Stress Interpolation Research Articles

Computation of incompressible and weakly-compressible viscoelastic liquids flow: finite element/volume schemes

In this study, we analyse viscoelastic numerical solution for an Oldroyd-B model under incompressible and weakly-compressible liquid flow conditions. We consider flow through a planar four-to-one contraction, as a standard benchmark, throughout a range of Weissenberg numbers up to critical levels. At the same time, inertial and creeping flow settings are also addressed. Within our scheme, we compare and contrast, two forms of stress discretisation, both embedded within a high-order pressure-correction time-marching formulation based on triangles. This encompasses a parent-cell finite element/SUPG scheme, with quadratic stress interpolation and recovery of velocity gradients. The second scheme involves a sub-cell finite volume implementation, a hybrid fe/fv-scheme for the full system. A new feature of this study is that both numerical configurations are able to accommodate incompressible, and low to vanishing Mach number compressible liquid flows. This is of some interest within industrial application areas. We are able to provide parity between the numerical solutions across schemes for any given flow setting. Close examination of flow patterns and vortex trends indicates the broad differences anticipated between incompressible and weakly-compressible solutions. Vortex reduction with increasing Weissenberg number is a common feature throughout. Compressible solutions provide larger vortices (salient and lip) than their incompressible counterparts, and larger stress patterns in the re-entrant corner neighbourhood. Inertia tends to reduce such phenomena in all instances. The hybrid fe/fv-scheme proves more robust, in that it captures the stress singularity more tightly than the fe-form at comparable Weissenberg numbers, reaching higher critical levels. The sub-cell structure, the handling of cross-stream numerical diffusion, and corner discontinuity capturing features of the hybrid fe/fv-scheme, are all perceived as attractive additional benefits that give preference to this choice of scheme.

Sub-cell approximations for viscoelastic flows—filament stretching

The accuracy, stability and consistency of new stress interpolation schemes are investigated, based upon sub-cell approximations. This includes the contrast of two alternative hybrid spatial discretisations: a cell-vertex finite element/volume (fe/fv) scheme and a finite element equivalent (fe). Here, the interest is to explore the consequences of utilizing conventional methodology and to demonstrate resulting drawbacks in the presence of complex stress equation source terms. Alternative strategies worthy of consideration are presented for a constant shear viscosity model, that of Oldroyd-B, with strain-hardening and unbounded extensional properties. We demonstrate how high-order accuracy may be achieved by respecting consistency in our algorithmic constructions. Both fe- and fv-spatial discretisations are embedded within this methodology. Linear interpolation for stress, of either fe- or fv-form on triangular sub-cells, is referenced within parent triangular finite elements in two dimensions. Finite element discretisation is employed for the momentum and continuity system, via a second-order pressure-correction scheme. In this regard, a complex filament-stretching flow with a free-surface is selected to compute the transient evolution of kinematic and stress fields. Shortcomings of various upwinding schemes are discussed, whilst dealing with such free-surface type problems and appropriate strategies for dealing with these types of problems are outlined.

Natural Vibrations of Laminated and Sandwich Plates

An exact analytical solution based on the propagator matrix method and a semianalytical solution based on a higher-order mixed approach (displacement and stress interpolation) have been presented in this paper to evaluate the natural frequencies as well as the stress and displacement mode shapes of simply supported, cross-ply laminated and sandwich plates. Continuity of the transverse stresses and displacements has been maintained at the laminae interfaces. Results have been presented for orthotropic plates, symmetric as well as nonsymmetric cross-ply composite and sandwich laminates. Results from the propagator matrix agree well with the published results for frequencies as well as for displacement and stress mode shapes. Furthermore, the frequencies and displacement and stress eigenvectors obtained from the proposed layerwise mixed method are in excellent agreement with those obtained by three-dimensional elasticity theory. Results obtained from the present equivalent single layer theory are in good agreement with those obtained from the displacement based higher order methods. The high accuracy of the present methods is further confirmed by comparing the response of a sandwich plate with significantly different layer properties for which the conventional displacement based formulations yield inaccurate solutions.

Simple constitutive relationship for C–Mn steels deformed at elevated temperatures

The strain, strain rate and temperature dependence of a C-Mn steel (BS grade 430) deformed under plane strain compression conditions have been investigated in the temperature range 1173-1373 K, at strain rates of 0.25-25 s-1. It has been determined that the stress-strain behaviour of this material can be modelled satisfactorily in a rational manner by combining the exponential saturation law advanced by Sah et al., for description of the evolution of the flow stress, and modified appropriately to account for dynamic recrystallisation effects, with the kinetic models advanced by Kocks and Sellars-Tegart-Garofalo. The Kocks model has been introduced for description of the temperature and strain rate dependence of both the initial flow stress, at the beginning of plastic flow, and the apparent saturation stress, achieved at elevated strains. The Sellars-Tegart-Garofalo model has been employed for the description of the actual saturation stress, achieved after the work softening transient brought about by the operation of dynamic recrystallisation, also in terms of temperature and strain rate. The relaxation strain, in the Sah et al. law has been found to be dependent on the effective strain rate and it has been determined that such a dependency can be described by means of a simple parametric power law relationship. Also, it has been found that the optimised value for activation energy for deformation at elevated temperatures is very close to the activation energy of self-diffusion of iron in austenite, which emphasises the need to introduce into this kind of analysis scaling parameters as the temperature dependent shear modulus of the material. The stress-strain/strain rate/temperature relationship that has been developed in the present work is able to characterise satisfactorily both the flow stress and work hardening rate of the material under the deformation conditions explored, allowing a valid interpolation of the flow stress at any strain, temperature, and strain rate.

Mechanical behaviour of 31 6L stainless steel under warm working conditions

A rational description of the constitutive behaviour of 316L stainless steel deformed between 873 K and 1123 K, over a wide range of strain rates, is proposed. Such a description involves the use of the Sah et al. evolution law for the characterisation of change in flow stress, the strain applied and the interpolation of saturation stress at any temperature and strain rate from a matrix derived from experimental flow stress curves. In addition, the initial work hardening rate is assumed to be a constant, independent of deformation temperature and strain rate. It is shown that this simple constitutive description of the material is able to provide a satisfactory description of both the flow stress and work hardening rate under most deformation conditions. Results regarding the change in saturation stress with temperature and strain rate indicate that dynamic strain aging effects, associated with negative values of the strain rate sensitivity of the saturation stress, are present in the temperature range 873-1073 K and strain rates of the order 0.1 -10 s-1.

Two hybrid stress membrane finite element families with drilling rotations

AbstractTwo new assumed stress membrane finite element families with two translational and one rotational degree of freedom per node are presented. The families, denoted by 8β(M) and 8β(D), are rank sufficient and invariant, and are derived using a unified Hu–Washizu like variational formulation. A recent stress mode classification method is used to illustrate the construction of the stress interpolation matrices. In each family, a number of new formulations are derived. Numerical results are presented. Copyright © 2001 John Wiley & Sons, Ltd.

Aliasing inertial oscillations in a 1/6° Atlantic circulation model: impact on the mean meridional heat transport

A 1/6° model simulation of the Atlantic ocean forced with daily fluxes from ECMWF (re-analysis 1979–1993 and analysis 1994–1999) has been carried out within the Clipper project. A storage strategy which filters out inertial oscillations is defined: five-day mean fields are continuously stored at five-day intervals. It is shown that aliasing errors on the monthly mean meridional heat transport (MHT, a second-order moment) are negligible in that case. These errors are of the order of 0.8 PW in the tropics in the case of a sampling strategy based on instantaneous fields stored every five days, even in the case where step-like variations in the forcing are avoided by an interpolation of the daily wind stress to the model time step. It is also shown that aliasing errors on the annual mean MHT can be as large as 0.2 PW in the tropics in the case of sub-sampling with instantaneous fields.

H-adaptive FE analysis of elasto-plastic non-homogeneous soil with large deformation

Abstract Error estimation and h -adaptive finite element procedures are implemented for large deformation analyses of foundations on soil where the strength increases with depth. Errors are estimated by comparing strains at Gauss points with more accurate estimates using Superconvergent Patch Recovery (SPR). Mesh refinement using subdivision concept is then used iteratively to obtain optimal meshes, satisfying a minimum element size criterion. The Remeshing and Interpolation Technique with Small Strain (RITSS) approach is then used for large deformation analysis, with stress interpolation using either a modified form of the unique element method (MUEM), or the stress-SPR approach. Example analyses are then presented illustrating the three main aspects of mesh refinement, stress interpolation and large deformation response. Criteria are given for minimum element size and displacement increment for strip and circular foundations bearing on soil with varying degree of non-homogeneity, and computed bearing capacities are shown to compare well with lower bound estimates. The effect of soil weight on deep penetration of a strip foundation is discussed, with particular reference to the pattern of soil heave adjacent to the foundation, and the magnitude of the bearing capacity.

Stress Intensity Factor Analysis of an Interface Crack between Dissimilar Materials under Thermal Stress Condition by Virtual Crack Extension Method.

A new method is presented for stress intensity factor analysis of interface crack problems under thermal stress conditions. The virtual crack extension method, which is used with the finite element method, is a powerful tool for estimation of the energy release rate. The virtual crack extension method is applied to stress intensity factor analyses of thermal interface crack problems. The energy release rate obtained by the virtual crack extension method is separated into individual stress intensity factors, K1 and KII, by the principle of superposition. The presented method is applied to several thermal stress interface crack problems. The results are compared with those obtained by the displacement and stress interpolation methods. It is found from these analyses that the present method gives very accurate results which are insensitive to the size of finite elements.

Development of finite elements for two-dimensional structural analysis using the integrated force method

The integrated force method has been developed in recent years for the analysis of structural mechanics problems. In the intgrated force method all independent forces are treated as unknown variables, which are calculated by simultaneously imposing equations of equilibrium and compatibility conditions. The development of a finite element library for the analysis of two-dimensional problems using the integrated force method is presented in this paper. Elements of triangular and quadrilateral shapes, capable of modeling arbitrary domain configurations are developed. The element equilibrium and flexibility matrices are derived by discretizing expressions for corresponding potential and complementary energies, respectively. Independent approximations of displacement and stress fields within finite elements are performed. Interpolation of the displacement field is done similarly as in the standard displacement method. The stress field is approximated using full polynomials of correct orders. A procedure for deriving the stress interpolation polynomials that utilizes the definitions of stress components in terms of Airy's stress function is developed. Such derived stress fields identically satisfy equations of equilibrium, and the resulting element matrices are insensitive to the orientation of local coordinate systems. A method to calculate the number of rigid body modes is devised, and it is shown that the present elements do not possess spurious zero energy modes. A number of example problems are solved using the present library and the results are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. A good agreement of the results, and better performance of the integrated force method, compared to the displacement method, in stress calculations, is observed.

Three-dimensional structural analysis by the integrated force method

The “integrated force method”, which has been recently developed for analyzing structures, is extended in this paper for three-dimensional structural analysis. A general formulation to generate the stress interpolation matrix in terms of complete polynomials of the required order is developed first. The formulation is based on the definitions of stress tensor components in terms of stress functions. The stress functions are written as complete polynomials and substituted into expressions for stress components. After eliminating dependent coefficients, the expressions for stress components are obtained as complete polynomials, where coefficients are defined as generalized independent forces. Such derived components of the stress tensor identically satisfy Navier's equations of equilibrium. The resulting element matrices are invariant with respect to coordinate transformation and are free of spurious zero energy modes. The formulation provides a rational way to calculate the exact number of independent forces for the required order of approximation with complete polynomials. The reduction in the number of independent forces and its influence on the accuracy of the response are also analyzed. The stress fields derived here are next used to develop a comprehensive finite element library for the three-dimensional structural analysis using the “integrated force method”. Elements of both tetrahedral and hexahedral shapes, capable of modeling arbitrary geometric configurations, are developed. A number of example problems with available analytical solutions are solved using the present developments and a good agreement of results with the analytical solutions is observed. The responses obtained using the “integrated force method” are also compared with those generated with the standard displacement method. A better overall performance of the “integrated force method” is observed.

A new mixed finite element method for viscoelastic flows governed by differential constitutive equations

A new mixed finite element method is presented that has improved numerical stability and has similar numerical accuracy compared to the EVSS/FEM developed by Rajagopalan et al. (J. Non-Newtonian Fluid Mech. 36 (1990) 159–192). The new method, denoted EVSS-G/FEM, is based on separate bilinear interpolation of all relevant components of the velocity gradient tensor. Lagrangian bilinear approximations of this variable are formed by least-squares interpolation of the derivatives of a Lagrangian biquadratic approximation to the velocity field that is generated by solution of the momentum and continuity equations using a standard mixed formulation for velocity and pressure. Elastic components of the stress are computed by solving the constitutive equation using either the streamline upwinding (SU) or streamline upwind Petrov-Galerkin (SUPG) methods with bilinear interpolation. The motivation for the interpolation of the velocity gradients and for the stress interpolation is to force the compatibility between these variables that is needed to satisfy differential viscoelastic constitutive equations in the limit where the velocity field vanishes, as it does near solid boundaries. The enhanced numerical stability of the EVSS-G/FEM over the EVSS/FEM formulation is demonstrated for the calculation of the linear stability of planar Couette flow of a UCM fluid, where closed form results exist for the linear stability of the steady-state flow; hence, any instability detected in the numerical solution is a result of either the finite element approximation or the time integration method. The second possibility is eliminated by using a fully implicit time integration method. Calculations with the EVSS-G/FEM are stable for De>100, whereas calculations with the EVSS/FEM become numerically unstable for De>5. The EVSS-G formulation is combined with finite element discretizations that incorporate local adaptive mesh refinement for increasing the accuracy of the calculations in regions where the stress and velocity gradients change rapidly. The accuracy of the EVSS-G/FEM is demonstrated by steady-state calculations for the flow between eccentric rotating cylinders, flow through a wavy-walled tube, and for flow through a square array of cylinders; each has been used before as a test problem for the accuracy of viscoelastic flow calculations. As in previous calculations, discretization of the constitutive equation by SUPG gives superior accuracy at high values of De compared to the application of SU.

On the use of finite element programs in structural evaluation and development of design charts

In many phases of structural design and analytical evaluation, the solution of stress and strain distribution in an elastic continuum is required. Special cases of such problems may range from two-dimensional plane stress or plain strain distribution, plate bending to analysis of fully three-dimensional solids. The finite element programs are often used to predict the critical regions for stress analysis and design. Stresses are generally of greater practical importance than displacements for structural design and evaluation. Most of the finite element computer programs calculate element stresses at the centroids, integration points, or nodes of elements. In this paper, examples of bridge deck analysis are used to illustrate the stress interpretation using the finite element programs. It is demonstrated that the stresses at nodes calculated by some finite element programs violate the equilibrium conditions and do not converge to the correct answers. These calculated stresses at nodes are usually too low and lead to unsafe designs and evaluations. Key words: finite element method, finite element programs, structural analysis, least square smoothing, stress interpolation.

An effective curved composite beam finite element based on the hybrid-mixed formulation

A laminated curved-beam finite element with six displacement degrees of freedom and three stress parameters is derived and evaluated. Both thermal and hygrothermal effects are included. The element is based on the Hellinger-Reissner principle and the hybrid-mixed formulation. The Timoshenko beam theory and classical lamination theory are employed in the finite element description. Within an element linear displacement interpolation is used; the generalized stresses are interpolated by either stress functions based on the equilibrium equations ( P 1) or constant stress approximation ( P 2). The beam element stiffness is obtained explicitly and numerical results show very good displacement prediction compared to analytical solutions. Generalized stresses are predicted accurately at the mid-point of the finite element only for constant stress interpolation. The P 1-type element yields more accurate displacement and stress prediction.

Analysis of bonding problems under thermal loading using the combined mixed functional

A new mixed model based on the combined mixed functional has been developed for the analysis of problems with bonding conditions and applied to the bimetallic thermostat problem. The mixed formulation has satisfied equilibrium equations, the displacement and the traction boundary conditions, as well as the interfacial continuities of traction in an average sense by virtue of the variational condition of the proposed mixed functional. A mixed finite element (QC 4/8), having no stability problem, has been proposed and successfully applied for the analysis of bonded structures with the selected combination coefficient. A bonding element has been developed based on a patch test-passed quadratic approximation for the relative displacements and bonding tractions in an average sense. Along all the bonding interfaces the results of the mixed method with the continuous stress interpolation were in close agreement with those of the stress formulation based on the principle of complementary virtual work.

On stress interpolation for hybrid models

AbstractA technique is proposed for the selection of stress interpolations for hybrid models. The present paper applies this approach to plane problems. The stiffness matrix is derived using the Hellinger–Reissner variational principle. This formulation uses infinitesimal equilibrium relationships and divides the assumed stress into its lower‐order and higher‐order parts. The patch test can be passed and the resulting elements are generally invariant. A plane four‐node quadrilateral element is described and compared with existing elements. Numerical studies show that the accuracy of the element is generally good.

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.

An efficient eight-node incompatible solid element with stress interpolation

A new eight-node solid element based on Hellinger-Reissner's energy functional modified by constant stress multipliers is presented. To ensure frame invariance, isoparametric coordinates are adopted in the interpolation of both stresses and displacements. Besides possessing the minimum number of stress parameters needed to avoid the occurrence of zero energy deformation modes, the eight-node element exhibits a low distortion sensitivity. Selected examples providing a broad range of analytical stress and displacement distributions are used to examine the performance of the element. Numerical results show that the element being suggested yields good predictions for both displacements and stresses when compared with other three-dimensional elements.

A finite element formulation for highly anisotropic incompressible elastic solids

AbstractA numerical approach is developed for the solution of problems of materials with extremely strong directions. Small deformations of a transversely isotropic linear elastic solid, reinforced by a single family of inextensible fibres, are considered. The kinematic constraint equations of incompressibility and inextensibility in the fibre direction lead to the appearance of an arbitrary hydrostatic pressure and an arbitrary tension stress in the constitutive equations. A Galerkin approach is used to discretize the virtual work and weak form of the constraint equations. Independent interpolation of the displacement, pressure and tension fields leads to a mixed system of equations, with characteristic zero‐diagonal terms. The assumption of plane stress conditions in the plane of the fibres results in a simplified displacement‐tension formulation, analogous to the primitive‐variable formulation of Stokes flow. A mixed penalty approximation is then employed to solve for displacement and tension stress fields. Computations are carried out using a biquadratic displacement element with discontinuous bilinear tension stress interpolation. The formulation is used to solve a number of simple beam problems and the results compared to closed‐form solutions.

A class of mixed assumed strain methods and the method of incompatible modes

AbstractA three‐field mixed formulation in terms of displacements, stresses and an enhanced strain field is presented which encompasses, as a particular case, the classical method of incompatible modes. Within this frame‐work, incompatible elements arise as particular ‘compatible’ mixed approximations of the enhanced strain field. The conditions that the stress interpolation contain piece‐wise constant functions and be L2‐ortho‐gonal to the enhanced strain interpolation, ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation. The preceding conditions are formulated in a form particularly convenient for element design. As an illustration of the methodology three new elements are developed and shown to exhibit good performance: a plane 3D elastic/plastic QUAD, an axisymmetric element and a thick plate bending QUAD. The formulation described herein is suitable for non‐linear analysis.