damage temperature for the friction coefficient is chosen as θ_d = 700K and the initial friction coefficient is F₀ = 0.3. The reference temperature is defined as θ0 = 293K.

Again, the time integration factor for the One-Step-θ scheme for the thermal field is chosen toθ_t = 0.5and the structural problem is assumed to be quasi-static. The time step size is defined as∆t= 0.01s and the overall simulation time isT = 3.5s. In the time interval0s< t≤0.1s, the block is completely fixed at its top surface and the hollow cylinders inner surface is subjected to an angular accelerationωˆ = 4.0^rad_s2. Afterwards, in the time interval0.1s< t≤3.5s the block is continuously pressed into the cylinder with a prescribed displacement amplitude ofuˆ= 1.75mm and the angular velocity of the cylinder is kept constant. The mortar contact is characterized by the block being the slave side and the cylinder is defined to act as master side, respectively. In addition, only the block is assumed to lose material due to wear.

The resulting temperatures are given in Figure 5.30. Here, the temperatures are shown for a

Figure 5.30: Temperature distribution for block on rotating disc example: results without wear (left) and when wear is included (right).

5.6. Thermo-structure-contact-wear interaction with finite shape changes

Figure 5.31: Material displacements for the block: The wire frame on the left-hand side repre-sents the reference configuration and the solid reprerepre-sents the material configuration at the end of the simulation. On the right-hand side, the pure material configuration is shown with highlighting the material displacement contours.

simulation without any included wear phenomena and with considered wear effects. It can nicely be seen that the temperatures are much lower for the simulation with wear compared to the sim-ulation with wear effects being ignored. This is due to the mass loss and corresponding finite shape change which naturally leads to decreased interface stresses compared to the simulation without wear. Of course, this effect results from the purely Dirichlet-controlled setting with pre-scribed displacements at the top of the block. In contrast, a setting with a force being applied at the top of the block would not lead to decreased contact traction to such an extent. However, this decrease in interface stresses causes a reduced dissipation at the contact interface and, accord-ing to (5.92) and (5.93), lower heat fluxes occur. Duraccord-ing the entire simulation, no oscillations in the temperatures for the problem with wear being included are noticed. This shows that the developed explicit algorithm performs in a robust and stable manner for the chosen time step size.

Finally, the resulting shape change due to the material loss at the contact interface is shown in Figure 5.31. It can be seen that the material loss at the contact interface of the block corresponds to nearly three element layers. But again, the proposed algorithm, including the developed shape evolution procedure, guarantees for an excellent mesh quality despite the huge material loss.

6. Mortar Methods for Volume