Dissipative Redistribution Research Articles

A unified residual-based thermodynamic framework for strain gradient theories of plasticity

A unified thermodynamic framework for gradient plasticity theories in small deformations is provided, which is able to accommodate (almost) all existing strain gradient plasticity theories. The concept of energy residual (the long range power density transferred to the generic particle from the surrounding material and locally spent to sustain some extra plastic power) plays a crucial role. An energy balance principle for the extra plastic power leads to a representation formula of the energy residual in terms of a long range stress, typically of the third order, a macroscopic counterpart of the micro-forces acting on the GNDs (Geometrically Necessary Dislocations). The insulation condition (implying that no long range energy interactions are allowed between the body and the exterior environment) is used to derive the higher order boundary conditions, as well as to ascertain a principle of the plastic power redistribution in which the energy residual plays the role of redistributor and guarantees that the actual plastic dissipation satisfies the second thermodynamics principle. The (nonlocal) Clausius–Duhem inequality, into which the long range stress enters aside the Cauchy stress, is used to derive the thermodynamic restrictions on the constitutive equations, which include the state equations and the dissipation inequality. Consistent with the latter inequality, the evolution laws are formulated for rate-independent models. These are shown to exhibit multiple size effects, namely (energetic) size effects on the hardening rate, as well as combined (dissipative) size effects on both the yield strength (intrinsic resistance to the onset of plastic strain) and the flow strength (resistance exhibited during plastic flow). A friction analogy is proposed as an aid for a better understanding of these two kinds of strengthening effects. The relevant boundary-value rate problem is addressed, for which a solution uniqueness theorem and a minimum variational principle are provided. Comparisons with other existing gradient theories are presented. The dissipation redistribution mechanism is illustrated by means of a simple shear model.

Fluorescence from SF6 in highly excited vibrational states: An overview

The spectroscopy of SF 6 , excited in its quasi-continuum by IR multiphoton absorption, has been recorded by IR fluorescence from ν = 700 to 1300 cm −1 with a highly sensitive device. The far resonance fluorescence exhibits structures directly connected to the intramolecular couplings. A model of couplings has been developed, the parameters of which are fixed by the experimental results. The strongest couplings (orders k = 3 and 4) have a density so low that they produce only a perturbative redistribution of the ν 3 mode. The dissipative redistribution is produced by the highest orders k ⩾5 and can be represented by a half-linewidth γ d ∼1.5 cm −1 for E vib ∼ 15,000 cm −1 . The onset of the SF 6 quasi-continuum is also predicted by the model, with characteristics in full agreement with published data.

Analysis of CF3I quasicontinuum states. III

CF3I quasicontinuum (QC) states are analyzed in the framework used previously for SF6 in papers I and II of this series. The existence of a hierarchy of intramolecular couplings Vk, decreasing vs the number k of vibrational quanta exchanged, with an effective density of coupled states ρk, is pointed out. The oscillator strength redistribution is described taking into account isolated resonances and perturbative redistribution associated to the lowest orders k, dissipative redistribution associated to higher orders, multiple transitions, and mode inhomogeneous spreading. Published homogeneous spectra of 12CF3I and 13CF3I at E=19 000 cm−1, near the dissociation threshold, are reinvestigated in this framework and a new assignment of the resonances is proposed. Using fluorescence data at 〈E〉=3200 cm−1, an interpolation of the position and weight of the main resonances ν1, ν4, ν2+ν3, 2ν5, in the frequency range 900–1200 cm−1, becomes possible. This interpolation allows to extract from IR multiphoton absorption data, at the laser frequency νL=1072 cm−1, an average Lorentzian half-width γ̄ (〈n〉) vs the average number of photons absorbed 〈n〉. γ̄ increases from 1 to 5 cm−1 for 〈n〉 varying from 5 to 10, and varies slowly around 4–5 cm−1 for 10≤〈n〉≤18. Finally, the onset of the QC, near E∼6000 cm−1, is explained using a modelization of the hierarchy Vk, ρk, without arbitrary parameters, and an intramolecular relaxation time ∼20 ps is predicted. This analysis leads to a very good agreement between the different sets of data concerning the CF3I molecule.

Analysis of SF6 quasicontinuum states. II

The IR fluorescence data of SF6 excited in its quasicontinuum (QC), reported in paper I [C. Angelié, J. Chem. Phys. 96, 8072 (1992)], are analyzed in comparison with IR multiphoton absorption (IRMPA) data, either in the nanosecond or in the picosecond regimes. A general description of the QC transition matrix elements is first derived. All the following features must be taken into account: multiple rovibrational transitions, perturbative redistribution onto secondary resonances, dissipative redistribution onto the whole energy shell, and intramolecular mode inhomogeneity producing a frequency dispersion. The intramolecular couplings obey a hierarchy Vk versus the number k of quanta exchanged, with an effective density of couplings ρk. A model without arbitrary parameters settles that the parameter γ=πρ<V2≳ can be shared between a dissipative part γd∼1–1.5 cm−1 (for E∼10 000–15 000 cm−1), corresponding to the orders k≥5, and a perturbative part γp∼2–15 cm−1, corresponding to the lowest orders k≂3 and 4, i.e., the strongest couplings cannot produce the dissipative relaxation. The same model explains that the QC threshold is at E∼4000 cm−1, as demonstrated by Raman experiments, with a half-width γd∼0.21 cm−1, in excellent agreement with picosecond spectroscopy giving a T1 relaxation time of ∼11 ps. The parameter γ is also extracted from IRMPA cross sections. It is found from these data that γ(E) increases from 0.25 to 15 cm−1 when E increases from 4000 to 34000 cm−1. Finally, picosecond data, showing a spectacular enhancement of the number of photons absorbed for short pulses of duration τL∼30 ps, are fully explained by the previous QC description: Energy is mainly absorbed selectively in the ν3 mode, producing a dramatic enhancement of the Rabi width. Then, all known data on the SF6 QC can be incorporated in a unique framework, likely generalizable to other molecules.

Out of resonance infrared fluorescence of SF6, 12CF3I, 13CF3I in quasicontinuum states. I

IR fluorescence of SF6 has been monitored in the spectral range 700–1300 cm−1 with a high sensitivity device, after IR multiphoton excitation at several vibrational energies up to 17 000 cm−1. Preliminary results have also been obtained for 12CF3I and 13CF3I. The sensitivity achieved allows us to focus on the out of resonance fluorescence which turns out to be directly proportional to the local intramolecular couplings represented by the parameter γ(E→E′)=πρ(E′)‖V(E→E′)‖2. The method is partly free from an inhomogeneous contribution and the result can be compared to the Lorentzian model of intramolecular redistribution. It is shown that the γ(E→E′) parameter exhibits a hierarchy of structures, the strongest couplings leading mainly to a perturbative redistribution, while the weakest couplings are responsible for the dissipative redistribution. The homogeneous dissipative part γd appears to be smaller than 1–1.5 cm−1.