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This article focuses on the buckling response of columns formed by pre-twisting a flat, narrow and straight material strip of a rectangular cross section. The experimental analysis accomplished for birch plywood strips covers twisting up to 90 degrees. The corresponding computational analysis covers the twisting process (geometrically nonlinear analysis) and the subsequent compression (linear buckling analysis, including the residual stresses from the twisting process) and includes comparisons for 3D solid and 2D shell finite elements within orthotropic linear elasticity. A further finite element analysis provides findings up to twisting angles of 400 degrees and includes nonlinear post-buckling analyses for the compression subsequent to twisting, in addition to the linear buckling analyses. These sets of analyses reveal some new findings regarding this classical problem. Most importantly, there are four characteristic twisting angles which are associated to mode jumps in buckling, to a plateau in the curve relating the critical load and the twisting angle and, finally, to a loss of stability during the twisting process. The influence of the twisting-induced stresses on the buckling response is investigated as well. • Compression response of twisted (by up to 90 degrees) plywood strips is studied experimentally. • 3D solid and shell FE models confirmed significant increase in critical buckling load. • FE analyses (for twisting up to 400 degrees) revealed four characteristic angles associated to: • (1) Mode jumps in buckling, (2) A plateau in the curve relating critical load and twisting angle, (3) A loss of stability during the twisting process. [ABSTRACT FROM AUTHOR]

The fundamental equations for Form II of Mindlin's second strain gradient elasticity theory for isotropic materials are first derived. A corresponding simplified formulation is then proposed, with six and two higher-order material parameters for the strain and kinetic energy, respectively. This simplified model is still capable of accounting for free surface effects and surface tension arising in second strain gradient continua. Within the simplified model, at first, surface tension effects appearing in nano-scale solids near free boundaries are analyzed. Next, a thin strip under tension and shear is considered and closed-form solutions are provided for analyzing the free surface effects. Expressions for effective Poisson's ratio and effective shear modulus are proposed and found to be size-dependent. Most importantly, for each model problem a stability analysis is accomplished disallowing non-physical solutions (befallen but not exclusively disputed in a recent Form I article). Finally, a triangular macro-scale lattice structure of trusses is shown, as a mechanical metamaterial, to behave as a second strain gradient continuum. In particular, it is shown that initial stresses prescribed on boundaries can be associated to one of the higher-order material parameters, modulus of cohesion, giving rise to surface tension. For completeness, a numerical free vibration eigenvalue analysis is accomplished for both a fine-scale lattice model and the corresponding second-order continuum via standard and isogeometric finite element simulations, respectively, completing the calibration procedure for the higher-order material parameters. The eigenvalue analysis confirms the necessity of the second velocity gradient terms in the kinetic energy density. [ABSTRACT FROM AUTHOR]

This paper is devoted to a gradient-elastic stress analysis of an infinite plate weakened by a cylindrical hole and subjected to two perpendicular and independent uni-axial tensions at infinity. The problem setting can be considered as an extension and generalization of the well-known Kirsch problem of the classical elasticity theory which is here extended with respect to the external loadings and generalized with respect to the continuum framework. A closed-form solution in terms of displacements is derived for the problem within the strain gradient elasticity theory on plane stress/strain assumptions. The main characters of the total and Cauchy stress fields are analyzed near the circumference of the hole for different combinations of bi-axial tensions and for different parameter values. For the original Kirsch problem concerning a uni-axially stretched plate, the analytical solution fields for stresses and strains are compared to numerical results. These results are shown to be in a full agreement with each other and, in particular, they reveal a set of new qualitative findings about the scale-dependence of the stresses and strains provided by the gradient theory, not common to the classical theory. Based on these findings, we finally consider the physicalness of the concepts total and Cauchy stress appearing in the strain gradient model. [ABSTRACT FROM AUTHOR]

This article is devoted to isogeometric analysis of higher-order strain gradient elasticity by user element implementations within a commercial finite element software Abaqus. The sixth-order boundary value problems of four parameter second strain gradient-elastic bar and plane strain/stress models are formulated in a variational form within an H 3 Sobolev space setting. These formulations can be reduced to two parameter first strain gradient-elastic problems of H 2 variational forms. The implementations of the isogeometric C 2 - and C 1 -continuous Galerkin methods, for the second and first strain gradient elasticity, respectively, are verified by a series of benchmark problems. With the first benchmark problem, a clamped bar in static tension, the convergence properties of the method in the energy norm are shown to be optimal with respect to the NURBS order of the discretizations. For the second benchmark, a clamped bar in extensional free vibrations, the analytical frequencies are captured by the numerical results within the classical and the first strain gradient elasticity. With three examples for the plane stress/strain elasticity, the convergence properties are shown to be optimal, the stress fields of different models are compared to each other, and the differences between the eigenfrequencies and eigenmodes of the models are analyzed. The last example, the Kraus problem, analyses the stress concentration factors within the different models. [ABSTRACT FROM AUTHOR]

The present work investigates the mechanical and thermomechanical bending response of beam structures possessing a triangular lattice microarchitecture. The validity of generalized continuum models, in general, and the associated dimensionally reduced models for functionally step-wise-graded microarchitectural beams, in particular, is approved by full-field finite element simulations. Most importantly, the necessity of the temperature gradient in the Helmholtz free energy is substantiated. The corresponding strong and weak forms for the associated Bernoulli–Euler and Timoshenko models of functionally graded beams are derived. The effective classical thermoelastic properties of a metamaterial with a triangular lattice microarchitecture are defined by means of computational homogenization. The additional length scale parameter involved in the generalized beam models, and associated to the particular triangular microarchitecture, is calibrated by fitting the mechanical bending responses of a series of lattice beams to the analytical solutions of the corresponding theoretical models. Strongly size-dependent mechanical and size-independent thermal bending responses are observed for both thin and thick beams with triangular lattice microarchitectures. Finally, different lattice beams with varying microarchitectures are introduced and shown to behave as generalized functionally step-wise-graded beams with respect to the higher-order elastic modulus, i.e., the length scale parameter varying in the direction of the beam axis. [ABSTRACT FROM AUTHOR]