Hermite Finite Element Method for Vibration Problem of Euler-Bernoulli Beam on Viscoelastic Pasternak Foundation

Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Euler-Bernoulli beam on viscoelastic Pasternak foundation can be used to analyze the deformation and response of buildings under complex geological conditions. In this paper, we use Hermite finite element method to get the numerical approximation scheme for the vibration equation of viscoelastic Pasternak foundation beam. Convergence and error estimation are rigourously established. We prove that the fully discrete scheme has convergence order O(τ2+h4), where τis time step size and his space step size. Finally, we give four numerical examples to verify the validity of theoretical analysis.

Finite element modeling of the viscoelastic responses of the eye during microvolumetric changes

A linear viscoelastic finite element model was built to investigate factors that influenced the intraocular pressure (IOP) elevations due to micro-volumetric changes in the eye at three different rates. The viscoelastic properties of the cornea and the sclera, including the instantaneous modulus, equilibrium modulus, and relaxation time constants, parametrically varied to examine their effects on IOP elevations at different rates of volumetric changes. The simulated responses were in good agreement with the previously reported experimental results obtained from porcine globes, showing the general trend of higher IOP elevations at faster rates. The simulations showed that all viscoelastic properties influenced the profile of the dynamic IOP due to volumetric changes, and the relative significance of a specific parameter was highly dependent on the rate of change.

Spatially-graded 3D-printed viscoelastic truss metamaterials for impact trajectory control and energy absorption

This study demonstrates that two-and three-dimensional spatially graded,truss-based polymeric-material metamaterials can be designed for beneficial impact mitigation and energy absorption capabilities.Through a combination of numerical and experimental techniques,we highlight the broad property space of periodic viscoelastic trusses,realized using 3D printing via selective laser sintering.Extending beyond periodic designs,we investigate the impact response of spatially variant viscoelastic lattices in both two and three dimensions.Our result reveal that introducing spatial variations in lattice topology allows for redirecting of the impact trajectory,opening new opportunities for engineering and tailoring lightweight materials with target impact functionality.This is achieved through the combined selection of base material and metamaterial design.

A mixed finite element scheme for viscoelastic flows with XPP model

A mixed finite element formulation for viscoelastic flows is derived in this paper, in which the FIC （finite incremental calculus） pressure stabilization process and the DEVSS （discrete elastic viscous stress splitting） method using the Crank-Nicolson-based split are introduced within a general framework of the iterative version of the fractional step algorithm. The SU （streamline-upwind） method is particularly chosen to tackle the convective terms in constitutive equations of viscoelastic flows. Thanks to the proposed scheme the finite elements with equal low-order interpolation approximations for stress-velocity-pressure variables can be successfully used even for viscoelastic flows with high Weissenberg numbers. The XPP （extended Pom-Pom） constitutive model for describing viscoelastic behaviors is particularly integrated into the proposed scheme. The numerical results for the 4：1 sudden contraction flow problem demonstrate prominent stability, accuracy and convergence rate of the proposed scheme in both pressure and stress distributions over the flow domain within a wide range of the Weissenberg number, particularly the capability in reproducing the results, which can be used to explain the ＂die swell＂ phenomenon observed in the polymer injection molding process.

Seismic wave modeling in viscoelastic VTI media using spectral element method

Spectral element method（SEM） for elastic media is well known for its great flexibility and high accuracy in solving problems with complex geometries.It is an advanced choice for wave simulations.Due to anelasticity of earth media,SEM for elastic media is no longer appropriate.On fundamental of the second-order elastic SEM,this work takes the viscoelastic wave equations and the vertical transversely isotropic（VTI） media into consideration,and establishes the second-order SEM for wave modeling in viscoelastic VTI media.The second-order perfectly matched layer for viscoelastic VTI media is also introduced.The problem of handling the overlapped absorbed corners is solved.A comparison with the analytical solution in a twodimensional viscoelastic homogeneous medium shows that the method is accurate in the wave-field modeling.Furtherly,numerical validation also presents its great flexibility in solving wave propagation problems in complex heterogeneous media.This second-order SEM with perfectly matched layer for viscoelastic VTI media can be easily applied in wave modeling in a limited region.

Tensile Shock Physics in Compressible Thermoviscoelastic Solid Medium

This paper addresses tensile shock physics in thermoviscoelastic (TVE) solids without memory. The mathematical model is derived using conservation and balance laws (CBL) of classical continuum mechanics (CCM), incorporating the contravariant second Piola-Kirchhoff stress tensor, the covariant Green’s strain tensor, and its rates up to order n. This mathematical model permits the study of finite deformation and finite strain compressible deformation physics with an ordered rate dissipation mechanism. Constitutive theories are derived using conjugate pairs in entropy inequality and the representation theorem. The resulting mathematical model is both thermodynamically and mathematically consistent and has closure. The solution of the initial value problems (IVPs) describing evolutions is obtained using a variationally consistent space-time coupled finite element method, derived using space-time residual functional in which the local approximations are in hpk higher-order scalar product spaces. This permits accurate description problem physics over the discretization and also permits precise a posteriori computation of the space-time residual functional, an accurate measure of the accuracy of the computed solution. Model problem studies are presented to demonstrate tensile shock formation, propagation, reflection, and interaction. A unique feature of this research is that tensile shocks can only exist in solid matter, as their existence requires a medium to be elastic (presence of strain), which is only possible in a solid medium. In tensile shock physics, a decrease in the density of the medium caused by tensile waves leads to shock formation ahead of the wave. In contrast, in compressive shocks, an increase in density and the corresponding compressive waves result in the formation of compression shocks behind of the wave. Although these are two similar phenomena, they are inherently different in nature. To our knowledge, this work has not been reported in the published literature.

Finite Element Modelling of Car Seat with Hyperelastic and Viscoelastic Foam Material Properties to Assess Vertical Vibration in Terms of Acceleration

Primary objective of automobile seats is to offer adequate level of safety and comfort to the seated human occupant, primarily against vibration. Ideally, any sort of automotive seat is constructed by mechanical framework, cushion, backrest and headrest. The frame structures are made of metallic alloys, while the cushion, backrest and headrest are made of polyurethane foam material. During the design phase of automotive seat, the greatest challenge is to assign realistic material properties to foam material;as it is non-linear in nature and exhibit hysteresis at low level stress. In this research paper, a car seat has been modelled in finite element environment by implementing both hyperelastic and viscoelastic material properties to polyurethane foam. The car seat has been excited with the loads due to car acceleration and human object and the effects of vibration in terms of vertical acceleration at different locations have been measured. The aims of this simulation study are to establish a car seat with the foam material properties as accurately as possible and provide a finite element set up of car seat to monitor the vertical acceleration responses in a reasonable way. The RMS acceleration values for headrest, backrest and cushion have been found to be 0.91 mm/sec2, 0.54 mm/sec2 and 0.47 mm/sec2, respectively, which showed that the car seat foam can effectively be modelled through combined hyperelastic and viscoelastic material formulations. The simulation outputs have been validated through real life testing data, which clearly indicates that this computerized simulation technique is capable of anticipating the acceleration responses at different car seat segments in a justified way.

Acoustic viscoelastic modeling by frequency-domain boundary element method

Earth medium is not completely elastic, with its viscosity resulting in attenuation and dispersion of seismic waves. Most viscoelastic numerical simulations are based on the finite-difference and finite-element methods. Targeted at viscoelastic numerical modeling for multilayered media, the constant-Q acoustic wave equation is transformed into the corresponding wave integral representation with its Green＇s function accounting for viscoelastic coefficients. An efficient alternative for full-waveform solution to the integral equation is proposed in this article by extending conventional frequency-domain boundary element methods to viscoelastic media. The viscoelastic boundary element method enjoys a distinct characteristic of the explicit use of boundary continuity conditions of displacement and traction, leading to a semi-analytical solution with sufficient accuracy for simulating the viscoelastic effect across irregular interfaces. Numerical experiments to study the viscoelastic absorption of different Q values demonstrate the accuracy and applicability of the method.

Superconvergence of a Nonconforming Finite Element Approximation to Viscoelasticity Type Equations on Anisotropic Meshes

The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solution and the optimal error estimate of its derivative with respect to time are derived by using some novel techniques. Moreover, employing a postprocessing technique, the global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is studied.

Hermite Finite Element Method for a Class of Viscoelastic Beam Vibration Problem

Beam equation can describe the deformation of beams and reflect various bending problems and it has been widely used in large engineering projects, bridge construction, aerospace and other fields. It has important engineering practice value and scientific significance for the design of numerical schemes. In this paper, a scheme for vibration equation of viscoelastic beam is developed by using the Hermite finite element. Based on an elliptic projection, the errors of semi-discrete scheme and fully discrete scheme are analyzed respectively, and the optimal L2-norm error estimates are obtained. Finally, a numerical example is given to verify the theoretical predictions and the validity of the scheme.

Hybrid natural element method for viscoelasticity problems

A hybrid natural element method（HNEM） for two-dimensional viscoelasticity problems under the creep condition is proposed. The natural neighbor interpolation is used as the test function, and the discrete equation system of the HNEM for viscoelasticity problems is obtained using the Hellinger–Reissner variational principle. In contrast to the natural element method（NEM）, the HNEM can directly obtain the nodal stresses, which have higher precisions than those obtained using the moving least-square（MLS） approximation. Some numerical examples are given to demonstrate the validity and superiority of this HNEM.

Study on Postseismic Impact of Wenchuan Earthquake on the Sichuan-Yunnan Region Based on Three-Dimensional Viscoelastic Finite Element Method

Based on the lateral segmentation and vertical stratification characteristics of the crustal medium in Sichuan-Yunnan region,and the asymmetry of the static dislocations on the coseismic fault plane of the Wenchuan M_S8.0 earthquake,we built a three-dimensional viscoelastic finite element model of the crust in the Sichuan-Yunnan region.The postseismic impact of the Wenchuan M_S8.0 earthquake on the Sichuan-Yunnan region was studied.The results show that:①The far-field horizontal deformation caused by the viscoelastic relaxation of the medium in the 10 years after the earthquake is about 0-20 mm within the Sichuan-Yunnan diamond-shaped block,which has a greater influence on north side and smaller on south side.②In the 10 years after the earthquake,the far-field vertical deformation caused by the viscoelastic relaxation effect of the medium is small,and it shows an increase of about 0-4 mm in most areas of the Sichuan-Yunnan diamond-shaped block.③The Xianshuihe fault and the eastern segment of the East Kunlun fault,which are close to the seismogenic fault,show a high gradient on deformation fields after the earthquake.④In order to compare with the strong earthquake activity in the SichuanYunnan region after the Wenchuan earthquake,the horizontal stress state and the Coulomb failure stress change of the active block boundary are also calculated.From the spatial distribution of the coseismic and postseismic displacement field,the fault activity characteristics reflected by the stress state and the stress loading of the fault layer reflected by the Coulomb failure stress change,there is a certain correlation with the spatial distribution of strong earthquake activity in this region.

Reflective cracking viscoelastic response of asphalt concrete under dynamic vehicle loading

In order to investigate the mechanical response of reflective cracking in asphalt concrete pavement under dynamic vehicle loading, a finite element model is established in ABAQUS. The viscoelastic behavior is described by a prony series which is calculated through nonlinear fitting to the creep test data obtained in the laboratory. Based on the viscoelastic theory, the time-temperature equivalence principle, fracture mechanics and the dynamic finite element method, both the Jintegral and the mix-mode stress intensity factor are utilized as fracture evaluation parameters, and a half-sine dynamic loading is used to simulate the vehicle loading. Finally, the mechanical response of the pavement reflective cracking is analyzed under different vehicle speeds, different environmental conditions and various damping factors. The results indicate that increasing either the vehicle speed or the structure damping factor decreases the maximum values of fracture parameters, while the structure temperature has little effect on the fracture parameters. Due to the fact that the vehicle speed can be enhanced by improving the road traffic conditions, and the pavement damping factor can become greater by modifying the components of materials, the development of reflective cracking can be delayed and the asphalt pavement service life can be effectively extended through both of these ways.

Numerical Simulation of Viscoelastic Extrudate Swell Through Elliptical Ring Die

The numerical simulation of extrudate swell is significant in extrusion processing.Precise prediction of extrudate swell is propitious to the control of melt flow and the quality of final products.A mathematical model of three-dimensional（3D）viscoelastic flow through elliptical ring die for polymer extrusion was investigated.The penalty function formulation of viscoelastic incompressible fluid was introduced to the finite element model to analyze 3D extrusion problem.The discrete elastic viscous split stress（DEVSS）and streamline-upwind PetrovGalerkin（SUPG）technology were used to obtain stable simulation results.Free surface was updated by updating the streamlines which needs less memory space.According to numerical simulation results,the effect of zero-shear viscosity and elongation parameter on extrudate swell was slight,but with the increase of volumetric flow rate and relax time the extrudate swell ratio increased markedly.Finally,the numerical simulation of extrudate swell flow for low-density polyethylene（LDPE）melts was investigated and the results agreed well with others’work.These conclusions provided quantitative basis for the forecasting extrudate swell ratio and the controlling of extrusion productivity shape.

VISCOELASTIC CONSTITUTIVE MODEL RELATED TO DEFORMATION OF INSECT WING UNDER LOADING IN FLAPPING MOTION

Flexible insect wings deform passively under the periodic loading during flapping flight. The wing flexibility is considered as one of the specific mechanisms on improving insect flight performance. The constitutive relation of the insect wing material plays a key role on the wing deformation, but has not been clearly understood yet. A viscoelastic constitutive relation model was established based on the stress relaxation ex- periment of a dragonfly wing （in vitro）. This model was examined by the finite element analysis of the dynamic deformation response for a model insect wing under the action of the periodical inertial force in flapping. It is revealed that the viscoelastic constitutive relation is rational to characterize the biomaterial property of insect wings in contrast to the elastic one. The amplitude and form of the passive viscoelastic deformation of the wing is evidently dependent on the viscous parameters in the constitutive relation.

Stress boundary layers in the viscoelastic flow past a cylinder in a channel:limiting solutions

The flow past a cylinder in a channel with the aspect ratio of 2 ： 1 for the upper convected Maxwell （UCM） fluid and the Oldroyd-B fluid with the viscosity ratio of 0.59 is studied by using the Galerkin/Least-square finite element method and a p-adaptive refinement algorithm. A posteriori error estimation indicates that the stress-gradient error dominates the total error. As the Deborah number, De, approaches 0.8 for the UCM fluid and 0.9 for the Oldroyd-B fluid, strong stress boundary layers near the rear stagnation point are forming, which are characterized by jumps of the stress-profiles on the cylinder wall and plane of symmetry, huge stress gradients and rapid decay of the gradients across narrow thicknesses. The origin of the huge stress-gradients can be traced to the purely elongational flow behind the rear stagnation point, where the position at which the elongation rate is of 1/2De approaches the rear stagnation point as the Deborah number approaches the critical values. These observations imply that the cylinder problem for the UCM and Oldroyd-B fluids may have physical limiting Deborah numbers of 0.8 and 0.9, respectively.

Error Estimates of H^1-Galerkin Mixed Methods for the Viscoelasticity Wave Equation

H1-Galerkin mixed methods are proposed for viscoelasticity wave equation.Depending on the physical quantities of interest,two methods are discussed.The optimal error estimates and the proof of the existence and uniqueness of semidiscrete solutions are derived for problems in one space dimension.And the methods don＇t require the LBB condition.

Viscoelastic relaxation of the upper mantle and afterslip following the 2014 M_(W)8.1 Iquique earthquake

An improved understanding of postseismic crustal deformation following large subduction earthquakes may help to better understand the rheological properties of upper mantle and the slip behavior of subduction interface.Here we construct a three-dimensional viscoelastic finite element model to study the postseismic deformation of the 2014 M_(W)8.1 Iquique,Chile earthquake.Elastic units in the model include the subducting slab,continental and oceanic lithospheres.Rheological units include the mantle wedge,the oceanic asthenosphere and upper mantle.We use a 2 km thick weak shear zone attached to the subduction fault to simulate the time-dependent stress-driven afterslip.The viscoelastic relaxation in the rheological units is represented by the Burgers rheology.We carry out grid-searches on the shear zone viscosity,thickness and viscosity of the asthenosphere,and they are determined to be 10^(17)Pa s,110 km and 2×10^(18)Pa s,respectively.The stress-driven afterlsip within the first two years is up to~47 cm and becomes negligible after two years(no more than 5 cm/yr).Our results suggest that a thin,low-viscosity oceanic asthenosphere together with a weak shear zone attached to the fault are required to better reproduce the observed postseismic deformation.

A NEW ALGORITHM OF TIME STEPPING IN DYNAMIC VISCOELASTIC PROBLEMS

A new scheme of time stepping for solving the dynamic viscoelastic problems are presented. By expanding variables at a discrete time interval, FEM based recurrent formulae are derived. A self-adaptive algorithm for different sizes of time steps can be carried out to improve computing accuracy. Numerical validation shows satisfactory performance.

Application of Viscoelastic Material in Structures Control

The factors influencing mechanical performances of viscoelastic material are studied.The proper finite element model for dynamical calculating the passive control of wind-earthquake resistance is constructed.A combined element stiffness matrix of damper-brace system is deduced.At last,the theoretical deduction is verified by comparing the theoretical results with experimental ones.