Stiff Inclusion Research Articles

Monte Carlo simulation of spatial frequency domain imaging for breast tumors during compression.

Near-infrared optical imaging methods have shown promise for monitoring response to neoadjuvant chemotherapy (NAC) for breast cancer, with endogenous contrast coming from oxy- and deoxyhemoglobin. Spatial frequency domain imaging (SFDI) could be used to detect this contrast in a low-cost and portable format, but it has limited imaging depth. It is possible that local tissue compression could be used to reduce the effective tumor depth. To evaluate the potential of SFDI for therapy response prediction, we aim to predict how changes to tumor size, stiffness, and hemoglobin concentration would be reflected in contrast measured by SFDI under tissue compression. Finite element analysis of compression on an inclusion-containing soft material is combined with Monte Carlo simulation to predict the measured optical contrast. When the effect of compression on blood volume is not considered, contrast gain from compression increases with the size and stiffness of the inclusion and decreases with the inclusion depth. With a model of reduction of blood volume from compression, compression reduces imaging contrast, an effect that is greater for larger inclusions and stiffer inclusions at shallower depths. This computational modeling study represents a first step toward tracking tumor changes induced by NAC using SFDI and local compression.

Effective Elastic Properties and Micro-mechanical Damage Evolution of Composite Granular Rocks: Insights from Particulate Discrete Element Modelling

The mechanical behavior of composite granular rocks is a multifaceted phenomenon with broad relevance in various geomechanical applications. Traditional homogenization models and continuum mechanics-based numerical methods often fall short of accurately capturing the intricacies of granular materials. Granular materials exhibit heterogeneity and arching mechanisms governing the force networks that ensure system stability. Unlike continuum-based approaches, discrete element methods (DEM) have an advantage in assessing effective material properties by considering material heterogeneity and grain-level physical interactions. This study evaluates effective elastic properties using DEM with flat-joint contact law for composite binary mixtures with a stiff inclusion embedded within a matrix material. We examine variations in the elastic properties across different structural and laminated geometrical distributions of inclusion materials. Our findings closely adhere to the Voigt-Reuss and Hashin–Strikman bounds within their specific conditions, demonstrating the promising application of DEM in the analysis of composite materials. In addition, our research provides an in-depth analysis of the distinctive stress-evolution and damage-evolution characteristics exhibited by various geometrical configurations of inclusions under unconfined compressive loading. These results offer invaluable insights into the mechanical behavior of composite granular rocks and underscore the potential applications of DEM in addressing rock physics modeling problems encountered in petroleum engineering.

The role of the nucleus for cell mechanics: an elastic phase field approach.

The nucleus of eukaryotic cells typically makes up around 30% of the cell volume and has significantly different mechanics, which can make it effectively up to ten times stiffer than the surrounding cytoplasm. Therefore it is an important element for cell mechanics, but a quantitative understanding of its mechanical role during whole cell dynamics is largely missing. Here we demonstrate that elastic phase fields can be used to describe dynamical cell processes in adhesive or confining environments in which the nucleus acts as a stiff inclusion in an elastic cytoplasm. We first introduce and verify our computational method and then study several prevalent cell-mechanical measurement methods. For cells on adhesive patterns, we find that nuclear stress is shielded by the adhesive pattern. For cell compression between two parallel plates, we obtain force-compression curves that allow us to extract an effective modulus for the cell-nucleus composite. For micropipette aspiration, the effect of the nucleus on the effective modulus is found to be much weaker, highlighting the complicated interplay between extracellular geometry and cell mechanics that is captured by our approach. We also show that our phase field approach can be used to investigate the effects of Kelvin-Voigt-type viscoelasticity and cortical tension.

Numerical Simulation on Crack–Inclusion Interaction for Rib-to-Deck Welded Joints in Orthotropic Steel Deck

Weld defects such as porosity, inclusion, burn-through, and lack of penetration are difficult to detect and control effectively in an orthotropic steel deck (OSD), which will be a fatigue crack initiation site and lead to several fatigue cracking. The crack growth behavior in defective welded joints is different from that of defect-free joints. This study investigates crack–inclusion interaction for rib-to-deck welded joints in OSDs based on numerical simulation and linear elastic fracture mechanics (LEFM). A refined finite element model of a half U-rib with cracks and inclusions was established by using the FRANC3D-ABAQUS interactive technology. The full processes of the crack–inclusion interaction from approaching and penetrating were accurately simulated. Critical parameters, including the stress intensity factor (SIF), the shape factor, the growth rate, and the growth direction were analyzed. The stiff and soft inclusions amplify and shield the SIF of cracks when the crack grows to the local area of inclusions. During the entire process of crack growth, the soft and stiff inclusion accelerate and inhibit the crack growth, respectively. The stiff inclusion will lead to asymmetric growth of the crack shape, where the portion of the crack away from the inclusions has a higher growth rate. The soft and stiff inclusions will attract and repel the direction of crack growth at the proximal point, respectively.

Online Multiscale Finite Element Simulation of Thermo-Mechanical Model with Phase Change

This paper presents a thermo-mechanical model with phase transition considering changes in the mechanical properties of the medium. The proposed thermo-mechanical model is described by a system of partial differential equations for temperature and displacements. In the model, soil deformations occur due to porosity growth caused by ice and water density differences. A finite-element approximation of this model on a fine grid is presented. The linearization from the previous time step is used to handle the nonlinearity of the problem. For reducing the size of the discrete problem, offline and online multiscale approaches based on the Generalized Multiscale Finite Element Method (GMsFEM) are proposed. A two-dimensional model problem simulating the heaving process of heterogeneous soil with a stiff inclusion was considered for testing the mathematical model and the multiscale approaches. Numerical solutions depict the process of soil heaving caused by changes in porosity due to the phase transition. The movement of the phase transition interface was observed. The change of medium properties, including the elastic modulus, was traced and corresponds to the phase transition interface. The proposed multiscale approaches significantly reduce the size of the discrete problem while maintaining reasonable accuracy. However, the online multiscale approach achieves better accuracy than the offline approach with fewer degrees of freedom.

Role of micro fibres in tailoring the specific heat capacity of cementitious composites at elevated temperatures: Experimental characterisation and micromechanical modelling

The temperature dependency of specific heat capacity is crucial for thermo-mechanical analysis of fibre reinforced cementitious composites. In this study, a combined experimental and theoretical analysis was conducted to characterise the role of micro fibres in tailoring the specific heat capacity of cementitious composites at elevated temperatures. The specific heat capacity of cement mortar reinforced with four types of fibres, including polypropylene, basalt, carbon and glass fibres, was measured by means of a transient method at up to 400 °C. Based on the effective medium theory, a multiscale homogenization model was developed to predict the specific heat capacity evolution of cementitious composite with temperature from micro to macro level. The results indicate that within the measured temperature range, the addition of 2.0 vol% polypropylene and glass fibres lead to the maximum rise or drop in specific heat capacity of mortar by 8.7% and 14.1%, respectively. In addition, the thermal expansion of polymer fibre was proved to effectively enhance the specific heat capacity of mortar due to thermal expansion coupling effect. The parametric analysis suggests that stiffer inclusions with high thermal expansion govern the effective specific heat capacity coupled with thermal expansion, while the composite incorporated with softer inclusions is insensitive to the variation in thermal properties of inclusions.

Elasticity imaging using physics-informed neural networks: Spatial discovery of elastic modulus and Poisson's ratio.

Elasticity imaging is a technique that discovers the spatial distribution of mechanical properties of tissue using deformation and force measurements under various loading conditions. Given the complexity of this discovery, most existing methods approximate only one material parameter while assuming homogeneous distributions for the others. We employ physics-informed neural networks (PINN) in linear elasticity problems to discover the space-dependent distribution of both elastic modulus (E) and Poisson's ratio (ν) simultaneously, using strain data, normal stress boundary conditions, and the governing physics. We validated our model on three examples. First, we experimentally loaded hydrogel samples with embedded stiff inclusions, representing tumorous tissue, and compared the approximations against ground truth determined through tensile tests. Next, using data from finite element simulation of a rectangular domain containing a stiff circular inclusion, the PINN model accurately localized the inclusion and estimated both E and ν. We observed that in a heterogeneous domain, assuming a homogeneous ν distribution increases estimation error for stiffness as well as the area of the stiff inclusion, which could have clinical importance when determining size and stiffness of tumorous tissue. Finally, our model accurately captured spatial distribution of mechanical properties and the tissue interfaces on data from another computational model, simulating uniaxial loading of a rectangular hydrogel sample containing a human brain slice with distinct gray matter and white matter regions and complex geometrical features. This elasticity imaging implementation has the potential to be used in clinical imaging scenarios to reliably discover the spatial distribution of mechanical parameters and identify material interfaces such as tumors. STATEMENT OF SIGNIFICANCE: Our work is the first implementation of physics-informed neural networks to reconstruct both material parameters - Young's modulus and Poisson's ratio - and stress distributions for isotropic linear elastic materials by having deformation and force measurements. We comprehensively validate our model using experimental measurements and synthetic data generated using finite element modeling. Our method can be implemented in clinical elasticity imaging scenarios to improve diagnosis of tumors and for mechanical characterization of biomaterials and biological tissues in a minimally invasive manner.

Variability, Validity and Operator Reliability of Three Ultrasound Systems for Measuring Tissue Stiffness: A Phantom Study

Introduction Ultrasound elastography is a method of measuring soft tissue stiffness to detect the presence of pathology. There are several ultrasound elastography devices on the market. The aim of this study was twofold. Firstly, to determine the validity of three different ultrasound systems used to measure tissue stiffness. Secondly, to determine the operator reliability and repeatability when using these three systems. Materials and methods Two observers undertook multiple stiffness measurements from a phantom model using three different ultrasound systems; the LOGIQ E9, the Aixplorer, and the Acuson S2000. The phantom model had four cylindrical-shaped inclusions (Type 1-4) of increasing stiffness values and diameter embedded within. The background phantom stiffness was fixed. The mean, standard deviation, and coefficient of variation (CV) were calculated from measured stiffness readings per diameter per inclusion. Intra-observer variability was assessed. The validity of the measured stiffness value was assessed by calculating the difference between the measured elasticities and actual phantom elasticities. Bland-Altman plots with limits of agreement were used to display the inter-observer agreement. The intraclass correlation coefficients (ICC) were used to measure intra-observer, inter-observer, and inter-system reliability. Results Each observer undertook 1020 measurements. All three systems generally underestimated the stiffness values for the inclusions; the higher the actual stiffness value, the more significant the underestimation. The percentage difference between measured stiffness and actual stiffness varied from -79.1% to 12.7%. The intra-observer variability was generally less than 5% for observers using the LOGIQ E9 and the Aixplorer systems but more than 10% over the stiffer inclusions (Types 3 and 4) for the Acuson system. There was 'almost perfect' intra-observer reliability and repeatability for both the LOGIQ E9 and the Aixplorer systems; this was 'moderate' for the Acuson system over specific inclusions. For all systems, there was 'almost perfect' inter-observer reliability and repeatability between Observer A and Observer B. The inter-system reliability and repeatability were 'almost perfect' between the LOGIQ E9 system and the Aixplorer system but 'poor' and 'moderate' when the Acuson system was matched with the LOGIQ E9 system and the Aixplorer system, respectively. Conclusion This study has demonstrated that the Acuson, LOGIQ E9, and Aixplorer ultrasound systems have low variability, high reproducibility, and good intra-observer and inter-observer reliability when used to measure tissue stiffness. However, they all underestimated the stiffness values during this in vitro study. This study also revealed that not all ultrasound systems are comparable when measuring tissue stiffness, with some having better inter-system reliability than others. Ongoing standardization of technology is required at the manufacturer level.

A multi-resolution approach to hydraulic fracture simulation

We present a multi-resolution approach for constructing model-based simulations of hydraulic fracturing, wherein flow through porous media is coupled with fluid-driven fracture. The approach consists of a hybrid scheme that couples a discrete crack representation in a global domain to a phase-field representation in a local subdomain near the crack tip. The multi-resolution approach addresses issues such as the computational expense of accurate hydraulic fracture simulations and the difficulties associated with reconstructing crack apertures from diffuse fracture representations. In the global domain, a coupled system of equations for displacements and pressures is considered. The crack geometry is assumed to be fixed and the displacement field is enriched with discontinuous functions. Around the crack tips in the local subdomains, phase-field sub-problems are instantiated on the fly to propagate fractures in arbitrary, mesh independent directions. The governing equations and fields in the global and local domains are approximated using a combination of finite-volume and finite element discretizations. The efficacy of the method is illustrated through various benchmark problems in hydraulic fracturing, as well as a new study of fluid-driven crack growth around a stiff inclusion.

Virtual element method for modeling the deformation of multiphase composites

In this paper, we study applications of the virtual element method (VEM) for simulating the deformation of multiphase composites. The VEM is a Galerkin approach that is applicable to meshes that consist of arbitrarily-shaped polygonal and polyhedral (simple and nonsimple) elements. In the VEM, the basis functions are defined as the solution of a local elliptic partial differential equation, and are never explicitly computed in the implementation of the method. The stiffness matrix of each element is built by using the elliptic projection operator of the internal virtual work (bilinear form) and it consists of two terms: a consistency term that is exactly computed (linear patch test is satisfied) and a correction term (ensures stability) that is orthogonal to affine displacement fields and has the right scaling. The VEM simplifies mesh generation for a multiphase composite: a stiff inclusion can be modeled using a single polygonal or polyhedral element. Attributes of the virtual element approach are highlighted through comparisons with Voronoi-cell lattice models, which provide discrete representations of material structure. The comparisons involve a suite of two-dimensional linear elastic problems: patch test, axisymmetric circular inclusion problem, and the deformation of a three-phase composite. The simulations demonstrate the accuracy and flexibility of the virtual element method.

Low-strain effective Young’s modulus model and validation for multi-layer vocal fold-based silicone specimens with inclusions

A model of the effective low-strain elastic Young’s modulus of multi-layer stacked composites is proposed, which is capable to account for an arbitrary stacked inclusion. Geometrical and discretization-based model results are validated against measured effective Young’s moduli (from 10 up to 40 kPa) on 14 molded silicone specimens embedding a stiff (298 kPa) inclusion with variable size, position, and stacking. Specimens without inclusion represent the muscle, superficial, and epithelium layers in a human vocal fold with Young’s moduli between 4 and 65 kPa. The proposed model allows to predict the influence of a stiff inclusion, mimicking a structural abnormality or pathology somewhere within the vocal fold, on the low-strain effective Young’s modulus. Quantifying the influence of an inclusion or local stiffening on the vocal fold bio-mechanics is a necessary step toward the understanding and mitigation of structural vocal fold pathologies and associated voice disorders.

Coupling BEM and VEM for the Analysis of Composite Materials with Damage

Numerical tools which are able to predict and explain the initiation and propagation of damage at the microscopic level in heterogeneous materials are of high interest for the analysis and design of modern materials. In this contribution, we report the application of a recently developed numerical scheme based on the coupling between the Virtual Element Method (VEM) and the Boundary Element Method (BEM) within the framework of continuum damage mechanics (CDM) to analyze the progressive loss of material integrity in heterogeneous materials with complex microstructures. VEM is a novel numerical technique that, allowing the use of general polygonal mesh elements, assures conspicuous simplification in the data preparation stage of the analysis, notably for computational micro-mechanics problems, whose analysis domain often features elaborate geometries. BEM is a widely adopted and efficient numerical technique that, due to its underlying formulation, allows reducing the problem dimensionality, resulting in substantial simplification of the pre-processing stage and in the decrease of the computational effort without affecting the solution accuracy. The implemented technique has been applied to an artificial microstructure, consisting of the transverse section of a circular shaped stiff inclusion embedded in a softer matrix. BEM is used to model the inclusion that is supposed to behave within the linear elastic range, while VEM is used to model the surrounding matrix material, developing more complex nonlinear behaviors. Numerical results are reported and discussed to validate the proposed method.

Dynamic response of three-dimensional rigid and flexible foundations on layered soils with local inhomogeneities

The dynamic response of rigid and flexible three-dimensional (3D) foundations of arbitrary shape, placed on a homogeneous or layered halfspace, which may contain a finite, longitudinally invariant structure or inhomogeneity, is determined numerically. A 2.5D coupled Integral Transform Method (ITM) - Finite Element Method (FEM) approach is used to compute the dynamic compliance at the surface of the stratified soil with inclusion, whereas 3D ITM fundamental solutions are applied in case of a homogeneous or layered subgrade. The foundation is modelled by 3D finite elements in both cases and coupled to the underlying ground by enforcing the compatibility conditions at the interface. Compliance functions at the soil foundation interface are presented and compared with existing solutions for verification as well as displacement distributions within the underlying ground are illustrated. The influence of the foundation stiffness on the total response as well as on the frequency dependent power transmission into the soil due to different load types is studied. Finally, the method is applied to evaluate the influence of a tunnel and a stiff cylindrical inclusion on the dynamic response of the foundation, thereby demonstrating the importance of taking into account the Structure-Soil-Structure-Interaction (SSSI).

Introducing regularization into the virtual fields method (VFM) to identify nonhomogeneous elastic property distributions

The identification of nonhomogeneous elastic property distributions has been traditionally achieved with well acknowledged optimization based inverse approaches, but when full-field displacement measurements are available, the virtual fields method (VFM) can be computationally more efficient by converting the large-scale optimization problem into multiple small-scale optimization problems. A possible downside of the VFM so far was not to take into account prior knowledge, which is often available and needed when there is a very large number of unknowns and the inverse problem is ill-posed. In this work, different approaches are proposed for introducing regularization into the VFM, aiming to penalize the local variations of identified stiffness properties in order to reduce the effects of uncertainty in the inverse problem resolution. The feasibility and accuracy of the regularized VFM are tested through several numerical and experimental datasets. It is shown that the main advantage of the novel VFM approaches is the low computational cost, as large-scale inverse problems with 10,000 unknown parameters can be solved within several seconds using a standard personal computer. Although the regularized VFM can successfully detect a stiff inclusion in a soft solid with high accuracy, regularization also introduces unexpected spurious effects in the results, blurring the interface between soft and stiff regions. We also observed that the regularization did not improve the smoothness significantly due to local effects of the small-scale optimization problem introduced in the proposed VFM method. Therefore, traditional regularization, which penalizes local variations of identified stiffness properties, can be combined with the VFM to solve inverse problems with a high computational efficiency, but supplemental regularization conditions will need to be adapted in the future to better delineate soft-stiff interfaces with this methodology.

Deep Convolutional Neural Networks for Displacement Estimation in ARFI Imaging.

Ultrasound elasticity imaging in soft tissue with acoustic radiation force requires the estimation of displacements, typically on the order of several microns, from serially acquired raw data A-lines. In this work, we implement a fully convolutional neural network (CNN) for ultrasound displacement estimation. We present a novel method for generating ultrasound training data, in which synthetic 3-D displacement volumes with a combination of randomly seeded ellipsoids are created and used to displace scatterers, from which simulated ultrasonic imaging is performed using Field II. Network performance was tested on these virtual displacement volumes, as well as an experimental ARFI phantom data set and a human in vivo prostate ARFI data set. In the simulated data, the proposed neural network performed comparably to Loupas's algorithm, a conventional phase-based displacement estimation algorithm; the rms error was [Formula: see text] for the CNN and 0.73 [Formula: see text] for Loupas. Similarly, in the phantom data, the contrast-to-noise ratio (CNR) of a stiff inclusion was 2.27 for the CNN-estimated image and 2.21 for the Loupas-estimated image. Applying the trained network to in vivo data enabled the visualization of prostate cancer and prostate anatomy. The proposed training method provided 26 000 training cases, which allowed robust network training. The CNN had a computation time that was comparable to Loupas's algorithm; further refinements to the network architecture may provide an improvement in the computation time. We conclude that deep neural network-based displacement estimation from ultrasonic data is feasible, providing comparable performance with respect to both accuracy and speed compared to current standard time-delay estimation approaches.

Shear Wave Velocity Estimation Using the Real-Time Curve Tracing Method in Ultrasound Elastography

The estimation of shear wave velocity is very important in ultrasonic shear wave elasticity imaging (SWEI). Since the stability and accuracy of ultrasonic testing equipment have been greatly improved, in order to further improve the accuracy of shear wave velocity estimation and increase the quality of shear wave elasticity maps, we propose a novel real-time curve tracing (RTCT) technique to accurately reconstruct the motion trace of shear wave fronts. Based on the curve fitting of each frame shear wave, the propagation velocity of two-dimensional shear waves can be estimated. In this paper, shear wave velocity estimation and shear wave image reconstruction are implemented for homogeneous regions and stiff spherical inclusion regions with different elasticity, respectively. The experimental result shows that the proposed shear wave velocity estimation method based on the real-time curve tracing method has advantages in accuracy and anti-noise performance. Moreover, by eliminating artifacts of shear wave videos, the velocity map acquired can restore the shape of inclusions better. The real-time curve tracing method can provide a new idea for the estimation of shear wave velocity and elastic parameters.

Handheld volumetric manual compression‐based quantitative microelastography

Compression optical coherence elastography (OCE) typically requires a mechanical actuator to impart a controlled uniform strain to the sample. However, for handheld scanning, this adds complexity to the design of the probe and the actuator stroke limits the amount of strain that can be applied. In this work, we present a new volumetric imaging approach that utilizes bidirectional manual compression via the natural motion of the user's hand to induce strain to the sample, realizing compact, actuator-free, handheld compression OCE. In this way, we are able to demonstrate rapid acquisition of three-dimensional quantitative microelastography (QME) datasets of a tissue volume (6 × 6 × 1 mm3 ) in 3.4 seconds. We characterize the elasticity sensitivity of this freehand manual compression approach using a homogeneous silicone phantom and demonstrate comparable performance to a benchtop mounted, actuator-based approach. In addition, we demonstrate handheld volumetric manual compression-based QME on a tissue-mimicking phantom with an embedded stiff inclusion and on freshly excised human breast specimens from both mastectomy and wide local excision (WLE) surgeries. Tissue results are coregistered with postoperative histology, verifying the capability of our approach to measure the elasticity of tissue and to distinguish stiff tumor from surrounding soft benign tissue.

A Comparative Study of Direct and Iterative Inversion Approaches to Determine the Spatial Shear Modulus Distribution of Elastic Solids

This paper presents a comparative study of two typical inverse algorithms, i.e., direct and iterative inversion methods, to reconstruct the shear modulus distribution of linearly elastic solids. Both approaches are based on the finite element framework and compared utilizing both the simulated and experimental data. The reconstruction results demonstrate that both approaches are capable of identifying the nonhomogeneous shear modulus distribution of solids well. It can also be found that the direct inversion method is much faster than the iterative inversion method, whereas the iterative inversion method is capable of yielding better shear modulus ratio between the stiff inclusion and the soft background even with very high noise levels. Afterwards, a thorough comparison on the advantages and disadvantages of these two approaches has been performed. This comparative study provides useful information on the selection of the proper inverse scheme in estimating nonhomogeneous elastic property distribution of soft solids nondestructively.

A modulus gradient model for inhomogeneous materials with isotropic linear elastic constituents

The one-dimensional modulus gradient (E-grad) model proposed in Gülaşık et al. (2018) is extended to more general three-dimensional inhomogeneous materials with isotropic linear elastic constituents. In addition to the constitutive equations and balance relations, length scale dependent differential relations for the material parameters of isotropic linear elasticity are provided. The finite element formulation for axisymmetric problems is derived and a model problem of a soft cylindrical rod with a stiff spherical inclusion is solved. It is seen that discontinuities and/or very sharp changes in the modulus, displacement, strain and stress fields that exist in local formulations are smoothed with the proposed gradient model. Furthermore, a graded interphase region around the stiff inclusion is obtained by the model as reported in the literature of polymer nanocomposites. The thickness of the graded interphase region is interpreted as a characteristic length scale of the polymer nanocomposite. It is also shown that increasing the internal length scale parameter increases the stiffness of the model. The proposed model is compared with a micromechanical model from literature and experiments conducted with polyimide/silica nanocomposites. The results obtained by the proposed approach capture the experimentally measured values of the nanocomposite modulus. Finally, the model is extended to obtain anisotropic macroscopic response by choosing different length scale parameters in different directions.

Numerical investigations of strain-gradient plasticity with reference to non-homogeneous deformations

In this work, a higher-order irrotational strain gradient plasticity theory is studied in the small strain regime. A detailed numerical study is based on the problem of simple shear of a non-homogeneous block comprising an elastic-plastic material with a stiff elastic inclusion. Combinations of micro-hard and micro-free boundary conditions are used. The strengthening and hardening behaviour is explored in relation to the dissipative and energetic length scales. There is a strong dependence on length scale with the imposition of micro-hard boundary conditions. For micro-free conditions there is marked dependence on dissipative length scale of initial yield, though the differences are small in the post-yield regime. In the case of hardening behaviour, the variation with respect to energetic length scale is negligible. A further phenomenon studied numerically relates to the global nature of the yield function for the dissipative problem; this function is given as the least upper bound of a function of plastic strain increment, and cannot be determined analytically. The accuracy of an upper-bound approximation to the yield function is explored, and found to be reasonably sharp in its prediction of initial yield.