Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.
Since its original introduction in structural design, density-based topology optimization has been applied to a number of other fields such as microelectromechanical systems, photonics, acoustics and fluid mechanics. The methodology has been well accepted in industrial design processes where it can provide competitive designs in terms of cost, materials and functionality under a wide set of constraints. However, the optimized topologies are often considered as conceptual due to loosely defined topologies and the need of postprocessing. Subsequent amendments can affect the optimized design performance and in many cases can completely destroy the optimality of the solution. Therefore, the goal of this paper is to review recent advancements in obtaining manufacturable topology-optimized designs. The focus is on methods for imposing minimum and maximum length scales, and ensuring manufacturable, well-defined designs with robust performances. The overview discusses the limitations, the advantages and the associated computational costs. The review is completed with optimized designs for minimum compliance, mechanism design and heat transfer.
One of the research direction of Horst Lippmann during his whole scientific career was devoted to the possibilities to explain complex material behavior by generalized continua models. A representative of such models is the Cosserat continuum. The basic idea of this model is the independence of translations and rotations (and by analogy, the independence of forces and moments). With the help of this model some additional effects in solid and fluid mechanics can be explained in a more satisfying manner. They are established in experiments, but not presented by the classical equations. In this paper the Cosserat-type theories of plates and shells are debated as a special application of the Cosserat theory.
Crack initiation in brittle materials is not covered by classical fracture mechanics that deals only with the growth of pre-existing cracks. In order to overcome this deficiency, the Finite Fracture Mechanics concept assumes the instantaneous formation of cracks of finite size at initiation. Within this framework, a coupled criterion was proposed at the beginning of the 2000’s requiring two necessary conditions to be fulfilled simultaneously. The first one compares the tensile stress to the tensile strength, while the other uses an energy balance and the material toughness. The present analysis is restricted to the 2D case, and, through a wide list of references, it is shown that this criterion gives predictions in agreement with experiments in various cases of stress concentration, which can be classified in two categories: the singularities, i.e. indefinitely growing stresses at a point, and the non-singular stress raisers. It is applied to different materials and structures: notched specimens, laminates, adhesive joints or embedded inclusions. Of course, a lot of work remains to do in these domains but also in domains that are almost not explored such as fatigue loadings and dynamic loadings as well as a sound 3D extension. Some ideas in these directions are issued before concluding that FFM and the coupled criterion have filled a gap in fracture mechanics.L’approche classique de la mécanique de la rupture des matériaux fragiles n’aborde pas les problèmes d’initiation de nouvelles fissures, elle ne traite que la croissance de fissures préexistantes. Afin de surmonter cette déficience, l’approche connue sous la désignation anglo-saxonne de Finite Fracture Mechanics suppose la formation instantanée de fissures de taille finie à l’initiation. Développé dans ce cadre, le critère couplé requiert la vérification simultanée de deux conditions nécessaires : la première compare la contrainte de traction à la résistance en traction du matériau tandis que l’autre utilise une équation de conservation de l’énergie et fait appel à la ténacité. La présente analyse se limite au cas 2D et, à travers une longue liste de références, il est montré que le critère couplé donne, aux points de concentration de contraintes, des prédictions d’amorçage de fissures qui sont en accord avec les expériences. On peut distinguer deux catégories : les singularités, lorsque les contraintes croissent indéfiniment en s’approchant d’un point, et les simples concentrations de contraintes, lorsque celles-ci tout en étant élevées restent bornées. Le critère est appliqué à différents matériaux et structures: éprouvettes entaillées, composites stratifiés, joints adhésifs, inclusions. Bien sûr, beaucoup de travail reste à faire dans ces domaines, mais il existe aussi des sujets qui ne sont pratiquement pas explorés comme la fatigue, les chargements dynamiques ainsi qu’une généralisation aux situations tridimensionnelles. Quelques idées sont émises dans ces directions avant de conclure que la FFM et le critère couplé ont comblé une lacune en mécanique de la rupture.Die Initiierung von Rissen wird in der klassischen Bruchmechanik nicht umfasst, da sich diese auf die Beschreibung des Verhaltens vorhandener Risse beschränkt. In der Bruchmechanik finiter Risse wird diese Einschränkung durch die Betrachtung der instantanen Entstehung von Rissen endlicher Länge aufgehoben. Im Rahmen dieses Konzeptes wurde ein gekoppeltes Kriterium vorgeschlagen, das eine hinreichende Versagensbedingung in Form zweier gleichzeitig zu erfüllender notwendiger Bedingungen darstellt: eine Bedingung der Festigkeitsmechanik und eine Energiebedingung für den Bruchprozess. Die gegenwärtige Formulierung ist auf zweidimensionale Modelle beschränkt und ist, wie anhand zahlreicher Referenzen dargestellt, geeignet experimentelle Ergebnisse über Versagen an unterschiedlichsten Spannungskonzentratoren korrekt abzubilden. Diese lassen sich klassifizieren in singuläre Spannungskonzentratoren mit lokal unendlich hohen Spannungen und in nicht-singuläre Spannungskonzentratoren mit lokal stark erhöhten aber endlichen Spannungen. Das Kriterium wurde auf verschiedenste Struktursituationen und Materialen angewendet: gekerbte Bauteile, Laminate, Klebverbindungen oder auch Materialeinschlüsse. Jedoch verbleiben weitere unerschlossene Felder für weitere Untersuchungen wie etwa die Betrachtung von zyklischen und dynamischen Lasten oder eine gründliche Erweiterung auf dreidimensionale Risse. Mögliche Ansätze für solche Erweiterungen werden vorgestellt bevor der Schluss gezogen wird, dass die Bruchmechanik finiter Risse mit dem gekoppelten Kriterium eine Lücke in der Bruchmechanik geschlossen hat.
In this paper, the self-adjointness of Eringen’s nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen’s model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever’s fundamental vibration frequency with respect to increasing Eringen’s small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen’s nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.
In continuum mechanics, there exists a unique theory for elasticity, which includes the first gradient of displacement. The corresponding generalization of elasticity is referred to as strain gradient elasticity or higher gradient theories, where the second and higher gradients of displacement are involved. Unfortunately, there is a lack of consensus among scientists how to achieve the generalization. Various suggestions were made, in order to compare or even verify these, we need a generic computational tool. In this paper, we follow an unusual but quite convenient way of formulation based on action principles. First, in order to present its benefits, we start with the action principle leading to the well-known form of elasticity theory and present a variational formulation in order to obtain a weak form. Second, we generalize elasticity and point out, in which term the suggested formalism differs. By using the same approach, we obtain a weak form for strain gradient elasticity. The weak forms for elasticity and for strain gradient elasticity are solved numerically by using open-source packages-by using the finite element method in space and finite difference method in time. We present some applications from elasticity as well as strain gradient elasticity and simulate the so-called size effect.
In this paper, the self-adjointness of Eringen's nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen's model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever's fundamental vibration frequency with respect to increasing Eringen's small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen's nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.
In this paper, an overview on a class of materials with high actual research interest will be given. Magnetic hybrid materials, i.e. liquid or elastic matrices filled with magnetic nano- or micro-particles provide the possibility to influence their mechanical behaviour by application of technically easily realizable magnetic fields. In particular, the viscous and elastic behaviour of magnetic hybrid materials can be influenced by magnetic fields. The physical reason for these changes are field-induced reconfigurations of the microstructure formed by the magnetic particles. Experimental techniques to observe these changes and their relation to changes in the mechanical behaviour of magnetic hybrid materials will be in focus of this review.
Rubber-like materials consist of chain-like macromolecules that are more or less closely connected to each other via entanglements or cross-links. As an idealisation, this particular structure can be described as a completely random three-dimensional network. To capture the elastic and nearly incompressible mechanical behaviour of this material class, numerous phenomenological and micro-mechanically motivated models have been proposed in the literature. This contribution reviews fourteen selected representatives of these models, derives analytical stress-stretch relations for certain homogeneous deformation modes and summarises the details required for stress tensors and consistent tangent operators. The latter, although prevalently missing in the literature, are indispensable ingredients in utilising any kind of constitutive model for the numerical solution of boundary value problems by iterative approaches like the Newton-Raphson scheme. Furthermore, performance and validity of the models with regard to the classical experimental data on vulcanised rubber published by Treloar (Trans Faraday Soc 40:59-70, 1944) are evaluated. These data are here considered as a prototype or worst-case scenario of highly nonlinear elastic behaviour, although inelastic characteristics are clearly observable but have been tacitly ignored by many other authors.
In this investigation, the free in-plane vibration analysis of orthotropic rectangular plates with non-uniform boundary conditions and internal line supports is performed with a modified Fourier solution. The exact solution for the problem is obtained using improved Fourier series method, in which both two in-plane displacements of the orthotropic rectangular plates are represented by a double Fourier cosine series and four supplementary functions, in the form of the product of a polynomial function and a single cosine series expansion, introduced to remove the potential discontinuities associated with the original displacement functions along the edges when they are viewed as periodic functions defined over the entire x–y plane. The unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the two sets of the boundary springs at the all boundary of the orthotropic rectangular plates without the need of making any change to the solutions. The excellent accuracy of the current result is validated by comparison with those obtained from other analytical approach as well as the finite element method (FEM). Numerical results are presented to illustrate the current method that is applied not only to the homogeneous boundary conditions but also to other interesting and practically important boundary restraints on free in-plane vibrations of the orthotropic rectangular plates, including varying stiffness of boundary springs, point supported, partially supported boundary conditions and internal line supports. In addition to this, the effects of locations of line supports are also investigated and reported. New results for free vibration of moderately orthotropic rectangular plates with various edge conditions and internal line supports are presented, which may be used for benchmarking of researchers in the field.
In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.
In view of the significant influences of multi-source uncertainties on structural safety, which generally exist in practical engineering (such as the dispersion of material, the uncertainty of external load and the error of processing technology), more academic research and engineering applications had paid attention to uncertainty in recent years. However, due to the complexity of the structural problems, there may be multiple uncertain parameters. Traditional methods of optimal design by single-source uncertainty model, particularly the one derived from probability theory, may no longer be feasible. This paper investigates a new formulation and numerical solution of reliability-based design optimization (RBDO) of structures exhibiting random and uncertain-but-bounded (interval and convex) mixed uncertainties. Combined with the non-probabilistic set-theory convex model and the classical probabilistic approach, the mathematical definition of hybrid reliability is firstly presented for a quantified measure of the safety margin. The reliability-based optimization incorporating such mixed reliability constraints is then formulated. The PSO algorithm is employed to improve the convergence and the stability in seeking the optimal global solution. Additionally, by introducing the general concept of the safety factor, the compatibility between the proposed hybrid RBDO technique and the safety factor-based model is further discussed. By virtue of the above two methods, two numerical examples of typical components (the cantilever structure and the truss structure) as well as one complex engineering example (the hypersonic wing structure) are performed, subjected to the strength or stiffness criteria. The accuracy and effectiveness of the present method are then demonstrated.
This paper introduces a new coordinate formulation for the kinematic and dynamic analysis of planar multibody systems composed of rigid bodies. The methodology presented in this work is called planar reference point coordinate formulation (RPCF) with Euler parameters. In the planar RPCF with Euler parameters, the rotational coordinates used for describing the body orientation are the redundant components of a two-dimensional unit quaternion that identify a planar set of Euler parameters. It is shown in the paper that the planar RPCF with Euler parameters allows for obtaining consistent kinematic and dynamic descriptions of two-dimensional rigid bodies. In the numerical solution of the equations of motion, the well-known generalized coordinate partitioning method can be effectively utilized to stabilize the violation of the algebraic constraints at the position and velocity levels leading to physically correct and numerically stable dynamic simulations. Furthermore, a standard numerical integration procedure can be employed for calculating an approximate solution of the equations of motion resulting from the planar RPCF with Euler parameters. In the paper, the computer implementation of the proposed formulation approach is demonstrated considering four rigid multibody systems which serve as simple benchmark problems.
This paper deals with the Lagrange multipliers corresponding to the intrinsic constraint equations of rigid multibody mechanical systems. The intrinsic constraint equations are algebraic equations that are associated with nonminimal sets of orientation parameters employed for the kinematic representation of large finite rotations. Two coordinate formulations are analyzed in this investigation, namely the reference point coordinate formulation (RPCF) with Euler parameters and the natural absolute coordinate formulation (NACF). While the RPCF with Euler parameters employs the four components of a unit quaternion as rotational coordinates, the NACF directly uses the orthonormal set of nine direction cosines for describing the orientation of a rigid body in the three-dimensional space. In the multibody approaches based on the RPCF with Euler parameters and on the NACF, the use of a nonminimal set of rotational coordinates facilitates a general and systematic formulation of the differential–algebraic equations of motion. Considering the basic equations of classical mechanics, the fundamental problem of constrained motion is formalized and solved in this paper by using a special form of the Udwadia–Kalaba method. By doing so, the Udwadia–Kalaba equations are employed for obtaining closed-form analytical solutions for the Lagrange multipliers associated with the intrinsic constraint equations that appear in the differential–algebraic dynamic equations developed by using the RPCF with Euler parameters and the NACF multibody approaches. Two simple numerical examples support the analytical results found in this paper.
A novel meshfree discretization technique in terms of the reproducing kernel particle method is presented for accurately evaluating mixed-mode intensity factors of cracked shear-deformable plates. Mindlin–Reissner plate theory is adopted to solve the cracked plates problem in the Galerkin formulation, considering transverse shear deformation. The diffraction method, visibility criterion and enriched basis are included in the generation of meshfree interpolants for the modeling of fracture. In this work, numerical integration is treated using the stabilized conforming nodal integration (SCNI) and subdomain stabilized conforming integration (SSCI). The J-integral (contour integral) is employed to analyze the fracture mechanics parameters. SCNI/SSCI is thus adopted to evaluate the contour integral and to split the original J-integral into symmetric and asymmetric J-integral values. They are calculated by decomposing the smoothed displacement, moment and shear force quantities into symmetric/asymmetric parts. In addition, a displacement ratio method is introduced to divide the asymmetric J-integral value into corresponding moment and shear force intensity factors. The accuracy of the intensity factors and the path-independent properties in mixed-mode fracture problems are critically examined through several numerical examples.
A micro-scale free vibration analysis of composite laminated Timoshenko beam (CLTB) model is developed based on the new modified couple stress theory. In this theory, a new anisotropic constitutive relation is defined for modeling the CLTB. This theory uses rotation–displacement as dependent variable and contains only one material length scale parameter. Hamilton’s principle is employed to derive the governing equations of motion and boundary conditions. This new model can be reduced to composite laminated Bernoulli–Euler beam model of the couple stress theory. An example analysis of free vibration of the cross-ply simply supported CLTB model is adopted, and an explicit expression of analysis solution is given. Additionally, the numerical results show that the present beam models can capture the scale effects of the natural frequencies of the micro-structure.
Small-scale effects in nanobeams are effectively described by the Eringen model of nonlocal elasticity. The nonlocal elastostatic problem of Bernoulli–Euler nanobeams is here formulated in variational terms by recognizing that the nonlocality effect is equivalent to a bending curvature distortion prescribed on a corresponding local nanobeam, subjected to the same kinematic boundary constraints and applied loads. The conditions to be imposed for the kinematic integrability of the bending curvature field are also provided to evaluate the bending moment solution in statically indeterminate nonlocal nanobeams. Since the curvature distortion describing the nonlocality effect is kinematically integrable in statically determinate structures, bending moments do not exhibit small-scale effects in non-redundant nanobeams. The equivalence method illustrated in the present paper is resorted to for solving the nonlocal elastostatic problem of nanobeams under constant transversal load distributions.
In this paper, formulation of the thin cylindrical shell via the modified couple stress theory by taking account of shear deformation and rotary inertia is obtained. To do this, the study developed the first shear deformable cylindrical shell theory by considering the size effects via the couple stress theory and the equations of motion of shell with classical and non-classical boundary conditions were extracted through Hamilton's principle. In the end, as an example, free vibrations of the single-walled carbon nanotube (SWCNT) were investigated. Here, the SWCNT was modeled as a simply supported shell, and the Navier procedure was used to solve the vibration problem. The results of the new model were compared with those of the classical theory, pointing to the conclusion that the classical model is a special case of the modified couple stress theory. The findings also demonstrate that the rigidity of the nano-shell in the modified couple stress theory compared with that in the classical theory is greater, resulting in the increase in natural frequencies. In addition, the effect of the material length scale parameter on the vibrations of the nano-shell in different lengths and thickness was investigated.
Mechanical properties are investigated for a class of microstructured materials with promising applications. Specifically, we consider a composite material with orthogonal, mutually interconnected fibers building a pantographic substructure. In order to predict the behavior of such a system in three-dimensional continuum, a reduced-order model is introduced by means of a bi-dimensional elastic surface accurately describing large deformations. The properties of this reduced-order model are characterized by an elastic energy density that involves second space derivatives of the displacement for capturing the resistance of twisted and bent fibers in plane as well as out of plane. For determining the coefficients in the elastic energy of the reduced-order model, we utilize a numerical inverse analysis and make use of ad hoc computational experiments performed by a direct numerical simulation on the microscale with detailed modeling of the pantographic substructure. This reduced-order model represents a homogenized material on macro-scale with its substructure on microscale. The homogenized model is capable of describing materials response at a significantly less computational cost than the direct numerical simulations.
The objective of this paper is to investigate the mechanical properties of samples of MS1 Maraging Steel (untreated and heat treated), which were produced by additive technology in various orientations in the working area of the building machine. MS1 steel (European 1.2709 and German X3NiCoMoTi 18-9-5) is well known for its high strength, high fracture toughness, good weldability, and dimensional stability during aging. The literature review, related to the mechanical properties and fracture of MS1 steel, found that there are no available studies of the effects of both building direction and heat treatment on the mechanical properties of MS1 steel. The authors decided to address this omission and present this entirely new research in this article. The uniaxial tensile tests to fracture were completed at two of the authors' workplaces. The results were statistically assessed using Grubbs' test for outliers, and then the data were processed using box plots to be easily comparable from the point of view of print direction, heat treatment, and the values declared by the metal powder producer or in the tables (for conventionally produced steel). Scanning electron microscopy was used to analyze the fracture surfaces obtained after tensile testing cylindrical samples. The results showed that there was an impact on the mechanical properties depending on the sample orientation within the same heat treatment type; there was also significant influence of heat treatment, while the possibility of the natural aging effect on mechanical properties was also noted.