Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua

This work elaborates upon two robust models of gradient elasticity and gradient plasticity, and one gradient model of heat transfer, as originally advocated by the second author in the 1980’s. The objective is, after recalling the links between these models and existing generalized continuum theories as developed in the 1960’s and subsequently, to apply the same methodology to the case of diffusion with a view to establishing generalized transport equations. Aifantis double diffusivity and conductivity theory that provides generalized mass or heat transfer equations is compared to micromorphic-type hyper-temperature and micro-entropy proposals. The double temperature and the micromorphic thermal models are shown to lead to equations more general that Cattaneo’s. The sign of the coefficient of the second time-derivative of temperature is found to differ according to both approaches. The double temperature model contains a fourth space derivative term not present in the micromorphic models. Such generalized equations can be useful, for example, in the interpretation of recent femtosecond laser experiments on metals.     

Non-linearity, periodicity and patterning in plasticity and fracture

The role and implications of microstructure heterogeneity and non-linearity in interpreting a variety of experimental observations during plastic flow and fracture are outlined. Traditionally, these observations were not related to theoretical modelling efforts as recent techniques in non-linear dynamics were not available and usual applications were limited to standard materials and processes where micro- and macroscales do not interact and can be treated independently. Advanced technology has imposed the need for the development of models accounting for scale coupling effects. One class of such models, the so-called gradient models, will be described in the paper in connection with the phenomena of dislocation and strain patterning, as well as with existing observations on oscillatory fracture.     

The role of interfaces in enhancing the yield strength of composites and polycrystals

A deformation-theory version of strain-gradient plasticity is employed to assess the influence of microstructural scale on the yield strength of composites and polycrystals. The framework is that recently employed by Fleck and Willis (J. Mech. Phys. Solids 52 (2004) 1855-1888), but it is enhanced by the introduction of an interfacial “energy” that penalises the build-up of plastic strain at interfaces. The most notable features of the new interfacial potential are: (a) internal surfaces are treated as surfaces of discontinuity and (b) the scale-dependent enhancement of the overall yield strength is no longer limited by the “Taylor” or “Voigt” upper bound. The variational structure associated with the theory is developed in generality and its implications are demonstrated through consideration of simple one-dimensional examples. Results are presented for a single-phase medium containing interfaces distributed either periodically or randomly.     

DYNAMIC ANALYSIS OF A GRADIENT ELASTIC POLYMERIC FIBER

A dynamic analysis of an elastic gradient-dependent polymeric fiber subjected to a periodic excitation is considered.A nonlinear gradient elasticity constitutive equation with strain-dependent gradient coefficients is first derived and the dispersive wave propagation properties for its linearized counterpart are briefly discussed.For the linearized problem a variational formulation is also developed to obtain related boundary conditions of both classical(standard)and non-classical(gradient)type.Analytical solutions in the form of Fourier series for the fiber’s displacement and strain fields are provided.The solutions depend on a dimensionless scale parameter(the diameter to length radio d = D/L)and,therefore,size effects are captured.     

FREE TRANSVERSE VIBRATIONS OF A DOUBLE-WALLED CARBON NANOTUBE：GRADIENT AND INTERNAL INERTIA EFFECTS

This is a modest contribution on higher-order continuum theory for predicting size effects in small-scale objects. It relates to a preceding article of the journal by the same authors（AMSS, 2013, 26： 9-20） which considered the longitudinal dynamical analysis of a gradient elastic fiber but, in addition to an internal length, an internal time parameter is also introduced to model delay/acceleration effects associated with the underlying microstructure. In particular, the free transverse vibration of a double-walled carbon nanotube（DWNT） is studied by employing gradient elasticity with internal inertia. The inner and outer carbon nanotubes are modeled as two individual elastic beams interacting with each other through van der Waals（vdW） forces. General explicit expressions are derived for the natural frequencies and the associated inner-to-outer tube amplitude ratios for the case of simply supported DWNTs. The effects of internal length（or scale）and internal time（or inertia） on the vibration behavior are evaluated. The results indicate that the internal length and time parameters of the adopted strain gradient-internal inertia generalized elasticity model have little influence on the lower order coaxial and noncoaxial vibration modes,but a significant one on the higher order modes.     

On non-singular GRADELA crack fields

A brief account is provided on crack-tip solutions that have recently been published in the literature by employing the so-called GRADELA model and its variants. The GRADELA model is a simple gradient elasticity theory involving one internal length in addition to the two Lame' constants, in an effort to eliminate elastic singularities and discontinuities and to interpret elastic size effects. The non-singular strains and non-singular (but sometimes singular or even hypersingular) stresses derived this way under different boundary conditions differ from each other and their physical meaning in not clear. This is discussed which focus on the form and physical meaning of non-singular solutions for crack-tip stresses and strains that are possible to obtain within the GRADELA model and its extensions.