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Over the past 50 years the Ogden model has been widely used in material modelling owing to its ability to match accurately the experimental data on elastomers at large strain, as well as its mathematical properties, such as polyconvexity. In this paper, these characteristics are exploited to formulate a finite-strain model that incorporates, through the phase-field approach recently proposed by Wu (Wu 2017 J. Mech. Phys. Solids 103 , 72–99) for small strains, a cohesive damage mechanism which leads to the progressive degradation of the material stiffness and to failure under tension. By properly tailoring the constitutive parameters, the model is capable of encompassing a wide range of effects, from brittle to pseudo-ductile failure modes. A plane stress problem is formulated to test the model against experiments on double-network elastomers, which display a pseudo-ductile damage behaviour at large strain, and on conventional rubber compounds with brittle failure. The results show that the proposed model is applicable to fracture coalescence and propagation in a wide range of materials. This article is part of the theme issue ‘The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity’.

This paper introduces a model for the mechanical response of anisotropic soft materials undergoing large inelastic deformations. The material is considered made by a isotropic matrix with embedded fibers, each component having its own relaxation dynamics. The constitutive equations are provided in terms of the free energy density and the dissipation density, which are both required to be thermodynamically consistent and structural frame-indifferent, i.e., independent of a rotation overimposed on the intermediate natural state of both matrix and fibers. This is in contrast to many of the currently used anisotropic inelastic models, which do not deal with the lack of uniqueness of the intermediate state. This issue is thoroughly discussed and in terms of two possible choices satisfying structural-frame indifference and leading to different flow rules of the inelastic processes. It is shown that different models from the literature can be incorporated in the proposed formulation including anisotropic viscoelasticity and growth.

Starting from a two-dimensional theory of magneto-elasticity for fiber-reinforced magnetic elastomers we carry out a rigorous dimension reduction to derive a rod model that describes a thin magneto-elastic strip undergoing planar deformations. The main features of the theory are the following: a magneto-elastic interaction energy that manifests itself through a distributed torque; a penalization term that prevents local interpenetration of matter; a regularization term that depends on the second gradient of the deformation and models microstructure-induced size effects. As an application, we consider a problem involving magnetically-induced buckling and we study how the intensity of the field at the onset of the instability increases if the length of the rod is decreased. Finally, we assess the accuracy of the deduced model by performing numerical simulations where we compare the two-dimensional and the one-dimensional theories in some special cases, and we observe excellent agreement.

In this work, a computational model for simulating the relevant mechanisms governing the pull-out of a discrete reinforcement embedded into cement based materials is described. The model accounts for the material and geometric properties of the reinforcement, which can include an anchored end, the interface between reinforcement and surrounding medium, and the relative inclination of the reinforcement to the crack plane. The reinforcement is modelled as a Timoshenko beam resting on a cohesive-like foundation that allows all the failure modes seen in the experiments to be accounted for, namely: debonding at the interface between the reinforcement and the concrete, cracking and spalling of the concrete matrix, rupture of the reinforcement. A comprehensive comparison with the experimental data available in the literature highlights the good predicting capabilities of the proposed model in terms of both peak force and dissipated energy. Furthermore, since the model is capable of simulating a discrete reinforcement of any direction towards the crack plane, complex mechanisms like micro-spalling of the matrix at the exit point of the reinforcement can be captured conveniently. By carrying out parametric analysis is possible to optimize the geometry of the anchored ends for maximizing the peak force and/or the energy dissipation in the pull-out process. Therefore, the developed model constitutes a relevant numerical tool for the optimization of discrete and continuous reinforcements of concrete structures including Fibre Reinforced Polymer (FRP) systems and Steel Fibre Reinforced Concrete (SFRC)., FEDER through the Operational Program for Competitiveness Factors – COMPETE and from the Portuguese Foundation for Science and Technology (FCT) under the project FCOMP-01-0124-FEDER-014844

Modal curvatures have been claimed to contain local information on damage and to be less sensitive to environmental variables than natural frequencies. However, simply using the difference between modal curvatures in the undamaged and damaged states can result into localization errors, due to the complex pattern that this quantity presents when considering broad damages or higher order modes. In this paper, we consider weakly damaged continuous beam and we exploit a perturbative solution of the beam equation of motion to obtain an analytical expression of the modal curvature variations in terms of damage distribution. The solution is then used to introduce a filtering technique of the modal curvature variation and to set up the inverse problem of damage localization based on modal curvatures only. Using numerical examples and experimental tests, we show that modal curvatures can be used for precise damage localization, once properly filtered.

The addition of metallic fibres in concrete is a widely used technique to increase its resistance to crack opening, energy absorption capacity, and durability. The properties of the resulting composite strongly depend on the geometry of the fibres, which usually dispose of an hooked end, as well as on the mechanical properties of both the fibre and the concrete. The optimization of the fibre reinforcement mechanisms can only be achieved by an approach that takes into account adequately these aspects. Therefore, in this work, a computational model incorporating the key features of a hooked end fibre embedded in a cement matrix is proposed. The fibre is modelled as a Timoshenko beam, whereas a cohesive interface is used to model the interaction with the surrounding concrete. Different failure mechanisms are defined including fibre debonding or fibre tensile rupture and concrete spalling at fibre exit point. The model is calibrated by using the results of an experimental campaign conducted by authors. A multi-step optimization algorithm is used to find the optimal geometry and model’s constitutive parameters that maximize the peak pull-out force and the energy absorption capacity in a fibre pull-out test. The analysis suggests that the use of concrete with high strength has the potential to increase both peak force and energy absorption capacity by designing the proper geometry of the fibre, FEDER through the Operational Program for Competitiveness Factors - COMPETE and Internationalization Program (POCI), under the project NG_ TPfib POCI-01-0247-FEDER-033719

We study the inverse problem which arises when designing thin magneto-elastic actuators with bespoken deformation modes. By using the nonlinear model of magneto-elastic rods which we have recently proposed, we formulate the design problem as a PDE-constrained minimization whose solution gives to the optimal distribution of the magnetization profile necessary to achieve the desired shape. The same problem is extended to control multiple deformed configurations which would allow a controlled motion of the actuator to be realized.

This paper investigates damage identification techniques based on the difference of modal frequencies, shapes and curvatures in the damaged and undamaged states of the structure. The sensitivity of the identification algorithm with respect to damage parameters is discussed and the minimum number of measured quantities to identify the damage is assessed. It is shown that modal curvatures can be effectively used to pre-localise the damage and to add a penalty term in the objective function which weighs the difference between natural frequencies and modal displacements. Such a term improves the local convexity of the objective function and enhances the convergence rate of the minimization algorithm. The procedure is validated against the results of the experiments on a parabolic arch carried out by the authors. The advantages of such an approach compared to techniques solely based on frequencies are that the ill-conditioning of the inverse problem is reduced and a more accurate estimate of the damage parameters is achieved.

The hybridisation of fibres reinforced laminates, i.e., the combined use of two or more families of fibres, is an effective technique to achieve a pseudo-ductile response and overcome the inherent brittleness which limits the wider use of composite materials. In this paper, a one-dimensional analytical model for unidirectional hybrid laminates is derived. The model considers two elastic–brittle layers bonded together by a cohesive elasto–plastic–brittle interface. This formulation is applied to the study of the debonding and fracture of laminates under uniaxial loading and the results compared to experiments available from the open literature. This study shows that the proposed model provides a close fit to the experimental data and it is able to match accurately the crack patterns seen in the experiments. The model predicts four different failure mechanisms and is able to discriminate among them according to the geometrical and mechanical properties of the layers.

One of the main theoretical issues in developing a theory of anisotropic viscoelastic media at finite strains lies in the proper definition of the material symmetry group and its evolution with time. In this paper the matter is discussed thoroughly and addressed by introducing a novel anisotropic remodelling equation compatible with the principle of structural frame indifference, a requirement that every inelastic theory based on the multiplicative decomposition of the deformation gradient must obey to. The evolution laws of the dissipative process are %obtained by introducing a novel (remodelling) balance equation which is completely determined by two scalar functions, the elastic strain energy and the dissipation densities. The proper choice of the dissipation function allows us to reduce the proposed model to the Ericksen anisotropic fluid, when deformation is sufficiently slow, or to the anisotropic hyperelastic solid for fast deformations. Finally, a few prototype examples are discussed to highlight the role of the relaxation times in the constitutive response., Comment: 43 pages,, 9 figures

In the present work, a multiscale model for fibre reinforced concrete (FRC) beams failing in bending is presented. At the microstructural level, the fibre is modelled as a one-dimensional continuum with axial, shear and bending deformability, with cohesive-like interfaces to simulate the interaction with the surrounding concrete. At the macroscopic level, the response of the beam is simulated by discretising the cross-section into layers and by enforcing the proper compatibility conditions between the layers. In the post-cracking stage, the tensile capacity is assured by the fracture energy of the concrete and the fibre resisting mechanisms simulated by the fibre pull-out constitutive laws determined at the microstructural level. The model can account for fibre distribution and orientation, controlled by the casting conditions and geometry of the mould. By using experimental data available from the open literature, it is proved that such an integrated approach is able to derive, by inverse analysis, the stress-crack width relationship of FRC, which is the fracture mode I information in the material nonlinear analysis of FRC structures with approaches based on the finite element method.

We introduce a continuum model for a fibre reinforced material in which the reference orientation of the fibre may evolve with time, under the influence of external stimuli. The model is formulated in the framework of large strain hyperelasticity and the kinematics of the continuum is described by both a position vector and by a remodelling tensor which, in the present context, is an orthogonal tensor representing the fibre reorientation process. By imposing suitable thermodynamical restrictions on the constitutive equation, we obtain an evolution equation of the remodelling tensor governed by the Eshelby torque, whose stationary solutions are studied in absence of any external source terms. It is shown that the fibres reorient themselves in a configuration that minimises the elastic energy and get aligned along a direction that may or may not be of principal strain. The explicit analysis of the Hessian of the strain energy density allows us to discriminate among the stationary solutions, which ones are stable. Examples are given for passive reorientation processes driven by applied strains or external boundary tractions. % Applications of the proposed theory to biological tissues, nematic or magneto-electro active elastomers are foreseen., 23 pages, 4 figures

The tremendous advances in high-resolution and high-speed cameras have allowed improving optical measurement techniques enabling extraction of dynamic characteristics of vibrating structures. These techniques turned out to be particularly useful for vibration observations as they are contactless, and they do not interfere with the structural response (as is the case of accelerometers installed on a light-weight set-up, with non-negligible additional masses). The paper reports the results of an experimental campaign, carried out to verify the accuracy of the dynamic displacements measurement obtained with a high-speed camera and the Hybrid Lagrangian Particle Tracking software, a software based on the solution of the Optical Flow equation. The experimental setup includes a 4-dof small-scale steel frame structure excited on a shaking table, a high-speed camera able to record images of the motion and five accelerometers. From images, a two-dimensional moving field is recorded, and the absolute and relative displacements are determined. Confidence in the novel developed measuring methodology is determined by direct comparison with acceleration measurements which confirm that the image analysis measurements turned out to be sufficiently accurate for damage identification purposes.

The localization of stiffness variation in damaged beams through modal curvatures, i.e., second derivative of mode shapes, is studied by exploiting a perturbative solution of the Euler–Bernoulli equation. It is shown that for low order modes the difference between undamaged and damaged modal curvatures has only one distinct peak if the damage is localized in a narrow region. This phenomenon is independent of the presence of experimental noise and of the technique used to reconstruct the curvature mode shapes from the displacement mode shapes. Broader damages cause the modal curvature difference to have several peaks outside the damage region that could result in a false damage localization. The same effect is present at higher modes for both narrow and broad damages. As a result, modal curvatures can be effectively used to localize structural damages only once they have been properly filtered. Here the perturbative solution is used to introduce an effective damage measure able to localize correctly narrow and broad damages and also single and multiple damages cases.

We report about the experimental identification of viscoelastic constitutive models for frequencies ranging within 0–10 Hz. Dynamic moduli data are fitted for several materials of interest to medical applications: liver tissue ( Chatelin et al., 2011 ), bioadhesive gel ( Andrews et al., 2005 ), spleen tissue ( Nicolle et al., 2012 ) and synthetic elastomer ( Osanaiye, 1996 ). These materials actually represent a rather wide class of soft viscoelastic materials which are usually subjected to low frequencies deformations. We also provide prescriptions for the correct extrapolation of the material behavior at higher frequencies. Indeed, while experimental tests are more easily carried out at low frequency, the identified viscoelastic models are often used outside the frequency range of the actual test. We consider two different classes of models according to their relaxation function: Debye models, whose kernel decays exponentially fast, and fractional models, including Cole–Cole, Davidson–Cole, Nutting and Havriliak–Negami, characterized by a slower decay rate of the material memory. Candidate constitutive models are hence rated according to the accurateness of the identification and to their robustness to extrapolation. It is shown that all kernels whose decay rate is too fast lead to a poor fitting and high errors when the material behavior is extrapolated to broader frequency ranges.

Fibre-reinforced elastomers are a class of elastomeric materials with peculiar mechanical properties. If the fibres are metallic or metallic-coated their orientation can be controlled by applying a magnetic field during the curing phase of the elastomer so to tailor the material anisotropy. Once the elastomer is cured, the application of a magnetic field may be used to control the deformation of the solid. In this paper, we aim at modelling both the pre-curing and post-curing phases within a unifying framework compatible with nonlinear elasticity and growth theory. The coupling between elasticity and magnetic field is obtained by considering a proper form of the free energy density which accounts for the mutual orientation between the magnetic field and the fibres. By considering some asymptotic limits, it is shown that the proposed approach incorporates both the usual magneto-elastic model of solids as well as those models used to study the magnetic reorientation of fibres in viscous solutions.

In this work, we investigate the effectiveness of a damage identification technique recently proposed in [1] and assess how it is affected by the number and position of the sensors used. Mode shapes and curvatures have been claimed to contain local information on damage and to be less sensitive to environmental variables than natural frequencies. It is known that notch-type damage produces a localized and sharp change in the curvature that unfortunately could be difficult to detect experimentally without the use of an adequate number of sensors. However, we have recently shown that even a coarse description of the modal curvature can still be employed to identify the damage, provided that it is used in combination with other modal quantities. Here, by exploiting the perturbative solution of the Euler-Bernoulli equation, we consider the inverse problem of damage localisation based on modal curvatures only and we ascertain the feasibility of their sole use for recostructing the damage shape. To do so, we set up a filtering procedure acting on modal curvatures which are expressed in a discrete form enabling further investigation on the effect of using a reduced number of measurement points. The sensitivity of the procedure to damage extension is further assessed.

A technique for damage localization and assessment based on measurements of both eigenvectors curvatures and eigenfrequencies is proposed. The procedure is based on two successive steps: a model independent localization, based on changes of modal curvatures, and the solution of a one-dimensional minimization problem to evaluate damage intensity. The observability properties of damage parameters is discussed and, accordingly, a suitable change of coordinates is introduced. The proposed technique is illustrated with reference to a cantilever Euler beam endowed with a set of piezoelectric transducers. To assess the robustness of the algorithm, a parametric study of the identification errors with respect to the number of transducers and to the number of considered modal quantities is carried out with both clean and noisecorrupted data.

Fading memory is a distinguishing characteristic of viscoelastic solids. Its assessment is often achieved by measuring the stress due to harmonic strain histories at different frequencies: from the experimental point of view, the storage and loss moduli are, hence, introduced. On the other side, the mathematical modeling of viscoelastic materials is usually based on the consideration of a kernel function whose decay rate is sufficiently fast. For several different solid materials, we have collated experimental evidence showing an high sensitivity to frequency variations of both the storage and loss moduli. By contrast, we prove that the commonly employed viscoelastic kernels (Prony series, continuous kernel, etc.) cannot reproduce this experimental behavior, as the resulting frequency sensitivity of the storage modulus is always zero when assessed at low frequency. This leads to identification problems of the material parameters which are strongly ill conditioned. However, we identify the specific kernel property which is responsible for this misbehavior: the long-term material memory must not decrease too fast. Some viscoelastic kernels, showing the correct memory’s rate of decay, are introduced and their improved ability to match the experimental data analyzed.

Carbon black-filled rubber and soft biological tissues are only two examples of materials which undergo large deformation processes and exhibit relevant dissipation and hysteresis losses. Nonlinear viscoelasticity encompass a wide class of constitutive models aimed at describing the behavior of such materials. The main goal of the present paper is to draw a comparison between the many viscoelastic constitutive relations recently proposed. To this aim the current stress value is expressed through a single hereditary integral of the deformation history; this choice, although generalizable, yet permits the introduction of an unifying formulation in which hereditary, strain rate and fractional-derivatives models can all be included. The models comparison is based on several compression tests with cylindrical carbon black-filled rubber specimens carried out at Dipartimento di Ingegneria Chimica e Materiali di Sapienza Universita di Roma and in cooperation with the Bridgestone Technical Center Europe SpA. In particular relaxation, creep, loading–unloading cycles at different speeds were considered. For each test case and for each model under consideration, a nonlinear optimization problem is solved to identify the optimal constitutive parameters; the robustness of the identified parameters to perturbations in the experimental data is also computed. The predictive capabilities of the models are then compared in terms of the stress response and the energy dissipation. Finally, the ability of each model in the cross-prediction of the relaxation/creep behavior when identified via cyclic loading and viceversa is examined.

Alignment of magnetic particles into a viscous fluid by a homogeneous magnetic field has been studied both experimentally and theoretically, but very few studies have investigated the effects of viscosity changes on the capability of achieving complex fibre assemblies in composite materials. In this paper, we study the alignment of short carbon fibres into a viscous matrix whose viscosity is made dependent on the time, in order to simulate the curing process of the composite. A simple model is derived which gives the evolution of fibre position and orientation in terms of the external field and shows good agreement with the experimental evidence on nickel coated carbon fibres embedded in PDMS. The model is used to predict the fibre distribution at curing and hence could be a useful tool to predict the mechanical, electrical and magnetic properties of the composite. A closed form expression of the minimum magnetic field intensity necessary to achieve the desired orientation is derived in terms of the magnetic and geometric properties of the fibres. This equation can be used to guide experiments, and offers a way to discriminate which matrix/fibre configuration can be most proficiently used to have a highly ordered composite.

Polydimethylsiloxane (PDMS) films reinforced with short Nickel-coated Carbon Fibres (NiCF) were successfully fabricated, with the fibres aligned along different directions using an external magnetic field. The fibres were dispersed in the host matrix using sonication and mechanical mixing before being cured for 48 hours in the magnetic field; thanks to the nickel functionalisation, the fibre orientation was achieved by a low intensity field ( < 0.2 T) which required an inexpensive experimental set-up. The main focus of this study was looking at the actuation potential of this magnetic composite material; successful actuation was achieved, showing its large displacement capability. The results confirm the presence of an instability controlled by the magnetic torque, as predicted by the introduced model. The composite films undergo a transition from a bending-only deformed configuration for the 0° fibre specimen, to a twisting-only configuration, achieved for fibres at 90°, whereas all the intermediate angles show both bending and twisting. This behaviour mirrors that which is used to propel a selection of marine mammals.

We derive a model for the finite motion of a magneto-elastic rod reinforced with isotropic (spherical) or anisotropic (ellipsoidal) inclusions. The particles are assumed weakly and uniformly magnetised, rigid and firmly embedded into the elastomeric matrix. We deduce closed form expressions of the quasi-static motion of the rod in terms of the external magnetic field and of the body forces. The dependences of the motion on the shape of the inclusions, their orientation, their anisotropic magnetic properties and the Young modulus of the matrix are analysed and discussed. Two case studies are presented in which the rod is used as an actuator suspended in a cantilever configuration. This work can foster new applications in the field of soft-actuators.

Structure evolution during deformation of isotropic auxetic foams is investigated by simple compression experiments. It is shown that the main feature observed in the experimental data can be accurately described by an isotropic hyperelastic model with an Ogden-type strain energy function. The model can be easily implemented in a finite element code and, hence, can be used to simulate foams undergoing complex structural deformations.

The mechanical properties of natural rubber reinforced with exfoliated graphene oxide (GO) nanoplatelets have been studied to evaluate the reinforcement efficiency of the filler. Tensile tests on rectangular specimens have been carried out at different strain rates for deformations up to 600%. Tensile tests showed a significant increase in the Young’s modulus of the composites even at low filler loading (+50% @ 1.00 wt%) that corroborates the strong interfacial interaction between GO and natural rubber. The viscoelastic properties are investigated through relaxation tests and loading–unloading cycles at different strain rates. It is shown that the Young’s modulus increases with the strain rate and the dissipation increases (+250% @ 0.75 wt%) for increased GO content because of the increased friction caused by the presence of high aspect-ratio GO platelets. These findings indicate significant improvements in the mechanical properties at much lower filler concentration than those usually achieved with carbon black or carbon nanotubes.

The large strain behaviour of a short fibre-reinforced composite is studied through numerical simulations. The reinforcing fibres yield the macroscopic response transversely isotropic which is indeed the case of many reinforcements currently used in composites: short carbon fibres, cellulose whiskers, carbon nanotubes. As a result of the analysis, it is shown that the reorientation of the fibres that takes place at large strain has a significant effect on the overall material response by changing the axis of isotropy. This behaviour can be adequately described by using a transversely isotropic model whose strain energy function depends on three invariants: two isotropic and one representing the stretch along the direction of the fibres. To assess its capabilities, the model is compared to the results of experiments carried out by the authors on nickel-coated chopped carbon fibres in a vulcanised natural rubber matrix for which the fibre orientation is achieved by controlling an external magnetic field prior to curing. Possible applications include micro-sized propulsion devices and actuators.Copyright © 2014 by ASME

The small sensitivity to local variations of mechanical characteristics turns out to be the major limit of indirect identification techniques based on frequency response measurements. To overcome this limit, the use of sensitivity enhancement techniques has recently proposed: the monitored structure is coupled to an auxiliary system, the constitutive parameters of which are then suitably tuned to enhance the sensitivities relevant to the identification process. The damage identification problem in these “augmented” structures is here introduced, and its main advantages and drawbacks are discussed. A beam-like structure coupled to a network of piezoelectric patches supplies an enlightening application of the proposed technique.

We propose a simple mathematical model to describe isotropic auxetic materials in the framework of the classical theory of nonlinear elasticity. The model is derived from the Blatz–Ko constitutive equation for compressible foams and makes use of a non-monotonic Poisson function. An application to the modelling of auxetic foams is considered and it is shown that the material behaviour is adequately described with only three constitutive parameters.

Predictions of the QLV (Quasi-Linear Viscoelastic) constitutive law are compared with those of the ABAQUS viscoelastic model for two simple motions in order to highlight, in particular, their very different dissipation rates and certain shortcomings of the ABAQUS model.