Plastic Deformation in Rolling Contact
It will have been evident from the analysis of rolling contact of elastic bodies that the contact zone is one of high stress concentration. With metallic solids, high loads would be expected to lead to plastic yielding and permanent deformation in the hig
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K.L. Johnson Cambridge University, Cambridge, UK
B. Jacobson et al. (eds.), Rolling Contact Phenomena © Springer-Verlag Wien 2000
164
K.L Johnson
LECTURE 1: ONSET of' PLASTIC YIELD
It will have been evident from the analysis of rolling contact of ela~tic bodies that the contact zone is one of high stress concentration. With metallic solids, high loads would be expected to lead to plastic yielding and permanent deformation in the highly stressed zone, which is a precursor to failure by wear and fatigue. This group of four lectures will be devoted to the mechanics of inelastic deformation in rolling contact. Before considering rolling contact, the relevant basic principles of the theory of plasticity will be reviewed and applied to the stationary contact of curved bodies. 1. Elements of' plastic behaviour.
1. I Simple tension and compression The characteristic behaviour of a ductile material loaded and unloaded in tension and compression is well known, and is illustrated in Fig.I.I. The response in tension and compression is linear and elastic up to a yield stress Oy, followed by plastic deformation and strain hardening. When the strains become large a distinction must be made between the nominal stress (load/initial cross-section area) and true or Cauchy stress (load/current area) and also between the nominal strain (e;o:tension or compressionloriginallength) and the logarithmic strain (loge{ current length / original length}). For an isotropic material, whose properties are independent of direction, the yield strengths in tension and compression have equal magnitudes and the curves of true stress and logarithmic strain are anti-symmetrical. However for most of the topics covered in these lectures the strains will be assumed to be sufficiently small for the difference between true and nominal stress and strain to be negligible. During unloading the initial response is elastic with the same modulus E. When followed by reverse loading many materials yield at a stress of lower magnitude than in the initial loading, followed by strain hardening at a steeper rate. This is known as the' Bauchinger effect' and is due to locked-in stresses in the micro structure of the material.
STRAIN E
Bauchinger effect
Fig.1.1 --True stress.log. strain; - - - Nominal stress-nominal strain
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Plastic Deformation in Rolling Contact
For the purpose of carrying out calculations in the plastic range it is valuable to have simple idealised models of this behaviour. A hierarchy of such models. chosen to represent different aspects of the above behaviour. is presented in Fig. 1.2.
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i
' I
_..1
I I
I
Eo
_1---1..
(c)
(a)
niT.
E:
..1--
I
" (e)
(d)
Fig.1.2 -
Loading; - - -
Unloading
(a) Elastic-perfectly plastic: Equal yield in tension and compression; zero strain hardening. (b) Isotropic hardening: Equal strain hardening in tension and compression. (c) Linearkinematichardening: Constant range between yield in tension and compression: equal strain hardening rates. (d) Non-linear kinematic hardening: Constant ran
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