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## On the statics of No-Tension masonry-like vaults and shells: solution domains, operative treatment and numerical validation

### Annals of Solid and Structural Mechanics (2011-12-01) 2: 107-122 , December 01, 2011

In the paper, structural analysis of masonry vaults is based on the fundamental assumption that the material cannot resist tensile stresses. For the vault to work as a No-Tension (NT) structure, it is recognized that a membrane surface completely included in the thickness of the vault exists [Heyman (1977) Equilibrium of shell structures. Oxford University Press, Oxford], designed in way to resist applied loads by purely compressive membrane forces. After reducing the problem to a plane-stress problem, the stress function Ψ(*x*,*y*) is introduced, like in the classical Pucher’s approach. Statically admissible solutions are proven to be ruled by the non-homogeneous Monge-Ampere equation. Anyway purely admissible stress fields often yield non-compatible and non-credible results with reference to fractures and strains. A path aimed at generating sets of solutions related to given load shapes is outlined, as a preliminary basis for solutions fully accomplishing not only equilibrium and admissibility, but also compatibility with strains and fractures. Further analytical developments, set up of operative procedures, validation by means of numerical campaign, investigation of special characters of approximations when dealing with particular load shapes, i.e. all topics related to operative features of the proposed approach are presented in the second part of the paper. Starting from the path outlined for identifying solution domains, a constrained optimization is set up to minimize a suitably defined square error function calculated between the assigned objective load function and the membrane surface expression, under the condition that the solution membrane function is contained in the vault profile. Some analytical management is needed in order to reduce the dependence of the problem on the number of parameters, and to partially simplify and expedite the subsequent numerical simulation. Thereafter, ad hoc calculus codes are built up for implementing the problem and performing the numerical investigation. Final validation of the approach is given by demonstrating, with some numerical applications, the effectiveness of the method even for load shapes difficult to be handled.

## Micromechanical analysis of periodic composites by prescribing the average stress

### Annals of Solid and Structural Mechanics (2010-12-01) 1: 117-137 , December 01, 2010

A new method for the micromechanical finite element analysis of unidirectional composites is presented. The method is especially effective in time-dependent analyses, as those involving viscoplasticity and viscoelasticity. The microstructure of the composites under consideration is characterized by periodicity and central symmetry, which allow analysing half of a unit cell for solving the micromechanical problem. Half of the unit cell contains only half of the continuous fibre and this makes the numerical analyses inexpensive in terms of computer memory usage and processing time. New boundary conditions are presented in order to prescribe the average stress on half of the unit cell of hexagonal and rectangular distributions of heterogeneities in the case of static analysis. Then, the new boundary conditions are used to prescribe the precise rate of the average stress in time-dependent analyses. With respect to other existing procedures, the proposed method is easy to adopt in commercial software and it does not require the modification of parts of the source code that are not usually accessible to the user. The proposed method is applied to the interesting case of composites with aligned long fibres imperfectly bonded to a viscoelastic matrix. The numerical simulations carried out in this work provide the loci of the average stress and strain corresponding to the initiation of the fibre-matrix debonding, which determines a considerable decay of the composite stiffness and strength. The influence of the geometrical properties of the microstructure is evaluated by analysing both hexagonal and rectangular distributions of fibres. The numerical results show how the inelastic behaviour of the matrix affects the loci corresponding to the initiation of the debonding.

## Frequency-locking in nonlinear forced oscillators near 3:1 and 4:1 resonances

### Annals of Solid and Structural Mechanics (2012-12-01) 4: 15-23 , December 01, 2012

In this paper, frequency-locking phenomenon in self-excited nonlinear oscillators subjected to harmonic excitation is investigated near the 3:1 and 4:1 subharmonic resonances. Analytical treatment based on perturbation techniques is performed to study the quasiperiodic modulation domain and the frequency-locking area in the vicinity of these resonances. It is shown that this analytical method, based on a double averaging procedure, is efficient to capture the modulation domain of the quasiperiodic response as well as the threshold of synchronization area near the considered subharmonic resonances.

## Numerical studies of ductile failure including the third stress invariant

### Annals of Solid and Structural Mechanics (2015-12-01) 7: 59-69 , December 01, 2015

This work addresses an extended version of the well known GTN isotropic hardening model and its numerical integration within a finite element code. The pre-existing yield function of the proposed constitutive model possesses the distinctiveness to be more accurate for arbitrary void volume fraction and especially to explicitly depend upon the third stress invariant. A fully implicit stress integration procedure, based on the return-mapping algorithm, with calculation of the consistent tangent operator is developed for the proposed model. In order to demonstrate the global accuracy and stability of the numerical solution, finite element damage simulations accounting for finite strain and using both the proposed model and, for the purpose of comparison, the GTN isotropic hardening model are performed for the traditional ductile solid problem of necking of a round tensile bar and the two-dimensional simple dynamic shearing problem. The numerical results highlight similarities, good agreement as long as softening initiation of specimen is not reached, and discrepancy as soon as failure of specimen starts, between the proposed model and the GTN model.

## Elastic–plastic analysis of rotating disks having non-linearly variable thickness: residual stresses by overspeeding and service stress state reduction

### Annals of Solid and Structural Mechanics (2010-05-01) 1: 87-102 , May 01, 2010

Two models for evaluation of elastic and elastic–plastic stress and strain in non-linear variable thickness rotating disks, either solid or annular, subjected to such a rotational speed to determine stresses beyond yielding are presented. The first model regards the elastic field and the second concerns the elastic–plastic field, assuming that material hardening is isotropic and follows a most general hardening power-law. In the elastic region, a non-homogeneous hypergeometric differential equation is obtained. Such differential equation, which solves the elastic problem in closed form for non-linear variable thickness disks, with thickness given by the power of a linear function which may define a fourfold infinity of profiles, is integrated in closed form. As concerns the elastic–plastic analysis, first of all the introduction is made of a correlation between equivalent plastic strain and equivalent stress according to Von Mises, which is more general than those known from literature. In the case of isotropic hardening a second-order, non-homogeneous, non-linear differential equation is found, which governs the stress state in the plastic region of the disk. The procedure allows to calculate stress and displacement states in the two regions—plastic and elastic—of the disk subjected to prestressing by overspeeding. Several examples of disks, also prestressed by overspeeding, are considered (annular and solid, convex or concave, and linear tapered disks); the matching of results of the theoretical model and those obtained by means of FEA is very good. Lastly, the residual stress state can be found in a prestressed disk by overspeeding and the stress state in the actual operating condition in the same disk is evaluated.

## On a simple cyclic plasticity modeling with implicit kinematic hardening restoration

### Annals of Solid and Structural Mechanics (2012-12-01) 4: 33-42 , December 01, 2012

This paper presents a fully three-dimensional plastic constitutive modeling framework suitable for the prediction of cyclic loading at large number of cycles. It can require only one yield surface and it is motivated by a simple rheological model where a restoration of the kinematic hardening is introduced. The classical kinematic hardening rules are then simply adapted leading to time-dependent evolution laws that are consistent with continuum thermodynamics requirements. The resulting behavior is physically motivated by many man-made materials of engineering interest such as bituminous material. This framework allows all types of yield functions to be easily implemented numerically. This is first illustrated with algorithmic details through a simple associative pressure-insensitive model example of the von Mises type. Then a more elaborated model is given where the present framework is applied to the description of bituminous materials submitted to triaxial static creep and to large number of cyclic loadings. Of particular interest is the ratcheting and the mean stress relaxation. The responses agree well with some experimental test results found in the literature.

## Identification of a planar crack in Zener type viscoelasticity

### Annals of Solid and Structural Mechanics (2010-01-01) 1: 3-8 , January 01, 2010

The paper addresses the identification of a planar crack for Zener type linear viscoelastic solids. Under the condition of low frequency, the Zener model of viscoelasticity establishes the equivalence between viscoelasticity and elasticity and the equations are reduced to a Helmholtz type problem for time harmonic loadings. The solutions to the crack identification problems are then obtained from the corresponding solutions in elasticity, using only one frequency.

## Towards nonlinear imperfect interface models including micro-cracks and smooth roughness

### Annals of Solid and Structural Mechanics (2017-12-01) 9: 13-27 , December 01, 2017

The present paper deals with a general asymptotic theory aimed at deriving some imperfect interface models starting from thin interphases. The novelty of this work consists in taking into account some non-standard constitutive behaviors for the interphase material. In particular, micro-cracks, surface roughness and geometrical nonlinearity are included into the general framework of the matched-asymptotic-expansion theory. The elastic equilibrium problem of a three-composite body comprising two elastic adherents and an adhesive interphase is investigated. Higher order interface models are derived within the cases of *soft* and *hard* interphase materials. Simple FEM-based numerical applications are also presented.

## The Dirichlet–Neumann problem revisited after modelling a new class of non-smooth phenomena

### Annals of Solid and Structural Mechanics (2014-12-01) 6: 29-36 , December 01, 2014

Generalized Standard Materials are governed by maximal cyclically monotone operators and modelled by convex potentials. Géry de Saxcé’s Implicit Standard Materials are modelled by biconvex bipotentials. Analyzing the intermediate class of n-monotone materials governed by maximal n-monotone operators and modelled by Fitzpatrick’s functions, we find out that n-monotonicity is a relevant criterion for the materials characterisation and classification. Additionally, the Fitzpatrick’s functions allow to describe the thermal or mechanical equilibrium equations of n-monotone materials by primal–dual two-fields variational principles. In doing so, we are led to Dirichlet–Neumann problems that we solve by Uzawa-type algorithms.

## Bending and free vibration of functionally graded piezoelectric microbeams based on the modified couple stress theory

### Annals of Solid and Structural Mechanics (2017-11-08): 1-16 , November 08, 2017

A size-dependent model for bending and free vibration of functionally graded piezoelectric (FGP) microbeam is developed by using modified couple stress theory and a unified higher order beam theory. This model can be specialized to various beam models, such as Euler–Bernoulli, Timoshenko as well as Reddy beam ones and vice versa. The governing equations of motion and associated boundary conditions are derived from Hamilton’s principle. Only one material length scale parameter is introduced to capture the size effect. The analytical solutions of simply supported FGP microbeam are presented by using Navier approach to bring out the effect of the material length scale parameter on the bending and free vibration of microbeam. Numerical simulations are presented to account for the effect of various parameters, such as material length scale parameters, volume fraction indexes, and slenderness ratios on the responses of static bending and free vibration of FGP microbeam.