Y.L. Bai

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.

Damage Field Equation and Criterion for Damage Localization

For heterogeneous materials with distributed microcracks or microvoids, damage evolution should be described in terms of a system of damage field and continuum mechanics equations. It was found that the dynamic function of damage f=f (D, σ), i.e. the intrinsic damage evolution rate and the macroscopic formulation of the nucleation, growth and coalescence of microdamages, plays a key role in the evolution. The population of microdamages has a tendency to form localized damage, namely a precursor to failure. The increase of the relative gradient of damage signifies the occurrence of damage localization. Under quasistatic small deformation in one dimensional strain state, this leads to the following criteria f D >f/D+θ and f D >f/D in Eulerian and Lagrangian co-ordinates respectively, where θ is dilatation rate. Whereas under the same assumptions the criterion for maximum stress is f;=θ Clearly, damage localization is a distinct feature of solids. It is relevant to the attainment of maximum stress via the dynamic function of damage f.

Damage Localization as a Possible Precursor of Earthquake Rupture

Based on the concepts of statistical mesoscopic damage mechanics, the rupture of a heterogeneous medium is investigated in terms of numerical simulations of a network model, subjected to simple shear loading. The heterogeneities are simulated by varying the sizes and fracture strains of the elements of the network. Progressive damage is governed by a damage field equation and a dynamic function of damage (DFD). From the damage field equation, a criterion for damage localization can be derived, and the DFD can be extracted from the simulations of the network. Importantly, the DFD intrinsically governs the damage localization. Both stress-free and periodic boundary conditions for the network are examined. It is found that damage localization may be the underlying mechanism of eventual rupture and thus could be used as a possible precursor of earthquake rupture.

Statistical microdamage mechanics and damage field evolution

Discussed are the underlying background of statistical microdamage mechanics, the fundamental partial differential equation of evolution of microdamage number density, two basic solutions, and the saturation of microdamage number density evolution. Knowledge of microdamage number density evolution is applied to engineering practice by using the field equations of microdamage number density and continuum damage. The addition of continuum equations renders a complete system of field equations of deformation and damage. However, they are open-ended in character at the continuum level although the dynamic damage function is completed from the meso- to the macro-scale level. Once decoupling of the function is made, the system of equations can be connected in an approximate manner. This provides a reasonable approximation to the continuum field of deformation and damage. The open literature prediction based on damage evolution relies on assuming arbitrary critical damage states. In this work, use is made of the criterion for damage localization. Several applications of statistical microdamage mechanics are made. This includes damage evolution in a heterogeneous medium and failure forecast under impact. The results show that statistical microdamage mechanics and the derived closed approximate continuum formulations are physically sound and practically effective.

Critical Sensitivity and Trans-scale Fluctuations in Catastrophic Rupture

Rupture in the heterogeneous crust appears to be a catastrophe transition. Catastrophic rupture sensitively depends on the details of heterogeneity and stress transfer on multiple scales. These are difficult to identify and deal with. As a result, the threshold of earthquake-like rupture presents uncertainty. This may be the root of the difficulty of earthquake prediction. Based on a coupled pattern mapping model, we represent critical sensitivity and trans-scale fluctuations associated with catastrophic rupture. Critical sensitivity means that a system may become significantly sensitive near catastrophe transition. Trans-scale fluctuations mean that the level of stress fluctuations increases strongly and the spatial scale of stress and damage fluctuations evolves from the mesoscopic heterogeneity scale to the macroscopic scale as the catastrophe regime is approached. The underlying mechanism behind critical sensitivity and trans-scale fluctuations is the coupling effect between heterogeneity and dynamical nonlinearity. Such features may provide clues for prediction of catastrophic rupture, like material failure and great earthquakes. Critical sensitivity may be the physical mechanism underlying a promising earthquake forecasting method, the load-unload response ratio (LURR).

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.

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.

A self-closed system of equations of damage evolution

This paper introduces a statistical mesomechanical approach to the evolution of damage. A self-closed formulation of the damage evolution is derived.

Robust Trajectory Tracking Control of Autonomous Quad-Rotor UAV Based on Double Loop Frame

This paper deals with the under-actuated characteristic of a quad-rotor unmanned aerial vehicle (UAV). By designing the double loop configuration, the autonomous trajectory tracking is realized. The model uncertainty, external disturbance and the senor noise are also taken into consideration. Then the controller is put forward in the inner loop. An optimal stability augmentation control (SAC) method is used to stabilize the horizon position and keep it away from oscillation. By calculating the nonlinear decouple map, control quantity is converted to the speeds of the four rotors. At last some simulation results and the prototype implementation prove that the control method is effective.

“Deborah numbers”, coupling multiple space and time scales and governing damage evolution to failure

Two different spatial levels are involved concerning damage accumulation to eventual failure.n nnucleation and growth rates of microdamage nN* and V*.nIt is found that the trans-scale length ratio c*/L does not directly affect the process. Instead,ntwo independent dimensionless numbers: the trans-scale one * * ( V*)including then* **5 *nN c V including mesoscopic parameters only, playnthe key role in the process of damage accumulation to failure.nThe above implies that there are three time scales involved in the process: the macroscopicnimposed time scale tim = /a and two meso-scopic time scales, nucleation and growth of damage,n(* *4)nN N t =1 n c and tV=c*/V*. Clearly, the dimensionless number De*=tV/tim refers to thenratio of microdamage growth time scale over the macroscopically imposed time scale. So,nanalogous to the definition of Deborah number as the ratio of relaxation time over external one innrheology. Let De be the imposed Deborah number while De represents the competition andncoupling between the microdamage growth and the macroscopically imposed wave loading. Innstress-wave induced tensile failure (spallation) De* < 1, this means that microdamage has enoughntime to grow during the macroscopic wave loading. Thus, the microdamage growth appears to benthe predominate mechanism governing the failure.nMoreover, the dimensionless number D* = tV/tN characterizes the ratio of two intrinsicnmesoscopic time scales: growth over nucleation. Similarly let D be the “intrinsic Deborahnnumber”. Both time scales are relevant to intrinsic relaxation rather than imposed one. Furthermore,nthe intrinsic Deborah number D* implies a certain characteristic damage. In particular, it isnderived that D* is a proper indicator of macroscopic critical damage to damage localization, likenD* ∼ (10–3~10–2) in spallation. More importantly, we found that this small intrinsic Deborah numbernD* indicates the energy partition of microdamage dissipation over bulk plastic work. Thisnexplains why spallation can not be formulated by macroscopic energy criterion and must bentreated by multi-scale analysis.