### Browsing by Department "Centre for Research in Computational and Applied Mechanics (CERECAM)"

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- ItemOpen AccessAnalysis of Nonlinear Dispersion of a Pollutant Ejected by an External Source into a Channel Flow(2010) Chinyoka, T; Makinde, O DThis paper focuses on the transient analysis of nonlinear dispersion of a pollutant ejected by an external source into a laminar flow of an incompressible fluid in a channel. The influence of density variation with pollutant concentration is approximated according to the Boussinesq approximation, and the nonlinear governing equations of momentum and pollutant concentration are obtained. The problem is solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, pollutant concentration, skin friction, and wall mass transfer rate for various parametric values. The model can be a useful tool for understanding the polluting situations of an improper discharge incident and evaluating the effects of decontaminating measures for the water body.
- ItemOpen AccessA computational neuromuscular model of the human upper airway with application to the study of obstructive sleep apnoea(2014) Pelteret, Jean-Paul; Reddy, B DayaNumerous challenges are faced in investigations aimed at developing a better understanding of the pathophysiology of obstructive sleep apnoea. The anatomy of the tongue and other upper airway tissues, and the ability to model their behaviour, is central to such investigations. In this thesis, details of the construction and development of a three-dimensional finite element model of soft tissues of the human upper airway, as well as a simplified fluid model of the airway, are provided. The anatomical data was obtained from the Visible Human Project, and its underlying micro-histological data describing tongue musculature were also extracted from the same source and incorporated into the model. An overview of the mathematical models used to describe tissue behaviour, both at a macro- and microscopic level, is given. Hyperelastic constitutive models were used to describe the material behaviour, and material incompressibility was accounted for. An active Hill three-element muscle model was used to represent the muscular tissue of the tongue. The neural stimulus for each muscle group to a priori unknown external forces was determined through the use of a genetic algorithm-based neural control model. The fundamental behaviour of the tongue under gravitational and breathing-induced loading is investigated. The response of the various muscles of the tongue to the complex loading developed during breathing is determined, with a particular focus being placed to that of the genioglossus. It is demonstrated that, when a time-dependent loading is applied to the tongue, the neural model is able to control the position of the tongue and produce a physiologically realistic response for the genioglossus. A comparison is then made to the response determined under quasi-static conditions using the pressure distribution extracted from computational fluid-dynamics results. An analytical model describing the time-dependent response of the components of the tongue musculature most active during oral breathing is developed and validated. It is then modified to simulate the activity of the tongue during sleep and under conditions relating to various possible neural and physiological pathologies. The retroglossal movement of the tongue resulting from the pathologies is quantified and their role in the potential to induce airway collapse is discussed.
- ItemOpen AccessComputational simulation of bone remodelling post reverse total shoulder arthroplasty(2017) Liedtke, Helen; McBride, Andrew Trevor; Reddy, B DayaBone is a living material. It adapts, in an optimal sense, to loading by changing its density and trabeculae architecture - a process termed remodelling. Implanted orthopaedic devices can significantly alter the loading on the surrounding bone. In addition, these devices rely on bone ingrowth to ensure secure implant fixation. In this project, a computational model that accounts for bone remodelling is developed and used to elucidate the response of bone following a reverse shoulder procedure. The reverse shoulder procedure investigated here is for rotary cuff deficient patients. In this procedure up to 75 % complications are reported in some clinical series. It is therefore necessary, for the design of successful implants, to understand the loading environment to promote bone growth in the correct areas. The physical process of remodelling is modelled using continuum scale, open system thermodynamics whereby the density of bone evolves isotropically in response to the loading it experiences. The fully-nonlinear continuum theory is solved approximately using the finite element method. The finite element library AceGEN forms the basis for the implementation. Several benchmark problems were implemented to validate the code and demonstrate features of the theory. These include several one-dimensional problems, the classical two-dimensional femur benchmark, and a series of three-dimensional examples. The three-dimensional examples include different loading scenarios on a rectangular block, as well as the investigation of the ASTM testing procedure of the glenoid side prosthesis implanted in a polyurethane foam block. The results clearly demonstrate the adaptive behaviour of the bone density in response to the magnitude and duration of the loading. The numerical implementation is also shown to be robust. The remodelling of the scapula post reverse shoulder arthroplasty is then investigated. A statistical shape model of the scapula was obtained from collaborators in the Division of Biomedical Engineering at the University of Cape Town. The finite element model was used to determine the density distribution in the scapula prior to surgery. A virtual surgery was then performed. The resulting geometry provides the input for the pre-processing phase of the post reverse shoulder arthroplasty model. The loading conditions for the reverse shoulder were provided by collaborators in the Division of Biomedical Engineering and the Leon Root Motion Analysis Laboratory at the Hospital for Special Surgery in New York City. The maximal loading condition at 90° abduction is used as the input for the simulation. It was found that the density increases in the vicinity of the screws, where the maximum stresses are concentrated, however, bone resorption is observed directly below and adjacent to the implant. No conclusive statement can be made, however, as only one loading scenario is considered and calibration of the model against experimental results is still outstanding. A unique feature of the code is that the upper and lower bounds of the density do not have to be enforced directly, as done in most bone remodelling theories in the literature. Rather, the bounds of the density are naturally enforced by calibrating the mass flux for the problem at hand. This project lays out the groundwork for a sound remodelling code, which can serve as a predictive tool in the field of orthopaedics.
- ItemOpen AccessConsistency and convergence of SPH approximations(2009) Penzhorn, KarlThis thesis is about a new approach to SPH. Instead of using a single kernel or shape function for approximation of a function and its derivatives, individual shape functions are used for each derivative. The investigation is carried out in one space dimension. After producing the conditions for consistency and convergence for the zeroth, first and second derivatives, a new set of linear or piecewise-linear shape functions which meet the minimum of these requirements are presented for each.
- ItemOpen AccessConstitutive modelling of the skin accounting for chronological ageing(2017) Pond, Damien; McBride, Andrew TrevorThe skin is the largest organ in the human body. It is the first line of contact with the outside world, being subject to a harsh array of physical loads and environmental factors. In addition to this, the skin performs numerous physiological tasks such as thermo-regualtion, vitamin D synthesis and neurotransduction. The skin, as with all biological tissue, is subject to chronological ageing, whereby there is a general breakdown of tissue function and a decline in mechanical properties. In addition to this, skin undergoes extrinsic forms of ageing through exposure to external factors such as ultraviolet radiation, air pollution and cigarette smoking. Skin modelling is an area of biomechanics that, although medical in nature, has expanded into areas such as cosmetics, military, sports equipment and computer graphics. Skin can be approximated at the macroscopic continuum scale as an anisotropic, nearly-incompressible, viscoelastic and non-linear material whose material properties are highly dependent on the ageing process. Through the literature, several phenomenologically based models have been satisfactorily employed to capture the behaviour inherent to the skin, but despite the intrinsic link to age, to date no constitutive model for the UV-induced ageing/damage of skin has been developed that is both capable of capturing the material and structural effects, and is embedded in the rigorous framework of non-linear continuum mechanics. Such a mechanistic model is proposed here. The macroscopic response of the skin is due to microscopic components such as collagen, elastin and the surrounding ground substance and the interaction between them. An overview on the structure of the skin helps motivate the form of the continuum model and identifies which aspects of the skin need to be captured in order to replicate the macroscopic response. Furthermore, the ageing process is explored and a firm understanding of the influence of ageing on the substructures is established. Over time, elastin levels tend to decrease which results in a loss of skin elasticity. Collagen levels drop with age, but tend to flatten out which results in an overall increase in skin stiffness and loss of anisotropy. A worm-like chain constitutive model, arranged in an 8-chain configuration, is employed to capture the mechanical response of the skin. The use of such a micro-structurally-motivated model attempts to connect the underlying substructures (collagen, elastin and ground substance) present in the skin to the overall mechanical response. The constitutive model is implemented within a finite element scheme. Simple uniaxial tests are employed to ascertain the validity of the model, whereby skin samples are stretched to elicit the typical anisotropic locking response. A more complex loading condition is applied through bulge tests where a pressure is applied to an in vitro skin specimen. This more complex test is subsequently used to conduct a series of ageing numerical experiments to ascertain the response of the model to changes in material properties associated with ageing. A modified model is then proposed to capture the ageing response of the skin. The key microscopic biophysical processes that underpin ageing are identified, approximated and adapted sufficiently to be of use in the macroscopic continuum model. Aspects of open-system thermodynamics and mixture theory are adapted to the context of ageing in order to capture a continuous ageing response.
- ItemOpen AccessDevelopment of a Micromorphic (Multiscale) Material Model aimed at Cardiac Tissue Mechanics(2020) Dollery, Devin; Skatulla, SebastianComputational cardiac mechanics has historically relied on classical continuum models; however, classical models amalgamate the behaviour of a material's micro-constituents, and thus only approximate the macroscopically observable material behaviour as a purely averaged response that originated on micro-structural levels. As such, classical models do not directly and independently address the response of the cardiac tissue (myocardium) components, such as the muscle fibres (myocytes) or the hierarchically organized cytoskeleton. Multiscale continuum models have developed over time to account for some of the micro-architecture of a material, and allow for additional degrees of freedom in the continuum over classical models. The micromorphic continuum [15] is a multiscale model that contains additional degrees of freedom which lend themselves to the description of fibres, referred to as micro-directors. The micromorphic model has great potential to replicate certain characteristics of the myocardium in more detail. Specifically, the micromorphic micro-directors can represent the myocytes, thus allowing for non-affine relative deformations of the myocytes and the extracellular matrix (ECM) of tissue constraining the myocytes, which is not directly possible with classical models. A generalized micromorphic approach of Sansour [73, 74, 75] is explored in this study. Firstly, numerical examples are investigated and several novel proofs are devised to understand the behaviour of the micromorphic model with regards to numerical instabilities, micro-director displacements, and macro-traction vector contributions. An alternative micromorphic model is developed by the author for comparison against Sansour's model regarding the handling of micro-boundary conditions and other numerical artifacts. Secondly, Sansour's model is applied to cardiac modelling, whereby a macro-scale strain measure represents the deformation of the ECM of the tissue, a micro-scale strain measure represents the muscle fibres, and a third strain measure describes of the interaction of both constituents. Separate constitutive equations are developed to give unique stiffness responses to both the ECM and the myocytes. The micromorphic model is calibrated for cardiac tissue, first using triaxial shear experiments [80], and subsequently, to a pressure-volume relationship. The contribution of the micromorphic additional degrees of freedom to the various triaxial shear modes is quantified, and an analytical explanation is provided for differences in contributions. The passive filling phase of the heart cycle is investigated using a patient-specific left ventricle geometry supplied by the Cape Universities Body Imaging Centre (CUBIC) [38].
- ItemOpen AccessFluid-structure interaction modelling of a patient-specific arteriovenous access fistula(2016) Guess, Winston; Reddy, B Daya; McBride, Andrew TrevorThis research forms part of an interdisciplinary project that aims to improve the detailed understanding of the haemodynamics and vascular mechanics in arteriovenous shunts that are required for haemodialysis treatments. A combination of new PCMRA imaging and computational modelling of in vivo blood flow aims to determine the haemodynamic conditions that may lead to the high failure rate of vascular access in these circumstances. This thesis focuses on developing a patient-specific fluid-structure interaction (FSI) model of a PC-MRA imaged arteriovenous fistula. The numerical FSI model is developed and simulated within the commercial multiphysics simulation package ANSYS® Academic Research, Release 16. The blood flow is modelled as a Newtonian fluid with the finite-volume method solver ANSYS® Fluent®. A pulsatile mass-flow boundary condition is applied at the artery inlet and a three-element Windkessel model at the artery and vein outlets. ANSYS® Mechanical™, a finite element method solver, is used to model the nonlinear behaviour of the vessel walls. The artery and vein walls are assumed to follow a third-order Yeoh model, and are differentiated by thickness and by material strength characteristics. The staggered FSI model is configured and executed in ANSYS® Workbench™, forming a semi-implicit coupling of the blood flow and vessel wall models. This work shows the effectiveness of combining a number of stabilisation techniques to simultaneously overcome the added-mass effect and optimise the efficiency of the overall model. The PC-MRA data, fluid model, and FSI model show almost identical flow features in the fistula; this applies in particular to a flow recirculation region in the vein that could potentially lead to fistula failure.
- ItemOpen AccessThe impact of thermophysical properties on nanofluid-based solar collector performance(2016) Gakingo, Godfrey Kabungo; Reddy, B Daya; Macdevette, MichelleNanofluids are a novel class of heat transfer fluids in which nanoparticles are dispersed in traditional heat transfer fluids. They offer enhanced thermophysical, rheological and radiative properties. These enhancements have resulted in recent research being centred on the application of nanofluids to various systems. An example of such systems is the solar volumetric flow receiver in which great efficiency improvements have been reported. To explain this efficiency increase, researchers have evaluated the impact of enhanced radiative properties of nanofluids while largely neglecting that of enhanced thermophysical properties. This study looks at the impact of enhanced thermophysical properties on the performance of nanofluid-based solar volumetric receivers. Particular focus is drawn to the impact of temperature dependent conductivity and volumetric specific heat capacity. Copper oxide - water nanofluid is employed as its temperature dependent properties have been characterised. [Please note: this thesis file has been deferred until June 2016]
- ItemOpen AccessMathematical and computational modelling of the dynamic behaviour of direct current plasma arcs(2010) Reddy, BatmanathanThe problem of direct-current plasma arc behaviour, interaction, and dynamics is considered in the context of metallurgical DC arc furnace applications. Particular attention is paid to the transient flow behaviour of arc systems. A mathematical formulation of the physics used to describe the arc system is presented, and includes the spatial and temporal evolution of fluid flow, heat transfer, and electromagnetism. Based on this formulation, a numerical model is developed using a finite difference approach on a regular cartesian grid in both two and three dimensions, with a special focus on robust stability, high resolution modelling, and high performance. A collection of results produced using the numerical model to study pilot plant-scale furnaces is then presented. These address a range of process and design variables and their effect on the numerical model's results. Where possible, the qualitative behaviour of the model is compared to available experimental data. A number of novel effects and phenomena are seen in the dynamic behaviour of the DC plasma arc model for both single and multiple arc systems, which may lead to improved understanding, control, and manipulation of such systems where they occur in industrial applications.
- ItemOpen AccessNumerical investigation of the edge profile in hot-rolling(1992) Veale, John; Mercer, Colin Douglas; Martin, J BDuring the hot-rolling of aluminium ingot into sheet, the material elongates in the rolling direction as it is reduced vertically. The spread which occurs in the lateral direction during the multiple pass schedules used in industry is minimal. However, the deformation on these edges is important. During the initial passes a concave profile develops - the material near the surfaces spreads outward while the material at the centre moves inward. The concave profile can lead to defects in the final product; these are the 'roll over' of material from the edges to the top and bottom surfaces, the fold over of material in the centre of the edge, and the formation of vertical edge cracks. To remove these the edges are trimmed at the end of the process. Research work in this area was motivated by the possibility of identifying means of reducing the amount of material that needs to be trimmed. The objectives of this thesis are to develop techniques of simulating the rolling, and to use these to investigate the deformation mechanisms which lead to the concave edge profile. Models of the rolling were developed using the general purpose, non-linear finite element code ABAQUS. To reproduce the edge profiles accurately requires large three-dimensional models, for which the explicit dynamic method was found to be the most suitable. The results of the analyses were used to investigate the mechanisms which lead to the concave edge profile. In the roll-gap the work-load arches through the ingot; and for roll passes with small reductions a stress pattern occurs which leads to the concave edge profile. In this pattern the stresses of highest magnitude at the surfaces are compressive stresses in the vertical direction, while in the centre of the ingot they are orientated in the rolling direction and are tensile. Thus deformation occurs by vertical compression near the surfaces, and by stretching in the rolling direction at the centre. At the edges the material is not constrained laterally; and due to the Poisson effect, the material spreads outward near the surfaces, and moves inward at the centre. The effect of certain variables on the edge profile were investigated with the modelling. The friction between the work-rolls and the ingot was found to have significant influence on the amount of lateral surface spread. Work hardening, strain rate and temperature effects in the material lead to variations in the yield stress through the height of the ingot. These effects were included in the modelling and were found to affect the shape of the profile, but to a lesser extent than the friction.
- ItemOpen AccessA patient-specific FSI model for vascular access in haemodialysis(2017) De Villiers, Anna Magdalena; Reddy, B Daya; McBride, Andrew TrevorThis research forms part of an interdisciplinary project that aims to improve the understanding of haemodynamics and vascular mechanics in arteriovenous shunting. To achieve the high flow rates that enable patients with renal disease to receive haemodialysis, a fistula is created between an artery and a vein. The patency rate of fistulas, especially those located in the upper arm, is low. The approach adopted here makes use of new magnetic resonance image (MRI) technology and computational modelling of blood flow, with a view to improving therapeutic strategies of disease requiring vascular interventions. This thesis presents the construction and development of a 3D finite element model of the fluid-structure interaction in a brachial–cephalic patient–specific fistula. An overview of the mathematical models that describe the vessel wall and fluid behaviour as well their interaction with each other is given. An Arbitrary Lagrangian- Eulerian (ALE) framework is used together with a transversely isotropic hyperelastic constitutive model for the vessel walls, while blood flow is modelled as a Newtonian fluid. A three-element Windkessel model is used to allow the fluid to move through the outlets of the computational domain without causing non–physical reflections. Flow data acquired from MRI is used to prescribe the flow at the inlet. The parameters of the Windkessel-model at the two outlets are calibrated to resemble the flow acquired from the 2D MRI. The model is validated against the flow patterns acquired from the 4D MRI. The flow patterns of the blood, and stress present in the vessel are investigated. Of special significance are the flow and wall shear stress at the anastomosis. An area of very high velocity in the anastomosis is followed by an area of recirculation and low velocity. The propagation of pressure waves and their reflection at the anastomosis are studied. Areas that are subjected to low wall shear stress, high oscillatory wall shear stress or flow circulation are identified as areas where intimal hyperplasia may develop. The flow results from the simulation show good qualitative agreement with the MRI data.
- ItemOpen AccessProper orthogonal decomposition with interpolation-based real-time modelling of the heart(2017) Rama, Ritesh Rao; Skatulla, Sebastian; Reddy, DayaSeveral studies have been carried out recently with the aim of achieving cardiac modelling of the whole heart for a full heartbeat. However, within the context of the Galerkin method, those simulations require high computational demand, ranging from 16 - 200 CPUs, and long calculation time, lasting from 1 h - 50 h. To solve this problem, this research proposes to make use of a Reduced Order Method (ROM) called the Proper Orthogonal Decomposition with Interpolation method (PODI) to achieve real-time modelling with an adequate level of solution accuracy. The idea behind this method is to first construct a database of pre-computed full-scale solutions using the Element-free Galerkin method (EFG) and then project a selected subset of these solutions to a low dimensional space. Using the Moving Least Square method (MLS), an interpolation is carried out for the problem-at-hand, before the resulting coefficients are projected back to the original high dimensional solution space. The aim of this project is to tackle real-time modelling of a patient-specific heart for a full heartbeat in different stages, namely: modelling (i) the diastolic filling with variations of material properties, (ii) the isovolumetric contraction (IVC), ejection and isovolumetric relation (IVR) with arbitrary time evolutions, and (iii) variations in heart anatomy. For the diastolic filling, computations are carried out on a bi-ventricle model (BV) to investigate the performance and accuracy for varying the material parameters. The PODI calculations of the LV are completed within 14 s on a normal desktop machine with a relative L₂-error norm of 6x10⁻³. These calculations are about 2050 times faster than EFG, with each displacement step generated at a calculation frequency of 1074 Hz. An error sensitivity analysis is consequently carried out to find the most sensitive parameter and optimum dataset to be selected for the PODI calculation. In the second phase of the research, a so-called "time standardisation scheme" is adopted to model a full heartbeat cycle. This is due to the simulation of the IVC, ejection, and IVR phases being carried out using a displacement-driven calculation method which does not use uniform simulation steps across datasets. Generated results are accurate, with the PODI calculations being 2200 faster than EFG. The PODI method is, in the third phase of this work, extended to deal with arbitrary heart meshes by developing a method called "Degrees of freedom standardisation" (DOFS). DOFS consists of using a template mesh over which all dataset result fields are projected. Once the result fields are standardised, they are consequently used for the PODI calculation, before the PODI solution is projected back to the mesh of the problem-at-hand. The first template mesh to be considered is a cube mesh. However, it is found to produce results with high errors and non-physical behaviour. The second template mesh used is a heart template. In this case, a preprocessing step is required where a non-rigid transformation based on the coherent point drift method is used to transform all dataset hearts onto the heart template. The heart template approach generated a PODI solution of higher accuracy at a relatively low computational time. Following these encouraging results, a final investigation is carried out where the PODI method is coupled with a computationally expensive gradient-based optimisation method called the Levenberg- Marquardt (PODI-LVM) method. It is then compared against the full-scale simulation one where the EFG is used with the Levenberg-Marquardt method (EFG-LVM). In this case, the PODI-LVM simulations are 1025 times faster than the EFG-LVM, while its error is less than 1%. It is also observed that since the PODI database is built using EFG simulations, the PODI-LVM behaves similarly to the EFG-LVM one.
- ItemOpen AccessThe study of creep in machine elements using finite element methods(1990) Weber, Marc Anton; Penny, R KIn this thesis a simplified analysis procedure is developed, in which creep laws are decoupled from damage laws, for the purposb of constructing methods of use in the early stages of high temperature design. The procedure is based on the creep and damage laws proposed by Kachanov and Rabotnov. The creep laws are normalised. with respect to a convenient normalising stress. As a consequence of this normalisation, the dependence of the creep law on the stress constant, the time and temperature functions, and the actual load level is removed. In addition, if the reference stress of the component is chosen as the normalising stress, the creep law becomes insensitive to the stress exponent. The non-dimensional creep laws are then implemented in a standard finite element scheme, from which the results of a stationary state creep analysis are then in non-dimensional form. In order to estimate rupture times, the maximum stationary stresses in a component are used together with the damage laws. Conservative failure criteria are derived from the creep and damage laws to extend the method to residual life assessment and damage monitoring. The procedure is illustrated and tested against simple examples and case studies.
- ItemOpen AccessThree-field mixed finite element approximations for problems in elasticity(2013) Chama, Abdoulkadri; Reddy, DayaThis thesis is concerned with three-field mixed methods for elasticity (often referred to as Hu-Washizu formulations) in which the variables are, for small-strain problems, the displacement, stress and strain. For problems in nonlinear elasticity the corresponding variables are the displacement, first Piola-Kirchhoff stress, and deformation gradient. Of particular interest is the design and analysis of mixed formulations that are uniformly stable in the incompressible limit. The first part of the thesis deals with problems in linear elasticity. Lamichhane, Reddy and Wohlmuth (Numer. Math., 104 (2006)) have shown that the conditions for stability and uniform convergence include an ellipticity condition and, secondly, a condition that the displacement together with a discrete pressure, suitably defined, constitute a stable Stokes pair. The latter condition implies that the inf-sup condition for the three-field formulation is satisfied. In the thesis, families of new stable mixed elements are generated by the following approach. First, a stable Stokes pair is chosen. Then, the space of discrete stresses is defined such that the associated discrete pressure corresponds to that of the Stokes pressure. The space of strains is defined such that it forms a superset of the space of stresses. The final task is that of showing that the spaces chosen in this way satisfy the discrete ellipticity condition. A number of new families of mixed elements are designed and analyzed in this way, and numerical examples in two and three space dimensions are presented to illustrate the theory. The second part of the thesis comprises a short chapter in which the displacement-dilatation- pressure formulation of Taylor (Int. J. Numer. Meth. Engng, 47 (2000)) is shown to be a special case of the general three-field formulation, and is then shown to be uniformly convergent. The final part of the thesis is concerned with the extension of the earlier approach to problems of nonlinear elasticity. The problem considered is the incremental or linearized version, of the kind that forms part of a Newton-Raphson process in numerical implementations, with the unknown variables being the increments in displacement, first Piola-Kirchhoff stress, and deformation gradient. In the discrete formulation the elasticity tensor (that is, the second derivative of the strain energy with respect to deformation gradient) is approximated by its mean value on each element. Conditions are established for the resulting incremental formulation to be stable and uniformly convergent, assuming that the continuous problem is stable. The analysis is illustrated through selected numerical examples.
- ItemOpen AccessTime integration schemes for piecewise linear plasticity(1991) Rencontre, LJ; Martin, JBThe formulation of a generalized trapezoidal rule for the integration of the constitutive equations for a convex elastic-plastic solid is presented. This rule, which is based on an internal variable description, is consistent with a generalized trapezoidal rule for creep. It is shown that by suitable linear extrapolation, the standard backward difference algorithm can lead to this generalized trapezoidal rule or to a generalized midpoint rule. In either case, the generalized rules retain the symmetry of the consistent tangent modulus. It is also shown that the generalized trapezoidal and midpoint rules are fully equivalent in the sense that they lead to the establishment of the same minimum principle for the increment. The generalized trapezoidal rule thus inherits the notion of B-stability and both rules offer the opportunity to exploit the second order rate of convergence for a = ½. However, in the generalized trapezoidal rule, the equilibrium. and constitutive equations are fully satisfied at the end of the time increment. This may be more convenient than the generalized midpoint rule, in which equilibrium and plastic consistency are satisfied at the generalized midpoint. A backward difference return algorithm for piecewise linear yield surfaces is then formulated, with attention restricted to an associated flow rule and isotropic material behavior. Both the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are considered in detail. The algorithm has the advantage of being fully linked to the governing principles and avoids the inherent problems associated with corners on the yield surface. It is fully consistent in that no heuristic assumptions are made. The algorithm is extended to include the generalized trapezoidal rule in such a way that the general structure of the backward difference algorithm is maintained. This allows both for the computational advantages of the generalized trapezoidal rule to be utilized, and for a basis for comparison between this algorithm and existing backward difference algorithms to be established. Using this fully consistent algorithm, the return paths in stress space for the Tresca and Mohr-Coulomb yield surfaces with perfectly plastic and linear hardening rules are identified. These return paths thus provide a basis against which heuristically developed algorithms can be compared.