Elastic Constitutive Law Research Articles

Periodic Solutions of an Impact-Driven Frequency Up-Conversion Piezoelectric Harvester

Frequency up-conversion has been proved to be an effective approach to increase the output power of a piezoelectric energy harvester (PEH). The proposed system can convert low-frequency vibration from ambient sources to the resonant vibration of the PEH hence can improve the output power efficiency. Frequency up-conversion technologies are introduced via impact or nonimpact magnetic forces to initiate the repeated free oscillations of the piezoelectric generator. No matter impact- or nonimpact-driven PEHs, most studies focus on either finite element simulation or experimental demonstration of PEHs electric power generations. Few, if any, study the effects of the impact-induced discontinuous dynamics on power generation efficiency. In this work, the energy harvesting performance of a piezoelectric beam upon interaction with a softer driving beam was studied. The discontinuous dynamics behind this impact-driven PEH was investigated, and strategies exploited to further improve the power efficiency of the frequency up-conversion process. Based on the linear elastic and linear mechanical-electrical constitutive laws, the lumped parameter models were built for both the driving beam and the piezoelectric driven beam. The numerical solution of the output power is obtained based on the vibration amplitude, frequency, and the electrical load. The soft beam is subjected to a sinusoidal base excitation, and the piezoelectric beam was excited via impacting with the soft driving beam. Based on the discontinuous dynamics theory, the performance of the energy harvesting of the impact-driven system was studied for period-1 and period-2 motions. Based on the stability and bifurcation analysis of periodic solutions, bifurcation diagrams of impact velocities, times, displacements and harvested power versus the frequency of the base excitation were also obtained, and compared to the power generation of a piezoelectric beam with base excitation.

Bifurcation and post-bifurcation of an inflated and extended residually-stressed circular cylindrical tube with application to aneurysms initiation and propagation in arterial wall tissue

A numerical procedure to analyze bifurcation and post-bifurcation of a finite deformation boundary-value problem for a residually-stressed elastic body is studied. In particular, the problem is the combined extension and inflation of a circular cylindrical tube subject to radial and circumferential residual stresses. The material model, given by a residual-stress dependent nonlinear elastic constitutive law in terms of invariants, is implemented in a finite element code. A numerical procedure to analyze the bifurcation and post-bifurcation of the finite deformation boundary-value problem at hand is developed based on the modified Riks method. The dependence of bifurcation and post-bifurcation behavior of tubes under the loading at hand on residual stresses is shown and compared with results when there is no residual stress. The finite deformation boundary-value problem is described mainly in terms of the inflation pressure, as well as the axial and azimuthal stretches of the tube. The dependence of these quantities on bifurcation is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular, the strength of the residual stress. It is found that bulging bifurcation is expected for sufficiently large values of the axial stretch. On the other hand, for small values of the axial stretch (close to the non-extended configuration), the onset of bifurcation is found to be the bending mode. Furthermore, for the latter case in subsequent motion, i.e. post-bifurcation, it is shown that bending triggers bulging as opposed to the situation in which bending is not allowed and the onset of bifurcation is associated with bulging. In addition, the bulge, or the abnormal enlargement, that is formed during post-bifurcation after the onset of bending bifurcation appears on one side of the tube showing an irregular shape which is consistent with the development of abdominal aortic aneurysms (AAA).

Modelling and Experimental Testing of Thick CFRP Composites Subjected to Low Velocity Impacts

The present paper investigates a modelling approach of experimentally tested thick panels made of Carbon Fibre Reinforced Polymers (CFRP). The coupons were made of 24 unidirectional (UD) laminae with a layup [45/0/-45/90]3s. The specimens were subjected to low velocity impact using a drop tower system. Several sensors, including a load cell and strain gauge, were utilized both for analysing the behaviour of the material against the impact and for performing a validation of the numerical models. Three energy levels were adopted: 8J, 10J and 12J.Numerical models were implemented into the finite element (FE) software LS-DYNA. A linear - elastic constitutive law with an instantaneous failure material was selected for mimicking the intralaminar behaviour of the carbon fibre composite. Enhanced Chang – Chang was adopted as the onset-of-failure criterion. This criterion is able to capture damage in different directions and permits the consideration (or not) of the shear behaviour in the failure equations. The capability of the model to capture the correct interface failure process was particularly emphasized and therefore cohesive elements with a bilinear traction – separation law were chosen for the reproduction of delamination. Finally, the experimental – numerical results were compared using first and foremost the overall delamination area and the curves force – time, force – displacement and absorbed energy – time as well as the strain measures obtained by the sensors.

Second gradient homogenization of multilayered composites based on the method of oscillating functions

The present paper aims at developing a homogenization scheme for the identification of strain-gradient elastic moduli, based on the mathematical approach of homogenization introduced by L. Tartar. We expose in the first part of this paper the needed mathematical apparatus in view of the derivation of the effective first and second gradient mechanical properties of two-phase composite materials, focusing on a one-dimensional situation. Each of the two phases is supposed to obey a second gradient linear elastic constitutive law. Application of the method of oscillating functions to the homogenization of multilayer materials showing strong strain gradients like composites with a strong contrast of properties of their constituents leads to closed form expressions of the effective first and second gradient moduli, including the coupling coefficients between first and second gradient terms. These results are then applied to the situations of multilayers with interfaces considered as the locus of strong strain gradients, and stratified materials. Analytical expressions of the effective moduli versus the first and second elasticity properties of the plies within the unit cell are obtained in both situations.

Determination of closed form expressions of the second-gradient elastic moduli of multi-layer composites using the periodic unfolding method

The present paper aims at introducing a homogenization scheme for the identification of strain–gradient elastic moduli of composite materials, based on the unfolding mathematical method. We expose in the first part of this paper the necessary mathematical apparatus in view of the derivation of the effective first- and second-gradient mechanical properties of two-phase composite materials, focusing on a one-dimensional situation. Each of the two phases is supposed to obey a second-gradient linear elastic constitutive law. Application of the unfolding method to the homogenization of multi-layer materials provides closed form expressions of all effective first- and second-gradient elastic moduli as well as coupling moduli between first- and second-gradient elasticity. A comparison between the unfolding method and the method of oscillating functions shows that both methods, despite their differences, deliver the same effective second-gradient elastic constitutive law for stratified materials.

Elastic shear modulus constitutive law found from entropy considerations

Isothermal and adiabatic shear moduli have been measured countless times. Adiabatic shear moduli which are generally considered to be the same as isothermal shear moduli are shown here to be unstable, so they cannot exist. A constitutive law that avoids adiabatic moduli is found to apply to polycrystalline isotropic solids. The equation is a power law in the volume per unit mass time the shear modulus. The volume is in turn a function of temperature T and pressure p in the system. The adiabatic moduli are established in shear stress versus strain space using a construction from isothermal elastic shear moduli. The derivation of this law replaces adiabatic with isochoric conditions in linear elastic, sheared solids. It is widely reported that the shear thermal expansion coefficient is zero since isotropic solids do not change the shape when heated. All solids with applied shear stresses and temperature dependent shear moduli change the shape and have shear thermal expansion coefficients. The thermal expansion coefficient is coupled into the shear component of the entropy and the T dependence of the isothermal shear modulus. In a sheared solid, only the single point τ = 0, γ = 0 has no shear entropy contributions and no thermal expansion and is mechanically unstable in adiabatic constructions. The exact form of the proposed constitutive law is based on solutions from a second order differential equation based on shear contributions from the entropy.

Elastoplastic nonlinear FEM analysis of FGM shells of Cosserat type

The paper is a continuation of [1] where the formulation of the elastic constitutive law for functionally graded materials (FGM) on the grounds of nonlinear 6-parameter shell theory with the 6th parameter (the drilling degree of freedom) was presented. Here the formulation is extended to the elasto-plastic range. The material law is based on J2 Cosserat plasticity and employs the well-known Tamura-Tomota-Ozawa (TTO) [2] mixture model with additional formulae for Cosserat material parameters. Formulation is verified by solving a set of demanding analyses of plates, curved and multi-branched shells, including geometry, thickness and material distribution variation parameter analyses.

Nonlinear In-Plane Stability of Deep Parabolic Arches Using Geometrically Exact Beam Theory

In this paper, we investigate the in-plane stability and post-buckling response of deep parabolic arches with high slenderness ratios subjected to a concentrated load on the apex, using the finite element implementation of a geometrically exact rod model and the cylindrical version of the arc-length continuation method enabled with pivot-monitored branch-switching. The rod model used here includes geometrically exact kinematics of the cross-section, exact kinetics, and a linear elastic constitutive law; and advantageously employs quaternion parameters to treat the cross-sectional rotations and to compute the exponential map in the configurational update of rotations. The evolution of the Frenet frame along the centroidal curve is used to determine the initial curvature of the rod. Three sets of boundary conditions, i.e. fixed–fixed (FF), fixed–pinned (FP) and pinned–pinned (PP), are considered, and arches with a wide range of rise-to-span ratios are analyzed for each set. Complete post-buckling response has been derived for all cases. The results reveal that although all the PP arches and all the FF arches (with the exception of FF arches with rise-to-span ratio less than 0.3) considered in this study buckle via bifurcation, the nature of stability of the different solution branches in the post-buckling regime differs from case to case. All FP slender parabolic arches exhibit limit-point buckling, again with several markedly different post-buckling behaviors. Also, some arches in the FF and PP case are shown to exhibit a clear bistable behavior in the post-buckled state.

Variational interface zone model for modeling of fluid induced fracture propagation

AbstractA novel variational framework for an interface zone model is developed and extended to poroelasticity. As was previously promoted in [1,2], the total energy of the system is composed by the bulk potential and fracture surface energy. In contrast to the phase‐field method, the fracture surface is approximated directly along the edges of the finite elements in terms of interface zero‐thickness finite elements. By introducing a new degree of freedom c (damage field) on the interface level, the solution is found by the minimization of the total potential energy with respect to the displacements and the damage field. An elastic interface constitutive law allowing for a normal and tangential displacement opening is adopted in the pre‐fracture regime. Assuming, that a crack propagates according to the Griffith's criterion of brittle fracture, fracture initiates and propagates in normal opening mode. Biot's theory is applied both to the bulk and interface elements for the simulation of fluid driven fracture in fully saturated materials. The pressure field within the interfaces is averaged between the pressure at the bulk element faces. Pressure continuity is enforced by means of a penalty functional. The flow within the fracture is modeled by the cubic law taking the displacement and damage variables into account. A number of numerical benchmark tests, which include comparisons with experimental results and analytical solutions, are performed to demonstrate the performance of the model. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cohesive zone fracture modelling of asphalt pavements with applications to design of high-performance asphalt overlays

Mechanism of fracture in a viscoelastic heterogeneous composite with thermo-rheological properties such as, asphalt mixture is quite involved and cannot be correctly simulated with simpler linear elastic fracture mechanics constitutive laws. Over the last decade and half, a number of researchers have adopted use of cohesive zone (CZ) fracture models for simulation of fracture in asphalt mixtures. CZ interface elements are utilised in finite element (FE) models for representation of crack path, these elements follow traction–displacement relationships that allow for gradually degrading traction capabilities along the crack path with increasing level of crack opening. This paper presents a review of CZ modelling approach for simulation of asphalt pavement and overlay cracking performances. Suitability of CZ modelling approach for capturing discrete fracture in asphalt mixtures at low temperatures is presented through simulation of lab scale test. An example is also presented to demonstrate applicability of CZ-based modelling effort in capturing the crack initiation and propagation in asphalt mixtures at low temperatures. Thereafter, an FE-based pavement simulation approach is discussed that can be utilised in design of asphalt overlays to lower the propensity of reflective cracking. A case study of designing asphalt overlay systems for four real-life pavements in Minnesota is presented to demonstrate the applicability of the CZ-based modelling approach in conducting mechanistic design of asphalt overlays.

Nonlocal elasticity tensors in dislocation and disclination cores

Nonlocal elastic constitutive laws are introduced for crystals containing defects such as dislocations and disclinations. In addition to pointwise elastic moduli tensors adequately reflecting the elastic response of defect-free regions by relating stresses to strains and couple-stresses to curvatures, elastic cross-moduli tensors relating strains to couple-stresses and curvatures to stresses within convolution integrals are derived from a nonlocal analysis of strains and curvatures in the defects cores. Sufficient conditions are derived for positive-definiteness of the resulting free energy, and stability of elastic solutions is ensured. The elastic stress/couple stress fields associated with prescribed dislocation/disclination density distributions and solving the momentum and moment of momentum balance equations in periodic media are determined by using a Fast Fourier Transform spectral method. The convoluted cross-moduli bring the following results: (i) Nonlocal stresses and couple stresses oppose their local counterparts in the defects core regions, playing the role of restoring forces and possibly ensuring spatio-temporal stability of the simulated defects, (ii) The couple stress fields are strongly affected by nonlocality. Such effects favor the stability of the simulated grain boundaries and allow investigating their elastic interactions with extrinsic defects, (iii) Driving forces inducing grain growth or refinement derive from the self-stress and couple stress fields of grain boundaries in nanocrystalline configurations.

Energy Harvesting with a Piezoelectric Thunder

In this work the energy harvesting performance of a piezoelectric curved energy generator (THUNDER) is studied via experimental and analytical methods. The analytical model of the THUNDER is created based on the linear mechanical electrical constitutive law of the piezoelectric material, the linear elastic constitutive law of the substrate, and the Euler-Bernoulli beam theory. With these linear modal functions, the Rayleigh-Ritz approach was used to then obtain the reduced mechanical electrical coupled modulation equations. With above analytical model, two types of energy harvest circuit are proposed and compared: 1) directly charging mode at low level excitation, and 2) memory stored optimal duty cycle step-down converter mode at high level excitation. The value of the optimal duty cycle is determined based on the characteristics of the vibration signals of the ambient vibration source. To reduce the energy consumption of the microcontroller, the optimum duty cycle values are stored in the microprocessor instead of doing onsite computing during energy harvesting process. For the purpose of designing a low power cost mechanical switch to control the operation of both modes, the threshold voltage between these two operation modes is converted into threshold displacement.

Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory

The paper presents the formulation of the elastic constitutive law for functionally graded materials (FGM) on the grounds of nonlinear 6-parameter shell theory with the 6th parameter being the drilling degree of freedom. The material law is derived by through-the-thickness integration of the Cosserat plane stress equations. The constitutive equations are formulated with respect to the neutral physical surface. The influence of the power-law exponent, micropolar characteristic length is evaluated in geometrically nonlinear FEM analyses. The results obtained with the neutral physical surface approach are compared with those computed with the middle surface approach. The influence of choice of the reference surface is observed especially in nonlinear stability analysis.

Energy harvesting efficiency optimization via varying the radius of curvature of a piezoelectric THUNDER

In this work the energy harvesting performance of a piezoelectric curved energy generator (THin layer UNimorph DrivER (THUNDER)) is studied via experimental and analytical methods. The analytical model of the THUNDER is created based on the linear mechanical electrical constitutive law of the piezoelectric material, the linear elastic constitutive law of the substrate, and the Euler–Bernoulli beam theory. With these linear modal functions, the Rayleigh-Ritz approach was used to obtain the reduced mechanical–electrical coupled modulation equations. The analytical model is verified by the experimental results. Both the experimental and analytical results of the THUNDER’s AC power output, DC power output with Rectifier Bridge and a capacitor, as well as the power output with a microcontroller energy harvesting circuit are reported. Based on the theoretical model, the analytical solution of the DC power is derived in terms of the vibration amplitude, frequency, and the electrical load. To harvest energy from low-frequency vibration source by a piezoelectric generator requires the piezoelectric device possessing low resonance frequency and good flexibility. The THUNDER developed by Langley Research Center exhibits high power when it is used as an energy generator and large displacement when it is used as an actuator. Compared to the less flexible PZT, although THUNDER is more difficult to model, THUNDER has better vibration absorption capacity and higher energy recovery efficiency. The effect of the THUNDER’s radius of curvature on energy harvesting efficiency is mainly investigated. We set the THUNDER’s radius of curvature as a dynamic tuning parameter which can tune the piezoelectric generators’ frequency with the source excitation frequency.

Bracing systems in masonry buildings. Part 1: Simple application of FE programs / Aussteifungssysteme im Mauerwerksbau. Teil 1: Einfache Anwendung von FE‐Programmen

AbstractDue to the low tensile strength of masonry perpendicular to the bed joint, masonry wall panels have non‐linear material properties. Assuming simple elastic constitutive laws, this article presents two modelling variants, which consider the lack of tensile strength in a simple manner. Both variants are investigated for their advantages and disadvantages. In a second part of the article, the application of the methods will be illustrated through the example of a four‐storey building.

Numerical homogenization of elastic and thermal material properties for metal matrix composites (MMC)

A two-scale material modeling approach is adopted in order to determine macroscopic thermal and elastic constitutive laws and the respective parameters for metal matrix composite (MMC). Since the common homogenization framework violates the thermodynamical consistency for non-constant temperature fields, i.e., the dissipation is not conserved through the scale transition, the respective error is calculated numerically in order to prove the applicability of the homogenization method. The thermomechanical homogenization is applied to compute the macroscopic mass density, thermal expansion, elasticity, heat capacity and thermal conductivity for two specific MMCs, i.e., aluminum alloy Al2024 reinforced with 17 or 30 % silicon carbide particles. The temperature dependency of the material properties has been considered in the range from 0 to $$500{\,}^\circ \mathrm {C}$$ , the melting temperature of the alloy. The numerically determined material properties are validated with experimental data from the literature as far as possible.

Combined bone internal and external remodeling based on Eshelby stress

Models for the internal and external bone remodeling are developed in the framework of the thermodynamics of irreversible processes. Internal remodeling is essentially accounted for by an evolution of the internal bone density versus the trace of the Eshelby stress. The growing surface is endowed with a specific mechanical behavior elaborated from a surface potential, depending upon the elastic surface strain and the external normal vector. The surface remodeling velocity is related to the driving force for growth identified as the surface divergence of Eshelby stress. The developed formalism is applied to simulate bone internal and external remodeling for 2D geometries, showing the influence of external mechanical stimuli on the evolution of the external shape of bone. Simulations of the evolution of the shape and density of bone structures are carried out successively at mesoscopic, microscopic and macroscopic levels, the former representing cellular level simulations of the trabecular unit cell. At the macroscopic level, a linear elastic constitutive law for trabecular bone is derived, relying on the discrete homogenization approach, whereby the continuum parameters such as stiffness evolve with morphology as remodeling occurs within bone. The developed numerical platform is able to simulate combined bone internal and external remodeling at both trabecular and macroscopic scales.

Phase-field analysis of finite-strain plates and shells including element subdivision

With the theme of fracture of finite-strain plates and shells based on a phase-field model of crack regularization, we introduce a new staggered algorithm for elastic and elasto-plastic materials. To account for correct fracture behavior in bending, two independent phase-fields are used, corresponding to the lower and upper faces of the shell. This is shown to provide a realistic behavior in bending-dominated problems, here illustrated in classical beam and plate problems. Finite strain behavior for both elastic and elasto-plastic constitutive laws is made compatible with the phase-field model by use of a consistent updated-Lagrangian algorithm. To guarantee sufficient resolution in the definition of the crack paths, a local remeshing algorithm based on the phase-field values at the lower and upper shell faces is introduced. In this local remeshing algorithm, two stages are used: edge-based element subdivision and node repositioning. Five representative numerical examples are shown, consisting of a bi-clamped beam, two versions of a square plate, the Keesecker pressurized cylinder problem, the Hexcan problem and the Muscat-Fenech and Atkins plate. All problems were successfully solved and the proposed solution was found to be robust and efficient.

Beam-Column Joint Model for Nonlinear Analysis of Non-Seismically Detailed Reinforced Concrete Frame

An efficient and simplified plane beam-column joint model that can describe the strength deterioration, stiffness degradation, and pinching effect was developed for the nonlinear analysis of non-seismically detailed reinforced concrete frames. The proposed beam-column joint model is a super-element consisting of eight spring components and one panel zone component, representing the bond-slip mechanism of the longitudinal reinforcement and the shear deformation mechanism of the joint concrete core region, respectively. In order to represent the dynamic response at the system level, the elastic constitutive law is applied to the eight connector springs, while the Bouc-Wen-Baber-Noori (BWBN) model is adopted to describe the hysteretic behavior of the panel zone component. For the implementation of the finite element analysis, the algorithmically consistent tangent of the BWBN model is derived as a uni-axial constitutive model, while the initial stiffness of the panel zone component is determined by the concrete compression strut assumption. The accuracy and efficiency of the proposed beam-column joint model were calibrated at both the component and structural levels by comparing the simulated results with the experimental data for non-seismically detailed joint sub-assemblages and a reinforced concrete plane frame.

Heterogeneous and anisotropic long-term concrete damage of the dez arch dam using thermal inverse analysis

AbsractIn this paper, an assumption used in the recent work of the author and his contributors considers the long-term concrete damage of the Dez arch dam as a homogeneous and isotropic process, was investigated in more detail and was adjusted. To this end, the vertical dam sections were divided into nine and six subsections along the thickness and height directions of the dam, respectively. In each subsection, a transversely isotropic damaged elastic constitutive law was considered for diagnostic analyses. Following the previous authors’ mentioned work, an optimization procedure (minimization of a certain error function) accompanied with an inverse thermal analysis was carried out. That analysis was performed within the framework of finite element (FE) method. Mentioned error function was defined as the sum of the squared residuals. Residuals were set as difference between nodal recorded displacements with inverse pendulums of the dam and the corresponding computed ones with the proposed model. Parameters considered as unknowns in the present inverse analysis, which have contributions in that above-mentioned error function, were the five independent elastic moduli presented in the formulation of the earlier mentioned transversely isotropic damaged elastic constitutive law: two Young’s moduli: one in the planes parallel to the up and down stream surfaces of the dam, called in-plane and the other in the perpendicular planes to those planes, named out-of-plane Young’s modulus; two Poisson’s ratios (in-plane and out-of-plane); and one shear modulus (out-of-plane). These parameters were identified performing numerous thermal inverse analyses. Obtained results revealed that the long-term damage of concrete of the Dez dam is a heterogeneous and anisotropic phenomenon because that the magnitude of the mentioned error function was obtained smaller than the corresponding value in the previous study which had been performed before based on the homogeneous and isotropic damage evolution assumption.