连续胶体材料化学-力学耦合有限元理论及其在ABAQUS中UEL程序实现

基于哈密顿原理,得到水凝胶的化学-力学耦合控制方程的等效积分形式和有限元形式。在整体坐标系下推导出用形函数表示的化学-力学耦合应变矩阵和单元刚度矩阵,并且得到在局部坐标系下的离散化形式。结合ABAQUS软件,编制了用户单元子程序UEL,通过数值算例验证了所开发单元的正确性,为在ABAQUS软件中实现各种耦合问题的有限元UEL编程提供了参考依据。     

A nonlinear,transient finite element method for coupled solvent diffusion and large deformation of hydrogels

Hydrogels are capable of coupled mass transport and large deformation in response to external stimuli. In this paper, a nonlinear, transient finite element formulation is presented for initial boundary value problems associated with swelling and deformation of hydrogels, based on a nonlinear continuum theory that is consistent with classical theory of linear poroelasticity. A mixed finite element method is implemented with implicit time integration. The incompressible or nearly incompressible behavior at the initial stage imposes a constraint to the finite element discretization in order to satisfy the Ladyzhenskaya–Babuska–Brezzi (LBB) condition for stability of the mixed method, similar to linear poroelasticity as well as incompressible elasticity and Stokes flow; failure to choose an appropriate discretization would result in locking and numerical oscillations in transient analysis. To demonstrate the numerical method, two problems of practical interests are considered: constrained swelling and flat-punch indentation of hydrogel layers. Constrained swelling may lead to instantaneous surface instability for a soft hydrogel in a good solvent, which can be regulated by assuming a stiff surface layer. Indentation relaxation of hydrogels is simulated beyond the linear regime under plane strain conditions, in comparison with two elastic limits for the instantaneous and equilibrium states. The effects of Poisson’s ratio and loading rate are discussed. It is concluded that the present finite element method is robust and can be extended to study other transient phenomena in hydrogels.     

Effect of interphase properties on the damping response of polymer nano-composites

In this paper a finite element (FE) based unitcell model has been devised with periodic boundary conditions to calculate the effective tan δ of a composite as a function of frequency. Using this method, it is demonstrated that the tan δ of the nano-composite manifests important information about the extent and properties of the interphase region that surrounds each nano-particle. Thus we imply that tan δ measurements provide a simple yet useful experimental tool to understand more about the interphase. However, severe agglomeration or wide size distribution of the nano-particles may introduce errors in measurement.     

Influence of skin/core debonding on free vibration behavior of foam and honeycomb cored sandwich plates

The dynamic behavior of partially delaminated at the skin/core interface sandwich plates with flexible cores is studied. The commercial finite element code ABAQUS is used to calculate natural frequencies and mode shapes of the sandwich plates containing a debonding zone. The influence of the debonding size, debonding location and types of debonding on the modal parameters of damaged sandwich plates with various boundary conditions is investigated. The results of dynamic analysis illustrated that they can be useful for analyzing practical problems related to the non-destructive damage detection of partially debonded sandwich plates.     

Harmonic behavior of silicon nanowire by molecular dynamics

Silicon nanowires (SiNW) have attracted increasing interests as potential core parts for nanoscale devices due to their distinguished mechanical and electrical properties. Using molecular dynamics (MD) method, we investigated the vibration behaviors of SiNW on the atomic scale, including the fundamental mode frequency and quality factor. The quality factor as low as 120 is attributed to the energy loss process coming from atomic friction and nonlinearity. We also found that the energy of lattice vibration (phonon energy) is much larger than the macroscale beam vibration energy. And both the energies are less than the surface relaxation energy of the beam. The comparison between MD and FEM has been made to discuss the validity of continuum approximation for the study of SiNW.     

A 3D direct coupling method for steady ship hydroelastic analysis

Asides from the influence of incoming waves, ships can experience steady motions, such as rigid-body sinkage and trim motions, and flexible-body vertical bending motions, due to a constant forward speed even under calm water conditions. In this paper, a novel approach to analyze steady-ship hydroelasticity, particularly for the steady-ship motions and surrounding steady-wave disturbances, is proposed using a three-dimensional (3D) direct coupling method, based on a higher-order boundary element method (HOBEM) and a higher-order shell finite element method (FEM). Within the linearized framework, a solution method is proposed based on a two-step procedure, using two types of Neumann–Kelvin (NK) linear flow models for the fluid part and a virtual work equilibrium equation for the structural part. The first step is to compute a mean position wave-resistance problem using the modified NK equation, the second step is to solve a perturbed position wave-resistance problem, by employing a classical NK model and a virtual work equation based on the first step’s solution. Detailed mathematical formulation and numerical procedures are described, and a few numerical results are illustrated. These include both rigid and flexible steady-ship motions, Von-Mises stress distributions, and wave-resistance coefficients for Froude numbers ranging from 0.15 to 0.5. Furthermore, the numerical results obtained using the present direct coupling method and a modal-based one are compared.     

Comparison of methods for vibration analysis of electrostatic precipitators

The paper presents two methods for the formulation of free vibration analysis of collecting electrodes of precipitators.The first,called the hybrid finite element method, combines the finit element method used for calculations of spring deformations with the rigid finite element method used to reflect mass and geometrical features,which is called the hybrid finite element method.As a result,a model with a diagonal mass matrix is obtained.Due to a specific geometry of the electrodes,which are long plates of complicated shapes,the second method proposed is the strip method which is a semi-analytical method.The strip method allows us to formulate the equations of motion with a considerably smaller number of generalized coordinates.Results of numerical calculations obtained by both methods are compared with those obtained using commercial software like ANSYS and ABAQUS.Good compatibility of results is achieved.     

An enhanced coupling of PD grids to FE meshes

The paper presents a new and simple way to couple FEM meshes to Peridynamic grids with different grid sizes. The excellent performance of the coupling technique is illustrated by means of 2D examples involving rigid body motions, uniform strain conditions and a case of crack branching. The method paves the way to an increased use of peridynamics within FEM software.     

An orthogonal mapping technique for the computation of a viscous free-surface flow

In this paper we describe finite element computations of the free-surface flow of a viscous fluid down an undulating inclined plane. The technique developed here employs an orthogonal mapping that is computed along with the velocity and pressure. This is allied to a technique to compute symbolically the Jacobian and other derivatives required for numerical continuation methods. The solutions obtained are compared with laboratory experiments and finite element computations reported by Pritchard and co-workers. The finite element computational method used by these authors employs spines to represent the free surface. An excellent agreement is shown to exist between the new computations and the laboratory experiments, and with the numerical solutions of Pritchard and co-workers.     

Identification of the plastic behavior of aluminum plates under free air explosions using inverse methods and full-field measurements

This article describes an inverse method for the identification of the plastic behavior of aluminum plates subjected to sudden blast loads. The method uses full-field optical measurements taken during the first milliseconds of a free air explosion and the finite element method for the numerical prediction of the blast response. The identification is based on a damped least-squares solution according to the Levenberg–Marquardt formulation. Three different rate-dependent plasticity models are examined. First, a combined model based on linear strain hardening and the strain rate term of the Cowper–Symonds model, secondly, the Johnson–Cook model and finally, a combined model based on a bi-exponential relation for the strain hardening term and the strain rate term of the Cowper–Symonds model. A validation of the method and its sensitivity to measurement uncertainties is first provided according to virtual measurements generated with the finite element method. Next, the plastic behavior of aluminum is identified using measurements from real free air explosions obtained from a controlled detonation of C4. The results show that inverse methods can be successfully applied for the identification of the plastic behavior of metals subjected to blast waves. In addition, the material parameters identified with inverse methods enable the numerical prediction of the material’s response with increased accuracy.     

A Taylor-Galerkin Method for Modeling Unsteady Fluid Flow Under its Own Weight

The free fluid-surface of incompressible creeping flows is analyzed using a finite element method. A pseudo-concentration (PC) function is introduced to determine the position of the free surface. The Taylor-Galerkin finite element method (TGFEM) is applied to solve the equation of the PC function. Nine-node quadratic interpolation is used for both PC and velocity. The unsteady flows of fluids moving of their own weight are analyzed using the proposed method.     

Application of streamwise diffusion to time-dependent free convection of liquid metals

A numerical analysis is given for the application of streamwise diffusion to high-intensity flows with marginal spatial resolution. Terms are added to the momentum equation which are similar to those used in the Petrov-Galerkin, Taylor-Galerkin and balancing tensor diffusivity methods. Values for the streamwise viscosity are obtained from mesh refinement studies. An illustration is given for the time-dependent free convection of a liquid metal in a cavity with differentially heated sided walls. The spatial problem is solved with the Galerkin finite element method and the time integration is performed with the backward Euler method. Solution quality and computation time are compared for results with and without added streamwise diffusion. For the cases presented, streamwise diffusion eliminates spurious oscillations and saves computation time without compromising the solution.     

A finite element analysis of a free surface drainage problem of two immiscible fluids

A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two-dimensional model and solved numerically. This problem is a transient two-phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time-stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.     

Finite element analysis of axisymmetric free jet impingement

A numerical algorithm to determine the impingement of an axisymmetric free jet upon a curved deflector is presented. The problem is considered within the potential flow theory with the allowance of gravity and surface tension effects. The primary dependent variable is the Stokes streamfunction, which is approximated through finite elements using the isoparametric Hermite Zienkiewicz element. To find the correct position of the free boundaries, a trial-and-error method is employed which amounts to solving a boundary value problem (BVP) for the Stokes streamfunction at each iteration step. An efficient method is proposed to solve this BVP. The algorithm to find the correct position of the free boundaries is tested by computing the impingement upon an infinite disc and a hemispherical deflector. To confirm the correctness of the solution, each problem has been solved using several different mesh gradings. A comparison between the Zienkiewicz and the other standard C⁰ finite elements is also given.     

An efficient three-dimensional semi-implicit finite element scheme for simulation of free surface flows

An efficient semi-implicit finite element model is proposed for the simulation of three-dimensional flows in stratified seas. The body of water is divided into a number of layers and the two horizontal momentum equations for each layer of water are first integrated vertically. Nine-node Lagrangian quadratic isoparametric elements are employed for spatial discretization in the horizontal domain. The time derivatives are approximated using a second-order-accurate semi-implicit time-stepping scheme. The distinguishing feature of the proposed numerical scheme is that only nodal values on the same vertical line are coupled. Two test cases for which analytic solutions are available are employed to test the proposed scheme. The test results show that the scheme is efficient and stable. A numerical experiment is also included to compare the proposed scheme with a finite difference scheme.     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

Equilibrium morphology of misfit particles in elastically stressed solids under chemo-mechanical equilibrium conditions

This paper studies the effects of chemical, elastic and interfacial energies on the equilibrium morphology of misfit particles due to phase separation in binary alloys under chemo-mechanical equilibrium conditions. A continuum framework that governs the chemo-mechanical equilibrium of the system is first developed using a variational approach by treating the phase interface as a sharp interface endowed with interfacial excess energy. An extended finite element method (XFEM) in conjunction with the level set method is then developed to simulate the behaviors of the coupled chemo-mechanical system. The coupled chemo-mechanics model together with the numerical techniques developed here provides an efficient simulation tool to predict the equilibrium morphologies of precipitates in phase separate alloys.     

On the coupling of plastic slip and deformation-induced twinning in magnesium: A variationally consistent approach based on energy minimization

The present paper is concerned with the analysis of the deformation systems in single crystal magnesium at the micro-scale and with the resulting texture evolution in a polycrystal representing the macroscopic mechanical response. For that purpose, a variationally consistent approach based on energy minimization is proposed. It is suitable for the modeling of crystal plasticity at finite strains including the phase transition associated with deformation-induced twinning. The method relies strongly on the variational structure of crystal plasticity theory, i.e., an incremental minimization principle can be derived which allows to determine the unknown slip rates by computing the stationarity conditions of a (pseudo) potential. Phase transition associated with twinning is modeled in a similar fashion. More precisely, a solid-solid phase transition corresponding to twinning is assumed, if this is energetically favorable. Mathematically speaking, the aforementioned transition can be interpreted as a certain rank-one convexification. Since such a scheme is computationally very expensive and thus, it cannot be applied to the analysis of a polycrystal, a computationally more efficient approximation is elaborated. Within this approximation, the deformation induced by twinning is decomposed into the reorientation of the crystal lattice and simple shear. The latter is assumed to be governed by means of a standard Schmid-type plasticity law (pseudo-dislocation), while the reorientation of the crystal lattice is considered, when the respective plastic shear strain reaches a certain threshold value. The underlying idea is in line with experimental observations, where dislocation slip within the twinned domain is most frequently seen, if the twin laminate reaches a critical volume. The resulting model predicts a stress-strain response in good agreement with that of a rank-one convexification method, while showing the same numerical efficiency as a classical Taylor-type approximation. Consequently, it combines the advantages of both limiting cases. The model is calibrated for single crystal magnesium by means of the channel die test and finally applied to the analysis of texture evolution in a polycrystal. Comparisons of the predicted numerical results to their experimental counterparts show that the novel model is able to capture the characteristic mechanical response of magnesium very well.     

Vibrations of a cylinder in a uniform flow in the presence of a no-slip side-wall

A circular cylinder placed in a uniform flow, and that spans the entire length between two side walls, may experience either parallel or oblique vortex shedding depending on the end conditions. It was shown by Mittal and Sidharth (2014) that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by the cylinder. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. The free vibrations of a circular cylinder, in the presence of a wall, are investigated. For comparison, computations with end walls, where a slip condition on velocity is specified, are also carried out. The Reynolds number, based on the diameter of the cylinder and free-stream speed of the flow, is Re=100. The initial condition for the free vibrations is the fully developed unsteady flow past a stationary cylinder with oblique shedding. It is found that as the amplitude of vibration of the cylinder builds up, the vortices shed from the cylinder align with its axis leading to parallel shedding. The response of the cylinder is associated with two branches: initial and lower. On the lower branch, the response of the cylinder is virtually identical from two- and three-dimensional computations. The flow as well as the response is different on the initial branch and outside the synchronization regime. Forced vibrations confirm the phenomena.