Spallation analysis with a closed trans-scale formulation damage evolution

A closed, trans-scale formulation of damage evolution based on the statistical microdamage mechanics is summarized in this paper. The dynamic function of damage bridges the mesoscopic and macroscopic evolution of damage. The spallation in an aluminium plate is studied with this formulation. It is found that the damage evolution is governed by several dimensionless parameters, i.e., imposed Deborah numbers De* and De, Mach number M and damage number S. In particular, the most critical mode of the macroscopic damage evolution, i.e., the damage localization, is deter-mined by Deborah number De+. Deborah number De* reflects the coupling and competition between the macroscopic loading and the microdamage growth. Therefore, our results reveal the multi-scale nature of spallation. In fact, the damage localization results from the nonlinearity of the microdamage growth. In addition, the dependence of the damage rate on imposed Deborah numbers De* and De, Mach number M and damage number S is discussed.

Characteristic Dimensionless Numbers in Multi-Scale and Rate-Dependent Processes

For this sake, the macroscopic equations of mechanics and the kinetic equations of the microstructural transformations should form a unified set that be solved simultaneously. As a case study of coupling length and time scales, the trans-scale formulation

Experimental Evidence of Critical Sensitivity in Catastrophe

The paper presents an experimental study on critical sensitivity in rocks. Critical sensitivity means that the response of a system to external controlling variable may become significantly sensitive as the system approaches its catastrophic rupture point. It is found that the sensitivities measured by responses on three scales (sample scale, locally macroscopic scales and mesoscopic scale) display increase prior to catastrophic transition point. These experimental results do support the concept that critical sensitivity might be a common precursory feature of catastrophe. Furthermore, our previous theoretical model is extended to explore the fluctuations in critical sensitivity in the rock tests.

Microstructure and mechanical properties of slowly cooled Cu47.5Zr47.5Al5

Cu47.5Zr47.5Al5 was prepared by arc melting and solidified in situ by suction casting into 2-5-mm-diameter rods under various cooling rates (200-2000 K/s). The microstructure was investigated along the length of the rods by electron microscopy, differential scanning calorimetry and mechanical properties were investigated under compression. The microstructure of differently prepared specimens consists of macroscopic spherical shape chemically inhomogeneous regions together with a low volume fraction of randomly distributed CuZr B2 phase embedded in a 2-7 nm size clustered "glassy-martensite" matrix. The as-cast specimens show high yield strength (1721 MPa), pronounced work-hardening behavior up to 2116 MPa and large fracture strain up to 12.1-15.1%. The fracture strain decreases with increasing casting diameter. The presence of chemical inhomogenities and nanoscale "glassy-martensite" features are beneficial for improving the inherent ductility of the metallic glass.

Damage evolution, localization and failure of solids subjected to impact loading

In order to reveal the underlying mesoscopic mechanism governing the experimentally observed failure in solids subjected to impact loading, this paper presents a model of statistical microdamage evolution to macroscopic failure, in particular to spallation. Based on statistical microdamage mechanics and experimental measurement of nucleation and growth of microcracks in an Al alloy subjected to plate impact loading, the evolution law of damage and the dynamical function of damage are obtained. Then, a lower bound to damage localization can be derived. It is found that the damage evolution beyond the threshold of damage localization is extremely fast. So, damage localization can serve as a precursor to failure. This is supported by experimental observations. On the other hand, the prediction of failure becomes more accurate, when the dynamic function of damage is fitted with longer experimental observations. We also looked at the failure in creep with the same idea. Still, damage localization is a nice precursor to failure in creep rupture.

Mechanical effects of the self-organization of dislocations and related critical characteristics

The effects of the dislocation pattern formed due to the self-organization of the dislocations in crystals on the macroscopic hardening and dynamic internal friction (DIF) during deformation are studied. The classic dislocation models for the hardening and DIF corresponding to the homogeneous dislocation configuration are extended to the case for the non-homogeneous one. In addition, using the result of dislocation patterning deduced from the non-linear dlislocation dynamics model for single slip, the correlation between the dislocation pattern and hardening as well as DIF is obtained. It is shown that in the case of the tension with a constant strain rate, the bifurcation point of dislocation patterning corresponds to the turning point in the stress versus strain and DIF versus strain curves. This result along with the critical characteristics of the macroscopic behavior near the bifurcation point is microscopically and macroscopically in agreement with the experimental findings on mono-crystalline pure aluminum at temperatures around 0.5T(m). The present study suggests that measuring the DIF would be a sensitive and useful mechanical means in order to study the critical phenomenon of materials during deformation.

Elastic-plastic constitutive relation of particle reinforced composites

In this paper, a systematic approach is proposed to obtain the macroscopic elastic-plastic constitutive relation of particle reinforced composites (PRC). The strain energy density of PRC is analyzed based on the cell model, and Che analytical formula for the macro-constitutive relation of PRC is obtained. The strength effects of volume fraction of the particle and the strain hardening exponent of matrix material on the macro-constitutive relation are investigated, the relation curve of strain versus stress of PRC is calculated in detail. The present results are consistent; with the results given in the existing references.

Implication of scaling hierarchy associated with nonequilibrium: field and particulate

The advent of nanotechnology has necessitated a better understanding of how material microstructure changes at the atomic level would affect the macroscopic properties that control the performance. Such a challenge has uncovered many phenomena that were not previously understood and taken for granted. Among them are the basic foundation of dislocation theories which are now known to be inadequate. Simplifying assumptions invoked at the macroscale may not be applicable at the micro- and/or nanoscale. There are implications of scaling hierrachy associated with in-homegeneity and nonequilibrium. of physical systems. What is taken to be homogeneous and equilibrium at the macroscale may not be so when the physical size of the material is reduced to microns. These fundamental issues cannot be dispensed at will for the sake of convenience because they could alter the outcome of predictions. Even more unsatisfying is the lack of consistency in modeling physical systems. This could translate to the inability for identifying the relevant manufacturing parameters and rendering the end product unpractical because of high cost. Advanced composite and ceramic materials are cases in point. Discussed are potential pitfalls for applying models at both the atomic and continuum levels. No encouragement is made to unravel the truth of nature. Let it be partiuclates, a smooth continuum or a combination of both. The present trend of development in scaling tends to seek for different characteristic lengths of material microstructures with or without the influence of time effects. Much will be learned from atomistic simulation models to show how results could differ as boundary conditions and scales are changed. Quantum mechanics, continuum and cosmological models provide evidence that no general approach is in sight. Of immediate interest is perhaps the establishment of greater precision in terminology so as to better communicate results involving multiscale physical events.

Contingent sensitivity to configuration in a nonlinear evolution system

A trans-scopic sensitivity of macroscopic failure to slight differentiation in the meso-scopic structure of a system with nonlinear evolution is reported. A periodical chain following a non-local load-sharing evolution was applied as a propotype in failure study. The results demonstrate that there is a transition region composed of globally stable (GS) and evolution induced catastrophic (EIC) modes. That is different from a critical threshold as predicted by percolation and renormalization group theories. Moreover, the EIC mode shows a distinctive sample specific behaviour. For instance, some neighbouring initial states may evolve into completely different final states, though different initial states can evolve into the same final states. As an example, a marginal configuration of EIC mode, a quasi-Fibonacci skeleton, is constructed.

Simulation of the Load-Unload Response Ratio and Critical Sensitivity in the Lattice Solid Model

The Load-Unload Response Ratio (LURR) method is an intermediate-term earthquake prediction approach that has shown considerable promise. It involves calculating the ratio of a specified energy release measure during loading and unloading where loading and unloading periods are determined from the earth tide induced perturbations in the Coulomb Failure Stress on optimally oriented faults. In the lead-up to large earthquakes, high LURR values are frequently observed a few months or years prior to the event. These signals may have a similar origin to the observed accelerating seismic moment release (AMR) prior to many large earthquakes or may be due to critical sensitivity of the crust when a large earthquake is imminent. As a first step towards studying the underlying physical mechanism for the LURR observations, numerical studies are conducted using the particle based lattice solid model (LSM) to determine whether LURR observations can be reproduced. The model is initialized as a heterogeneous 2-D block made up of random-sized particles bonded by elastic-brittle links. The system is subjected to uniaxial compression from rigid driving plates on the upper and lower edges of the model. Experiments are conducted using both strain and stress control to load the plates. A sinusoidal stress perturbation is added to the gradual compressional loading to simulate loading and unloading cycles and LURR is calculated. The results reproduce signals similar to those observed in earthquake prediction practice with a high LURR value followed by a sudden drop prior to macroscopic failure of the sample. The results suggest that LURR provides a good predictor for catastrophic failure in elastic-brittle systems and motivate further research to study the underlying physical mechanisms and statistical properties of high LURR values. The results provide encouragement for earthquake prediction research and the use of advanced simulation models to probe the physics of earthquakes.

Evolution induced catastrophe in a nonlinear dynamical model of material failure

In order to study the failure of disordered materials, the ensemble evolution of a nonlinear chain model was examined by using a stochastic slice sampling method. The following results were obtained. (1) Sample-specific behavior, i.e. evolutions are different from sample to sample in some cases under the same macroscopic conditions, is observed for various load-sharing rules except in the globally mean field theory. The evolution according to the cluster load-sharing rule, which reflects the interaction between broken clusters, cannot be predicted by a simple criterion from the initial damage pattern and even then is most complicated. (2) A binary failure probability, its transitional region, where globally stable (GS) modes and evolution-induced catastrophic (EIC) modes coexist, and the corresponding scaling laws are fundamental to the failure. There is a sensitive zone in the vicinity of the boundary between the GS and EIC regions in phase space, where a slight stochastic increment in damage can trigger a radical transition from GS to EIC. (3) The distribution of strength is obtained from the binary failure probability. This, like sample-specificity, originates from a trans-scale sensitivity linking meso-scopic and macroscopic phenomena. (4) Strong fluctuations in stress distribution different from that of GS modes may be assumed as a precursor of evolution-induced catastrophe (EIC).

Dynamic function of damage and its implications

Dynamic function of damage is the key to the problem of damage evolution of solids. In order to understand it, one must understand its mesoscopic mechanisms and macroscopic formulation. In terms of evolution equation of microdamage and damage moment, a dynamic function of damage is strictly defined. The mesoscopic mechanism underlying self-closed damage evolution law is investigated by means of double damage moments. Numerical results of damage evolution demonstrate some common features for various microdamage dynamics. Then, the dynamic function of damage is applied to inhomogeneous damage field. In this case, damage evolution rate is no longer equal to the dynamic function of damage. It is found that the criterion for damage localization is closely related to compound damage. Finally, an inversion of damage evolution to the dynamic function of damage is proposed.

A Lagrangian lattice Boltzmann method for Euler equations

A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution function consists of macroscopic Lagrangian variables at time steps n and n + 1. It is different from the standard lattice Boltzmann method. In this method the element, instead of each particle, is required to satisfy the basic law. The element is considered as one large particle, which results in simpler version than the corresponding Eulerian one, because the advection term disappears here. Our numerical examples successfully reproduce the classical results.

Weibull modulus for diverse strength due to sample-specificity

By sample specificity it is meant that specimens with the same nominal material parameters and tested under the same environmental conditions may exhibit different behavior with diversified strength. Such an effect has been widely observed in the testing of material failure and is usually attributed to the heterogeneity of material at the mesoscopic level. The degree with which mesoscopic heterogeneity affects macroscopic failure is still not clear. Recently, the problem has been examined by making use of statistical ensemble evolution of dynamical system and the mesoscopic stress re-distribution model (SRD). Sample specificity was observed for non-global mean stress field models, such as the duster mean field model, stress concentration at tip of microdamage, etc. Certain heterogeneity of microdamage could be sensitive to particular SRD leading to domino type of coalescence. Such an effect could start from the microdamage heterogeneity and then be magnified to other scale levels. This trans-scale sensitivity is the origin of sample specificity. The sample specificity leads to a failure probability Phi (N) with a transitional region 0 < (N) < 1, so that globally stable and catastrophic modes could co-exist. It is found that the scatter in strength can fit the Weibull distribution very well. Hence, the Weibull modulus is indicative of sample specificity. Numerical results obtained from the SRD for different non-global mean stress fields show that Weibull modulus increases with increasing sample span and influence region of microdamage.

Weak Zone Related Seismic Cycles in Progressive Failure Leading to Collapse in Brittle Crust

Until quite recently our understanding of the basic mechanical process responsible for earthquakes and faulting was not well known. It can be argued that this was partly a consequence of the complex nature of fracture in crust and in part because evidence of brittle phenomena in the natural laboratory of the earth is often obliterated or obscured by other geological processes. While it is well understood that the spatial and temporal complexity of earthquakes and the fault structures emerge from geometrical and material built-in heterogeneities, one important open question is how the shearing becomes localized into a band of intense fractures. Here the authors address these questions through a numerical approach of a tectonic plate by considering rockmass heterogeneity both in microscopic scale and in mesoscopic scale. Numerical simulations of the progressive failure leading to collapse under long-range slow driving forces in the far-field show earthquake-like rupture behavior. $En Echelon$ crack-arrays are reproduced in the numerical simulation. It is demonstrated that the underlying fracturing induced acoustic emissions (or seismic events) display self-organized criticality------from disorder to order. The seismic cycles and the geometric structures of the fracture faces, which are found greatly depending on the material heterogeneity (especially on the macroscopic scale), agree with that observed experimentally in real brittle materials. It is concluded that in order to predict a main shock, one must have extremely detailed knowledge on very minor features of the earth's crust far from the place where the earthquake originated. If correct, the model proposed here seemingly provides an explanation as to why earthquakes to date are not predicted so successfully. The reason is not that the authors do not understand earthquake mechanisms very well but that they still know little about our earth's crust.