Narrow Results

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.

Reaction–diffusion (RD) equation behavior has been studied in various fields, including biology, bioengineering and chemistry. Their solution leads to the formation of patterns which are stable in time and unstable in space, especially when RD system parameters are in the Turing space. Such patterns can be changed due to the growth of the domain where the reaction takes place. This article presents RD equations concerning growing domains in 2D and 3D. Several numerical examples have been solved using different geometries to study the effect of growth on pattern formation. The finite element method was used in conjunction with the Newton–Raphson method for the numerical solution to approximate nonlinear partial differential equations. It was found that growth affected Turing pattern formation, thereby generating complex structures in the domain.

We analyze the variational limit of one-dimensional next-to-nearest neighbours (NNN) discrete systems as the lattice size tends to zero when the energy densities are of multiwell or Lennard–Jones type. Properly scaling the energies, we study several phenomena as the formation of boundary layers and phase transitions. We also study the presence of local patterns and of anti-phase transitions in the asymptotic behaviour of the ground states of NNN model subject to Dirichlet boundary conditions. We use this information to prove a localization of fracture result in the case of Lennard–Jones type potentials.

We consider a class of processes in which the flow of an incompressible fluid through a porous matrix is accompanied by a mass exchange between the constituents. This paper is in two parts. In the first part we use an upscaling procedure to derive a macroscopic law for the flow, starting from the analysis of the phenomena at the pore scale. To this end we utilize a simplified geometry of the solid matrix. The resulting governing equation is of Darcy type, as expected, but, owing to mass exchange, it contains a complicated nonlocal dependence on the hydraulic conductivity. In the second part we look directly for the formulation of a macroscopic model in the framework of mixture theory. We emphasize the basic role of energy dissipation: the two methods lead to the same conclusions, provided that the same dissipation rates are postulated.

Using ideas from continuum mechanics we construct a theory of gravity. We show that this theory is equivalent to Einstein's theory of general relativity; it is also a much faster way of reaching general relativity than the conventional route. Our approach is simple and natural: we form a very general model and then apply two physical assumptions supported by experimental evidence. This easily reduces our construction to a model equivalent to general relativity. Finally, we suggest a simple way of modifying our theory to investigate nonstandard spacetime symmetries.

We extend Padmanabhan's entropy functional formalism to show that, in addition to the Gauss–Bonnet (GB) or the entire series of Lanczos–Lovelock Lagrangians already obtained, more general higher-order corrections to General Relativity, i.e. the so-called modified gravity theories, also emerge naturally from this formalism. This extension shows that the formalism constitutes a valuable tool to investigate, at each order in the curvature, the possible structure the higher-order modified gravity theories might have. As an application, the extended formalism is used to evaluate the horizon entropy in a modified gravity theory of the second-order in the curvature. Our findings are in agreement with previous results from the literature.

We consider a modification of General Relativity motivated by the treatment of anisotropies in Continuum Mechanics. The Newtonian limit of the theory is formulated and applied to galactic rotation curves. By assuming that the additional structure of spacetime behaves like a Newtonian gravitational potential for small deviations from isotropy, we are able to recover the Navarro–Frenk–White profile of dark matter halos by a suitable identification of constants. We consider the Burkert profile in the context of our model and also discuss rotation curves more generally.

The paper proposes an amendment to the relativistic continuum mechanics which introduces the relationship between density tensors and the curvature of spacetime. The resulting formulation of a symmetric stress–energy tensor for a system with an electromagnetic field leads to the solution of Einstein Field Equations indicating a relationship between the electromagnetic field tensor and the metric tensor. In this EFE solution, the cosmological constant is related to the invariant of the electromagnetic field tensor, and additional pulls appear, dependent on the vacuum energy contained in the system. In flat Minkowski spacetime, the vanishing four-divergence of the proposed stress–energy tensor expresses relativistic Cauchy’s momentum equation, leading to the emergence of force densities which can be developed and parameterized to obtain known interactions. Transformation equations were also obtained between spacetime with fields and forces, and a curved spacetime reproducing the motion resulting from the fields under consideration, which allows for the extension of the solution with new fields.

The semi-inverse method is adopted to establish a family of fractal variational principles of the one-dimensional compressible flow under the microgravity condition, and Cauchy–Lagrange integral is successfully derived from the obtained variational formulation. A suitable application of the Lagrange multiplier method is also elucidated.

During fetal development the morphology and function of the organs and tissues is determined. An example occurs with the formation of the cerebral cortex. On the external surface of the brain there are numerous folds (gyri, sulci, and fissures) that determine brain function. The exact cause for the formation of patterns of these folds is unknown. This article proposes a reaction-diffusion model in conjunction with a process of surface mechanical strain to explain the morphogenesis of the superficial structure of the brain.The model is solved using finite elements. There have been tests done on the formation of brain patterns through the reaction-diffusion equations with parameters in the space of Turing and by random mechanical strain. Several numerical examples have been developed that show an acceptable correlation between the results and clinical reality. With the model we can represent, qualitatively, the formation of the cerebral cortex by the proposed model. The model can approximate, and explain, lissencephaly and polymicrogyria, diseases that develop in the cerebral cortex and lead to medical complications to sufferers.

An integrated crystal viscoplastic modeling process has been developed to account for the effect of microstructure in the mechanical response of polycrystalline materials. Grain distributions, including size, shape and orientation, are generated automatically based on probability theories using VGRAIN software. For each set of control parameters (average, maximum and minimum grain size) used in the micro-film simulations, six grain orientation patterns were generated randomly for a micro-film based on a gamma distribution; a large number of analyses have been carried out to account for statistical variations in the spatial pattern of grain orientations. The simulations are used to investigate the effects of grain size and orientation on necking and flow stress in stainless steel under uniaxial tension, and to quantify the extent that variability in the spatial distribution of orientations affects the predictions. Based on the numerical studies, a map was generated indicating under what circumstances macro-mechanics theory can be used and when Crystal Plasticity (CP) theory must be used to ensure the accuracy of the analysis; if the theories are not used appropriately, huge errors can be expected.

Conducting polymer (CP) is an electroactive polymer that displays specific electronic properties, including conductivity. The utility of CP-based soft actuators in various biomedical applications has recently been motivated due to their low voltage-driven specialty compared to the widely used high voltage-driven dielectric elastomers. In some biomedical applications, highly delicate CP actuators may be torn or damaged for unknown reasons. In this regard, this study develops a tear fracture model for fiber-reinforced CP actuators to investigate a specific fracture test of mode III, namely, the trousers test, which involves pulling two legs of a cut specimen horizontally apart. The development of the tear fracture model adopts a well-known Griffith criterion along with the thermodynamically consistent continuum mechanics approach. Additionally, prominent strain energy capturing elastomer strain-stiffening at a moderate strain range is used in conjunction with an empirically established correlation to couple the two internal phenomena, ion diffusion and mechanical deformation of the CP actuators. Later, the effects of various electrical and geometrical parameters on the tearing of the actuator are also addressed.

Sensor application for detection of acetone molecule(s) present in human breath is developed for cantilevered single walled-boron nitride nanotube (SW-BNNT) and analyzed in the present work. The same can be used for continuous monitoring of diabetes. Biocompatibility nature of BNNTs justify their use in biomedical applications. The possible use of the BNNT as nanomechanical resonators is explored in the present study. An atomistic three dimensional (3D) space frame model of fixed-free SW-BNNT-based nanoresonator is developed. The proposed model investigates the feasibility of SW-BNNT for sensing acetone molecules present in human breath for detecting diabetes. Dynamic analysis of fixed-free SW-BNNT for variable aspect ratios of nanotubes is carried out. Presence of acetone molecule(s) causes a shift in the resonant frequency of SW-BNNT. It is observed that this frequency shift is quite significant with presence of more acetone molecules and shows mass sensitivity of SW-BNNT toward acetone molecules. Continuum mechanics-based analytical approach has been used to validate the newly developed sensor equations as the results are found to be in close proximity. The result thus paves new path for the application of SW-BNNTs as biosensor for detection of acetone molecule(s) present in human breath.

Soft matter with hyperelastic behavior may be harnessed for novel applications. However, it is not achievable if the mechanical behaviors of soft matter are not well understood. At present, various traditional extensometers have been used to measure the engineering strain of materials to determine the mechanical properties. The basic assumption of extensometers is that the strain is assumed to be uniform over the gage length. However, this assumption does not hold good in case of experimental specimens having significant nonuniform strain distribution, for example, tensile tests on notched specimens or materials that undergo localized deformations. Hence, it is imperative to adopt a new method which enables us to capture the actual strain field on the surface of a material. Digital image correlation (DIC) technique is an adequate approach that has been widely used in many fields of science and engineering. In this paper, we have presented a mapping algorithm for hyperelastic materials, translating the strain field provided by DIC to the stress field based on continuum mechanics. It overcomes the limitation of extensometers and captures the real stress field for such materials. This method will not only improve the measuring accuracy of stress and strain fields in current experiments, but also greatly promote the study of the localized characteristic for nonlinear and inhomogeneous materials.

A continuum theory for the mechanical response of solid bodies subjected to potentially finite deformation is further developed and applied to solve several new problems in the context of the micromechanics of crystalline solids. The theory invokes concepts from Finsler differential geometry, and it provides a diffuse interface description of fracture surfaces. The director or internal state vector is associated with an order parameter describing degradation of the solid. Here, the deformation gradient between pseudo-Finsler reference and spatial configuration spaces is decomposed into a product of two terms, neither necessarily integrable to a vector field. The first is the recoverable elastic deformation, the second is the residual deformation attributed to changes in free volume in failure zones. The latter is restricted to spherical or isotropic symmetry; resulting Euler–Lagrange equations for mechanical and state variable equilibrium are derived. Metric tensors and volume elements depend on the internal state via a conformal transformation, i.e., Weyl scaling. This version of the theory is first applied to tensile fracture of magnesium. Analytical solutions demonstrate the model's capability to predict ductile versus brittle fracture depending on incorporation of Weyl scaling, with results aligned with molecular dynamics (MD) simulations. The second application is shear fracture in boron carbide: solutions depict weakening and tensile pressure in conjunction with structural collapse in shear transformation zones, as suggested by experiments, quantum mechanics, and/or MD simulations.

It is an established fact that multiscale modeling is an effective way of studying materials over a realistic length scale. In this work, we demonstrate the use of sequential and concurrent multiscale modeling to study the effect of cyclic loading on both the atomic and continuum regions, of graphene, a material which comes with its own set of unique properties. Moreover, to further strengthen this work, we have studied the temperature effects during the cyclic loading, by analyzing the effect of loading and varying temperature gradients.

A constitutive framework based on concepts from phase field theory and pseudo-Finsler geometry is exercised in numerical simulations of deformation and fracture of ceramic polycrystals. The material system of interest is boron carbide, a hard but brittle ceramic. Some microstructures are enabled with thin layers of a secondary amorphous phase of boron nitride between grains of boron carbide. The constitutive theory accounts for physical mechanisms of twinning, crystal-to-glass phase transformations, cleavage fracture within grains and separation and cavitation at grain boundaries (GBs). According to the generalized Finsler approach, geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors, that enter the energy potential in a manner similar to order parameters of phase field models. A partially linearized version of the theory is invoked in finite element simulations of polycrystals, with and without GB layers, subjected to pure shear loading. Effects of grain size and layer properties — thickness, shear modulus and surface energy — are studied parametrically. Results demonstrate that twinning and amorphization occur prominently in nanocrystals but less so in aggregates with larger grains that tend to fail earlier by fracture. Structural changes occur readily in the latter at smaller applied strains only in conjunction with elastic shear softening in localized degraded or damaged regions. Hall–Petch scaling of peak shear strength with grain size is observed. Strength is increased via addition of amorphous layers that shift the failure mode from transgranular to intergranular and further by cavity expansion in layers that induces local elastic compression and suppresses crack extension. Stiff layers provide the largest peak strength enhancement, while elastically compliant layers may improve toughness and strength in the softening regime.

Research on two-dimensional (2D) materials, such as graphene and molybdenum disulfide (MoS₂), now involves researchers worldwide, implementing cutting edge technology to study them. However, when considering using 2D materials in such promising applications, one of the major concerns is the mechanical failure, which remains heavily underexplored. In this work, we demonstrate the use of sequential and concurrent multiscale modeling to study the effect of various fracture modes on graphene and molybdenum disulfide (MoS₂). Some of the fracture modes explored are pure tensile loading, shear loading, and a combination of tensile and shear loading and cyclic loading to investigate fatigue.

The following sections are included:

• Continuum Mechanics
• The Continuity, Motion, and Energy Equations
• Constitutive Relations for Continuum Mechanics
• Structure of Steady Shockwaves – Direct Integration
• Shockwave Structure Using an Eulerian Grid
• Rayleigh-Bénard Flow Using an Eulerian Grid
• Smooth Particle Applied Mechanics
  • The Continuity Equation is an Identity with SPAM
  • Exact Conservation of Momentum with SPAM
  • Exact Conservation of Energy with SPAM
• Rayleigh-Bénard Flow with SPAM
  • Boundary Conditions with SPAM
  • Multiple Solutions with SPAM
• Equilibrating and Collapsing Fluid Columns
• Summary of Lecture 11
• References