On the statements of boundary conditions in models of the woven materials production

В статье обсуждаются проблемы постановка краевых задач при моделировании процессов аддитивного производства 3D материала, при учете наличия в нем дополнительных выделенных направлений (выкладки волокон в тканых материалах, арматуры в бетонных конструкциях, биоволокон в мышечной ткани и т.д.). Выводится общая форма тензорного соотношения на поверхности наращивания, при учете дополнительного выделенного направления. Определяется необходимая система независимых аргументов определяющей тензорной функции на поверхности наращивания в рассматриваемом случае. Определяется полный набор совместных рациональных инвариантов тензора напряжений и характерных директоров. Дается инвариантно-полная формулировка определяющих соотношений на поверхности наращивания. Предложены постановки краевых задач, моделирующих процессы синтеза тканых 3D материалов. Полученные дифференциальные ограничения конкретизируются для ортогональных систем координат, учитывающих геометрию процесса наращивания. The article discusses the problem of boundary value problems in models of the additive production processes of a 3D material, taking into account the presence of additional selected directions in it (laying out fibers in woven materials, reinforcement in concrete structures, biofibers in muscle tissue, etc.). The general form of the tensor relation on the growing surface is shown, taking into account the additional selected direction. The necessary system of independent arguments of the constitutive tensor function on the growing surface in the considered case is determined. A complete set of joint rational invariants of the stress tensor and characteristic directors is determined. An invariant-complete formulation of the constitutive relations on the growing surface is given. The formulation of boundary value problems that simulate the processes of synthesis of woven 3D materials are proposed. The resulting differential constraints are specified for orthogonal coordinate systems taking account of the geometry of the growing process.

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, $$3+11$$ 3 + 11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and $$1+6$$ 1 + 6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

Decomposition of third-order constitutive tensors

Third-order tensors are widely used as a mathematical tool for modeling the physical properties of media in solid-state physics. In most cases, they arise as constitutive tensors of proportionality between basic physical quantities. The constitutive tensor can be considered the complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be used as a tool for proper identification of natural materials, such as crystals, and for designing artificial nanomaterials with prescribed properties. In this paper, we study the algebraic properties of a general asymmetric third-order tensor relative to its invariant decomposition. In correspondence with different groups acting on the basic vector space, we present the hierarchy of different types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a general asymmetric third-order tensor, these features are described explicitly. In the case of special tensors with a prescribed symmetry, the decomposition is demonstrated to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of pair symmetry and the Hall tensor as an example of pair skew-symmetry.

Locally-synchronous, iterative solver for Fourier-based homogenization

We use the algebraic orthogonality of rotation-free and divergence-free fields in the Fourier space to derive the solution of a class of linear homogenization problems as the solution of a large linear system. The effective constitutive tensor constitutes only a small part of the solution vector. Therefore, we propose to use a synchronous and local iterative method that is capable to efficiently compute only a single component of the solution vector. If the convergence of the iterative solver is ensured, i.e., the system matrix is positive definite and diagonally dominant, it outperforms standard direct and iterative solvers that compute the complete solution. It has been found that for larger phase contrasts in the homogenization problem, the convergence is lost, and one needs to resort to other linear system solvers. Therefore, we discuss the linear system’s properties and the advantages as well as drawbacks of the presented homogenization approach.

Classification of primary constraints for new general relativity in the premetric approach

We introduce a novel procedure for studying the Hamiltonian formalism of new general relativity (NGR) based on the mathematical properties encoded in the constitutive tensor defined by the premetric approach. We derive the canonical momenta conjugate to the tetrad field and study the eigenvalues of the Hessian tensor, which is mapped to a Hessian matrix with the help of indexation formulas. The properties of the Hessian matrix heavily rely on the possible values of the free coefficients [Formula: see text] appearing in the NGR Lagrangian. We find four null eigenvalues associated with trivial primary constraints in the temporal part of the momenta. The remaining eigenvalues are grouped in four sets, which have multiplicity 3, 1, 5 and 3, and can be set to zero depending on different choices of the coefficients [Formula: see text]. There are nine possible different cases when one, two, or three sets of eigenvalues are imposed to vanish simultaneously. All cases lead to a different number of primary constraints, which are consistent with previous work on the Hamiltonian analysis of NGR by Blixt et al. (2018).

A generalization of the Einstein–Maxwell equations

The proposed modifications of the Einstein–Maxwell equations include: (1) the addition of a scalar term to the electromagnetic side of the equation rather than to the gravitational side, (2) the introduction of a four-dimensional, nonlinear electromagnetic constitutive tensor, (3) the addition of curvature terms arising from the non-metric components of a general symmetric connection and (4) the addition of a non-isotropic pressure tensor. The scalar term is defined by the condition that a spherically symmetric particle be force-free and mathematically well behaved everywhere. The constitutive tensor introduces two structure fields: One contributes to the mass and the other contributes to the angular momentum. The additional curvature terms couple both to particle solutions and to localized electromagnetic and gravitational wave solutions. The pressure term is needed for the most general spherically symmetric, static metric. It results in a distinction between the Schwarzschild mass and the inertial mass.

Anisotropic linear elastic parameter estimation using error in the constitutive equation functional

A modified error in the constitutive equation-based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored. Following an alternating minimization procedure associated with the underlying optimization problem, the new strategy results in an explicit material parameter update formula for general anisotropic material. This immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction. We consider a few common materials to derive such conditions. Then, we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space-dependent material orientations. Finally, we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest.

Equivalence principles, spacetime structure and the cosmic connection

After reviewing the meaning of various equivalence principles and the structure of electrodynamics, we give a fairly detailed account of the construction of the light cone and a core metric from the equivalence principle for photons (no birefringence, no polarization rotation and no amplification/attenuation in propagation) in the framework of linear electrodynamics using cosmic connections/observations as empirical support. The cosmic nonbirefringent propagation of photons independent of energy and polarization verifies the Galileo Equivalence Principle (Universality of Propagation) for photons/electromagnetic wave packets in spacetime. This nonbirefringence constrains the spacetime constitutive tensor to high precision to a core metric form with an axion degree and a dilaton degree of freedom. Thus comes the metric with axion and dilation. Constraints on axion and dilaton from astrophysical/cosmic propagation are reviewed. Eötvös-type experiments, Hughes–Drever-type experiments, redshift experiments then constrain and tie this core metric to agree with the matter metric, and hence a unique physical metric and universality of metrology. We summarize these experiments and review how the Galileo equivalence principle constrains the Einstein Equivalence Principle (EEP) theoretically. In local physics this physical metric gives the Lorentz/Poincaré covariance. Understanding that the metric and EEP come from the vacuum as a medium of electrodynamics in the linear regime, efforts to actively look for potential effects beyond this linear scheme are warranted. We emphasize the importance of doing Eötvös-type experiments or other type experiments using polarized bodies/polarized particles. We review the theoretical progress on the issue of gyrogravitational ratio for fundamental particles and update the experimental progress on the measurements of possible long range/intermediate range spin–spin, spin–monopole and spin–cosmos interactions.

Searches for the Role of Spin and Polarization in Gravity: A Five-Year Update

Searches for the role of spin in gravitation dated before the firm establishment of the electron spin in 1925. Since mass and spin, or helicity in the case of zero mass, are the Casimir invariants of the Poincaré group and mass participates in universal gravitation, these searches are natural steps to pursue. In this update, we report on the progress on this topic in the last five years after our last review. We begin with how is Lorentz/Poincaré group in local physics arisen from spacetime structure as seen by photon and matter through experiments/observations. The cosmic verification of the Galileo Equivalence Principle for photons/electromagnetic wave packets (Universality of Propagation in spacetime independent of photon energy and polarization, i.e. nonbirefringence) constrains the spacetime constitutive tensor to high precision to a core metric form with an axion degree and a dilaton degree of freedom. Hughes-Drever-type experiments then constrain this core metric to agree with the matter metric. Thus comes the metric with axion and dilation. In local physics this metric gives the Lorentz/Poincaré covariance. Constraints on axion and dilaton from polarized/unpolarized laboratory/astrophysical/cosmic experiments/observations are presented. In the end, we review the theoretical progress on the issue of gyrogravitational ratio for fundamental particles and the experimental progress on the measurements of possible long range/intermediate range spin-spin, spin-monopole and spin-cosmos interactions.