A personal, but quite general viewpoint on current morphogenic plasticity developments is given by elaborating on dislocation patterning and deformation bands with an eye on Nasr Ghoniem’s pioneering contributions to material instabilities. It is based on the gradient approach advocated by the author in the early 1980s and subsequently pursued by him and his co-workers, as well as other leading scientists, including Ghoniem himself. Since the physical origin of plastic flow in metals is due to the existence of dislocations, a brief discussion on the removal of classical elasticity singularities for this type of line defects is provided based on a gradient modification of Hooke’s law and its extension to also include fractional and fractal effects. The article will thus first briefly revisit early efforts on pattern-forming instabilities in plasticity and then discuss the need for combined gradient-stochastic models to capture plastic heterogeneity phenomena at small scales. It will continue with listing easy-to-use fractional/fractal dislocation solutions for potential implementation to respective dislocation-based computer simulations and conclude with a few remarks on possible extensions of the Laplacian-based gradient approach described herein to other multiscale/multiphysics phenomena. The last topic is an open issue that has not been pursued as yet in the material physics and mechanics community. Even though some of our new unpublished results are preliminary and sporadically shared with the community in conference presentations available at the internet, it is hoped that they can still inspire a much-needed collective collaboration and more elaborate interdisciplinary studies in the near future.

In this paper, we solve explicitly and analyze rigorously inhomogeneous initial-boundary-value problems (IBVP) for several fourth-order variations of the traditional diffusion equation and the associated linearized Cahn-Hilliard (C-H) model (also Kuramoto-Sivashinsky equation), formulated in the spatiotemporal quarter-plane. Such models are of relevance to heat-mass transfer phenomena, solid-fluid dynamics and the applied sciences. In particular, we derive formally effective solution representations, justifying a posteriori their validity. This includes the reconstruction of the prescribed initial and boundary data, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula is utilized to rigorously deduce the solution's regularity and asymptotic properties near the boundaries of the domain, including uniform convergence, eventual (long-time) periodicity under (eventually) periodic boundary conditions, and null noncontrollability. Importantly, this analysis is indispensable for exploring the (non)uniqueness of the problem's solution and a new counter-example is constructed. Our work is based on the synergy between: (i) the well-known Fokas unified transform method and (ii) a new approach recently introduced for the rigorous analysis of the Fokas method and for investigating qualitative properties of linear evolution partial differential equations (PDE) on semi-infinite strips. Since only up to third-order evolution PDE have been investigated within this novel framework to date, we present our analysis and results in an illustrative manner and in order of progressively greater complexity, for the convenience of readers. The solution formulae established herein are expected to find utility in well-posedness and asymptotics studies for nonlinear counterparts too.

Explicit integral representations consistent with the fundamental Ehrenpreis–Palamodov principle are herein constructed for the novel solution formulae, recently obtained in closed form via rigorous implementation of the Fokas unified transform method, for the Sobolev–Barenblatt pseudo-parabolic model of seepage in porous media and for the Rubinshtein–Aifantis double-diffusion system formulated on semi-bounded domains.

In a 1987 article of the last author dedicated to the memory of a pioneer of classical plasticity Aris Philips of Yale, the last author outlined three examples of self-organization during plastic deformation in metals: persistent slip bands (PSBs), shear bands (SBs) and Portevin Le Chatelier (PLC) bands. All three have been observed and analyzed experimentally for a long time, but there was no theory to capture their spatial characteristics and evolution in the process of deformation. By introducing the Laplacian of dislocation density and strain in the standard constitutive equations used for these phenomena, corresponding mathematical models and nonlinear partial differential equations (PDEs) for the governing variable were generated, the solution of which provided for the first time estimates for the wavelengths of the ladder structure of PSBs in Cu single crystals, the thickness of stationary SBs in metals and the spacing of traveling PLC bands in Al-Mg alloys. The present article builds upon the 1987 results of the aforementioned three examples of self-organization in plasticity within a unifying internal length gradient (ILG) framework and expands them in 2 major ways by: (i) introducing the effect of stochasticity and (ii) capturing statistical characteristics when PDEs are absent for the description of experimental observations. The discussion focuses on metallic systems, but the modeling approaches can be used for interpreting experimental observations in a variety of materials.

This study presents a peridynamic (PD) correspondence model with strain gradient elasticity (SGE) to capture size effect on strength of nano- and micro-scale structures. The classical elasticity theory lacking a length scale parameter does not account for the microstructure-dependent size effects. Strain Gradient (SG) elasticity is an extension of the classical elasticity with a length scale parameter(s) which can be linked to the microstructure of the material. Peridynamic theory introduces damage into the constitutive relations in a natural way by allowing for interactions of a material point within its horizon which is referred to as the internal length parameter. The correspondence model of the PD theory permits the incorporation of the constitutive relationship of SG theory. Therefore, the present approach combines the effects of PD and SGE length scale parameters on the stiffness and strength of the material. The resulting equations present two length parameters: the horizon of a material point in PD theory and the characteristic length in SGE theory. The PD equation of motion with SGE (PDSG) is free of spatial derivatives and allows for the imposition of classical and nonclassical boundary conditions. The PDSG is first applied to investigate the deformation response of a one-dimensional nanoscale film subjected to a quasi-static tensile load and a dynamic load through an initial constant strain. The static response is compared with the analytical solution for varying ratio of SG length parameter to PD horizon. The dynamic response is compared with a computational solution while satisfying the nonclassical boundary conditions. Subsequently, the PDSG is applied to study the deformation response of a two-dimensional nanoscale film subjected to a quasi-static axial load and a specified tangential end displacement. These two cases are considered under special displacement constraints to compare with the available analytical solutions. Subsequently, a two-dimensional nanoscale film with or without an internal crack is subjected to uniform tensile loading. Finally, crack propagation is simulated under incremental displacement loading. The results capture the analytical predictions and the expected increase in stiffness with increasing ratio of SG length parameter to PD horizon.

A novel method is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) on the half-line for a well-known coupled system of evolution partial differential equations. The so-called double-diffusion model, which is based on a simple, yet general, inhomogeneous diffusion configuration, describes accurately several important physical and mechanical processes and thus emerges in miscellaneous applications, ranging from materials science, heat-mass transport and solid–fluid dynamics, to petroleum and chemical engineering. For instance, it appears in nanotechnology and its inhomogeneous version has recently appeared in the area of lithium-ion rechargeable batteries. Our approach is based on the extension of the unified transform (also called the Fokas method), so that it can be applied to systems of coupled equations. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. The novel solution formulae are also utilized to rigorously deduce the solution’s regularity properties near the boundaries of the spatiotemporal domain. In particular, we prove uniform convergence of the solution to the data, its rapid decay at infinity as well as its smoothness up to (and beyond) the boundary axes, provided certain data compatibility conditions at the quarter-plane corner are satisfied. As a sample of important applications of our analysis and investigation of the boundary behavior of the solution and its derivatives, we both prove a novel uniqueness theorem and construct a ‘non-uniqueness counterexample’. These supplement the preceding ‘constructive existence’ result, within the framework of well-posedness. Moreover, one of the advantages of the unified transform is that it yields representations which are defined on contours in the complex Fourier λ -plane, which exhibit exponential decay for large values of λ . This important characteristic of the solutions is expected to allow for an efficient numerical evaluation; this is envisaged in future numerical-analytic investigations. The new formulae and the findings reported herein are also expected to find utility in the study of questions pertaining to well-posedness for nonlinear counterparts too. In addition, our rigorous approach can be extended to IBVPs for other significant models of mathematical physics and potentially also to higher-dimensional and variable-coefficient cases.

A novel technique is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) (Dirichlet, Neumann and Robin) on the half-line, for a well-known pseudo-parabolic partial differential equation. This so-called Barenblatt's equation arises in a plethora of important applications, ranging from heat-mass transfer, solid-fluid-gas dynamics and materials science, to mechanical, chemical and petroleum engineering, as well as electron physics, radiation and diffusive processes. Our approach is based on the extension of the Fokas method, so that it can be applied to problems with mixed derivatives. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each type of IBVP, the novel formulae are utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain and the problem's well-posedness. Furthermore, importantly, our solutions' numerical advantages are demonstrated and highlighted by way of a concrete and illustrative example. Our rigorous approach can be extended to IBVPs for other significant models.

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

This study couples peridynamic (PD) theory with strain gradient elasticity (SGE) theory to investigate the combined effect of PD and SGE length scale parameters on size effect. The SGE theory introduces a length scale parameter in the stress-strain relations to account for size effect. The solution to the classical form of SGE equation of motion requires two additional non-classical boundary conditions arising from the presence of fourth-order spatial derivatives. The wave dispersions level off as the wave number increases as observed in real materials. The PD theory allows for nonlocal interactions of a material point within its horizon which serves as the PD length scale parameter. The equation of motion emerges in the form of an integral equation and the internal forces are expressed through nonlocal interactions (bonds) between the material points within a continuous body. It is a reformulation of classical continuum mechanics which is free of spatial derivatives. The numerical results concern a previously considered carbon nanotube (CNT). Under quasi-static axial loading, the results indicate an increased strengthening effect along the length of the tube. Under dynamic loading, its longitudinal vibration response is captured through explicit time integration. The numerical predictions capture the reference solutions corresponding to the classical SGE equation.

The linear double-diffusivity (D-D) model of Aifantis, comprising two coupled Fick-type partial differential equations and a mass exchange term connecting the diffusivities, is a paradigm in modeling mass transport in inhomogeneous media, e.g., fissures or fractures. Uncoupling of these equations led to a higher order partial differential equation that reproduced the non-classical transport terms, analyzed independently through Barenblatt’s pseudoparabolic equation and the Cahn–Hilliard spinodal decomposition equation. In the present article, we study transport in a nonlinearly coupled D-D model and determine the regime-switching of the associated diffusive processes using a revised formulation of the celebrated Lux method that combines forward Fourier transform with a Laplace transform followed by an Inverse Fourier transform of the governing reaction–diffusion (R–D) equations. This new formulation has key application possibilities in a wide range of non-equilibrium biological and financial systems by approximating closed-form analytical solutions of nonlinear models.

In this study systematic numerical analyses are outlined searching for additional instabilities in the co-rotating Taylor–Couette system within the fully deterministic sequential approach of bifurcations (SBA) to turbulence. The main idea of the search strategy is the application of a forcing function, rotation, which has a direct physical interpretation, and that was realized in prior experimental work. The forcing induces disturbances that lead to bifurcations of new states. Thus, turbulence can be generated and observed in a rotating fluid without the imposing additional forcing sources. The imposition of thermoconvective forcing in the Taylor–Couette system will be discussed separately. Important findings include the discovery of the interplay of new and already known states, the transition of steady states to oscillatory ones and higher order states in the SBA via vortex merger/separation and re-allocation of symmetries for a more intensified mass transport. The results of the present work enhance the results of [ 1 ]. They will be revisited within an internal length gradient (ILG) framework accounting for weekly nonlocal effects as suggested in the concluding section of the paper.

Non-linear solutions and their stability are presented for homogeneously heated channel flows with a simple geometry under the influence of a constant pressure gradient or when the vanishing of the mass flux across any lateral cross-section of the channel is imposed. The critical Grashof number is determined by linear stability analysis for various values of the Prandtl number. In our numerical study the angle of inclination of the channel is taken into account. We found that in each case studied, with the exception of a horizontal layer of fluid and when the applied constant pressure gradient is zero, the basic flow looses stability through a Hopf bifurcation. Following the linear stability analysis our numerical studies are focused on the emerging secondary flows and their stability, in order to identify possible bifurcation points for tertiary flows. We conclude with a few comments on revisiting the present results within an internal length gradient (ILG) framework accounting for higher order velocity and temperature gradients.

A recent study analyzed the role of stochastic perturbations on the interface dynamics of two interacting species within a double-diffusivity framework, involving double diffusion models. The model relied on a restricted translation–reflection (TR) symmetry manifold, leading to a single variable description. The present study generalizes this model for a TR symmetry violating system that does not permit reduction to a single variable dynamics, leading to a hitherto unseen stochastic resonance (SR), a mechanism that indicates discrete, rather than a continuous, mode of energy transport. The SR exhibited by the model captures the signature fast transport observed in stochastically driven dynamics of nanopolycrystals, that previous deterministic models failed to emulate. We show that the speed of transfer relates to the strength of energy cross-correlations between the two diffusing species communicating through interface dynamics that eventually drives the energy throughput and identifies the role of stochasticity in nanopolycrystalline transport.

A new method to consider the formation and stability of dislocation patterns is outlined. It is based on the assumption of a fluid-like dislocation state which evolves within the lattice state. This evolution is specified by the balance laws of mass and momentum together with appropriate constitutive equations for the mechanical fields associated with the dislocation state. The approach leads to nonlinear partial differential equations of the diffusion reaction type. As in the case of dissipative structures and other self-organization phenomena, the competition between diffusive and reacting terms determines the occurence and persistence of spatial instabilities, the wavelength of periodic ones, and the shape of two and three dimensional dislocation patterns such as cell walls, vein, and ladder structures.

Starting from the simple assumption that the displacement field underneath a pyramidal (Berkovich/Vickers) tip for very shallow indents is described by the Boussinesq solution, while for deeper ones is dictated by the tip's specific geometry, a combination of the two leads to a combined displacement field for the material at contact with the tip. Continuum mechanics is then utilized for the calculation of the strain tensor, while gradient elasticity theory is adopted for the determination of the stress tensor underneath the tip. Gradient elasticity was assumed in order for the gradient term to be able to model, in this case, the inhomogeneously applied load due to the specific tip geometry, which is different than a flat punch. The thus calculated stress component along the loading (vertical) direction is a parametric function of the material's elastic constants (Poisson's ratio, Lame constants), as well as of the maximum elastic deformation and the gradient coefficient. By appropriately modifying these parameters, the proposed formulation seems to be able to predict the mechanical response of the material underneath the indenter for the specific pyramidal tip geometry (Berkovich or Vickers), without any approximations or empirical relations. The validity of the proposed formulation's predictions was checked against thin film delamination and shear band formation AFM/SEM micrographs, which showed a very good qualitative as well as quantitative comparison with the theoretical predictions.

Model analogies and exchange of ideas between physics or chemistry with biology or epidemiology have often involved inter-sectoral mapping of techniques. Material mechanics has benefitted hugely from such interpolations from mathematical physics where dislocation patterning of platstically deformed metals and mass transport in nanocomposite materials with high diffusivity paths such as dislocation and grain boundaries, have been traditionally analyzed using the paradigmatic Walgraef-Aifantis (W-A) double-diffusivity (D-D) model. A long standing challenge in these studies has been the inherent nonlinear correlation between the diffusivity paths, making it extremely difficult to analyze their interdependence. Here, we present a novel method of approximating a closed form solution of the ensemble averaged density profiles and correlation statistics of coupled dynamical systems, drawing from a technique used in mathematical biology to calculate a quantity called the basic reproduction number R0, which is the average number of secondary infections generated from every infected. We show that the R0 formulation can be used to calculate the correlation between diffusivity paths, agreeing closely with the exact numerical solution of the D-D model. The method can be generically implemented to analyze other reaction-diffusion models.

A deformation theory of gradient plasticity is employed to study elastoplastic axisymmetric boreholes subjected to far-field biaxial tension. The gradient dependence is introduced in the constitutive framework by assuming that the flow stress (i.e., an effective stress measure) depends not only on an effective strain measure but also on the Laplacian thereof. The classical theory of linear elasticity is adopted for the elastic deformations while strain softening is assumed for the plastic part, where an equivalent work hypothesis is used to associate the stress state to the final total strain. Representative stress distributions are illustrated and compared in the context of size effects, which originate from the presence of the gradient term in the governing equations. The influence of the borehole radius on the initiation of plastic deformation, the nominal stress-strain response and the evolution of the elastoplastic interface are examined. Moreover, a maximum principal strain failure criterion is employed to discuss size effects on the onset of macroscopic fracture. The obtained results show that narrower boreholes have higher plastic and failure strength.

A new method for interpreting Vickers microindentation data is proposed, based on continuum mechanics and, more precisely, the Gradient Elasticity framework. The main feature is the elastic properties’ calculation from the initial (elastic) region of the load vs. depth indentation data, which makes the calculation independent of the maximum indentation depth. This approach deviates significantly from the semi-empirical method of Oliver and Pharr, which calculates the material properties (such as elastic modulus and hardness) from the elastic unloading region, with the calculated values of the elastic modulus and hardness being strongly dependent on the indentation depth and, therefore, giving rise to the so-called indentation size effect (ISE). The proposed framework considers the Vickers indentation as a compression test with a complex geometry, as the pyramidal indenter tip applies load to directions perpendicular to its four faces. An elastic displacement field is initially assumed following Boussinesq’s solution before an indent is made, while afterwards the displacement of the material in contact with the tip is assumed to follow the Vickers tip’s geometry. The respective von Mises equivalent strain calculated through a continuum mechanics approach can qualitatively capture thin film delamination micrographs and shear band formation, showing the potential of the present formulation to model such micro-deformation problems. The traction vector calculated on each of the four sides of the Vickers tip, leads to the generation of load-displacement data, which compare well with experimental indentation data, with the elastic properties (i.e. elastic modulus) thus calculated being in accordance with the corresponding literature values.