Crack Tip Fields Under Non‐steady Creep Conditions—i. Estimates of the Amplitude of the Fields

Abstract

Under non-steady creep conditions, the stress and strain rate fields near the tip of a stationary crack can be described by the singular fields of Hutchinson, Rice and Rosengren for power-law creeping materials. Estimation formulae are presented for describing the amplitude of these fields under load and displacement controlled boundary conditions. For constant loading, the formulae reduce to the result of Riedel and Rice for short times after load application and to the steady state line integral C* for long times. At intermediate times, the estimate is validated by detailed finite-element computation. For displacement-controlled loading, the amplitude of the near-tip fields is shown to fall rapidly, consistent with finite-element analysis. The implications of the results for data collection and defect assessments are discussed in a companion paper. NOMENCLATURE A = area between contours B = constant in creep law of equation (2) B, = specimen thickness a, u =crack size, crack growth rate C ( a / w ) , C' = elastic compliance function, dC/d(a/w) C ( t ) = non-steady creep integral of equation (4) C* = steady state creep integral of equation ( 5 ) E, E' = Young's modulus, E/(l -v 2 ) e,, e,, = elastic strain, strain rate tensors h, = dimensionless function in equation (9) I, = dimensionless function of n in equations (7, 8) J = contour integral of equation (6) K = elastic stress intensity factor m ( a / w ) , m' = limit load ratio, dm/d(a/w) L, L , = characteristic lengths in equations (9, 11) n =creep stress index in equation (2) n, = unit normal r = distance from crack tip S, = deviatoric stress tensor ds = element of contour r t, to = time, normalising time P, Po, P, = applied load, normalising load, collapse load u,, u, = displacement, displacement rate tensors cS,,L',, = creep strain rate, creep strain rate tensors L ' ,~ = creep strain rate at stress uXf w = section width [ = contour around the crack tip x, y = Cartesian co-ordinates A, A = specimen load-line displacement, displacement rate L,, C, , = strain tensor, dimensionless function in equation (8) A,,, At, = creep, elastic components of A 0 = polar angle K = I in plane stress, K = 0.75 in plane strain 263 264 R. A. AINSWORTH and P. J. BUDDEN p = constant in equation (26), estimated in the Appendix v = Poisson's ratio u,,, 8,, = stress tensor, function in equation (7) uEl, CT$ = reference stress in equation (10). value for load Po uo = yield stress t = dimensionless time of equations (22) and (24) d, = exponent in crack growth law of equation (32)

Three-dimensional numerical analysis of out-of-plane creep crack-tip constraint in compact tension specimens

“…The C (t)/C * et/t red curve of the Eq. (3) defined by Ainsworth and Budden [37] gives the best agreement with FEM results for the CT specimens with various thicknesses. 2 The FEM result shows that the linear relation between the redistribution time t red and C * on a logelog scale is almost independent on the specimen thickness (hence the out-ofplane creep constraint induced by the specimen thickness).…”

“…Ainsworth and Budden [37] also defined an approximate equation for C (t)/C * as a function of t/t red and n:…”

Singular fields near a sharp V-notch for power law creep material

“…For slowly growing crack in power law creep materials, the singular order and domination of asymptotic field were discussed at different power creep exponent (Hui and Riedel 1981). Ainsworth and Budden (1990) presented estimation formulae for describing the amplitude of these fields under load and displacement controlled boundary conditions with the formulae reduce and finite-element computation. Carpinteri and Paggi (2009) presented multi-material wedges and crack in non-homogeneous materials are discussed from the engineering point of view.…”

Numerical analysis of the effect of initial plasticity on transient creep in compact tension specimen under mechanical load

“…By simple interpolation, Riedel [1] proposed an estimation equation for C(t)/C*. Later, Ainsworth and Budden [2] developed another, slightly different approximation of this relationship. In both cases, C(t) converges to C* as t → ∞.…”

“…However, initial widespread plasticity at the crack tip invalidates this approach for estimation of C(t). For elastic-plastic-creep problems, Joch and Ainsworth [3] proposed an estimation equation for C(t), by extending the elastic-creep analysis [2]. Using finite element method (FEM), they analyzed the effect of plasticity on the magnitude of C(t) during the transition phase of creep.…”