A contact model for a creeping sphere and a rigid flat
Introduction
The creep phenomenon is a time dependent mechanical behavior of materials and is technically considered a form of visco-elasticity. Creep may occur with almost any material under certain operating conditions. These include metals at high temperature, polymers at room temperature, and any material under the effect of nuclear radiation. Contact creep is common in a variety of tribological applications e.g. rail and wheel [1], [2], MEMS RF switches [3], magnetic tapes [4] and various natural and artificial joints in the human body [5], [6], [7], [8]. This later application, which involves the creep of polymers or articular cartilage under compression, has generated substantial interest in testing and modeling of the behavior of creeping polymers and biological tissues [9], [10], [11], [12], [13]. One main reason for the growing interest in polymer and cartilage creep is the medical condition known as osteoarthritis, which affects tens of millions of people around the world. This condition involves the degradation of cartilage tissue inside synovial joints [14]. The state of the art treatment for severe condition of osteoarthritis today involves replacing the entire damaged joint with a new mechanical joint made of rigid, bio-compatible materials (see e.g. [11]) that may include polymers like UHMWPE. A typical structure of a knee joint replacement that contains both metallic and polymer materials can be seen in Fig. 1. Our interest in the contact creep problem stems from the phenomenon of increasing friction as a result of longer contact time under a constant normal load e.g. [10], [15], [16], and the potential effect of creep on wear [17], which may affect both natural and artificial joints.
The creep behavior usually depends on the temperature and the stress level to which the material is subjected, and obviously depends on the duration of application of these conditions. It is important to note that the creep effect is not elastic and hence, non-reversible upon removal of the load under normal operating conditions [18], [19]. Yet, some creeping materials may restore their original shape after some time under certain environmental conditions such as temperature or the presence of certain chemical solutions. The effects of the stress, time, and temperature on the deformation of a creeping specimen were studied so far mostly experimentally. These experiments include different measuring systems and different applied loads [20].
Two basic approaches can be found in the literature for modeling the behavior of creeping materials. The first is the mathematical approach (see e.g. Gittus [21], and also Refs. [18], [19], [22], [23]). Here, the models describe either a visco-elastic fluid with constant flow rate under a constant stress, or a solid that completely stop creeping after some time. The other approach to treat the creep phenomenon utilizes empirical formulas to relate the various creep parameters based on creep experiment results e.g. [24], [25].
As can be seen from Section 1, most of the information found in the literature concerning the contact of creeping polymers is empirical. The mathematical contact models that are based on a combination of springs and dashpots to describe the material behavior are unable to capture correctly the results of creep experiments presented in Ref. [20]. Hence, the goal of the present research is to develop a new creeping contact model for a single asperity of a polymeric material surface, using improved material modeling. More specifically, this research concentrates on the effect of contact time under constant normal load on the contact parameters such as displacement and contact area of a polymeric creeping sphere in contact with a rigid flat.
Section snippets
Background
The most relevant test to our present investigation is a compression experiment of a certain rod where the displacement of a dead weight pressing the rod over increasing contact times is measured [20]. This experiment is done for different stresses, different temperatures, or other environmental conditions (like radiation or humidity). The results are then transformed to the form of apparent creep modulus using Eq. (1).where σ is the resulting stress, ɛ is the measured strain, and E
The creep contact model
The schematic of a sphere loaded by a rigid flat is shown in Fig. 3. A commercial software ANSYS version 10.0 was used to analyze a 2D, axi-symmetric finite element model of the creeping sphere contacting a rigid flat. Fig. 4 shows one half of a hemisphere, which was divided into two different zones 1 and 2 to enable non-uniform meshing. The size of zone 2 was selected so that it will always contain the entire expected contact area.
The following boundary conditions are imposed: (1) No radial
Results and discussion
The results of interest for the present research are given in Eqs. (7), (8), (9). The parameters shown in these equations are: the creep strain (ɛcr), creep displacement (ωcr), and creep contact area (Acr).
According to the MTH material model, ɛcr(t) of Eq. (7) can also be expressed as (see Eq. (5)):
The values of the two contact parameters ω and A at t = 0 are according to the Hertz solution [31]:
A
Conclusion
This research concerns a contact problem between a creeping polymer sphere and a rigid flat. Existing models of creep behavior were examined, resulting in the selection of the modified time hardening (MTH) model to represent polymer materials. A numerical model was created using the commercial finite elements software ANSYS version 10.0. The numerical model was verified using both a static test to verify the mesh and element behavior, and a creep test of a rod under compression to verify the
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