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Additive manufacturing enables the production of lattice structures, which have been proven to be a superior class of lightweight mechanical metamaterials whose specific stiffness can reach the theoretical limit of the upper Hashin–Shtrikman bound for isotropic cellular materials. To achieve isotropy, complex structures are required, which can be challenging in powder bed additive manufacturing, especially with regard to subsequent powder removal. The present study focuses on the Finite Element Method simulation of 2.5D anisotropic plate lattice metamaterials and the investigation of their lightweight potential. The intentional use of anisotropic structures allows the production of a cell architecture that is easily manufacturable via Laser Powder Bed Fusion (LPBF) while also enabling straightforward optimization for specific load cases. The work demonstrates that the considered anisotropic plate lattices exhibit high weight-specific stiffnesses, superior to those of honeycomb structures, and, simultaneously, a good de-powdering capability. A significant increase in stiffness and the associated surpassing of the upper Hashin–Shtrikman bound due to anisotropy is achievable by optimizing wall thicknesses depending on specific load cases. A stability analysis reveals that, in all lattice structures, plastic deformation is initiated before linear buckling occurs. An analysis of stress concentrations indicates that the introduction of radii at the plate intersections reduces stress peaks and simultaneously increases the weight-specific stiffnesses and thus the lightweight potential. Exemplary samples illustrate the feasibility of manufacturing the analyzed metamaterials within the LPBF process.

With aid of the Stroh octet formalism, we obtain the elastic field in an infinite Kirchhoff laminated anisotropic plate containing a parabolic Eshelby inclusion undergoing uniform mid-plane in-plane eigenstrains and eigencurvatures. We prove the uniformity of the internal elastic field of membrane stress resultants, bending moments, total mid-plane in-plane strains, total mid-plane curvatures and in-plane rigid-body rotation inside the parabolic inclusion. The decaying and non-uniform elastic field in the exterior of the parabolic inclusion is also obtained. In addition, the internal uniform elastic field and the exterior elastic field at the vertex of the parabola are determined explicitly in terms of the integrals of the reduced elastic stiffnesses and mid-plane eigenstrains and eigencurvatures without solving the Stroh eigenvalue problem. Our solution is further employed to study the internal elastic field inside a through-thickness elliptical elastic inhomogeneity embedded within a parabolic inclusion.

We consider the problem of bending of an anisotropic plate of constant thickness with rigid inclusions. The algorithm of its solution is constructed on the basis of reduction of the problem of bending of a multiply connected anisotropic plate with inclusions to the evaluation of the Lekhniwskii potentials for an auxiliary plane problem of the theory of elasticity with properly chosen elastic constants. To solve this auxiliary problem, we use the method of singular integral equations in the complex form. We consider the problems of bending for plates with inclusions subjected either to the action of moments at infinity (for infinitely large plates) or to transverse loading (for finite plates). We also present some examples of evaluation of stresses acting in the plates containing rigid inclusions of different shapes or systems of inclusions of this kind.

A generalized eigenvalue formulation is developed for the free vibration analysis of anisotropic rectangular plates with rotationally restrained edges using the finite integral transform method. For free vibration problems, casting the governing equations into a generalized eigenvalue problem is particularly advantageous because it enables the direct and systematic extraction of multiple natural frequencies and their associated mode shapes within a unified framework, while avoiding the need for assumed trial functions or solution searching near initial guesses. In the present study, a two-dimensional sine integral transform is introduced into the governing equation of anisotropic plates with bending-twisting coupling, and the mechanical description of rotationally restrained boundary conditions is incorporated simultaneously, thereby converting the original partial differential boundary value problem into a generalized eigenvalue problem. The corresponding analytical solution is then established through the finite integral transform framework. The accuracy and reliability of the proposed method are verified through comparisons with finite element results and published data. Based on the obtained analytical solution, the effects of boundary conditions, rotational stiffness coefficients, aspect ratio, and key stiffness components on the vibration characteristics of anisotropic rectangular plates are further examined. The present study provides an effective analytical framework for free vibration analysis of anisotropic plates with nonclassical rotational restraints and offers theoretical support for the dynamic design and optimization of advanced composite plate structures.

An approximate method for solving the problem of linear viscoelasticity for thin anisotropic plates subject to transverse bending is proposed. The method of small parameter is used to reduce the problem to a sequence of boundary problems of applied theory of bending of plates solved using complex potentials. The general form of complex potentials in approximations and the boundary conditions for determining them are obtained. Problems for a plate with elliptic elastic inclusions are solved as an example. The numerical results for a plate with one, two elliptical (circular), and linear inclusions are analyzed.

The developing of the effective methodic of elastic orthotropic plates’ calculation and the research on the base of their state under different boundary conditions are of great importance nowadays. The representation of the received results in the form, convenient for practical use, is also important. For practical applications in engineering are important tables for determining deflections and internal forces of structures. Such tables for the isotropic case under various conditions of plate support on the contour are given in many works. As for the anisotropic plates, there are no such tables, with the exception of one Huber table compiled for a freely supported rectangular orthotropic plate, depending on the relationship between the stiffness values. Here is a method of calculating the non-homogeneous anisotropic rectangular plates with arbitrary fixation on the contour is set forth, which is reduced to a boundary value problem. The main idea of a calculated general methodic of linear marginal differential tasks calculation is based on underlying of the main part of a solution. Such approach is proved by means of development and some generalization of common positions of a variational method of marginal tasks of mathematical physics of self-conjugated tasks solution. To solve a system of equations in terms of displacements using finite difference method (FDM) in combination with different variations of analytical solutions. It is advisable to construct a numerical solution of the problem so that in difficult cases the support fixing and uploading solution sought, not directly, but in the form of amendments to the known solution for simple cases of reference to consolidate and uploading at finding the solutions which the analytical methods or the FDM with sparse mesh may be used. Given as examples are the results of calculation for a series of square orthotropic plates with a fixed boundary under the action of uniformly distributed load.

We consider an Eshelby’s inclusion of arbitrary shape with prescribed uniform mid-plane eigenstrains and eigencurvatures in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite Kirchhoff laminated anisotropic thin plates. The inclusion has the same extensional, coupling and bending stiffnesses as the surrounding material. The boundary of the semi-infinite plate can be described by free, rigidly clamped and simply supported edges. We derive solutions of simple form by using the new Stroh octet formalism for the coupled stretching and bending deformations of anisotropic thin plates and the method of analytic continuation. In particular, real solutions of the far-field elastic fields induced by an inclusion of arbitrary shape are obtained. Specific examples of an elliptical inclusion in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite anisotropic plates are presented to demonstrate the obtained general solutions.

Anisotropic materials are widely found in nature and engineering, and their crack paths often exhibit complex morphologies such as deflection, kinking, and zig-zag propagation. The formation of these patterns is governed not only by the anisotropic fracture toughness but also by the external loading, whose influence on the crack path is still not fully understood. In this work, we analysed crack propagation in a strongly anisotropic plate with an edge crack under three external loading modes: uniaxial tension (UT), a stationary K-field boundary (Stat-KB), and a surfing K-field boundary (Surf-KB) that moves with the nominal crack tip. A strongly anisotropic high-order phase-field model was used to simulate crack propagation, with the anisotropy degree controlled by a scalar χ . The results show that UT gives an almost straight crack, and Stat-KB leads to a single kinked but non-stationary path, whereas Surf-KB produces a quasi-steady zig-zag path whose kink angle and amplitude increase with χ .

This paper describes a method for determining the strain state of a thin anisotropic plate with elastic arbitrarily arranged elliptical inclusions. Complex potentials are used to reduce the problem to determining functions of generalized complex variables, which, in turn, comes down to an overdetermined system of linear algebraic equations, solved by singular expansions. This paper presents the results of numerical calculations that helped establish the influence of rigidity of elastic inclusions, distances between inclusions, and their geometric characteristics on the bending moments occurring in the plate. It is found that the specific properties of distribution of moments near the apexes of linear elastic inclusions, characterized by moment intensity coefficients, occur only in the case of sufficiently rigid and elastic inclusions.

Derived in this work are the explicit expressions of the three real matrices H, L and S in the Stroh-type formalism for the bending deformation of an anisotropic, linearly elastic plate based on the Kirchhoff theory. The plate is homogeneous in the thickness direction or symmetrically laminated about its mid-plane, and thus, the stretching and bending deformations are decoupled. The three real matrices are the counterparts of the Barnett–Lothe tensors in the Stroh formalism for generalized plane strain elasticity. Identities relating H, L and S are developed. Several applications are presented to demonstrate the usefulness of the derived expressions of H, L and S.