Showing papers in "Computational Mechanics in 1994"

Complex wavenumber Fourier analysis of the p-version finite element method

Abstract: High-order finite element discretizations of the reduced wave equation have frequency bands where the solutions are harmonic decaying waves. In these so called “stopping” bands, the solutions are not purely propagating (real wavenumbers) but are attenuated (complex wavenumbers). In this paper we extend the standard dispersion analysis technique to include complex wavenumbers. We then use this complex Fourier analysis technique to examine the dispersion and attenuation characteristics of the p-version finite element method. Practical guidelines are reported for phase and amplitude accuracy in terms of the spectral order and the number of elements per wavelength.

A constitutive formulation for anisotropic materials suitable for wave propagation computer programs. II

Abstract: The constitutive relationships for an anisotropic material are established for shock wave propagation and nonlinear, large deformation computer programs, commonly referred to as hydrocodes. Stresses are formulated in terms of strains; the procedure for separating material compressibility effects (equation of state) from strength effects is formulated which permits the consistent calculation of stresses in the elastic regime, and allows the mean pressure to be defined in accordance with their scalar interpretations. Futher, this procedure permits the computation of inelastic response by scaling of deviatoric stresses, so the equivalent stress resides on a yield or failure surface, without changing the pressure. The procedure for computing the equivalent plastic strain and non-radial return to the yield surface, which results from a calculated overstress, is developed. Also, the transformation matrices for large deformation (rotation), necessary for transformation between material and geometric coordinates, are presented.

Implementation of the transformation field analysis for inelastic composite materials

Abstract: The transformation field analysis is a general method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities. The various unit cell models, or the corrected inelastic self-consistent or Mori-Tanaka fomulations, the so-called Eshelby method, and the Eshelby tensor itself are all seen as special cases of this more general approach. The method easily accommodates any uniform overall loading path, inelastic constitutive equation and micromechanical model. The model geometries are incorporated through the mechanical transformation influence functions or concentration factor tensors which are derived from elastic solutions for the chosen model and phase elastic moduli. Thus, there is no need to solve inelastic boundary value or inclusion problems, indeed such solutions are typically associated with erroneous procedures that violate (62); this was discussed by Dvorak (1992). In comparison with the finite element method in unit cell model solutions, the present method is more efficient for cruder mesches. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. An addition to the examples shown herein, the method can be applied to many other problems, such as those arising in active materials with eigenstrains induced by components made of shape memory alloys or other actuators. Progress has also been made in applications to electroelastic composites, and to problems involving damage development in multiphase solids. Finally, there is no conceptural obstacle to extending the approach beyond the analysis of representative volumes of composite materials, to arbitrarily loaded structures.

Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates

Abstract: One objective of the present study is to use arbitrary complete 3-dimensional constitutive equations without reduction or manipulation in nonlinear plate and shell analysis. The obvious consequence, namely the extension of a conventional 5-parameter shell formulation with Reissner-Mindlin kinematics to a 6-parameter formulation including the full set of stress and strain state does not solve the problem because a significant error in bending dominated cases occurs. To avoid this error the transverse normal strain is allowed to vary linearly across the thickness. This so-called 7-parameter theory recently proposed in the group of the authors resorts to the Enhanced Assumed Strain concept and preserves the basic features of a displacement formulation.

Analysis of partly wrinkled membranes by the method of dynamic relaxation

Abstract: A version of the method of dynamic relaxation is developed to analyze equilibrium configurations of partly wrinkled membranes. In this method equilibria are regarded as long time limits of a damped dynamical problem. The membrane theory considered is based on the concept of a relaxed strain energy function that automatically incorporates the effects of wrinkling. For neo-Hookean materials, existence theorems of nonlinear elasticity are used to show that the relaxed potential energy possesses minimizers in a certain function space. Moreover, solutions of the equilibrium equations furnish global minima of the energy, for certain classes of boundary data. Such deformations are automatically stable according to the minimum-energy criterion. This result motivates the search for solutions of the equilibrium equations, although the existence theory does not guarantee that energy minimizers possess the degree of regularity required by these equations. Several examples of two-and three-dimensional deformations are presented.

Comparison of different finite deformation inelastic damage models within multiplicative elastoplasticity for ductile materials

Abstract: The new contribution of this study is to formulate two wellknown isotropic elastoplastic damage concepts for ductile materials in the framework of ‘geometrically exact’ finite multiplicative elastoplasticity. For the model originally proposed by Lemaitre the damage evolution follows from a dissipation potential and the hypothesis of general associativity. In contrast, the Gurson model takes into account the balance of mass separately to formulate damage evolution. In this contribution both formulations are based on logarithmic Hencky strains leading to a simple application of the so called ‘exponential map’ stress integrator which is the algorithmic counterpart of the multiplicative elastoplastic formulation adopted. Special emphasis is directed towards the numerical implementation of these models within the framework of finite element analysis of inelastic boundary value problems. To compare the results of numerical computations several standard examples within finite elastoplasticity are analysed with both damage models and the results are contrasted to the outcome of an analysis with the classical v. Mises model. thereby, the dramatic influence of damage on the behaviour within necking and localization computations is highlighted. The different behaviour of the two models considered within compression dominated problems is appreciated.

Microscale reduction error indicators and estimators for a periodic heterogeneous medium

Abstract: Quantification of idealization errors associated with homogenization of periodic heterogenous medium is presented. The proposed Microscale Reduction Error (MRE) estimators and indicators are based on estimating the uniform validity properties of the double scale asymptotic expansion. The technique leads to a reliable quality control of the microscopic response of interest which is obtained on the basis of the mathematical homogenization theory.

The inverse mapping and distortion measures for 8-node hexahedral isoparametric elements

Abstract: The inverse relations of the isoparametric mapping for the 8-node hexahedra are derived by using the theory of geodesics in differential geometry. Such inverse relations assume the form of infinite power series in the element geodesic coordinates, which are shown to be the skew Cartesian coordinates determined by the Jacobian of the mapping evaluated at the origin. By expressing the geodesic coordinates in turn in terms of the isoparametric coordinates, the coefficients in the resulted polynomials are suggested to be the distortion parameters of the element. These distortion parameters can be used to sompletely describe the inverse relations and the determinant of the Jacobian of the mapping. The meanings of them can also be explained geometrically and mathematically. These methods of defining the distortion measures and deriving the inverse relations of the mapping are completely general, and can be applied to any other two-or three-dimensional isoparametric elements.

An advanced direct time domain BEM formulation for general 3-D elastodynamic problems

Abstract: A new advanced time domain BEM formulation is presented for the study of general 3-D elastodynamic problems. The proposed method is based on the infinite space Stokes fundamental solutions, which, for the first time, are written in terms of body forces in the form of higher order B-spline time distributions. Higher order spatial and temporal discretization schemes are applied to the boundary integral equations of the elastodynamic system yielding a time marching solution for the characteristic response of the system due to an excitation with a B-spline distribution in time. This characteristic response due to a B-spline excitation forms the basis, within the framework of a general B-spline superposition scheme, for the calculation of the responses of the same elastodynamic system due to any transient forcing function.

Large-strain 3D-analysis of fibre-reinforced composites using rebar elements: hyperelastic formulations for cords

Abstract: The well-known finite element representation of reinforcing bars by means of overlay (“rebar”) elements is recast in the context of finite strain analyses of cord-reinforced composite materials. The variational formulation including the linearized forms is presented on the basis of hyperelasticity. Three material laws including two variants of the Neo-Hooekean model and the quadratic logarithmic model are investigated. An explicit formulation for uniaxial stress states is given for the Neo-Hooekean model. A comparative evaluation with regards to computational efficiency and physical plausibility shows that the logarithmic model is optimally suited for this class of problems and for moderately large strains. The rebar-element concept in conjunction with an incompressible finite element formulation for the representation of a rubber matrix material is applied to comparative finite strain FE-analyses of a cord-reinforced rubber sandwich panel, with different hyperelastic models used for modelling of the ply material.

A second-order Godunov algorithm for two-dimensional solid mechanics

Abstract: The second-order Godunov method is extended to dynamic wave propagation in two-dimensional solids undergoing nonlinear finite deformation. It is shown that this explicit method is linearly stable for timesteps satisfying the standard CFL condition, does not support the development of hourglass modes, and handles non-reflecting boundaries very naturally. The computational cost is essentially one evaluation of the kinetic equation of state per cell and timestep, the same as explicit finite element methods employing reduced quadrature.

Low-velocity impact of laminated composites using a layerwise theory

Abstract: A layer-wise theory is used to study the low velocity impact response of laminated plates. The forced-vibration analysis is developed by the modal superposition technique. Six different models are introduced for representation of the impact pressure distribution. The first five models, in which the contact area is assumed to be known, result in a nonlinear integral equation similar to the one obtained by Timoshenko in 1913. The resulting nonlinear integral equation is discretised using a time-finite-element scheme. Two different interpolation functions, namely: (i) Lagrangian and (ii) Hermite are used to express the impact force. The Hermitian-polynomials based representation, obviously, more sophisticated, is introduced to verify the Lagrangian based representation. Due to its modular nature the present numerical technique is preferable to the existing numerical methods in the literature. The final loading model, in which the time dependence of the contact area is taken into account according to the Hertzian contact law, resulted in a relatively more complicated but more relalistic, nonlinear integral equation. The analytical developments concerning this model are all new and reported for the first time in this paper. Also a simple, but accurate, numerical technique is developed for solving our new nonlinear integral equation which results in the time-history of the impact force. Our numerical results are first tested with a series of existing example problems. Then a detailed study concerning all the response quantities, including the in-plane and interlaminar stresses, is carried out for cross-ply laminates and important conclusions are reached concerning the usefulness and accuracy of the various plate theories.

An enthalpy formulation based on an arbitrarily deforming mesh for solution of the Stefan problem

Abstract: Traditionally schemes for dealing with the Stefan phase change problem are separated into fixed grif or front tracking (deforming grid) schemes. A standard fixed grid scheme is to use an enthalpy formulation and track the movement of the phase front via a liquid fraction variable. In this paper, an enthalpy formulation is applied on a continuously deforming finite element grid. This approach results in a general numerical scheme that incorporates both front tracking and fixed grid schemes. It is shown how on appropriate setting of the grid velocity a fixed or deforming grid solution can be generated from the general scheme. In addition an approximate front tracking scheme is developed which can produce accurate non-oscillatory predictions at a computational cost close to an efficient fixed grid scheme. The versatility of the general scheme and the approximate front tracking scheme are demonstrated on solution of a number of Stefan problems in both one and two dimensions.

Fracture analysis of multi-site cracking in fuselage lap joints

Abstract: A two-dimensional plane stress elastic fracture mechanics analysis of a cracked lap joint fastened by rigid pins is presented and results are applied to the problem of multi-site damage (MSD) in riveted lap joints of aircraft fuselage skins. Two problems are addressed, the problem of equal length MSD cracks and the problem of alternating length MSD cracks. For the problem of equal length cracks, two models of rivet/skin interactions are studied and the role of residual stresses due to the riveting process is explored. Stress intensity factors are obtained as a function of normalized crack length. Also, the load distribution among rivet rows and the compliance change of the joint due to MSD cracking are obtained. For the problem of alternating length cracks, attention is focussed on how load is distributed between columns of rivets and how this load shedding can alter crack tip stress intensity factors. The equal and alternating length crack analyses reveal no clear-cut mechanism to explain the relative uniformity of fatigue cracks emerging from lap joint rivet holes in actual aircraft and in mechanical lap joint tests.

Non-linear analysis of plates by the Analog Equation Method

Abstract: The Analog Equation Method is applied to large deflection analysis of thin elastic plates. The von-Karman plate theory is adopted. The deflection and the stress function of the non-linear problem are established by solving two linear uncoupled plate bending problems under the same boundary conditions subjected to “appropriate” (equivalent) fictitious loads. Numerical examples are presented which illustrate the efficiency and the accuracy of the proposed method.

A quasi-conforming triangular laminated composite shell element based on a refined first-order theory

Abstract: A “quasi-conforming” triangular laminated shell element based on a refined first-order shear deformation theory is presented. The Hu-Washizu variational principle, involving strain and displacement fields as variables, with stresses being considered as Lagrange multipliers, is used to develop the laminate composite shell element. Both strains and displacements are discretized in the element, while displacements alone are discretized at the boundary. The inter-element C 1 continuity is satisfied a posteriori in a weak form. Due to the importance of rotations and shear deformation in the geometrically non-linear analyses of shells, 7 degrees of freedom per node are chosen, viz. three displacements, two first-derivatives in the in-plane directions of the out-of-plane displacement, and two transverse shear strains at each node. To consider the effect of transverse shear deformation on the global behavior of the laminated composite shell, the Reissner-Mindlin first-order theory, with shear correction factors of Chow and Whitney, is adopted. The transverse shear stresses are obtained through the integration of the 3-D equilibrium equations; and the warping induced by transverse shear is considered in the calculation of the in-plane stresses to improve their accuracy. Numerical examples show that the element has good convergence properties and leads to highly accurate stresses.

Completed double layer boundary element method in elasticity and stokes flow: Distributed computing through PVM

Abstract: This paper reports a message-passing master/slave implementation of an indirect boundary element method under a distributed computing environment through Parallel Virtual Machine (PVM). The method involves a deflation of the spectral radius followed by a domain iteration which is most suitable to deal with particulate solids or flowing suspensions. The scalability, and the nice load balancing characteristics of the method afforded by the domain decomposition is illustrated by a number of large scale simulations, including the shear and elongational deformation of arrays of rigid inclusions in an elastic matrix, and the sedimentation of a sphere through an array of neutrally-buoyant particles. Our experience indicates that PVM provides an effective distributed computing environment for suspension research and its applications.

Advances in control-volume-based finite-element methods for compressible flows

Abstract: Recent developments in the application of a control-volume-based finite-element method, that has proven successful in solving incompressible flow problems, to the solution of compressible flow problems are presented. The finite Element Differential Scheme (FIELDS) is demonstrated to retain the pressure checker boarding problem for the case of Euler flow under certain conditions of flow. The source of this is investigated and remedies are provided that surmount this problem for all flows including Euler flows. Success is demonstrated for incompressible flow and a formulation is provided for extension to compressible flows. One dimensional testing on a supersonic converging-diverging nozzle exhibits extremely high accuracy of flow prediction including both shock strength and sharpness of resolution.

3-D finite element analysis of composite beams with parallel fibres, based on homogenization theory

Abstract: The theoretical model of a beam of unidirectional composites, based on the homogenization theory and a refined kinematical hypothesis is presented. Effective material coefficients for the constitutive equation are computed. Description of the stresses on the level of the periodic microstructure is given. The kinematical hypothesis for the beam type behaviour includes the independent shear rotations. The resulting modelling strategy is presented for which a finite element code has been developed. Application of the theory to superconducting coils is shown.

An analysis of an explicit algorithm and the radial return algorithm, and a proposed modification, in finite plasticity

Abstract: This paper presents an analysis of Nemat-Nasser's (1991) explicit algorithm and a radial return algorithm for isotropic Von Mises materials undergoing large deformations. It is found that the final-radial return algorithm can result in a simple scalar equation (for the constant of proportionality that defines the plastic flow), which can be easily solved. A new modified algorithm is also proposed. All the three algorithms are efficient. No iterative scheme is required. For proportional loading, all the three algorithms work well. But, in cases where the direction of stress cannot well follow the direction of the deformation rate, which occurs when spin effect is significant or when the direction of deformation rate keeps changing, the explicit algorithm has a problem with its convergence; the final-radial return algorithm oscillates for large time steps; while the presently proposed modified algorithm can provide reasonable results even for large time steps, without any convergence problems.

An analytical-numerical alternating method for elastic-plastic analysis of cracks

Abstract: A new algorithm based on the Schwartz-Neumann alternating technique is developed for the solution of elastic-plastic fracture mechanics problems. An analytical solution for an elastic crack, with arbitrary crack-face loading, is used inside an initial stress iterative procedure as an addition to the finite element solution for the uncracked body. Iteration processes of the alternating method and of the initial stress method are performed simultaneously. Numerical examples show that the proposed elastic-plastic alternating method in conjunction with the equivalent domain integral method provides reasonable values of the J-integral.

A two-dimensional theory for the analysis of laminated plates

Abstract: A new displacement-based two-dimensional theory for the analysis of multilayered plates is presented. The theory is based on the only kinematic constraint of transverse inextensibility, whereas no restrictions are imposed on the representation of the in-plane displacement components. A governing system of integral-differential equations is obtained which can be given a closed-form solution for a number of problems where no boundary layer are present. It is also shown that most of the 2-D plate models can be directly derived from the presented theory. The possibility of developing asymptotic solutions in the boundary layers is discussed with reference to the problem of a plate in cylindrical bending. Finally some numerical solutions are compared with those given by the plate model by Lo et al. (1977) and with F.E.M. solutions.

A numerical solution for dynamic contact problems satisfying the velocity and acceleration compatibilities on the contact surface

Abstract: For the numerical solution of dynamic contact problems, correct contact points and displacements are determined by iteratively reducing the displacement error vector monotonically toward zero. And spurious oscillations are prevented from the solution by enforcing the velocity and acceleration compatibilities of the contact points with the corresponding error vectors. Economic computation is possible because the accelerated iterative schemes are used and because decomposition of large matrix is not required in the iterations for the contact analysis of elastic bodies. Numerical simulations are conducted to demonstrate the accuracy of the solution and the necessity of the velocity and acceleration compatibilities on the contact surface.

Nonlinear analysis of bimodular composite plates under compression

Abstract: The present work deals with the buckling and post-buckling behaviour of bimodular laminated composite plates. The fiber-governed bimodular constitutive model is adopted for the analysis and the geometrical nonlinearities are accounted for by using the consistent small strain and moderate rotation theory for a shear deformable Mindlin plate. Governing equations of the problem are developed and the finite element formulation is given using lagrangian C°-elements. Some numerical examples are developed to analyze the buckling and post-buckling behaviour of laminated bimodular plates.

Dynamic analysis of inelastic plates by the D/BEM

Abstract: A direct boundary element method is developed for the dynamic analysis of thin inelastic flexural plates of arbitrary planform and boundary conditions. It employs the static fundamental solution of the associated elastic problem and involves not only boundary integrals but domain integrals as well. Thus boundary as well as interior elements are employed in the numerical solution. Time integration is accomplished by the explicit algorithm of the central difference predictor method. A viscoplastic constitutive theory with state variables is employed to model the material behaviour. Numerical results are also presented to illustrate quantitatively the proposed method of solution.

Limit analysis considering initial constant loadings and proportional loadings

Abstract: This paper deals with the limit analysis of rigid perefectly plastic body under the combined action of initial constant loadings and proportional loadings. Using the von Mises yielding condition and the finite element technique, we present an efficient algorithm based on the mathematical programming formula of the classical kinematic theorem of plasticity theory. This algorithm includes an iterative procedure which produces a monotonically decrescent sequence converging to an upper bound of the real limit load. The results of some numerical examples are presented and show the stable convergency of the new algorithm.

Boundary element analysis of dissimilar materials and interface crack

Abstract: The highly-accurate BEM elastostatic program, which is especially useful for the analysis of dissimilar materials and interface cracks, is introduced in brief By using this program, we can deal with the elastostatic poblems of isotropic or orthotropic dissimilar materials and also the bonded residual stress due to the mismatch of material constants This paper shows some applications of the BEM program to the analysis of dissimilar materials and interface cracks considering the residual stress quantitatively, and also shows the method to evaluate the strength of dissimilar materials based on the interfacial fracture mechanics Some experimental results and the evaluation on the strength of dissimilar materials are also presented

A boundary element method for calculating the K field for cracks along a bimaterial interface

Abstract: A multi-region boundary element method (BEM) based on the modified crack closure method (CCM) is developed to obtain the energy release rate G for cracks in homogeneous materials and along a bimaterial interface. The energy release rate obtained using the CCM are compared with that obtained using the crack opening displacement (COD) method. A combination of these methods allows us to determine the phase angle ψ and therefore the complex stress intensity factor K for crack problems. We access the accuracy of our BEM by comparing its results with known analytic solutions and previous FEM results in the literature. Computations are also carried out for the asymmetric double cantilever beam (ADCB) specimen, which has been used to determine fracture toughness of polymer/polymer and polymer/nonpolymer interfaces. An auxiliary KA field method to evaluate K is also discussed.

Dynamic shear band development in dipolar thermoviscoplastic materials

Abstract: We study thermomechanical deformations of a viscoplastic body deformed in plane strain compression at a nominal strain-rate of 5000 sec-1 We develop a material model in which the second order gradients of the velocity field are also included as kinematic variables and propose constitutive relations for the corresponding higher order stresses This introduces a material characteristic length l, in addition to the viscous and thermal lengths, into the theory It is shown that the computed results become mesh independent for l greater than a certain value Also, the consideration of higher order velocity gradients has a stabilizing effect in the sense that the initiation of shear bands is delayed and their growth is slower as compared to that for nonpolar (l=0) materials

The application of interface elements and enriched or rate-dependent continua to micro.mechanical analyses of fracture in composites

Abstract: Micromechanical analyses are presented of matrix fracture and fibre-matrix debonding in carbon-epoxy composites. The fibres and the matrix are connected by special interface elements which allow for a geometric discontinuity (debonding) to arise. Two different models for the fibres and the matrix of the composite structure are used: firstly an elasto-plastic Cosserat continuum and secondly a visco-plastic continuum. In the micro-mechanical analyses of matrix fracture it is demonstrated that, upon mesh-refinement, the load deflection curves converge to a unique solution when these regularised continua are applied, and that the width of the localisation band remains finite. When the interaction between matrix cracking and debonding is investigated the same observations hold.