Fundamentals on wear phenomena

5.1.3. Proposal for mortar based wear approaches

The most important ingredients and new scientific contributions of the presented wear approaches are given in the following:

• extension of the finite deformation dual mortar contact formulation towards 3D fretting wear simulations, see also Farah et al. [69].

• implementation of the first fully implicit finite wear algorithm in a non-steady-state, finite deformation regime based on dual mortar methods with an Arbitrary-Lagrangian-Eulerian approach, see also Farah et al. [74].

• extension of the developed finite wear algorithm towards thermo-structure-contact-wear interaction problems.

In summary, the methods proposed in this chapter bring together existing ideas of computational wear modeling and dual mortar finite element methods and extend them towards a comprehen-sive and more general framework for efficient treatment of wear problems than possible to date.

5.2. Fundamentals on wear phenomena

The main focus of this chapter is on the development of wear algorithms based on mortar meth-ods. For this purpose, any phenomenological wear law could be employed. Thus, the wear phe-nomena are considered in a macroscopic and very general form. However, the basic types of wear phenomena and their physical background are briefly outlined in the next subsection. Then, the commonly employed wear law of Archard is introduced, which acts as basis for the developed wear algorithms regardless of which wear type is considered, see Subsection 5.2.2. Finally, the continuum mechanical description of wear problems with considerably large material loss is given in Subsection 5.2.3.

5.2.1. Frictional wear phenomena

Wear phenomena can be generally described as volume loss due to frictional dissipation at the contact interface. Thus, according to Popov [209], friction and wear always appear together.

However, depending on the application, one could also model wear without friction and friction without wear. In the following of this thesis, the most general assumption of wear with frictional effects is modeled. Despite this very general wear description, it can be distinguished between many types of wear, see Popov [209] and Rabinowicz [223]. The most relevant wear phenomena arefretting wearandabrasive wear.

5.2.1.1. Fretting wear

Information concerning fretting wear and fretting fatigue can be found in the vast amount of literature existing on this topic, where the interested reader is exemplarily referred to Aldham et al. [5], Hills [108], Hurricks [124] and Waterhouse [284]. In this subsection, only a brief description of these phenomena is given.

Figure 5.1: Fretting wear: micro crack initiation (left) and fracture of micro cracks which leads to material removal (right).

Figure 5.2: Forms of abrasive wear: three body abrasive wear (left) and two body abrasive wear (right).

Fretting wear is a destructive process of interface degradation, which occurs between clamped contacts when subjected to repeated loading cycles and minute relative motion. In each of these cycles, no noticeable change in the surfaces appear, but after a certain number of cycles micro cracks could occur due to the occurring large shearing stresses. Depending on the employed material, the coating and the loading condition, these micro cracks could erode and lead to slight material removal, see Figure 5.1. It is also possible that the introduced cracks propagate and lead to bulk brittle fracture, which is then denoted as fretting fatigue. Distinction between fretting wear, fretting fatigue and other fatigue classifications can also be made based on the length and orientation of the cracks relative to the contact surface and the presence of plasticity near the interface. A detailed classification of these related effects is not required in order to construct a finite element algorithm for macroscopic wear modeling and is therefore beyond the scope of this thesis.

5.2.1.2. Abrasive wear

Abrasive wear is characterized as material loss when two distinctively hard bodies are in contact or when three body contact occurs with hard particles being involved, see Godet [89]. This is schematically visualized in Figure 5.2. During the process of abrasive wear, the harder material penetrates and cuts the softer body. In the left part of Figure 5.2, three body contact is shown

5.2. Fundamentals on wear phenomena for a zoom in view. The particle is harder than the two contact bodies and thus leads to material loss at them. Two body wear is shown in the right part of Figure 5.2. There, the interaction with the worn debris particles is negligible and the two bodies themselves cause the abrasive behavior. However, for the numerical algorithms developed in this thesis, only two body contact is explicitly modeled. The effect of three body contact could be included in the wear parameter later on (cf. Section 5.2.2), but the debris particles themselves are not modeled as additional bodies.

5.2.1.3. Further wear phenomena

In addition to the aforementioned fretting wear and abrasive wear, there are many more types available in the existing literature. For example, adhesive wear occurs when two bodies with comparable hardness are in contact. According to Popov [209], their interaction can be described as welding together of surfaces with micro-roughness followed by tearing processes of particles.

Characteristic for this type of wear is the plastic deformation of metals at the contact interface when reaching a certain stress level. Furthermore,corrosive wear is caused by chemical modi-fications of the surface which leads to erosion at the interface layer. Without frictional sliding, the corrosive processes form a film on the contact surfaces, which on the other hand slows down the corrosion. However, when the loading conditions cause frictional sliding, the corrosive layer will be worn away and the interface erosion continues.

5.2.2. Archard’s wear law

For the following derivation of the wear algorithms, the wear law from Archard [7] is employed as a phenomenological approach to relate kinematic quantities with the worn volume. The es-sential aspects of the proposed approach are however not limited to this specific type of law, but can also be applied with other (macroscopic) wear laws. Archard’s wear law is a general for-mulation, which is valid for various physical types of wear, but also ignores some fundamental effects such as thermal relationships and the interaction with worn material within the contact zone. However, when considering only relatively small sliding velocities it is a suitable basis for further algorithmic developments, see Rabinowicz [223]. According to Archard’s law, the worn volumeVwis globally expressed as

Vw=KP S

H , (5.1)

with the normal force P, the sliding length S, the hardness of the softer material H and the dimensionless wear coefficientK. Employing Archard’s law within a finite element formulation later on requires a local expression for the loss of material. For this purpose, the amount of worn volume Vw is expressed as wear depth w per area. Thus, the global form of Archard’s law is reformulated as rate of the wear depth in spatial configuration:

˙

w=kw|pn| ||v_τ,rel||, (5.2)

where the local wear coefficient in spatial configurationkwis not a dimensionless parameter, but has the dimension of an inverse pressure. It can be equivalently expressed in terms of the spatial

frictional dissipation rate densityd, which has been already introduced in Section 3.1. Based on˙ the considerations in Ramalho and Miranda [224], the rate of the spatial wear depth reads:

˙ w= kw

Fd˙=Kd.˙ (5.3)

Here, the original wear coefficient kw and the Coulomb friction coefficient F are combined to the so-called energy wear coefficientK. Experiments in Ramalho and Miranda [224] suggest that assuming a constant energy wear coefficient leads to better results than assuming a constant wear coefficient, thus indicating the strong coupling of wear effects and frictional sliding conditions.

In the following, a relation between spatial and material wear is derived based on the consid-erations in Lengiewicz and Stupkiewicz [157]. The transformation of the frictional dissipation rate densityd˙to the material configuration (i.e. toD) was already outlined in Section 3.1 and is˙ shortly reviewed as

D˙ =jad ,˙ (5.4)

with dAand dA₀ being the areas of an infinitesimal surface segment in spatial and material de-scription, respectively. The relation between the spatial wear rate and its material counterpartW˙ can be established by expressing wear as a volume loss per area. For this purpose, it is assumed that the loss of material is defined in the spatial and material unit normal direction, respectively.

Thus, mass conservation yields

Here, as introduced in Chapter 2, dV represents an infinitesimal volume element in the spatial configuration, whereas dV0 is the corresponding counterpart in the material configuration. Ad-ditionally,ρis the spatial density and ρ0 is the density in the material configuration. Enforcing mass conservation yields

jaw˙ =J W ,˙ (5.6)

whereJ is the determinant of the deformation gradient. Assuming a constant friction coefficient and having equations (5.4) and (5.6) at hand, it is obvious that the wear coefficients in spatial and material configuration are related by

kw=J Kw. (5.7)

To the author’s knowledge, there are no unambiguous and conclusive experimental data, which would justify the choice of a constant wear coefficient in material or spatial configuration as being more physically meaningful. Therefore, both approaches have been implemented and will be evaluated in the numerical examples later on. However, one could think of non-constant wear coefficients in both configurations, which depend on the current normal load or other quanti-ties. For this purpose, it should be mentioned that the work in Pearson and Shipway [199] has indicated that the wear coefficient is independent of the slip amplitude and thus independent from displacement quantities. Employing a backward Euler scheme for local time discretization of (5.2) results in the incremental formulation

∆w =kw|pn| ||vτ,rel||∆t =kw|pn| ||uτ,rel||, (5.8)

5.2. Fundamentals on wear phenomena with u_τ,rel being the relative tangential slip increment within one pseudo-time step. Assuming that both involved bodies may undergo a process of material loss at the interface, two wear quantitiesw⁽¹⁾andw⁽²⁾have to be defined for the slave and for the master surface, respectively.

Correspondingly, two generally different wear coefficients k_w⁽¹⁾ and k_w⁽²⁾ are introduced. As the kinematic quantitiespn anduτ,relare equivalent for slave and master side in the continuous set-ting, only the wear coefficients cause differences for slave-sided and master-sided wear. Thus, the relation between wear on the slave side and wear on the master side can be directly formu-lated as ratio of the wear coefficients:

∆w⁽¹⁾= k_w⁽¹⁾ kw⁽²⁾

∆w⁽²⁾. (5.9)

Considering the absolute wear depth without taking into account the different signed wear quan-tities due to opposed interface normals for slave and master side, the cumulative wear depth can be expressed as

∆w = (k_w⁽¹⁾+k⁽²⁾_w )|pn| ||u_τ,rel||. (5.10)

5.2.3. Continuum mechanics for finite wear

When taking into account finite wear, the mechanical configuration to which kinematic quan-tities are referred constantly changes, which introduces a major additional complexity in con-tinuum modeling as compared with finite deformation contact problems without removal of material due to wear, see Chapter 3. Therefore, an additional configuration of each involved body is introduced: the time-dependent undeformed material (worn) configuration Ω⁽ⁱ⁾_m ⊂ R³, which represents the unloaded worn state. This concept of a third configuration, which repre-sents the undeformed worn state was sucessfully employed in Lengiewicz and Stupkiewicz [157]

and Stupkiewicz [263] and thus it is the basis for the following considerations. The material con-figuration is occupied by all material pointsX˜ at a certain timet. The Dirichlet, Neumann and contact boundaries of the material configuration read Γ⁽ⁱ⁾_m,u, Γ⁽ⁱ⁾_m,σ and Γ⁽ⁱ⁾_m,c. It is assumed that reference, spatial and material configurations are coincident for the initial timet= 0. The inter-connections between these configurations are visualized for one exemplary body in Figure 5.3.

By introducing the material configurationΩ⁽ⁱ⁾_m , the reference configurationΩ⁽ⁱ⁾₀ can be interpreted as observer domain for the material motionφ, viz.

φ:

Here, the bijective mappingφconnects a material pointX˜ with its observer point in the reference configuration X. An additional bijective mapping ψ is introduced to describe the connection between observer pointsX and spatial pointsx, as

ψ :

Figure 5.3: Three continuum mechanical configurations and their interconnections: reference configuration, material (worn) configuration and spatial configuration. Figure taken from Farah et al. [74].

Accordingly, the mapping of the physical motion ϕ originally defined in (2.1) changes and is now expressed withφandψas

ϕ=ψ◦φ⁻¹ :







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].

Im Dokument Mortar Methods for Computational Contact Mechanics Including Wear and General Volume Coupled Problems  (Seite 127-132)