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In this paper, a strain gradient continuum model for a metamaterial with a periodic lattice substructure is considered. A second gradient constitutive law is postulated at the macroscopic level. The effective classical and strain gradient stiffness tensors are obtained based on asymptotic homogenization techniques using the equivalence of energy at the macro- and microscales within a so-called representative volume element. Numerical studies by means of finite element analysis were performed to investigate the effects of changing volume ratio and characteristic length for a single unit cell of the metamaterial as well as changing properties of the underlying material. It is also shown that the size effects occurring in a cantilever beam made of a periodic metamaterial can be captured with appropriate accuracy by using the identified effective stiffness tensors. [ABSTRACT FROM AUTHOR]

We present a review of the recent workshop "Micropolar Continua and beyond" which held in March 28–31, 2023, at Technische University of Berlin, Germany. [ABSTRACT FROM AUTHOR]

As there are different computational methods for simulating problems in generalized mechanics, we present simple applications and their closed‐form solutions for verifying a numerical implementation. For such a benchmark, we utilize these analytical solutions and examine three‐dimensional numerical simulations by the finite element method (FEM) using IsoGeometric Analysis (IGA) with the aid of open source codes, called tIGAr, developed within the FEniCS platform. A study for the so‐called wedge forces and double tractions help to comprehend their roles in the displacement solution as well as examine the significance by comparing to the closed form solutions for given boundary conditions. It is found that numerical results are in a good agreement with the analytical solutions if wedge forces and double tractions are considered. It is also presented how the wedge forces become necessary in order to maintain equilibrium in strain gradient materials. [ABSTRACT FROM AUTHOR]

In this paper, size effects exhibited by mechanical metamaterials have been studied. When the sizescale of the metamaterials is reduced, stiffening or softening responses are observed in experiments. In order to capture both the stiffening and softening size effects fully, a second-order asymptotic homogenization method based on strain gradient theory is used. By this method, the metamaterials are homogenized and become effective strain gradient continua. The effective metamaterial parameters including the classical and strain gradient stiffness tensors are calculated. Comparisons between a detailed finite element analysis and the effective strain gradient continua model have been made for metamaterials under different boundary conditions, different aspect ratios, different unit cells (closed or open cells) and different topologies. It shows that both stiffening and softening size effects can be captured by using the effective strain gradient continua models. [ABSTRACT FROM AUTHOR]

Based on the principles of rational continuum mechanics and electrodynamics (see Truesdell and Toupin in Handbuch der Physik, Springer, Berlin, 1960 or Kovetz in Electromagnetic theory, Oxford University Press, Oxford, 2000), we present closed-form solutions for the mechanical displacements and stresses of two different magnets. Both magnets are initially of spherical shape. The first (hard) magnet is uniformly magnetized and deforms due to the field induced by the magnetization. In the second problem of a (soft) linear-magnetic sphere, the deformation is caused by an applied external field, giving rise to magnetization. Both problems can be used for modeling parts of general magnetization processes. We will address the similarities between both settings in context with the solutions for the stresses and displacements. In both problems, the volumetric Lorentz force density vanishes. However, a Lorentz surface traction is present. This traction is determined from the magnetic flux density. Since the obtained displacements and stresses are small in magnitude, we may use Hooke's law with a small-strain approximation, resulting in the Lamé- Navier equations of linear elasticity theory. If gravity is neglected and azimuthal symmetry is assumed, these equations can be solved in terms of a series. This has been done by Hiramatsu and Oka (Int J Rock Mech Min Sci Geomech Abstr 3(2):89-90, 1966) before. We make use of their series solution for the displacements and the stresses and expand the Lorentz tractions of the analyzed problems suitably in order to find the expansion coefficients. The resulting algebraic system yields finite numbers of nonvanishing coefficients. Finally, the resulting stresses, displacements, principal strains and the Lorentz tractions are illustrated and discussed. [ABSTRACT FROM AUTHOR]

Highlights: [•] The constitutive equations for ductile damage of metals are presented. [•] Two normalized measures represent the kinetic process of strain-induced damage. [•] The evolution of damage is determined in uniaxial tensile and upsetting tests. [•] The equilibrium point for kinetics of the damage growth and healing is revealed. [•] The connection of the damage measure with the fracture locus is shown. [Copyright &y& Elsevier]

Abstract: A method based on Fourier series is presented, which allows to calculate the local stress–strain response of a three-dimensional periodic structure subjected to a spatial average of strain. The periodicity allows the reduction of the problem to that of a Representative Volume Element (RVE). The solution operator (which can easily be calculated in Fourier space) is defined for a simplified problem, and it is shown that this operator may also be used for the original problem. In order to illustrate the use of this procedure, an example problem is presented. A global error is defined and calculated for the example problem. [Copyright &y& Elsevier]

A numerical approach simulating the effects of coupled local-global mode interaction on the postbuckling of axially compressed sandwich beam columns is presented. The formulation is based on the weak form of the balance of linear momentum whilst large strains are considered. An open source software is used to solve the formulation and results are presented showing good phenomenological agreement with analytically derived solutions. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Strain gradient theory is an accurate model for capturing the size effect and localization phenomena. However, the challenge in identification of corresponding constitutive parameters limits the practical application of the theory. We present and utilize asymptotic homogenization herein. All parameters in rank four, five, and six tensors are determined with the demonstrated computational approach. Examples for epoxy carbon fiber composite, metal matrix composite, and aluminum foam illustrate the effectiveness and versatility of the proposed method. The influences of volume fraction of matrix, the stack of RVEs, and the varying unit cell lengths on the identified parameters are investigated. The homogenization computational tool is applicable to a wide class materials and makes use of open-source codes in FEniCS. We make all of the codes publicly available in order to encourage a transparent scientific exchange. [Display omitted] • We exploit asymptotic homogenization for determining all the parameters in the strain gradient theory. • An FEM based computational approach determines all parameters in rank four, five, and six tensors in generalized elasticity. • The proposed homogenization method identifies the effective parameters of composite materials with periodic microstructures. • Verification of the identified parameters is conducted in the case of a 3-D beam bending problem by using open-source packages in FEniCS. • We make all of the codes publicly available in order to encourage a transparent scientific exchange. [ABSTRACT FROM AUTHOR]