We propose an effective heat equation for an enhanced heat conduction problem that can account for thermal scale-size effects in miniaturized or nanofabricated metamaterials. The derivation of the proposed enhanced heat conduction problem is based on contributions to the global energy balance in the form of Gurtin’s microforce balance introduced in [M. E. Gurtin, Phys. D, 92 (1996), pp. 178–192] and was first proposed by S. Forest and M. Amestoy [C. R. Méc., 336 (2008), pp. 347–353]. The resulting effective heat equation retains its classical form with temperature as the macroscopic variable; however, the effective conductivity tensor is constructed using the local entropy and the local gradient of entropy instead of the local temperature.

Architected materials offer bright possibilities to achieve specialized mechanical properties that are not possible with most engineering and natural materials. For example, they allow the design of personalized metal medical implants with reduced stress shielding and improved implant longevity. However, the interplay between the characteristic length scales of a microscopic architecture and macroscopic dimensions can significantly alter the mechanical response of such implants. Here, we systematically investigate geometric size effects in cellular lattices and mechanical metamaterials under fundamental compressional, shear, and torsional loads for representative unit cell designs. Using finite element simulations, we analyze how variations in the size of a unit cell affect the effective elastic moduli of the lattices and validate our findings experimentally. We further apply these insights to analyze the mechanical behavior of a simplified metamaterial implant under complex loads and various interfacial conditions between the bone and the implant. Our results identify critical relations between the size effects parameters and the mechanical properties of cellular lattices and metamaterials. We highlight the importance of cell size in geometrically constrained applications and propose a procedure for a full-scale discrete analysis of size effects. Our findings lay the groundwork for the advancement of higher-order continuum theories and will help drive the adoption of metamaterials across a wide range of engineering applications.

We delineate a multiscale and multiphysics electromechanical system modelling the response of dielectric elastomer and quantify its macroscopic behaviour. Our approach involves formulating a precise theoretical framework within the main laws of thermodynamics. In the initial stage, constitutive laws are derived through a generalized Coleman–Noll procedure. Subsequently, the constitutive laws are refined using a linear approximation of the response function and homogenization theory. This assumes rapidly oscillating space-charges as a source term in a moderately strong electric field. The approach involves an indirect method that approximates the quasi-static Maxwell’s equations in free space, combined with the periodic homogenization asymptotic procedure. Employing the periodic unfolding operator and the averaging operator facilitates the transition to the homogenization limit, revealing the pivotal role of correctors in bridging the gap between microscopic and macroscopic aspects by quantifying disparities between heterogeneous and homogenized solutions.

We present a finite-volume based numerical scheme for a nonlocal Cahn-Hilliard equation which combines ideas from recent numerical schemes for gradient flow equations and nonlocal Cahn-Hilliard equations. The equation of interest is a special case of a previously derived and studied system of equations which describes phase separation in ternary mixtures. We prove the scheme is both energy stable and respects the analytical bounds of the solution. Furthermore, we present numerical demonstrations of the theoretical results using both the Flory-Huggins (FH) and Ginzburg-Landau (GL) free-energy potentials.

We establish the existence of a weak solution for a strongly coupled, nonlinear Stokes–Maxwell system, originally proposed by Nika and Vernescu (Z Angew Math Phys71(1):1–19, 2020) in the three-dimensional setting. The model effectively couplesthe Stokes equation with the quasi-static Maxwell’s equations through the Lorentzforce and the Maxwell stress tensor. The proof of existence is premised on: (i) theaugmented variational formulation of Maxwell’s equations, (ii) the definition of a newfunction space for the magnetic induction and the verification of a Poincar’e-typeinequality, and (iii) the deployment of the Altman–Shinbrot fixed point theorem whenthe magnetic Reynolds number, Rm, is small.

We derive two different effective models from a heterogeneous Cosserat continuum taking into account the Cosserat intrinsic length of the constituents. We pass to the limit using homogenization via periodic unfolding and in doing so we provide rigorous proof to the results introduced by Forest, Pradel, and Sab (Int. J. Solids Struct. 38 (26-27): 4585-4608 ’01). Depending on how different characteristic lengths of the domain scale with respect to the Cosserat intrinsic length, we obtain either an effective classical Cauchy continuum or an effective Cosserat continuum. Moreover, we provide some corrector type results for each case.

We derive effective models for a heterogeneous second-gradient elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities ℓ, the intrinsic lengths of the constituents ℓSG and ℓchiral, and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓSG, ℓchiral, ℓ, and L we obtain either an effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors’ structure rely on a suitable use of the periodic unfolding operator.

Accurate population data is crucial for assessing exposure in disaster risk assessments. In recent years,there has been a signifcant increase in the development of spatially gridded population datasets.Despite these datasets often using similar input data to derive population fgures, notable diferencesarise when comparing them with direct ground-level observations. This study evaluates the precisionand accuracy of food exposure assessments using both known and generated gridded populationdatasets in Sweden. Specifcally focusing on WorldPop and GHSPop, we compare these datasetsagainst ofcial national statistics at a 100 m grid cell resolution to assess their reliability in foodexposure analyses. Our objectives include quantifying the reliability of these datasets and examiningthe impact of data aggregation on estimated food exposure across diferent administrative levels.The analysis reveals signifcant discrepancies in food exposure estimates, underscoring the challengesassociated with relying on generated gridded population data for precise food risk assessments.Our fndings emphasize the importance of careful dataset selection and highlight the potential foroverestimation in food risk analysis. This emphasises the critical need for validations against groundpopulation data to ensure accurate food risk management strategies.

We offer an insight into our mathematical endeavors, which aim to advance the foundational understanding of energy systems in a broad context, encompassing facets such as charge transport, energy storage, markets, and collective behavior. Our working techniques include a combination of well-posed mathematical models (both deterministic and stochastic), mathematical analysis arguments (mostly concerned with model dimension reduction and averaging, periodic homogenization), and simulation tools (numerical approximation techniques, computational statistics, high-performance computing).