Efficient finite difference formulation of a geometrically nonlinear beam element

The article is focused on a two-dimensional geometrically nonlinear formulation of a Bernoulli beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which are combined with the kinematic equations and generalized material equations, leading to a set of three first-order differential equations. These equations are then discretized by finite differences and the boundary value problem is converted into an initial value problem using a technique inspired by the shooting method. Accuracy of the numerical approximation is conveniently increased by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant, which leads to high computational efficiency. The element has been implemented into an open-source finite element code. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions

A phase-field model for strain localization

Strain localization in quasi-brittle materials occurs when a material is subjected to a high level of mechanical solicitations and inelastic strains develop in relatively narrow zone where micro-cracks appear. The gradual evolution of the micro-cracks results in the formation of localized bands up to the development of stress-free cracks. The localized zone or plastic band is generally associated to a faster growth of strain and is characterized by inelastic phenomena such as opening and propaga- tion of cracks, initiation and growth of voids. Conversely, outside of this zone, the material unloads elastically. Extensive research has been carried out to address issues related to the modeling of localized defor- mation. A crucial point is the kinematic description of the band that has been addressed by three main models [1]. The first category of models considers the presence of a strong discontinuity in the displacement field and is typical of elastic fracture mechanics. The second approach considers the plastic band with finite thickness separated from the remaining part of the body by two weak discontinuities, as a zero-thickness interface characterized by its own tractions-displacement jumps law. Lastly, the third group considers constitutive enrichments with an internal length scale related to the width of the localization zone. Nonlocal and gradient theories that relate the constitutive be- havior of a material point with those in the neighboring region, fall into this category. Recently, phase-field models, usually adopted to describe gradual chemical changes from one phase to another, have been applied to model the transition between the fully broken and the sound material in a diffusive way. These models are characterized by the evolution of an auxiliary field (the phase field) that takes the role of an order parameter. They have been used to model brittle fracture [2] and ductile fracture [3]. The present paper presents a thermodynamically consistent formulation of the localization problem in quasi-static regime adopting the phase field approach. The introduction of the phase field vari- able enriches the solid kinematics, in this sense the proposed formulation can be categorized in the class of the regularized models where the plastic band is smeared on its neighboring volume with the order parameter assuming the unit value in middle surface of the band and zero value far from the same one. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of brittle fracture as reported in the experimental literature

A phase-field model for strain localization analysis in softening elastoplastic materials

The present paper deals with the localization of strains in those structures consisting of materials exhibit- ing plastic softening response. It is assumed that strain localization develops in a finite thickness band separated from the remaining part of the structure by weak discontinuity surfaces. In view of the small thickness of the band with respect to the dimensions of the structure, the interphase concept is used for the mechanical modeling of the localization phenomenon. We propose a formulation for the quasi- static modeling of strain localization based on a phase-field approach. In this sense, the localization band is smeared over the volume of the structure and a smooth transition between the fully broken and the sound material phases is introduced. The mechanical performance of the model is illustrated for the case of uniaxial tensile test, discussing the instability of the force-displacement response, and for the case of three-point bending test, comparing the analytical results with the experimental ones

Electromechanical impedance method to assess the stability of dental implants

In this paper we illustrate the application of the electromechanical impedance (EMI) technique, popular in structural health monitoring, to assess the stability of dental implants. The technique consists of bonding a piezoelectric transducer to the element to be monitored. When subjected to an electric field, the transducer induces low to high frequency structural excitations which, in turn, affect the transducer's electrical admittance. As the structural vibrations depend on the mechanical impedance of the host structure (in this case the implant secured to the jaw), the measurement of the PZT's admittance can infer the progress of the osseointegration process. In the study presented in this article we created a 3D finite element model to mimic a transducer bonded to the abutment of a dental implant placed in a host bone site. We simulated the healing that occurs after surgery by changing the Young's modulus of the bone-implant interface. The results show that as the Young's modulus of the interface increases, i.e. as the mechanical interlock of the implant within the bone is achieved, the electromechanical characteristic of the transducer changes. The model and the findings of this numerical study may be used in the future to predict and interpret experimental data, and to develop a robust and cost-effective method for the assessment of primary and secondary dental implant stability

On the FE·Meshless computational homogenization for the analysis of two-dimensional heterogeneous periodic materials

Over the last few years, substantial progresses have been made in the two-scales com- putational homogenization. This method is essentially based on the assessment of the macroscopic constitutive behavior of heterogeneous materials through the boundary value problem (BVP) of a statistically representative volume element, named as unit cell (UC). In this framework, the first-order method has now matured to a standard tool and several extensions have been addressed in the literature [1, 2]. In the present study, a first-order homogenization scheme based on a discontinuous- continuous approach is presented. At the mesoscopic level the formation and propagation of fracture is modeled employing a UC consisting of an elastic unit surrounded by elasto- plastic zero-thickness interfaces, characterized by a discontinuous displacement field. At the macroscopic level, instead, the model maintains the continuity of the displacement field. The inelastic effects are enclosed in a smeared way, introducing a strain localization band established on the basis of a spectral analysis of the UC acoustic tensor. Another key-point is the numerical solution of the UC BVP, which is obtained by means of a more cost-effectiveness mesh-free model. Both linear and periodic boundary conditions have been applied to the UC

Numerical and Experimental Assessment of FRP-Concrete Bond System

Fiber reinforced polymer (FRP) composite systems are widely used to repair structurally deficient constructions thanks to their good corrosion resistance, light weight and high strength. The quality of the FRP-substrate interface bond is a crucial parameter affecting the performance of retrofitted structures. In this study, ultrasonic testing have been used to assess the quality of the bonding. In the case of FRP laminates adhesively bonded to concrete, high scattering attenuation occurs due to the presence of concrete heterogeneities. The substrate material behaves almost like a perfect absorber generating a considerable number of short-spaced echo peaks that make the defect echo not distinguishable. In order to avoid scattering, waves longer than the discontinuity have to be used, but this expedient makes bonding defects undetectable. The presented technique is based on the energy distribution measurement of ultrasonic signals by means of a statistical parameter, named Equivalent Time Length (ETL). A preliminary numerical study involving a 1-D system with a material discontinuity was performed. 2D finite element (FE) analyses were also performed. The experimental study involved laboratory FRP reinforcements bonded to concrete substrates with imposed well-known defects, and seismic retrofitted concrete walls. The experimental and the numerical findings showed that the ETL is sensitive to the presence of bonding defects in the sense that lower values mean higher reflection of wave energy (low quality of bonding) and higher values mean lower reflection and higher penetration through the bonding (good quality of bonding)

Granular chains for the assessment of thermal stress in slender structures

Slender beams subjected to compressive stress are common in civil and mechanical engineering. The rapid in-situ measurement of this stress may prevent structural anomalies. In this paper, we describe the coupling mechanism between highly nonlinear solitary waves (HNSWs) propagating along an L-shaped granular system and a beam in contact with the granular medium. We evaluate the use of HNSWs as a tool to measure stress in thermally loaded structures and to estimate the neutral temperature, i.e. the temperature at which this stress is null. We investigated numerically and experimentally one and two L-shaped chains of spherical particles in contact with a prismatic beam subjected to heat. We found that certain features of the solitary waves are affected by the beam's stress. In the future, these findings may help developing a novel sensing system for the nondestructive prediction of neutral temperature and thermal buckling

Efficient formulation of a two-noded geometrically exact curved beam element

The article extends the formulation of a 2D geometrically exact beam element proposed by Jirasek et al. (2021) to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first-order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The article develops the theoretical framework based on the Navier-Bernoulli hypothesis, with a possible extension to shear-flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral-shaped beam, and a spring-like beam with a zig-zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, that is, the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the Appendix

Solitary Waves for the Assessment of Thermal Stress and Neutral Temperature in Slender Structures

Slender columns and continuous welded rails subjected to compressive stress are common in many civil structures. The rapid in-situ measurement of this stress may be of interest to prevent structural anomalies such as buckling. In this article, the authors describe the coupling mechanism between highly nonlinear solitary waves (HNSWs) propagating along an L-shaped granular system and a beam in contact with the granular medium. The aim is the evaluation of HNSWs as a tool to measure stress in thermally loaded structures and to estimate the neutral temperature, i.e. the temperature at which this stress is null. The authors investigated numerically and experimentally one chain of spherical particles in contact with a prismatic beam subjected to thermal stress. The effect of the beam\u2019s temperature on certain features of the solitary waves is investigated. In the future, the findings presented in this article may be used to develop a novel nondestructive evaluation tool for the computation of the neutral temperature and the quantification of stress