Uniqueness of the concentrated-load problem in the linear theory of couple-stress elasticity

Summary It is pointed out that there exist at least two different solutions of the problem of concentrated loads in the two-dimensional, linear couple-stress theory when the formulation is based on the usual uniqueness theorem. An extension of this uniqueness theorem is proved. A set of conditions sufficient for uniqueness is found and is used in a formulation of the concentrated load problem which results in a unique solution. The significant new condition is that the order of the stress singularity is limited to O(r^–1), where r is the distance from the concentrated load.

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Sommario Si fa notare che esistono almeno due soluzioni diverse del problema dei carichi concentrati nella teoria lineare, a due dimensioni, delle coppie di volume quando la formulazione è basata sul teorema di unicità.Si dimostra una estensione di questo teorema di unicità. Si trova un gruppo di condizioni sufficienti per l'unicità; queste condizioni vengono usate nella formulazione del problema del carico concentrato che dà luogo ad un'unica soluzione.La nuova condizione significativa è che l'ordine della singolarità dello sforzo è limitato a O(r^–1), dove r è la distanza dal carico concentrato.

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This work is a result of research sponsored by the Office of Naval Research, U.S. Navy, under Contract Nonr-610(06).     

Hysteresis loops predicted by isoenergy density theory for polycrystals. Part II: cyclic heating and cooling effects predicted from non-equilibrium theory for 6061-T6 aluminum, SAE 4340 steel and Ti–8Al–1Mo–1V titanium cylindrical bars

Mathematical tools and physical models should fit like hand and glove. Traditionally, such an understanding has worked surprisingly well in mechanics and physics. Differential calculus enabled the determination of motion of celestial bodies. Heisenberg's use of Hilbert space applied to calculations in quantum mechanics. These foot steps, however, failed to continue in the field of material science when attempting to address the evolution of material damage. On one hand, the mathematical formulation of the Riemann–Hilbert problem achieved great success for solving macrocrack problems but whether the same tool could be used for lower scale defects seems to be of less concern. This is a surprise because it is not obvious by any means whether microcracks can be distinguished from macrocrack by size difference alone, particularly when they are both treated in a single formulation.There are several issues of the fatigue problem that must be addressed. To begin with, local failure initiating near a surface should be distinguished from global failure that correspond to separation of a specimen into two or more pieces. The interim stages of initiation and termination would depend on the ductility and brittleness of the material in addition to the history of cyclic loading. Three different materials 6061-T6 aluminum, SAE 4340 steel and Ti–8Al–1Mo–1V titanium will therefore be analyzed to show how their uniaxial properties would compare with the dissipation energy density functions and oscillation in temperature. Hysteresis loops for the local material points are determined. Their averages for the ASTM standard hour-glass specimen are then found. They give the global hysteresis loops corresponding to measurements from the uniaxial tests. The local dissipative energy density loops represent the irreversible nature of the materials are also taken into consideration. They are calculated for the soft aluminum and the hard titanium with steel being in between.Obtained also are the cyclic cooling and heating characteristics of the material under fatigue; they fluctuate about the ambient temperature. The -function designating the order and disorder of the fatigue process is also found to be oscillatory in character. Unlike entropy, it is not non-increasing or non-decreasing. It can change sign in a given process. This implies that the order of a system can be enhanced and then impeded or vice versa. Such a behavior is shown to prevail during the initial stage of the fatigue.     

Optically-active dihydropyridines via lipase-catalyzed enantioselective hydrolysis

Several prochiral esters of 1,4-dihydropyridines were enantioselectively hydrolyzed by Pseudomonas lipases (AK, P-30, and K-10) and Candida cylindracea lipase (OF-360). The stereochemical preferences of the lipases P-30 and K-10 were found to be always 4-Pro R and that of OF-360 to be 4-Pro S. In contrast, the prochiral preference of the lipase AK varied depending on the substitution on the dihydropyridine ring. The N-methoxymethyl derivatives afforded the 4S isomers (95% ee) whereas the N-unsubstituted compounds yielded the 4R isomers (50–70% ee).     

Multiscale behavior of crack initiation and growth in piezoelectric ceramics

The multiscale nature of cracking in ferroelectric ceramics is explored in relation to the crack growth enhancement and retardation behavior when the direction of applied electric field is reversed with reference to that of poling. An a priori knowledge of the prevailing fracture behavior is invoked for the energy dissipated in exchange of the macro- and micro-crack surface. To avoid the formalism of developing a two-scale level model, a single dominant crack is considered where the effect of microcracking could be reflected by stable crack growth prior to macro-crack instability. This is accounted for via a length ratio parameter λ. Micro- and macro-crack damage region would necessarily overlap in the simplified approach of applying equilibrium mechanics solutions to different scale ranges that are connected only on the average over space and time. The strain energy density theory is applied to determine the crack growth segments for conditions of positive, negative and zero electric field. The largest and smallest crack segments were found to correspond, respectively, to the positive and negative field. All of the three piezoceramics PZT-4, PZT-5H and P-7 followed such a trend. This removes the present-day controversy arising from the use of the energy release rate concept that yields results independent of the sign of the electric field. Interaction of non-similar crack growth with the direction of electric field is also discussed in relation to Mode II cracking. The crack initiation angle plays a dominant role when the growth segment is sufficiently small. Otherwise, a more complex situation prevails where consideration should also be given to the growth segment length. Failure stresses of Modes I and II cracking are also obtained and they are found to depend not only on the electric field density but also on crack length and the extent of slow crack growth damage. These findings suggest a series of new experiments.     

Thermal/mechanical interaction of subcritical crack growth in tensile specimen

The mutually interacting thermal/mechanical effects are accounted for by retaining the rate of change of volume with surface dV/dA, in the surface/volume energy density theory. An exchange between surface and volume energy would thus prevail in accordance with the rates at which the loads are transmitted throughout the system. Obtained are results for a center-cracked specimen subjected to monotonically rising tensile load at the rate of . Eight load steps are taken with an even increment increase of 69 MPa starting from 276 MPa. The temperature is found to oscillate about the ambient condition as the crack grew incrementally in a stable fashion. What differed considerably from the plasticity theory are the local stress and strain distribution and the crack growth characteristics. This is particularly pronounced in regions close to the crack tip where the local strain rates and strain rate history change from element to element, an affect that is not accounted for in plasticity.     

Arrest characteristics of an impacted crack dissipating plastic energy

When a structural member is accidentally struck, an initial defect may grow and then arrest. An estimate of its size increase is made by considering the geometry of a centrally cracked panel subjected to a step function load in time. Two load amplitudes differing by a factor of five are considered. Under impact, the crack accelerates and then decelerates prior to arrest. The dynamic characteristics depend on the interaction of the elastic-plastic stress waves intervening with the physical boundaries. This effect is assessed quantitatively by computing for the energy stored in a unit volume of material and by incorporating sliding nodes in the finite element method. The energy dissipated by plastic deformation must be accounted for as it is no longer available for creating new macrocrack surface. Obtained are the near tip normal stresses that are found to change from compression to tension. Their magnitude is considerably larger than the corresponding static values. Increase in crack length changes from 0.87% to 20.6% when the magnitude of the impact load is raised five times. The rate of change of the strain energy density factor ΔS with crack growth Δa is found to be governed by the condition ΔS/Δa = const. during loading while dynamic relaxation corresponded to a nonlinear behavior. The physical implication of this remains to be clarified in view of the fact that plasticity theory may not adequately explain the near tip crack behavior.     

Impact response of a cracked layer embedded in a composite

The impact response of a laminate composite with a crack or flaw normal to the interface is studied in terms of the intensification of the dynamic stresses around the crack border. Analytically, the laminate is modeled by a single layer with the crack sandwiched between two other layers of dissimilar material. Fourier and Laplace transforms are employed such that the problem reduces to the solution of a system of dual integral equations. Numerical results for the dynamic stress intensity factor are obtained by solving a Fredholm integral equation. The dynamic stress intensity factors are shown to fluctuate as a function of time, reaching a maximum that depends on the elastic properties of the composite and the flaw size.Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, Pennysylvania 18015. Published in Mekhanika Polimerov, No. 5, pp. 835–840, September–October, 1978.     

Foreword

<正>Aviation safety is the theme given to the International Conference on Airworthiness and Fatigue:7thICSAELS Series Conference,Beijing,China,held on 25–27 March,2013.This event was initiated by the International Center of Sustainability,Accountability,and Eco-Affordability for Large and Small(ICSAELS)and the Institute of Mechanics,Chinese Academy of Sciences.