By G.A.C. Graham, J.R. Walton
The major emphasis of those Lecture Notes is on developing ideas to precise viscoelastic boundary worth difficulties; notwithstanding homes of the equations of viscoelasticity that supply the theoretical underpinnings for developing such ideas also are lined. specific recognition is paid to the answer of crack and get in touch with difficulties. This paintings is of curiosity within the context of polymer fracture, modelling of fabric behaviour, rebound checking out of polymers and the phenomenon of hysteretic friction.
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Extra info for Crack and Contact Problems for Viscoelastic Bodies
1 MIXED 80UNDARY CONDITIONS ON THE WEDGE FACES A natural generalisation of the above analysis is to investigate the situation when mixed boundary conditions are applied to the wedge faces. This has been done by Atkinson and Bourne (1990). The geometry is shown in Fig. 6 and the boundary conditions now take the form C. 13) 9 =0 where again f(r) denotes some applied Ioad whose Mellin transform exists and H(t) is the Heaviside unit step function and again more general Ioads could This problern could be thought of as be considered without difficulty.
See Atkinson and Bourne (1989) for more detailed arguments Note that the ideas of this section can be and a number of other results. applied to plane strain problems in a similar way. 4. PLANE STRAIN STRESS SINGULARITIES AT CORNERS The main idea of Section 3 was that the eigenvalue equation for stresssingularities in viscoelastic media may have the same form in the Laplace transform domain as that in an elastic medium, but with the major difference that the parameters in the eigenvalue equations are not constants but depend on s the Laplace transform variable through the Laplace transform of the 1 This Ieads to results for the field near a viscoelastic relaxation moduli.
S )] ' 'zl" (' )+"2 (, I ))[cos(zrr) 1 1 1 + [ 2 I I I 11. (s )+IJ. 22) s z[IJ. (s )+fJ. (s ))(cos(zrr)-b) I I I 2 I where ll (s ) - 11. 23) ll (s ) + ll (s ) I I 2 I The real time behaviour of the displacement and stress fields will follow on inverting the above transforms, and each is characterised by solutions to the eigenequation cos(zrr) Explicit solutions are available establishes the roots as . 2 =b to (3. 24) 1 z = ± .!.. 24) and one particular n = 0,±1,±2, ... 25) where the choice of branch of (b -1) 2 is made so that the root to be taken has positive imaginary part and where it is understood that the principal branch of the logarithm has been taken.