Quantitative characterization of the interfacial adhesion of Ni thin film on steel substrate: A compression-induced buckling delamination test

A compression-induced buckling delamination test is employed to quantitatively characterize the interfacial adhesion of Ni thin film on steel substrate. It is shown that buckles initiate from edge flaws and surface morphologies exhibit symmetric, half-penny shapes. Taking the elastoplasticity of film and substrate into account, a three-dimensional finite element model for an edge flaw with the finite size is established to simulate the evolution of energy release rates and phase angles in the process of interfacial buckling-driven delamination. The results show that delamination propagates along both the straight side and curved front. The mode II delamination plays a dominant role in the process with a straight side whilst the curved front experiences almost the pure mode I. Based on the results of finite element analysis, a numerical model is developed to evaluate the interfacial energy release rate, which is in the range of 250–315 J/m² with the corresponding phase angle from −41° to −66°. These results are in agreement with the available values determined by other testing methods, which confirms the effectiveness of the numerical model.     

Finite difference and finite element methods for mhd channel flows

A Galerkin finite element method and two finite difference techniques of the control volume variety have been used to study magnetohydrodynamic channel flows as a function of the Reynolds number, interaction parameter, electrode length and wall conductivity. The finite element and finite difference formulations use unequally spaced grids to accurately resolve the flow field near the channel wall and electrode edges where steep flow gradients are expected. It is shown that the axial velocity profiles are distorted into M-shapes by the applied electromagnetic field and that the distortion increases as the Reynolds number, interaction parameter and electrode length are increased. It is also shown that the finite element method predicts larger electromagnetic pinch effects at the electrode entrance and exit and larger pressure rises along the electrodes than the primitive-variable and streamfunction–vorticity finite difference formulations. However, the primitive-variable formulation predicts steeper axial velocity gradients at the channel walls and lower axial velocities at the channel centreline than the streamfunction–vorticity finite difference and the finite element methods. The differences between the results of the finite difference and finite element methods are attributed to the different grids used in the calculations and to the methods used to evaluate the pressure field. In particular, the computation of the velocity field from the streamfunction–vorticity formulation introduces computational noise, which is somewhat smoothed out when the pressure field is calculated by integrating the Navier–Stokes equations. It is also shown that the wall electric potential increases as the wall conductivity increases and that, at sufficiently high interaction parameters, recirculation zones may be created at the channel centreline, whereas the flow near the wall may show jet-like characteristics.     

An energy conserving co-rotational procedure for non-linear dynamics with finite elements

A new procedure is proposed for implicit dynamic analysis using the finite element method. The main aim is to give stable solutions with large time-steps in the presence of significant rigid body motions, in particular rotations. In contrast to most conventional approaches, the time integration strategy is closely linked to the element technology with the latter involving a form of co-rotational procedure. For the undamped situation, the solution procedure leads to an algorithm that exactly conserves energy when constant external forces are applied (i.e. with gravity loading).     

A coupled fluid-elasticity model for the wave forcing of an ice-shelf

A mathematical model for predicting the vibrations of ice-shelves based on linear elasticity for the ice-shelf motion and potential flow for the fluid motion is developed. No simplifying assumptions such as the thinness of the ice-shelf or the shallowness of the fluid are made. The ice-shelf is modelled as a two-dimensional elastic body of an arbitrary geometry under plane-strain conditions. The model is solved using a coupled finite element method incorporating an integral equation boundary condition to represent the radiation of energy in the infinite fluid. The solution is validated by comparison with thin-beam theory and by checking energy conservation. Using the analyticity of the resulting linear system, we show that the finite element solution can be extended to the complex plane using interpolation of the linear system. This analytic extension shows that the system response is governed by a series of singularities in the complex plane. The method is illustrated through time-domain simulations as well as results in the frequency domain.     

Computing lower and upper bounds on stress intensity factors in bimaterials

This paper presents a finite element approach for finding complementary bounds of stress intensity factors (SIFs) in bimaterials. The SIF is formulated as an explicit computable linear function of displacements by means of the two-point extrapolation method. An appropriate and computable form of the SIF plays a crucial role in the dual problem involved in the computing procedure of both lower and upper bounds. In our discussions, computable forms of stress intensity factors, K₀ and K_r, are derived, which are related to the energy release rate, and the ratio of the open mode and shear mode SIFs, respectively. Based on a posteriori finite element error estimation, a bounding procedure is used to compute the bounds on the two stress intensity factors. Finally, bounds on the SIFs in a bimaterial interface crack problem are provided to verify the procedure.     

A Dorodnitsyn finite element formulation for laminar boundary layer flow

The Dorodnitsyn boundary later formulation is given a finite element interpretation and found to generate very accurate and economical solutions when combined with an implicit, non-iterative marching scheme in the downstream direction. The algorithm is of order (Δ²u, Δx) whether linear or quadratic elements are used across the boundary layer. Solutions are compared with a Dorodnitsyn spectral formulation and a conventional finite difference formulation for three Falkner-Skan pressure gradient cases and the flow over a circular cylinder. With quadratic elements the Dorodnitsyn finite element formulation is approximately five times more efficient than the conventional finite difference formulation.     

Nonlinear elastic effects in graphite/epoxy: An analytical and numerical prediction of energy flux deviation

Manipulating acoustic wave propagation through a material have several interdisciplinary applications. Here we predict shift in energy flux deviation for acoustic waves propagating in unidirectional graphite/epoxy due to applied normal and shear stresses using both an analytical model, using acoustoelastic continuum theory, and a finite element discrete numerical model. The acoustoelastic theory predicts that the quasi-transverse (QT) wave exhibits larger shifts in energy flux deviation compared to quasi-longitudinal (QL) or the pure transverse (PT) due to an applied shear stress for fiber orientation angle ranging from 0° to 60°. Due to an applied shear stress the QT wave exhibits a shift in energy flux deviation at 0° fiber orientation angle as compared to normal stress case where the flux deviation and its load induced shift are both zero. A finite element model (FEM) is developed where equations of motion include the effect of nonlinear elastic coefficients. Element equations were integrated in time using Newmark’s method to determine the shift in energy flux deviations in graphite/epoxy for different loading cases. The energy flux shift of QT waves predicted by FEM for fiber orientation angles from 0° to 60° for applied shear stress case is in excellent agreement with acoustoelastic theory. Because energy shift magnitudes are not small, it is possible to experimentally measure these shifts and calibrate shifts with respect to load type (normal/shear) and magnitude.     

High-order time integration applied to metal powder plasticity

The present article treats two objectives. In the first investigation attention is focused on the application of time-adaptive finite elements formulated on the basis of a high-order time integration procedure on a constitutive model for compressible finite strain viscoplasticity for metal powder. In this connection, it has to be emphasized that the integration procedure is not only applied to the evolution equations on Gauss-point level but on the total system of differential–algebraic equations resulting from the application of the vertical line method on the quasi-static finite element equations. The specific application emerges from the field of metal powder compaction. Particular studies are carried out using stiffly accurate, diagonally implicit Runge–Kutta methods in combination with the Multilevel-Newton algorithm for solving the DAE-system. In this respect, the effort vs. accuracy behavior is investigated which is also related to order reduction known in elastoplasticity. The second topic treats the local stress algorithm for taking into account the yield function based finite strain viscoplasticity model, where the classical Newton–Raphson method fails. This is the reason why most constitutive models of powder materials are implemented into explicit finite element codes. Thus, the proposed investigations compare different methods in view of a stable and efficient integration process in implicit finite element formulations.     

Numerical solution of a first-order conservation equation by a least square method

Least square methods have been frequently used to solve fluid mechanics problems. Their specific usefulness is emphasized for the solution of a first-order conservation equation. On the one hand, the least square formulation embeds the first-order problem into equivalent second-order problem, better adapted to discretization techniques due to symmetry and positive-definiteness of the associated matrix. On the other hand, the introduction of a least square functional is convenient for finite element applications. This approach is applied to the model problem of the conservation of mass (the unknown is the density ρ) in a nozzle with a specified velocity field (u, v), possibly including jumps along lines simulating shock waves. This represent a preliminary study towards the solution of the steady Euler equations. A finite difference and a finite element method are presented. The choice of the finite difference scheme and of a continuous finite element representation for the groups of variables (ρu, ρv) is discussed in terms of conservation of mass flux. Results obtained with both methods are compared in two numerical tests with the same mesh system.     

Computational modeling of rate-dependent domain switching in piezoelectric materials

In this contribution a micromechanically motivated model for rate-dependent switching effects in piezoelectric materials is developed. The proposed framework is embedded into a three-dimensional finite element setting whereby each element is assumed to represent an individual grain. Related dipole (polarization) directions are thereby initially randomly oriented at the element level to realistically capture the originally un-poled state of grains in the bulk ceramics. The onset of domain switching processes is based on a representative energy criterion and combined with a linear kinetics theory accounting for time-dependent propagation of domain walls during switching processes. In addition, grain boundary effects are incorporated by making use of a macromechanically motivated probabilistic approach. Standard volume-averaging techniques with respect to the response on individual grains in the bulk ceramics are later on applied to obtain representative hysteresis and butterfly curves under macroscopically uniaxial loading conditions at different loading frequencies. It turns out that the simulations based on the developed finite element formulation nicely match experimental data reported in the literature.     

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.     

A new finite element method for strain gradient theories and applications to fracture analyses

A new compatible finite element method for strain gradient theories is presented. In the new finite element method, pure displacement derivatives are taken as the fundamental variables. The new numerical method is successfully used to analyze the simple strain gradient problems – the fundamental fracture problems. Through comparing the numerical solutions with the existed exact solutions, the effectiveness of the new finite element method is tested and confirmed. Additionally, an application of the Zienkiewicz–Taylor C1 finite element method to the strain gradient problem is discussed. By using the new finite element method, plane-strain mode I and mode II crack tip fields are calculated based on a constitutive law which is a simple generalization of the conventional J₂ deformation plasticity theory to include strain gradient effects. Three new constitutive parameters enter to characterize the scale over which strain gradient effects become important. During the analysis the general compressible version of Fleck–Hutchinson strain gradient plasticity is adopted. Crack tip solutions, the traction distributions along the plane ahead of the crack tip are calculated. The solutions display the considerable elevation of traction within the zone near the crack tip.     

A variational model for fracture mechanics: Numerical experiments

In the variational model for brittle fracture proposed in Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319-1342], the minimum problem is formulated as a free discontinuity problem for the energy functional of a linear elastic body. A family of approximating regularized problems is then defined, each of which can be solved numerically by a finite element procedure. Here we re-formulate the minimum problem within the context of finite elasticity. The main change is the introduction of the dependence of the strain energy density on the determinant of the deformation gradient. This change requires new, more general existence and Γ-convergence results. The results of some two-dimensional numerical simulations are presented, and compared with corresponding simulations made in Bourdin et al. [2000. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797-826] for the linear elastic model.     

Finite element analysis of V-ribbed belts using neural network based hyperelastic material model

A three-dimensional finite element model was built to study V-ribbed belt pulley contact mechanics. The model consists of a pulley and a segment of V-ribbed belt in contact with the pulley. A material model for the belt, including the rubber compound and the reinforcing cord is developed. Rubber is modeled as hyperelastic material. The hyperelastic strain energy function is approximated by neural network trained by rubber test data. Reinforcing cord is modeled as elastic rebar. The material model developed is implemented in the commercial finite element code ABAQUS to simulate the V-ribbed belt-pulley system. A study is then conducted to investigate the effect of belt pulley system parameters on the contact mechanics. The effects of temperature and aging on belt materials are also investigated. The information gained from the analysis can be applied to optimize V-ribbed belt and pulley design.     

Improved finite element forms for the shallow-water wave equations

This paper presents new finite element formulations of the shallow-water wave equations which use different basis functions for the velocity and height fields. These arrangements are analysed with the Fourier transform technique which was developed by Schoenstadt,¹ and they are also compared with other finite difference and finite element schemes. The new schemes are integrated in time for two initial states and compared with analytic solutions and numerical solutions from other schemes. The behaviour of the new forms is excellent and they are also convenient to apply in two dimensions with triangular elements.     

Fuzzy variational principle and its applications

Linear and non-linear peaky fuzzy numbers and their arithmetic operations are constructed for the analysis of engineering structures with fuzzy characteristic quantities. Fuzziness of the corresponding quantities is consistently incorporated into the functional of the total potential energy. A set of deterministic recursive equations is obtained as the alternative expressions of the fuzzy variational principle by means of the second-order perturbation technique. The fuzzy Ritz method and the fuzzy finite element method are presented as the applications of the fuzzy variational principle. Accordingly, the roundabout procedures frequently used in the formulations of the fuzzy finite element method are avoided. A benchmark problem of a bending beam with fuzzy Young's modulus under fuzzy external loading is solved by the developed fuzzy numerical methods. Numerical examples show that results determined by these two fuzzy methods are both little conservative, and are in good agreement with those obtained by the analytical method. Moreover, the fuzzy Ritz method or the fuzzy finite element method can provide more valuable information than the conventional deterministic methods.     

On the fracture toughness of ferroelastic materials

The toughness enhancement due to domain switching near a steadily growing crack in a ferroelastic material is analyzed. The constitutive response of the material is taken to be characteristic of a polycrystalline sample assembled from randomly oriented tetragonal single crystal grains. The constitutive law accounts for the strain saturation, asymmetry in tension versus compression, Bauschinger effects, reverse switching, and strain reorientation that can occur in these materials due to the non-proportional loading that arises near a propagating crack. Crack growth is assumed to proceed at a critical level of the crack tip energy release rate. Detailed finite element calculations are carried out to determine the stress and strain fields near the growing tip, and the ratio of the far field applied energy release rate to the crack tip energy release rate. The results of the finite element calculations are then compared to analytical models that assume the linear isotropic K-field solution holds for either the near tip stress or strain field. Ultimately, the model is able to account for the experimentally observed toughness enhancement in ferroelastic ceramics.     

An energy-based crack growth criterion for modelling elastic–plastic ductile fracture

A crack growth criterion is derived based on the Griffith energy concept and the cohesive zone model for modelling fracture in elastic–plastic ductile materials. The criterion is implemented in the finite element context by a virtual crack extension technique. An automatic modelling of the ductile fracture process is realised by combining a local remeshing procedure and the criterion. The validity of the derived criterion is examined by modelling a compact tension specimen.     

Trigonometric function used to formulate a multi-nodal finite tubular element

It is presented an alternative formulation to solve the problem of the deformation analysis for tubular element under pinching loads. The solution is based on a new displacement field defined from a total set of trigonometric functions. The solution is developed in a multi-nodal finite tubular ring element with a total of eight degrees of freedom per section considered. The purpose of this paper is to provide an easy alternative formulation when compared with a complex finite shell element or beam element analysis for the same application. Several case studies presented have been compared and discussed with numerical analyses results reported by other authors and the results obtained with a shell element from a Cosmos/M^® programme.