Nonlinear Finite Element Procedure Research Articles

Vibration of electrically actuated MEMS Timoshenko microbeams based on a hierarchical beam element

In this paper, vibration of Timoshenko microbeams with an axial force in micro-electromechanical systems (MEMS) is studied for the first time by using a nonlinear finite element procedure. Based on the von Kármán geometric nonlinearity and the modified couple stress theory (MCST), a beam element is formulated by employing hierarchical functions to interpolate the displacement field. Using the derived element, the discretized equation of motion for the microbeam is constructed and then solved by the Newton-Raphson iterative procedure in conjunction with the Newmark method. The natural frequencies, pull-in voltages and dynamic deflections are computed for a clamped-clamped microbeam under electrostatic actuation of a given direct current (DC) voltage. The numerical result reveals that the axial force and the microsize effect have a significant influence on the vibration, and the fundamental frequency of the microbeams is underestimated by ignoring the size effect. The effects of the axial force, the applied voltage and the material length scale parameter on the vibration of the beam are studied in detail and highlighted.

Performance of strengthened RC beams with torsional CFRP wraps in monolithic framed structures: Experimental and numerical investigation

The performance of externally applied wraps to simply supported reinforced concrete (RC) beams under pure torsion has been well-defined in the literature. In most cases, the previous research works utilised carbon fibre reinforced polymers (CFRP) wraps with different configurations to enhance the beams torsional capacities. While the effectiveness of these wraps in monolithic framed beams in RC buildings is less understood. Further, there is still a lack of information regarding the influence of interacted-combined loads on both the wrapped beams and the joined columns. This paper deals with the presence of different wrapping schemes on the response of framed beams that joined to columns, and how different levels of eccentric torsional forces combined with bending and shear forces can alter the damage state. The experimental programme of this study involved a series of beam-column assemblies tested under concentric and eccentric combined loads. Based on the test observations, a nonlinear finite element (FE) procedure is proposed and validated for simulating the full response of RC beam-column assemblies. This validated FE procedure is then used to explore critical parameters which govern the wraps performance and the failure modes. The outcomes of this study provide comprehensive insights into the role of the interacted forces and the wrapping configurations on the developed deformations and the members integrity. In particular, the results confirmed that the torsional force adversely affected the flexural capacities in proportion to the ratio of the torsion applied. Meanwhile, the interacted forces can induce substantial strain and damage levels that can be extended to the joined column. The contribution of CFRP wraps is found to be limited to the concrete strut’s capacity and the extent of torsional shear-flow in consistent with the imposed loads ratios and the developed truss mechanism.

Recent Advances in the Absolute Nodal Coordinate Formulation: Literature Review From 2012 to 2020

Abstract Absolute nodal coordinate formulation (ANCF) is a nonincremental nonlinear finite element procedure that has been successfully applied to the large deformation analysis of multibody systems for more than two decades. Although a comprehensive review on ANCF was conducted by Gerstmayr et al. (2013, “Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems,” J. Comput. Nonlinear Dyn., 8(3), p. 031016), significant theoretical developments have been made since then at a much faster pace to improve the element accuracy and computational efficiency. In order to overview recent advances in ANCF simulation capabilities that are not covered in the first review paper, this paper aims to conduct a comprehensive review of 259 papers concerning ANCF published from 2012 to 2020. It is shown that the ANCF element library has grown substantially for beam, plate/shell, solid elements, eliminating drawbacks of ANCF elements developed earlier. The application areas have extended, especially in the aerospace field, and the enhanced ANCF simulation capabilities have been demonstrated in solving challenging engineering problems. Research efforts have been made continually to integrate computer-aided design (CAD) and analysis with ANCF elements. Furthermore, computational improvements and multiphysics simulations have become major research topics for ANCF. It is also demonstrated that the accurate ANCF geometry description can be exploited to facilitate structural optimization of multibody systems.

Size-dependent behavior of a MEMS microbeam under electrostatic actuation

The size-dependent behavior of a silicon microbeam with an axial force in MEMS is studied using a nonlinear finite element procedure. Based on a refined third-order shear deformation theory and the modified couple stress theory (MCST), nonlinear differential equations of motion for the beam are derived from Hamilton’s principle, and they are transferred to a discretized form using a two-node beam element. Newton-Raphson based iterative procedure is used in conjunction with Newmark method to obtain the pull-in voltages and deflections of a clamped-clamped microbeam under electrostatic actuation. The influence of the axial force, applied voltage and material length scale parameter on the behavior of the beam is studied in detail and highlighted.

Biomechanical design framework for prosthetic feet: Experimentally validated non-linear finite element procedure

Performance evaluation of prosthetic feet during their design is typically performed experimentally, which may be time and cost intensive. This work presents a first-of-its-kind application of a numerical procedure for the a priori determination of various stance phase biomechanical parameters of a prosthetic foot, such as its roll-over characteristics, centre of pressure trajectory, ankle flexion moment arm and ankle range of motion, to aid in its design. The numerical procedure is based on finite element analysis, which includes geometric, material and contact non-linearity. Boundary conditions emulating the rocker-based inverted pendulum model were employed to evaluate the biomechanical parameters. The finite element model was validated by employing an inverted pendulum-based apparatus using the structurally complex Ottobock Solid Ankle Cushioned Heel (SACH) prosthetic foot as the test device.A comparison of the numerical and experimental results showed low magnitude of errors. For example, the percentage error of the radius of curvature of the roll-over shape was ~0.1%. The differences found appear to be clinically insignificant, which substantiates the reliability of the model. The proposed numerical model can be employed to obtain detailed a priori insights into the biomechanical parameters influencing a prosthetic foot's characteristics during gait, which can better inform the design, analysis and prescription of prosthetic feet.

Phase-field modeling of coupled anisotropic plasticity–ductile fracture in rate-dependent solids

In this paper, an anisotropic ductile fracture is studied using a phase-field model. This model is on the basis of the energy principle in which a scalar field is added as a phase-field variable to determine the probability paths of cracks. Therefore, the system of equations, included for the displacement and phase field, is obtained and solved in a nonlinear finite element procedure through a user element subroutine in ABAQUS commercial software. The effect of anisotropy on cracks’ paths for an anisotropic fracture is assessed according to the Hill criterion for ductile materials. Moreover, an implicit integration algorithm for an anisotropy model combined with a phase field is shown for the plane strain state. The proposed model can also be used for the combination of various constitutive models along with an anisotropy such as the anisotropic rate-dependent (visco-Hill) and the anisotropic rate-independent (elasto-Hill). Furthermore, the threshold parameter is introduced to examine the plastic deformation influence on cracks initiation and propagation in this study. Finally, the influence of the anisotropy on the responses of different specimens in various fracture processes is interrogated. The accuracy of the suggested model is demonstrated by juxtaposing the achieved and given experimental and numerical results in the way that it can be able to simulate the anisotropy effects on the gradual evolution of cracks from ductile to brittle.

Finite element analysis of coupled phase-field and thermoelasticity equations at large strains for martensitic phase transformations based on implicit and explicit time discretization schemes

In this paper, a staggered, nonlinear finite element procedure is developed to solve the large-strain based coupled system of time dependent Ginzburg-Landau (GL) and thermoelasticity equations for phase transformations at the nanoscale. Geometrical nonlinearities are included based on the total Lagrangian description where the total deformation gradient is defined as the multiplicative decomposition of elastic, transformational and thermal deformation gradient tensors. The principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion whose discretized form is solved using the iterative methods which gives the thermal and transformational deformation gradient tensors. Next, the elastic deformation gradient tensor and the first Piola-Kirchhoff stress are calculated and substituted in the GL equation which gives the phase order parameter. The weighted residual method is used to derive the corresponding finite element form which allows for the phase-dependent surface energy BCs. Explicit and different implicit methods belonging to the generalized trapezoidal family are used for time discretization of the GL equation. Various examples of phase transformations are simulated and discussed. The effect of different time integration schemes are also investigated. The current study allows for a proper modeling of various phase field problems at large and small strains.

On the stress state-based coupled plasticity – Ductile damage model for aluminum alloys considering the influence of high-rate impulsive preload

The purpose of this study is to produce a ductile damage model for 7075-T6 aluminum alloy by correlating the effect of stress - state parameters to the pre-mechanical working exited by an impulsive shock loading. A stress state-based ductile damage model is coupled implicitly with constitutive plasticity framework to anticipate a generalized true fracture locus of the Al-7075-T6. To simulate the constitutive equations, a fully nonlinear finite element procedure is developed to capture the corresponding stress state, hardening, and damage evolutions during the high- velocity impact preload as well as a subsequent progressive static deformation in the tensile tests. The results are calibrated by selected experimental tests representing different stress - states. Also, two conditions are investigated for the stress-strain curves obtained from the uniaxial tests; specimens with the high rate pre-mechanical workings and specimen without pre-mechanical workings. Moreover, by applying two fracture initiation criteria, i.e., the Hosford- Coulomb and Xue models, two types of the fracture locus are predicted for the Al-7075-T6 alloys in terms of the plastic strains and stress - state parameters under the conditions mentioned above. By considering the experimental data, a new evolutionary model of the ductile damage, which is coupled with the J2 plasticity is proposed to relate the initial rate - dependent stress state. By using the fracture criteria, our model may be reliable for a phenomenological scheme of the damage evolution and useful for comparisons with the Xue damage model. Finally, analysis of the fracture surfaces by Scanning Electron Microscope revealed that the fracture surfaces of all samples mainly contained the dimples, which indicates the presence of ductile fracture in all specimens before and after the shock loading.

Explicit nonlinear finite element approach to the Lagrangian-based coupled phase field and elasticity equations for nanoscale thermal- and stress-induced martensitic transformations

In this paper, a nonlinear finite element procedure is developed to solve the coupled system of phase field and elasticity equations at large strains for martensitic phase transformations at the nanoscale. The transformation is defined based on an order parameter which varies between 0 for austenite to one for martensite. The phase field equation relates the rate of change of the order parameter to the Helmholtz free energy. Both the Helmholtz free energy and the transformational strain tensor depend on the order parameter, and the coupling between the phase field and elasticity equations occurs due to the presence of elastic energy in the Helmholtz free energy and the transformational strain in the total strain. The deformation gradient tensor is introduced as a multiplicative decomposition of elastic and transformational gradient tensors. A staggered strategy is used to solve the coupled system of equations so that first, the principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion. It is then discretized and solved using the Newton–Raphson method which gives the displacements and consequently, the deformation gradient tensor. Next, since the order parameter is obtained from the phase field equation from the previous time step, the elastic deformation gradient tensor and the first Piola–Kirchhoff stress can be calculated and substituted in the phase field equation to find the order parameter for the current time step. The weighted residuals method is used to derive the finite element form of the phase field equation and the explicit method is used for its time discretization. The finite element procedure is well verified by comparing the obtained results with those from other simulations. Examples of thermal- and stress-induced phase transformations are presented, including austenite–martensite interface propagation and growth of preexisting martensitic regions. The developed algorithm and code can be advanced to solve kinetic problems coupled with mechanics at large strains for phase transformations and similar phenomena at the nanoscale.

Developed moving mesh method for high-speed railway pantograph-catenary interaction based on nonlinear finite element procedure

ABSTRACTIn modern electrified railway industries, the transmission quality of the electric power from the power supply to the locomotive is determined by the dynamic performance of the catenary-pantograph interaction. The mathematical modelling is a high-performance tool to study the dynamic behaviour of the catenary-pantograph system. In order to improve the calculation efficiency, a moving mesh method is implemented with a nonlinear finite element model of catenary. The contact wire is re-meshed according to the contact point between the pantograph and the catenary in each simulation time step. Addressing the unbalanced forces caused by re-meshing the contact wire, an iterative procedure is proposed based on the equilibrium equations of the flexible cable element. Through the analysis of simulation results, the proposed method shows good performance in improving the simulation efficiency and eliminating unbalanced forces. The comparison with the measurement data demonstrates the preliminary validation of the proposed method.

Mechanical Property Analysis of rubber-like materials under Large De-formation in Uniaxial Tension, Biaxial Tension and Expansion of Cylindrical Membrane

Based on the modified strain energy function from Gao, uniaxial tension and biaxial tension of rubber like materials has been researched. For verification purposes, the proposed constitutive model from Gao has been implemented in a general-purpose FEA software. And then the nonlinear finite element procedure is presented for the analysis of rubber-like materials. The volumetric incompressibility condition of the deformation from rubber like materials is included in the formulation by using the penalty method. By utilizing the FEA software ABAQUS, uniaxial tension and biaxial tension model from rubber like materials have been established. In order to compare the FEA results with theoretical results, curve stress-principal stretch has been plotted with different material parameters, which indicate the correctness and rationality of theoretical analysis. In the end, analysis of cylindrical rubber membrane has been proposed based on the proposed strain energy function. The results show that the constitutive parameter n has a strengthening effect on the rubber membrane material and instability of cylindrical rubber membrane will be occur as the circumferential or longitudinal principal stretch increases. This research has revealed the deformational mechanism and provided reasonable reference for the design of rubber like materials. DOI: http://dx.doi.org/10.5755/j01.mech.24.1.18377

Numerical Evaluation of Compressive Strain Capacity for API X100 Line Pipe

In the strain-based design, pipeline steels are required to satisfy not only the stress capacity but also the strain capacity. Thus, considerable efforts have been made to meet the various requirements of the pipe-material properties that might undergo large deformation during the installation stage in various hostile environments. An adequate compressive strain capacity can help effectively avoid local buckling. Hence, highly deformable steel becomes crucial in achieving the required strain capacity. This study proposes a nonlinear finite element procedure based on the commercial software ABAQUS combined with the User-defined Material Module (UMAT), which is created using Fortran language. The Gurson–Tvergaard–Needleman (GTN) model is implemented in UMAT to fully consider the nonlinear behavior of an API X100 pipe subjected to compressive loading. The numerical simulation results of the full-scale bending test were compared with the experimental results to verify the nonlinear finite element procedure equipped with the GTN model. The simulation results are in good agreement with the experimental results. The parametric studies reveal the effects of geometric imperfections and material properties on the compressive strain limit.

Study on Wind-Induced Vibration Behavior of Railway Catenary in Spatial Stochastic Wind Field Based on Nonlinear Finite Element Procedure

Due to its long-span structure and large flexibility, an electrified railway catenary is very sensitive to environmental wind load, especially the time-varying stochastic wind, which may lead to a strong forced vibration of contact line and deteriorate the current collection quality of the pantograph–catenary system. In this paper, in order to study the wind-induced vibration behavior of railway catenary, a nonlinear finite element procedure is implemented to construct the model of catenary, which can properly describe the large nonlinear deformation and the nonsmooth nonlinearity of dropper. The spatial stochastic wind field is developed considering the fluctuating winds in along-wind, vertical-wind, and cross-wind directions. Using the empirical spectra suggested by Kaimal, Panofsky, and Tieleman, the fluctuating wind velocities in three directions are generated considering the temporal and spatial correlations. Based on fluid-induced vibration theory, the model of fluctuating forces acting on catenary are developed considering the spatial characteristics of catenary. The time- and frequency-domain analyses are conducted to study the wind-induced vibration behavior with different angles of wind deflection, different angles of attack, as well as different geometries of catenary. The effect of spatial wind load on contact force of pantograph–catenary system is also investigated.

Sliding Mode Control with PD Sliding Surface for High-Speed Railway Pantograph-Catenary Contact Force under Strong Stochastic Wind Field

As is well known, the external disturbance (especially the stochastic wind load) has nonnegligible effect on the operation of pantograph-catenary system, which may cause the strong fluctuation in contact force as well as the increased occurrence of contact loss. In order to improve the current collection quality of a high-speed railway pantograph-catenary system under a strong stochastic wind field, a sliding mode controller with a proportional-derivative (PD) sliding surface for a high-speed active pantograph is proposed. The nonlinear finite element procedure is employed to establish the catenary model. The fluctuating wind speeds along catenary are simulated using empirical spectrums. The buffeting forces exerted on contact and messenger wires are derived to construct the stochastic wind field along the catenary. A PD sliding surface is properly determined to guarantee that the mechanical impedance of pantograph head at the dominant frequencies of contact force decreases when the sliding surface approaches zero. Through several numerical simulations with different wind velocities and wind angles, the control performance of two popular control laws (proportional switching law and constant switching law) is evaluated.

Anisotropic Swelling in Fiber-reinforced Hydrogels: An Incremental Finite Element Method and Its Applications in Design of Bilayer Structures

In this paper, we present a theoretical and finite element framework for the inhomogeneous swelling of fiber-reinforced anisotropic hydrogels and investigate the effects of their orientations and moduli on the swelling behaviors of hydrogel bilayers. The effects of fibers on swelling are incorporated into the constitutive equations by introducing the corresponding energy contribution, expressed as the invariants related to the deformation of fibers, into the Flory–Rehner model. Unified constitutive equations and corresponding tangent moduli for anisotropic hydrogels reinforced with two families of fibers are deduced. A nonlinear finite element procedure is developed to simulate the inhomogeneous steady anisotropic swelling, where the chemical potential is prescribed in an incremental way. The accuracy and effectiveness of the numerical procedure are demonstrated by the anisotropic swelling of hydrogel blocks with one or two families of fibers. Based on the numerical procedure, many kinds of swelling configurations for hydrogel bilayers with different fiber orientations and modulus coefficients can be obtained, ranging from pure bending, twisting or saddle-like to combinations of them, which provide systematic guidance for the design of anisotropic-hydrogel based bilayers.

Anisotropic Swelling in Fiber-Reinforced Hydrogels: An Incremental Finite Element Method and Its Applications in Design of Bilayer Structures

In this paper, we present a theoretical and finite element framework for the inhomogeneous swelling of fiber-reinforced anisotropic hydrogels and investigate the effects of their orientations and moduli on the swelling behaviors of hydrogel bilayers. The effects of fibers on swelling are incorporated into the constitutive equations by introducing the corresponding energy contribution, expressed as the invariants related to the deformation of fibers, into the Flory–Rehner model. Unified constitutive equations and corresponding tangent moduli for anisotropic hydrogels reinforced with two families of fibers are deduced. A nonlinear finite element procedure is developed to simulate the inhomogeneous steady anisotropic swelling, where the chemical potential is prescribed in an incremental way. The accuracy and effectiveness of the numerical procedure are demonstrated by the anisotropic swelling of hydrogel blocks with one or two families of fibers. Based on the numerical procedure, many kinds of swelling configurations for hydrogel bilayers with different fiber orientations and modulus coefficients can be obtained, ranging from pure bending, twisting or saddle-like to combinations of them, which provide systematic guidance for the design of anisotropic-hydrogel based bilayers.

A non-linear constitutive relation for the analysis of FRCM elements

Nowadays, the strengthening of existing buildings represents one of the most innovative fields within current research in civil engineering. Among the developed techniques, a recent solution consists in the use of FRCM composites (Fabric Reinforced Cementitious Matrix), which are obtained by placing a dry grid of fibers inside a cement-based material (mortar). In comparison with traditional systems, the use of FRCM seems to provide some advantages; however, a full understanding of the mechanical properties of each component (mortar and fibers) and of their interaction, as well as their effect on the strengthened structure, still represents an open research topic. This work aims to be a first attempt to numerically simulate the global behavior of FRCM composites through the development of a macroscopic constitutive model subsequently implemented into a Non-Linear Finite Element (NLFE) procedure. The effectiveness of the proposed procedure is verified through comparisons with significant experimental results available in technical literature, relative to FRCM tension ties. The influence exerted by the adoption of different materials (such as Poliparafenilenbenzobisoxazolo (PBO) and carbon) for the internal fiber grid on the global behavior is also analyzed and discussed.

The origin of in-plane stresses in axially moving orthotropic continua

• In-plane motion of an axially travelling orthotropic sheet was modelled. • Velocity difference between supports generates strain due to mass conservation. • In a travelling viscoelastic material the axial strain varies along the length. • Small deformations of free edges due to the Poisson effect affect the velocity. • Inertial effects lead to additional cross-directional contraction. In this paper, we address the problem of the origin of in-plane stresses in continuous, two-dimensional high-speed webs. In the case of thin, slender webs, a typical modelling approach is the application of a stationary in-plane model, without considering the effects of the in-plane velocity field. However, for high-speed webs this approach is insufficient, because it neglects the coupling between the total material velocity and the deformation experienced by the material. By using a mixed Lagrange–Euler approach in model derivation, the solid continuum problem can be transformed into a solid continuum flow problem. Mass conservation in the flow problem and the behaviour of free edges in the two-dimensional case are both seen to influence the velocity field. We concentrate on solutions of a steady-state type, and study briefly the coupled nature of material viscoelasticity and transport velocity in one dimension. Analytical solutions of the one-dimensional equation are presented with both elastic and viscoelastic material models. The two-dimensional elastic problem is solved numerically using a nonlinear finite element procedure. An important new fundamental feature of the model is the coupling of the driving velocity field to the deformation of the material, while accounting for small deformations of the free edges. The results indicate that inertial effects produce an additional contribution to elastic contraction in unsupported, free webs.

Effect of Soil Properties on the Response of Pile to Underground Explosion

This paper develops and presents a fully coupled nonlinear finite element (FE) procedure to treat the response of piles to ground shocks induced by underground explosions. The Arbitrary Lagrange Euler coupling formulation is used in the study with proper state material parameters and equations. Pile responses in four different soil types, viz, saturated soil, partially saturated soil, and loose and dense dry soils are investigated and the results compared. Numerical results are validated by comparing them with those from a standard design manual. Blast wave propagation in soils, horizontal pile deformations, and damages in the pile are presented. Pile damage presented through plastic strain diagrams will enable vulnerability assessment of the piles under the blast scenarios considered. The numerical results indicate that the blast performance of single pile foundations embedded in saturated soil and loose dry soil are more severe than those embedded in partially saturated soil and dense dry soil. The present findings can serve as a benchmark reference for future analysis and design.

Strain-Based Analysis for Geometrically Nonlinear Beams: A Modal Approach

A strain-based geometrically nonlinear finite element beam formulation for structural dynamic and aeroelastic analysis of slender beams and wings has been successfully applied to study the coupled nonlinear aeroelasticity and flight dynamics of different, very flexible aircraft. To improve the ability to use such aeroelastic framework in control design, a modal-based approach is developed in this paper to solve the strain-based nonlinear beam equations with a reduced number of degrees of freedom. The modal approach is applied to study geometrically nonlinear static and transient problems of constrained and free slender beams, subject to external structural or aerodynamic excitations. Accuracy and performance of the new formulation are compared with the direct nonlinear finite element procedure.