Nicola Luigi Rizzi - Academia.edu (original) (raw)

Papers by Nicola Luigi Rizzi

Materials Science Forum, 2009

In a key paper [1] W.T. Koiter proposed a nonlinear theory of shells that, although disregarded f... more In a key paper [1] W.T. Koiter proposed a nonlinear theory of shells that, although disregarded for long time, has been reconsidered in recents times. In fact, due also to the works of Ciarlet [2], as recorded by D. J. Steigman [3], it has been recognized to furnish the best all-around theory, despite the fact that it has not been obtained either as a gamma limit or an asymptotic limit of the three-dimensional theory. The aim of this paper is to exploit that theory in order to obtain a 1D model apt to describe the nonlinear behaviour of slender plates standing alone or assembled to form a folded one. In is shown that, due the assumption inherent in the Koiter theory and to possibile other fair ones, the procedure leads to a higher gradient 1D model. In the paper the features of that model are discussed giving particular attention to the treatment of the boundary condition that, as it is well known, are one the main point to deal with in higher gradient theory.

We investigate the buckling of a ‘Roorda’ frame by means of a direct one-dimensional beam model. ... more We investigate the buckling of a ‘Roorda’ frame by means of a direct one-dimensional beam model. The frame is acted upon by a ‘dead’ load at the joint and is constrained there by an out-of-plane linear elastic spring. The possibility of warping constraints at the beam ends is also considered; the spring simulates the presence of braces in actual 3D frames. The numerical results are compared with those already obtained by the authors for an infinitely stiff spring.

In this paper a 2D continuum model, thought as the homogenized limit of a microstructured pantogr... more In this paper a 2D continuum model, thought as the homogenized limit of a microstructured pantographic sheet, is studied. The microstructure is characterized by two families of parallel fibers, whose deformation measures account for bending, elongation and relative rotation of the fibers. The deformation energy density of the homogenized model depends on both first and second gradients of the displacement. Modal analysis is performed in order to assess the peculiarities of the dynamic behavior of higher gradient models, and in particular the difference, with respect to classical laminae, in the dependence of the eigenfrequencies on the stiffness.

Composites Part B: Engineering, 2017

We present a homogenized model for the analysis of a 2D continuum with two straight families of i... more We present a homogenized model for the analysis of a 2D continuum with two straight families of inextensible fibres embedded in it. The kinematics of the continuum is analyzed and, motivated by phenomenological observations, it is assumed that the strain energy depends on the shear deformation of the fibres and on their bending curvature. It is shown that in order to account for the latter deformation it is necessary to introduce second gradient strains. The problem is formulated as a nonlinear constrained minimization, after introducing a suitable discretization of the domain. Some deformation processes are simulated using different constitutive hypotheses, comparing the predictions obtained assuming the presence of only first gradient or second gradient deformations, or a combination of both. It is found that the first gradient model leads to the presence of discontinuities in the rotation of the fibres, while the second gradient model regularizes these discontinuities by means of boundary layers. In particular in some deformation processes an instability of geometrical nature is observed when the second gradient model is used, that can be suppressed by the first gradient contribution.

Composites Part B: Engineering, 2017

Several nanomechanics formulations based on common two and three-body bonding potentials are comp... more Several nanomechanics formulations based on common two and three-body bonding potentials are compared. Numerical simulations and analytical approaches are used to investigate the not negligible differences among the predictions of the in-plane elastic constants of graphene sheets in the literature, separately exploring the role of the potentials and that of the structural descriptions (beams and trusses) of the original molecular mechanics (MM) model. The energetic differences between the structural models and the MM model are highlighted through exact discrete homogenization procedures. In so doing, some theoretical expressions of the graphene elastic constants available in the literature are recovered and supported by the numerical experimentation. The results provide also an assessment of the accuracy of some potentials largely employed in the literature with respect to several ab-initio reference solutions and the experimental measurements available. Some suggestions towards a reparametrization of the modified Morse potential are consequently formulated.

Zeitschrift für angewandte Mathematik und Physik, 2016

In this paper, we determine numerically a large class of equilibrium configurations of an elastic... more In this paper, we determine numerically a large class of equilibrium configurations of an elastic two-dimensional continuous pantographic sheet in three-dimensional deformation consisting of two families of fibers which are parabolic prior to deformation. The fibers are assumed: i) to be continuously distributed over the sample, ii) to be endowed of bending and torsional sti↵nesses and iii) tied together at their points of intersection to avoid relative slipping by means of internal (elastic) pivots. This last condition characterizes the system as a pantographic lattice [1, 2, 34, 35]. The model that we employ here, developed by Steigmann and dell'Isola [108] and first investigated in [55], is applicable to fiber lattices in which three dimensional bending, twisting and stretching are significant as well as a resistance to shear distortion, i.e. to the angle change between the fibers. Some relevant numerical examples are exhibited in order to highlight the main features of the model adopted: in particular buckling and post-buckling behavior of pantographic parabolic lattices is investigated. The fabric of the metamaterial presented in this paper has been conceived to resist more e↵ectively in the extensional bias tests by storing more elastic bending energy and less energy in the deformation of elastic pivots: a comparison with a fabric constituted by beams which are straight in the reference configuration shows that the proposed concept is promising.

Zeitschrift für angewandte Mathematik und Physik, 2016

Hencky (¨U ber die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Ge... more Hencky (¨U ber die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Ph.D. thesis, Engelmann, 1921) proposed a discrete model for elasticae by introducing rigid bars and rotational springs. Hencky (Proc R Soc Lond A Math Phys Eng Sci 472(2185), 2016) approach has been introduced to heuristically motivate the need of second gradient continua. Here, we present a novel numerical code implementing directly the discrete Henckytype model which is robust enough to solve the problem of the determination of equilibrium configurations in the large deformation and displacement regimes. We apply this model to study some potentially applicable problems, and we compare its performances with those of the second gradient continuum model. The numerical evidence presented supports the conjecture that Hencky-type converges to second gradient model.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2016

Recently it has been proposed the study of a new class of bidimensional metamaterials, [33, 34] w... more Recently it has been proposed the study of a new class of bidimensional metamaterials, [33, 34] which have been called extensible pantographic sheets. Such bidimensional continua are the generalization of the continua introduced in [68, 69, 80] as they take simultaneously into account the elastic energy in the extensional deformation and geodesic bending of constituting fibers. In the present paper we consider in the deformation energy a term accounting for the effect of shear deformation. The phenomena which we highlight can be of relevance to model the mechanical behavior of composite fiber reinforcements and of some lattices of beams constrained by internal pivots. We compare the effect of different linear or cubic shear stiffnesses i) on the deformation of pairs of pantographic sheets suitably interconnected and ii) on the deformation of pantographic sheets deformed under the action of forces concentrated on points.

Thin-Walled Structures, 2013

An L frame made up by beam and column having channel cross sections, has been analyzed in a previ... more An L frame made up by beam and column having channel cross sections, has been analyzed in a previous work by two of the authors [14]. Depending on the aspect ratio and the joint configuration, it has been proved that the structure can exhibit two simultaneous buckling modes. Here using the asymptotic theory of elastic bifurcation that takes into account mode interaction, the initial slope of the bifurcated paths has been determined. Three cases of joint configurations, which are the more common used in welded connections, have been considered. For each case three admissible bifurcated paths have been found. Two of them show a slope having the same order of magnitude of the ones found in the absence of mode interaction while the remaining exhibits a slope largely steepest. Selecting, for each joint case, the bifurcated path with the higher slope and between them the smallest one, it is found that it is associated to the path which bifurcates at the higher critical load. This means that the stiffer structure is also the less imperfection sensitive. Finally for each one of the cases studied, the effect of initial imperfection has been considered and the real load carrying capacity of the frames has been determined. Finally some results have been compared with those obtained using the FE code ABAQUS.

Journal of Symbolic Computation, 1985

Computers & Structures, 1985

A perturbation procedure for the buckling and postbuckling analysis of elastic structures is show... more A perturbation procedure for the buckling and postbuckling analysis of elastic structures is shown to be well suited to be implemented as an automatic symbolic manipulation procedure. The postbuckling analysis of a circular arch is considered as an example, and the asymptotic description of the bifurcated equilibrium path is given. The main purposes of the automatic procedure are to generate the representation of the Frechet operator for the strain field and to perform integration by parts. This allows the manipulation of correct expressions of the basic relationships, as the strain-displacement one, without introducing any simplifying assumption or restriction. The perturbation equations are automatically generated and a solution procedure leads to parametric expressions for the coefficients of the asymptotic expansion of the bifurcated path. The symbolic manipulation system used is REDUCE.

International Journal of Solids and Structures, 2021

In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkab... more In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkable potentialities. Indeed it allowed for the synthesis of second gradient 2D (nonlinear) continua: i.e. 2D shells whose deformation energy depends also on the second derivatives of displacements in the tangent directions to the reference configuration. Moreover, pantographic architecture seems to be able to produce metamaterials whose macroscopic elongations are large, albeit remaining in the elastic regime. The theoretically shown potentialities have started to become of «practical» interest thanks to a series of experiments, which were made possible by the recent 3D additive manufacturing. The actual construction of pantographic architecture has been based on the design of two arrays of beams interconnected by small cylinders, whose behavior can be modeled in different ways: if they are very short they can be regarded as clamps, while if they are short enough as elastic (or inelastic for large rotations) cylindrical hinges connecting the beams of different arrays. Otherwise, they must be modeled as elastic (or inelastic) elements allowing for relative rotations and displacements. In this paper, we focus on this particular case and we introduce, after a homogenization based on heuristic arguments, a 2D generalized continuum model whose kinematics is characterized by two placement and rotation fields (one for each array of beams) and whose deformation energy depends on relative displacements and rotations. The offset between the two beams arrays is proven to be an essential tool for defining effective invariant kinematical deformation measures. In facts, one wants to postulate a deformation energy for the introduced 2D generalized continuum which gives predictions in agreement with those given by the more refined 3D model where the pantographic architecture is described with its maximum geometric complexity and where the constituting material is assumed to be modelable as a standard 3D first gradient continuum. In the present paper, in order to arrive at the correct conjecture for the postulated energy, we consider the concept of averages of rotations in SO(3) Lie group. The used enriched kinematics is seen to be a possible alternative to the adoption of second gradient 2D models. Some rather surprising deformation processes are studied, where interesting non-symmetric post-buckling phenomena are observed in both the models used. Mentioned post-buckling has been observed experimentally.

International Journal of Solids and Structures, 2016

In this paper a procedure for constructing a 1D continuum equivalent to a Koiter plate, is presen... more In this paper a procedure for constructing a 1D continuum equivalent to a Koiter plate, is presented. The result is used to build-up a coarse model of a Thin Walled Beam. The model results to be a nonlinear hyperelastic 1D second gradient continuum, endowed with an internal microstructure described by seven scalar kinematic parameters. The kinematic parameters are able to describe, although in a rough way, the deformation of the TWB in the plane of its cross-section. The nonlinear behaviour of a simply supported strut has been analyzed and the results compared with those given by the commercial finite element code ABAQUS. © 2016 Elsevier Ltd

Thin-Walled Structures, Jul 1, 2013

An L frame made up by beam and column having channel cross sections, has been analyzed in a previ... more An L frame made up by beam and column having channel cross sections, has been analyzed in a previous work by two of the authors [14]. Depending on the aspect ratio and the joint configuration, it has been proved that the structure can exhibit two simultaneous buckling modes. Here using the asymptotic theory of elastic bifurcation that takes into account mode interaction, the initial slope of the bifurcated paths has been determined. Three cases of joint configurations, which are the more common used in welded connections, have been considered. For each case three admissible bifurcated paths have been found. Two of them show a slope having the same order of magnitude of the ones found in the absence of mode interaction while the remaining exhibits a slope largely steepest. Selecting, for each joint case, the bifurcated path with the higher slope and between them the smallest one, it is found that it is associated to the path which bifurcates at the higher critical load. This means th...

Thin-Walled Structures, Jul 1, 2013

An L frame made up by beam and column having channel cross sections, has been analyzed in a previ... more An L frame made up by beam and column having channel cross sections, has been analyzed in a previous work by two of the authors [14]. Depending on the aspect ratio and the joint configuration, it has been proved that the structure can exhibit two simultaneous buckling modes. Here using the asymptotic theory of elastic bifurcation that takes into account mode interaction, the initial slope of the bifurcated paths has been determined. Three cases of joint configurations, which are the more common used in welded connections, have been considered. For each case three admissible bifurcated paths have been found. Two of them show a slope having the same order of magnitude of the ones found in the absence of mode interaction while the remaining exhibits a slope largely steepest. Selecting, for each joint case, the bifurcated path with the higher slope and between them the smallest one, it is found that it is associated to the path which bifurcates at the higher critical load. This means th...

Mathematics and Mechanics of Complex Systems

Recently growing attention has been paid to the particular class of metamaterials which has been ... more Recently growing attention has been paid to the particular class of metamaterials which has been called pantographic. Pantographic metamaterials have the following peculiar features: (i) their continuum model, at the macroscale, has to include a term of the deformation energy depending on the second gradient of placement, (ii) they can show an elastic behavior in large deformation regimes, and (iii) they are resilient and tough during rupture phenomena (dell'Isola et al. 2015). In order to predict pantographic metamaterials' mechanical behavior, it is possible to introduce a three-dimensional continuum micromodel, in which their internal geometrical microstructure is described in detail. However, the computational costs of this choice are presently prohibitive. In this paper, we introduce a reduced order model for pantographic sheets-which are an example of an elastic surface-whose kinematics include, for each of the two constituting families of fibers fully independent three-dimensional (i) placement and (ii) rotation fields. In this way it is possible to include, also in the reduced order model, (i) the initial and the actual offset between the fibers and (ii) the deformation energy of the interconnecting pivots. By postulating a simplified expression for the deformation energy we prove that also a reduced order model can describe some experimental observed buckling and postbuckling phenomena. The promising results which we present here motivate the quest of more general expressions for deformation energy capable of capturing the fully nonlinear behavior exhibited by pantographic sheets.

Materials Science Forum, 2009

In a key paper [1] W.T. Koiter proposed a nonlinear theory of shells that, although disregarded f... more In a key paper [1] W.T. Koiter proposed a nonlinear theory of shells that, although disregarded for long time, has been reconsidered in recents times. In fact, due also to the works of Ciarlet [2], as recorded by D. J. Steigman [3], it has been recognized to furnish the best all-around theory, despite the fact that it has not been obtained either as a gamma limit or an asymptotic limit of the three-dimensional theory. The aim of this paper is to exploit that theory in order to obtain a 1D model apt to describe the nonlinear behaviour of slender plates standing alone or assembled to form a folded one. In is shown that, due the assumption inherent in the Koiter theory and to possibile other fair ones, the procedure leads to a higher gradient 1D model. In the paper the features of that model are discussed giving particular attention to the treatment of the boundary condition that, as it is well known, are one the main point to deal with in higher gradient theory.

We investigate the buckling of a ‘Roorda’ frame by means of a direct one-dimensional beam model. ... more We investigate the buckling of a ‘Roorda’ frame by means of a direct one-dimensional beam model. The frame is acted upon by a ‘dead’ load at the joint and is constrained there by an out-of-plane linear elastic spring. The possibility of warping constraints at the beam ends is also considered; the spring simulates the presence of braces in actual 3D frames. The numerical results are compared with those already obtained by the authors for an infinitely stiff spring.

In this paper a 2D continuum model, thought as the homogenized limit of a microstructured pantogr... more In this paper a 2D continuum model, thought as the homogenized limit of a microstructured pantographic sheet, is studied. The microstructure is characterized by two families of parallel fibers, whose deformation measures account for bending, elongation and relative rotation of the fibers. The deformation energy density of the homogenized model depends on both first and second gradients of the displacement. Modal analysis is performed in order to assess the peculiarities of the dynamic behavior of higher gradient models, and in particular the difference, with respect to classical laminae, in the dependence of the eigenfrequencies on the stiffness.

Composites Part B: Engineering, 2017

We present a homogenized model for the analysis of a 2D continuum with two straight families of i... more We present a homogenized model for the analysis of a 2D continuum with two straight families of inextensible fibres embedded in it. The kinematics of the continuum is analyzed and, motivated by phenomenological observations, it is assumed that the strain energy depends on the shear deformation of the fibres and on their bending curvature. It is shown that in order to account for the latter deformation it is necessary to introduce second gradient strains. The problem is formulated as a nonlinear constrained minimization, after introducing a suitable discretization of the domain. Some deformation processes are simulated using different constitutive hypotheses, comparing the predictions obtained assuming the presence of only first gradient or second gradient deformations, or a combination of both. It is found that the first gradient model leads to the presence of discontinuities in the rotation of the fibres, while the second gradient model regularizes these discontinuities by means of boundary layers. In particular in some deformation processes an instability of geometrical nature is observed when the second gradient model is used, that can be suppressed by the first gradient contribution.

Composites Part B: Engineering, 2017

Several nanomechanics formulations based on common two and three-body bonding potentials are comp... more Several nanomechanics formulations based on common two and three-body bonding potentials are compared. Numerical simulations and analytical approaches are used to investigate the not negligible differences among the predictions of the in-plane elastic constants of graphene sheets in the literature, separately exploring the role of the potentials and that of the structural descriptions (beams and trusses) of the original molecular mechanics (MM) model. The energetic differences between the structural models and the MM model are highlighted through exact discrete homogenization procedures. In so doing, some theoretical expressions of the graphene elastic constants available in the literature are recovered and supported by the numerical experimentation. The results provide also an assessment of the accuracy of some potentials largely employed in the literature with respect to several ab-initio reference solutions and the experimental measurements available. Some suggestions towards a reparametrization of the modified Morse potential are consequently formulated.

Zeitschrift für angewandte Mathematik und Physik, 2016

In this paper, we determine numerically a large class of equilibrium configurations of an elastic... more In this paper, we determine numerically a large class of equilibrium configurations of an elastic two-dimensional continuous pantographic sheet in three-dimensional deformation consisting of two families of fibers which are parabolic prior to deformation. The fibers are assumed: i) to be continuously distributed over the sample, ii) to be endowed of bending and torsional sti↵nesses and iii) tied together at their points of intersection to avoid relative slipping by means of internal (elastic) pivots. This last condition characterizes the system as a pantographic lattice [1, 2, 34, 35]. The model that we employ here, developed by Steigmann and dell'Isola [108] and first investigated in [55], is applicable to fiber lattices in which three dimensional bending, twisting and stretching are significant as well as a resistance to shear distortion, i.e. to the angle change between the fibers. Some relevant numerical examples are exhibited in order to highlight the main features of the model adopted: in particular buckling and post-buckling behavior of pantographic parabolic lattices is investigated. The fabric of the metamaterial presented in this paper has been conceived to resist more e↵ectively in the extensional bias tests by storing more elastic bending energy and less energy in the deformation of elastic pivots: a comparison with a fabric constituted by beams which are straight in the reference configuration shows that the proposed concept is promising.

Zeitschrift für angewandte Mathematik und Physik, 2016

Hencky (¨U ber die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Ge... more Hencky (¨U ber die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Ph.D. thesis, Engelmann, 1921) proposed a discrete model for elasticae by introducing rigid bars and rotational springs. Hencky (Proc R Soc Lond A Math Phys Eng Sci 472(2185), 2016) approach has been introduced to heuristically motivate the need of second gradient continua. Here, we present a novel numerical code implementing directly the discrete Henckytype model which is robust enough to solve the problem of the determination of equilibrium configurations in the large deformation and displacement regimes. We apply this model to study some potentially applicable problems, and we compare its performances with those of the second gradient continuum model. The numerical evidence presented supports the conjecture that Hencky-type converges to second gradient model.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2016

Recently it has been proposed the study of a new class of bidimensional metamaterials, [33, 34] w... more Recently it has been proposed the study of a new class of bidimensional metamaterials, [33, 34] which have been called extensible pantographic sheets. Such bidimensional continua are the generalization of the continua introduced in [68, 69, 80] as they take simultaneously into account the elastic energy in the extensional deformation and geodesic bending of constituting fibers. In the present paper we consider in the deformation energy a term accounting for the effect of shear deformation. The phenomena which we highlight can be of relevance to model the mechanical behavior of composite fiber reinforcements and of some lattices of beams constrained by internal pivots. We compare the effect of different linear or cubic shear stiffnesses i) on the deformation of pairs of pantographic sheets suitably interconnected and ii) on the deformation of pantographic sheets deformed under the action of forces concentrated on points.

Thin-Walled Structures, 2013

An L frame made up by beam and column having channel cross sections, has been analyzed in a previ... more An L frame made up by beam and column having channel cross sections, has been analyzed in a previous work by two of the authors [14]. Depending on the aspect ratio and the joint configuration, it has been proved that the structure can exhibit two simultaneous buckling modes. Here using the asymptotic theory of elastic bifurcation that takes into account mode interaction, the initial slope of the bifurcated paths has been determined. Three cases of joint configurations, which are the more common used in welded connections, have been considered. For each case three admissible bifurcated paths have been found. Two of them show a slope having the same order of magnitude of the ones found in the absence of mode interaction while the remaining exhibits a slope largely steepest. Selecting, for each joint case, the bifurcated path with the higher slope and between them the smallest one, it is found that it is associated to the path which bifurcates at the higher critical load. This means that the stiffer structure is also the less imperfection sensitive. Finally for each one of the cases studied, the effect of initial imperfection has been considered and the real load carrying capacity of the frames has been determined. Finally some results have been compared with those obtained using the FE code ABAQUS.

Journal of Symbolic Computation, 1985

Computers & Structures, 1985

A perturbation procedure for the buckling and postbuckling analysis of elastic structures is show... more A perturbation procedure for the buckling and postbuckling analysis of elastic structures is shown to be well suited to be implemented as an automatic symbolic manipulation procedure. The postbuckling analysis of a circular arch is considered as an example, and the asymptotic description of the bifurcated equilibrium path is given. The main purposes of the automatic procedure are to generate the representation of the Frechet operator for the strain field and to perform integration by parts. This allows the manipulation of correct expressions of the basic relationships, as the strain-displacement one, without introducing any simplifying assumption or restriction. The perturbation equations are automatically generated and a solution procedure leads to parametric expressions for the coefficients of the asymptotic expansion of the bifurcated path. The symbolic manipulation system used is REDUCE.

International Journal of Solids and Structures, 2021

In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkab... more In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkable potentialities. Indeed it allowed for the synthesis of second gradient 2D (nonlinear) continua: i.e. 2D shells whose deformation energy depends also on the second derivatives of displacements in the tangent directions to the reference configuration. Moreover, pantographic architecture seems to be able to produce metamaterials whose macroscopic elongations are large, albeit remaining in the elastic regime. The theoretically shown potentialities have started to become of «practical» interest thanks to a series of experiments, which were made possible by the recent 3D additive manufacturing. The actual construction of pantographic architecture has been based on the design of two arrays of beams interconnected by small cylinders, whose behavior can be modeled in different ways: if they are very short they can be regarded as clamps, while if they are short enough as elastic (or inelastic for large rotations) cylindrical hinges connecting the beams of different arrays. Otherwise, they must be modeled as elastic (or inelastic) elements allowing for relative rotations and displacements. In this paper, we focus on this particular case and we introduce, after a homogenization based on heuristic arguments, a 2D generalized continuum model whose kinematics is characterized by two placement and rotation fields (one for each array of beams) and whose deformation energy depends on relative displacements and rotations. The offset between the two beams arrays is proven to be an essential tool for defining effective invariant kinematical deformation measures. In facts, one wants to postulate a deformation energy for the introduced 2D generalized continuum which gives predictions in agreement with those given by the more refined 3D model where the pantographic architecture is described with its maximum geometric complexity and where the constituting material is assumed to be modelable as a standard 3D first gradient continuum. In the present paper, in order to arrive at the correct conjecture for the postulated energy, we consider the concept of averages of rotations in SO(3) Lie group. The used enriched kinematics is seen to be a possible alternative to the adoption of second gradient 2D models. Some rather surprising deformation processes are studied, where interesting non-symmetric post-buckling phenomena are observed in both the models used. Mentioned post-buckling has been observed experimentally.

International Journal of Solids and Structures, 2016

In this paper a procedure for constructing a 1D continuum equivalent to a Koiter plate, is presen... more In this paper a procedure for constructing a 1D continuum equivalent to a Koiter plate, is presented. The result is used to build-up a coarse model of a Thin Walled Beam. The model results to be a nonlinear hyperelastic 1D second gradient continuum, endowed with an internal microstructure described by seven scalar kinematic parameters. The kinematic parameters are able to describe, although in a rough way, the deformation of the TWB in the plane of its cross-section. The nonlinear behaviour of a simply supported strut has been analyzed and the results compared with those given by the commercial finite element code ABAQUS. © 2016 Elsevier Ltd

Thin-Walled Structures, Jul 1, 2013

An L frame made up by beam and column having channel cross sections, has been analyzed in a previ... more An L frame made up by beam and column having channel cross sections, has been analyzed in a previous work by two of the authors [14]. Depending on the aspect ratio and the joint configuration, it has been proved that the structure can exhibit two simultaneous buckling modes. Here using the asymptotic theory of elastic bifurcation that takes into account mode interaction, the initial slope of the bifurcated paths has been determined. Three cases of joint configurations, which are the more common used in welded connections, have been considered. For each case three admissible bifurcated paths have been found. Two of them show a slope having the same order of magnitude of the ones found in the absence of mode interaction while the remaining exhibits a slope largely steepest. Selecting, for each joint case, the bifurcated path with the higher slope and between them the smallest one, it is found that it is associated to the path which bifurcates at the higher critical load. This means th...

Thin-Walled Structures, Jul 1, 2013

An L frame made up by beam and column having channel cross sections, has been analyzed in a previ... more An L frame made up by beam and column having channel cross sections, has been analyzed in a previous work by two of the authors [14]. Depending on the aspect ratio and the joint configuration, it has been proved that the structure can exhibit two simultaneous buckling modes. Here using the asymptotic theory of elastic bifurcation that takes into account mode interaction, the initial slope of the bifurcated paths has been determined. Three cases of joint configurations, which are the more common used in welded connections, have been considered. For each case three admissible bifurcated paths have been found. Two of them show a slope having the same order of magnitude of the ones found in the absence of mode interaction while the remaining exhibits a slope largely steepest. Selecting, for each joint case, the bifurcated path with the higher slope and between them the smallest one, it is found that it is associated to the path which bifurcates at the higher critical load. This means th...

Mathematics and Mechanics of Complex Systems

Recently growing attention has been paid to the particular class of metamaterials which has been ... more Recently growing attention has been paid to the particular class of metamaterials which has been called pantographic. Pantographic metamaterials have the following peculiar features: (i) their continuum model, at the macroscale, has to include a term of the deformation energy depending on the second gradient of placement, (ii) they can show an elastic behavior in large deformation regimes, and (iii) they are resilient and tough during rupture phenomena (dell'Isola et al. 2015). In order to predict pantographic metamaterials' mechanical behavior, it is possible to introduce a three-dimensional continuum micromodel, in which their internal geometrical microstructure is described in detail. However, the computational costs of this choice are presently prohibitive. In this paper, we introduce a reduced order model for pantographic sheets-which are an example of an elastic surface-whose kinematics include, for each of the two constituting families of fibers fully independent three-dimensional (i) placement and (ii) rotation fields. In this way it is possible to include, also in the reduced order model, (i) the initial and the actual offset between the fibers and (ii) the deformation energy of the interconnecting pivots. By postulating a simplified expression for the deformation energy we prove that also a reduced order model can describe some experimental observed buckling and postbuckling phenomena. The promising results which we present here motivate the quest of more general expressions for deformation energy capable of capturing the fully nonlinear behavior exhibited by pantographic sheets.