Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

The activity of non-basal slip systems and dynamic recovery at room temperature in fine-grained AZ31B magnesium alloys

Compression properties of a refractory multi-component alloy, Ta20Nb20Hf20Zr20Ti20, were determined in the temperature range of 296–1473 K and strain rate range of 10−1–10−5 s−1. The properties were correlated with the microstructure developed during compression testing. The alloy was produced by vacuum arc melting, and it was hot isostatically pressed (HIPd) and homogenized at 1473 K for 24 h prior to testing. It had a single-phase body-centered cubic structure with the lattice parameter a = 340.4 pm. The grain size was in the range of 100–200 μm. During compression at a strain rate of έ = 10−3 s−1, the alloy had the yield strength of 929 MPa at 296 K, 790 MPa at 673 K, 675 MPa at 873 K, 535 MPa at 1073 K, 295 MPa at 1273 K and 92 MPa at 1473 K. Continuous strain hardening and good ductility (e ≥ 50%) were observed in the temperature range from 296 to 873 K. Deformation at T = 1073 K and έ ≥ 10−3 s−1 was accompanied by intergranular cracking and cavitation, which was explained by insufficient dislocation and diffusion mobility to accommodate grain boundary sliding activated at this temperature. The intergranular cracking and cavitation disappeared with an increase in the deformation temperature to 1273 and 1473 K or a decrease in the strain rate to ~10−5 s−1. At these high temperatures and/or low-strain rates the alloy deformed homogeneously and showed steady-state flow at a nearly constant flow stress. Partial dynamic recrystallization, leading to formation of fine equiaxed grains near grain boundaries, was observed in the specimens deformed at 1073 and 1273 K and completed dynamic recrystallization was observed at 1473 K.

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Microstructure and Elevated Temperature Properties of a Refractory TaNbHfZrTi Alloy

Crack tip and associated domain integrals from momentum and energy balance

Cleavage, dislocation emission, and shielding for cracks under general loading

Fracture at high temperatures

The aim of the paper is to answer the question: which loading parameter determines the stress and strain fields near a crack tip, and thereby the growth of the crack, under creep conditions? As candidates for relevant loading parameters, the stress intensity factor K I , the path-independent integral C*, and the net section stress σ n e t have been proposed in the literature. The answer, which is attempted in this paper, is based on the time-dependent stress analysis of a stationary crack in Mode I tension. The material behavior is modeled as elastic-nonlinear viscous, where the nonlinear term describes power law creep. At the time t = 0, load is applied to the cracked specimen, and in the first instant the stress distribution is elastic. Subsequently, creep deformation relaxes the initial stress concentration at the crack tip, and creep strains develop rapidly near the crack tip. These processes may be analytically described by self-similar solutions for short times t. An important result of the analysis is that small-scale yielding may be defined. In creep problems, this means that elastic strains dominate almost everywhere except in a small "creep zone" which grows around the crack tip. If crack growth ensues while the creep zone is still small compared with the crack length and the specimen size, the stress intensity factor governs crack growth behavior. If, however, the calculated creep zone becomes larger than the specimen size, the stresses become finally time-independent and the elastic strain rates can be neglected. In this limiting case, the stress field is the same as in the fully-plastic limit of power law hardening plasticity that has been treated in the literature. The loading parameter that determines the near tip fields uniquely is then the path-independent integral C*. It should be emphasized that K 1 and C* characterize opposite limiting cases. Which case applies in a given situation can be decided by comparing the creep zone size with the specimen size and the crack length. Criteria for small-scale yielding are worked out in several alternative forms. Besides several methods of estimating the creep zone size, a convenient expression for a characteristic time is derived also, which characterizes the transition from small-scale yielding to extensive creep of the whole specimen.

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Tensile Cracks in Creeping Solids

The evaluation of crack growth tests under creep conditions must be based on the stress analysis of a cracked body taking into account elastic, plastic and creep deformation. In addition to the well-known analysis of a cracked body creeping in secondary (steady-state) creep, the stress field at the tip of a stationary crack is calculated for primary (strain-hardening) or tertiary (strain-softening) creep of the whole specimen. For the special hardening creep-law considered, a path-independent integral C ∗ h , can be defined which correlates the near-tip field to the applied load. It is also shown how, after sudden load application, creep strains develop in the initially elastic or, for a higher load level, plastic body. Characteristic times are derived to distinguish between short times when the creep-zones, in which creep strains are concentrated, are still small, and long times when the whole specimen creeps extensively in primary and finally in secondary and tertiary creep. Comparing the creep-zone sizes with the specimen dimensions or comparing the characteristic times with the test duration, one can decide which deformation mechanism prevails in the bulk of the specimen and which load parameter enters into the near-tip stress field and determines crack growth behavior. The governing load parameter is the stress intensity factor K 1 if the bulk of the specimen is predominantly elastic and it is the J -integral in a fully-plastic situation when large creep strains are still confined to a small zone. The C ∗ h -integral applies if the bulk of the specimen deforms in primary or tertiary creep, and C ∗ is the relevant load parameter for predominantly secondary creep of the whole specimen.

Creep deformation at crack tips in elastic-viscoplastic solids

Creep crack growth in two of the commonly used creep-resistant ferritic steels was investigated. Both steels were tested in the as-processed condition and after many years of service in electric power plants. The test temerature was 540°C (in one case also 500°C and 565°C) and test durations ranged from a few days to a year. In 1/2Cr-1/2Mo-1/4V steel, crack growth occurred intergranularly by grain boundary cavitation, while 21/4Cr-1Mo steel also exhibited transgranular growth. The deformation behavior of CT specimens was analyzed in detail. It was found that for typically 20 percent of the lifetime, the instantaneous elastic-plastic strains and, more importantly, primary creep determine the deformation response. The evolution of the crack length parallels that of the load-line deflection: Initially, the crack grows relatively fast, decelerates until a steady growth rate is reached, and finally accelerates due to the increasing stress intensity at the longer crack. This behavior can be described reasonably accurately by models for creep crack growth taking into account the elastic-plastic transient and primary creep. It is demonstrated that the Ct parameter correlates crack growth rates during the transients, whereas C* becomes the appropriate load parameter during steady-state creep. However, C* fails to describe crack growth in single-edge notched tension specimens with shallow cracks. This is tentatively ascribed to excessive crack-tip blunting and other geometry changes. Unloading and reloading frequently leads to accelerated crack growth.

Creep crack growth in ductile, creep-resistant steels

Plastic yield at crack tips on singular slip-planes, inclined to the crack plane, has been studied under plane-strain conditions for combined tension, hydrostatic stress, and in-plane shear. The singular integral equation, which represents the equilibrium condition of edge dislocations on the slip-planes, is transformed into a Fredholm integral equation in order to avoid difficulties that occur with its numerical solution. Results are presented for the slip-band length, the plastic crack-tip opening displacement, stress fields, and crack-opening contours. A series expansion of the results obtained numerically confirms approximate analytical expressions given by J.R. Rice (1974), up to the third-order in the applied stresses. The results of finite element methods agree with values of the crack-tip opening displacement obtained here to within 10 per cent. Ahead of the crack tip, the principal tensile stresses exceed the principal shear stresses by a factor of 10, approximately.

Hermann Riedel

Papers

Fracture at high temperatures

Tensile Cracks in Creeping Solids

Creep deformation at crack tips in elastic-viscoplastic solids

Creep crack growth in ductile, creep-resistant steels

Plastic yielding on inclined slip-planes at a crack tip