Mixed mode near-tip fields for cracks in materials with strain-gradient effects

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical KIII field; stresses have r -3/2 singularity near the crack tip, and are significantly larger than the classical K III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral.

The effective threshold for fatigue crack propagation: a plastic size effect?

Recent experiments have shown that the microscale material behavior is very different from that of bulk materials, and displays strong size effects when the characteristic length associated with the deformation is on the order of microns. Conventional continuum theories, however, can not predict this size dependence because they do not have an intrinsic length in their constitutive models. A new continuum theory, namely the strain gradient theory, has been proposed to investigate the deformation of solids at the microscale. For materials undergoing plastic deformation, the basis of strain gradient theory is the dislocation theory in materials science, and strain gradient plasticity has agreed remarkably well with experiments. For elastic materials with microstructures, it has also been established that the material behavior can be represented by an elastic strain gradient theory. A general approach to investigate fracture of materials with strain gradient effects is established. Both the near-tip asymptotic fields and the elastic full-field solutions are obtained in closed form. Due to stain gradient effects, stresses ahead of a crack tip are significantly higher than those in the classical K field. The plastic zone size surronunding a crack tip is estimated by elastic near-tip fields, as well as by the Dugdale model. It is established that the plastic zone is, in general, much more round and larger than that estimated from the classical K field.

Micromechanical modeling of damage evolution and fracture behavior in particle reinforced metal matrix composites based on the conventional theory of mechanism-based strain gradient plasticity

Modeling damage and fracture within strain-gradient plasticity

A comparative discrete-dislocation/nonlocal crystal-plasticity analysis of plane-strain mode I fracture

In a remarkable series of experiments, Elssner et al. (1994) and Korn et al. (2002) observed cleavage cracking along a bimaterial interface between Nb and sapphire. The stress required for cleavage cracking is around the theoretical strength of the material. Classical plasticity models fall short to reach such a high stress level. We use the conventional theory of mechanism-based strain gradient plasticity (Huang et al., 2004) to investigate the stress field around the tip of an interface crack between Nb and sapphire. The tensile stress at a distance of 0.1 µm to the interface crack tip reaches 13.3σY ,w hereσY is the yield stress of Nb. This stress is nearly 4 times of that predicted by classical plasticity theory (3.6σY ) at the same distance to the crack tip, and is high enough to trigger cleavage cracking in materials and interfaces. This is consistent with Elssner et al.'s (1994) and Korn et al.'s (2002) experimental observations.

A detailed finite element analysis of crack initiation and stable crack extension is performed under Mode I plane stress, small-scale yielding conditions. A small strain, J2 incremental plasticity theory is employed and both elastic-perfectly plastic materials and power law hardening materials are considered. Some issues pertaining to the stationary plane stress crack problem, such as the range of dominance of the asymptotic stress and deformation fields and the amount of non-proportional loading near the crack tip are addressed. Special attention is devoted to the perfectly plastic idealization, by performing a separate singular finite element analysis, to clarify some details about the asymptotic fields near the stationary crack tip. The full-field numerical solution is used to simulate synthetic (optical) caustic patterns at different distances from the crack tip, which are compared with experimental observations and with asymptotic analytical results. A nodal release procedure is used to simulate quasi-static crack extension. It is found that the asymptotic angular extent of the active plastic zone, surrounding the propagating crack tip, is from θ = 0 to about θ = 45° for the perfectly plastic case. The near-tip angular stress distribution within the active plastic zone is in good agreement with the variation in a centered fan, as predicted by a preliminary asymptotic analysis by Rice, for the perfectly plastic case. It is also observed that the σrr stress component has a strong radial variation within the active plastic zone. The angular extent of active yielding around the moving tip increases with hardening, while its maximum radial extent ahead of the tip decreases. Clear evidence of an elastic unloading region following the active plastic zone is found, but no secondary (plastic) reloading along the crack flank has been numerically observed for any level of hardening. The crack tip opening profile during growth is obtained for various levels of hardening. A ductile crack growth criterion is employed to investigate the nature of the J resistance curves under plane stress. Finally, the influence of hardening on the potential for stable crack growth is examined.

In the two parameter approaches, one of the second parameters, T-stress, Q and A2 is used together with J-integral to describe the constraint loss near the crack tip under large scale yielding states. Among the second parameters, Q seems to be most promising from a practical point of view because it applies for three-dimensional geometries. In order to determine the Q for a given geometry and load, however, three-dimensional elastic-plastic finite element analysis, which is time consuming and costly, has had to be performed so far. In this work an experimental method to measure Q-parameter insitu is described. The basic idea comes from the fact that side necking near a crack tip indicates the loss of stress triaxiality, which can be scaled by Q. From the out-of-plane displacement and the in-plane strain measured on the surface of side necking near the crack tip, stress field averaged through thickness is calculated and then Q is determined from the difference between the stress filed and the HRR field corresponding to the identical J-integral. To prove the validity, three-dimensional finite element simulation has been performed for a CT configuration with side-grooves. Q-value which was calculated directly from the near-tip stress field is compared with that determined by simulating the experimental procedure according to the proposed method, that is, the Q-value determined from the lateral displacement and the in-plane strain. Also, the effect of location where the displacement and strain are measured is explored. Moreover, an easy way for measuring the displacement and strain simultaneously is described. That is based on Stereoscopic Digital Photography and high resolution Digital Image Correlation (DIC) software, and can be performed along with conventional fracture tests. A case study for a CT specimen of ferritic steel is presented.

Small scale, grain size and substrate effects in nano-indentation experiment of film–substrate systems

Evaluation of crack tip stress and strain fields under nonproportional loading in a viscoelastic material

Most engineering structural metallic alloys are used in polycrystalline form. The nature of the mechanical response of these systems is complex and hierarchical, spanning a range of scales. Lattice strains, distortions and defects (notably, dislocations) nucleate, interact, pile up at grain boundaries and self-organize at the (sub)micrometre scale. Individual grains experience strong interactions with their neighbours and geometric features (cracks, notches). Groups of grains sharing common orientation find themselves embedded within large ensembles possessing certain statistical properties (size distributions, preferred orientation, etc.). Ultimately, the macroscopic properties of grain aggregates are determined by this hierarchy of interactions. Notably, while collective properties such as stiffness are relatively well represented by averages, strength properties associated with fracture, fatigue crack propagation, creep and damage show a strong dependence on the local microscopic conditions of the ‘weakest link’. Ongoing improvements in the spatial resolution of X-ray imaging and tomography and the availability of micro-focused X-ray beams open up a number of opportunities for the study of the structure and deformation at (sub)micrometre scales. Fundamental questions concerning the scale dependence and strain gradient effects in solids can now be tackled by the combination of synchrotron X-ray methods and suitably refined deformation modelling. In this study, a range of methodologies and experimental configurations are presented that have allowed us to develop improved insight into the physical mechanisms of plastic deformation in ductile metallic alloys. Examples include white-beam energy-dispersive diffraction, micro-beam Laue diffraction, scanning micro-beam diffraction topography, high-resolution reciprocal space mapping and imaging. Connections are established with advanced numerical models of polycrystal deformation using strain gradient plasticity and discrete dislocation dynamics modelling.

With introducing the strain gradient into plasticity theory, the crack field singularity at the crack tip changes. Characterization of the crack in strain-gradient dependent materials has not found a common consensus. In the present work the crack tip field is systematically investigated within the frame of the strain-gradient plasticity using the element-free Galerkin method. Under both small scale yielding and finite yielding conditions the crack field in strain-gradient plasticity consists of three zones: The elastic K-field, the known plastic HRR-field dominated by the known energy release rate, \({\mathcal G}\), and the hyper-singular stress field which cannot be characterized by the known fracture mechanics parameters. Effects of the finite strains have been considered for both small scale yielding and finite cracked specimens. The computational results show that the finite strains do not change the characterization of the crack tip fields under the strain-gradient plasticity. In all investigated cases the hyper-singular zone is small (\({r < \mathcal G/\sigma_0}\)) and well contained by the HRR zone under the \({\mathcal G}\) dominance condition. The hyper-singular zone is in the size of \({r/(\mathcal G/\sigma_0)}\) and may grow with the material length, but decrease with applied load. The known elastic-plastic fracture mechanics parameter, \({\mathcal G}\), can be directly applied to characterize the crack under strain-gradient plasticity, within the intrinsic material length much smaller than the overall component size, l < 0.1%L0. In this case the strain-gradient term in the gradient plasticity will not affect description of cracks.

Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity

Anisotropic plastic stress field at a mixed-mode crack tip under plane and anti-plane strain