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A three dimensional distinct element model was developed for the multiblock Baalbek columns to investigate the response of the existing columns as well as 0.5m and 1m scaled models used in shaking table experiments. Periodic pulses of varying shapes, periods, and accelerations were applied and the critical combinations of acceleration and period that caused collapse were noted. The response and mode of collapse are shown to be dependent on the period and the amplitude of the pulse. Depending on the size of the column and the period of the applied critical pulse three modes of collapse are observed: (1) multiblock rocking/wobbling followed by toppling of one or more blocks; (2) rock and/or wobble then overturn along or outside the applied pulse plane; (3) overturn after one or two impacts along the plane of the applied pulse. The displacement history of the column was plotted alongside the ground motion to visualize and quantify the rocking and /or wobbling. All column sizes showed the same response qualitatively in the acceleration-period space and the wobbling was shown to be confined to a well-defined region. Stability charts delineating the safe-unsafe boundaries showed bi-modal behavior for all pulses tested but with a shift due to the different pulse energy contents. Numerical results compared well with experimental data from shaking table experiments on the scaled columns. The capabilities of the three dimensional numerical modeling approach are confirmed in this work for use as a predictive tool or an archaeoseismic analysis tool for columns and large masonry structures.

Reduced 1/20 and 1/40 models of the Baalbek columns were subjected to harmonic excitations of varying amplitudes and periods to investigate rocking and overturning. 2D UDEC numerical simulations an...

In order to improve the efficiency of fiber reinforced plastics (FRP) confinement as a method to repair and strengthen concrete structures, a parametric analysis was carried out to investigate the effects of cylinder slenderness and the stiffness of the confinement on the localization pattern, the stress&ndash, strain response and the effectiveness of the confinement. FRP-wrapped concrete cylinders under axial compression were modeled in a high-resolution finite element model. Concrete was modeled as a Mohr&ndash, Coulomb material. The bi-linear stress&ndash, strain structural responses concur with published experimental data. Localization along discrete shear planes results in a failure mechanism that causes non-uniform hoop stresses in the FRP wrap due to the movement of solid wedges in the mechanism. A characteristic length for localization was identified and found in agreement with published experimental observations. The confinement efficiency shows a clear dependence on the confinement level and a weak dependence on slenderness above the characteristic length. A simple mechanistic model is proposed for the second branch of the bi-linear stress&ndash, strain response curve. The results of this study can be used to estimate the confinement efficiency factor and refine the design recommendations of Equation 12.1 of ACI 440.2R17.

The application of the Hoek – Brown empirical strength criterion to actively confined concrete is investigated. A large database of experimental results on hydraulically and triaxially confined concrete is reviewed and analyzed. The normalized two-parameter Hoek – Brown criterion provides a simpler and more robust fit than theoretical and empirical models previously proposed in the literature for actively confined concrete. The Hoek – Brown curved failure surface is used to predict the failure angles that are experimentally observed in the brittle mode. A residual stress lower bound is proposed based on the brittle fracture mechanism. The analysis of this residual stress lower bound allows the formal identification of the transition between the brittle and the ductile failure modes in confined concrete. The modeling results are compared with the available published data and good agreement is found between the experimental observations and the predictions of the model. Recommendations for the integration of the Hoek – Brown model in the concrete design equations are also provided.

An innovative mechanistic based method for passive confinement efficiency estimation is proposed based on the extension to rectangular sections of the pulley model previously proposed by the writers. A refined finite element model was developed using a nonlinear concrete constitutive law in order to analyze stresses in columns passively confined with fiber reinforced polymer wraps. Rectangular and square cross sections of variable corner radii were investigated with reference to a circular cross section. Results showed an increase in corner stresses with sharper corner radii, a localization of failure at the corners, and a decrease in confinement effectiveness with an increase in the rectangularity of the cross section or an increase in corner sharpness. A rigorous numerical method for calculating geometric confinement efficiency factors is proposed and typical factors are calculated and compared with the predictions of the simple pulley model showing good agreement.

Numerical simulations of water flow over stepped spillways with different step configurations are presented. The finite element computational fluid dynamics module of the ADINA software was used to predict the main characteristics of the flow. This included the determination of the water surface, the development of skimming flow over corner vortices, and the determination of energy dissipation. Since the actual flow is turbulent, the [email protected] flow model was used. A two-phase solution process was adopted in order to optimize the overall simulation efficiency. In the first phase, a simple yet reasonable water surface consisting of three straight lines was used as an initial guess and was treated as a fixed wall. In the second phase, the results from the first phase were used as initial conditions and the water surface was treated as a free surface that evolved to attain a steady state configuration. For all the cases considered, the predicted water surface profile over the entire length of the spillway was in close agreement with the experimentally measured water surface profile. The predicted energy dissipation was also comparable to the experimentally attained values.

An experimental investigation of the corner sharpness effect in square columns with carbon fibre-reinforced polymer (CFRP) wraps is presented using traditional compressive testing of short square columns as well as an innovative set-up that allows testing directly for the CFRP strength when wrapped around the corner of a flat slab. Both types of test confirmed the observation that the confinement efficiency decreases as the corner sharpness increases, owing to stress concentrations that cause localised failure of the CFRP.

This paper presents an investigation into the hydraulics of regular ogee-profile spillways. The free-surfaces of the fluid for several flow heads as measured in the hydraulics laboratory are used as benchmarks. The finite element computational fluid dynamics software, ADINA, was used to predict the free surface over an ogee spillway and thus model the flow field. Since the actual flow is turbulent the k–e flow model was used. For the cases considered in this paper, ADINA predicted reasonable free surface results that are consistent with general flow characteristics over spillways. The results are also in close agreement with measured free-surface profiles over the entire length of the spillway.

We find the exact solution of three interacting friendly directed walks on the square lattice in the bulk, modelling a system of homopolymers that can undergo gelation by introducing two distinct interaction parameters that differentiate between the zipping of only two or all three walks. We establish functional equations for the model's corresponding generating function that are subsequently solved exactly by means of the \emph{obstinate kernel method}. We then proceed to analyse our model, first considering the case where triple-walk interaction effects are ignored, finding that our model exhibits two phases which we classify as free and gelated regions, with the system exhibiting a second-order phase transition. We then analyse the full model where both interaction parameters are incorporated, presenting the full phase diagram and highlighting the additional existence of a first-order gelation boundary., Comment: 28 pages, 10 figures

In certain penetration events it is proposed that the primary mode of deformation of the target can be approximated by known analytical expressions. In the context of an analysis code, this approximation eliminates the need for discretizing the target as well as the need for a contact algorithm. Thus, this method substantially reduces the computer time and memory requirements. In this paper a forcing function which is derived from a spherical-cavity expansion (SCE) analysis has been implemented in a transient dynamic finite element code. This irnplementation is capable of computing the structural and component responses of a projectile due to a three dimensional penetration event. Simulations are presented for 7.1 l-mm-diameter, 74.7-mm-long, spherical-nose, vacuum- arc-remelted (VAR) 4340 steel projectiles that penetrate 6061-T6511 aluminum targets. Final projectile configurations obtained from the simulations are compared with post-test radiographs obtained from the corresponding experiments. It is shown that the simulations accurately predict the permanent projectile deformation for three dimensional loadings due to incident pitch and yaw over a wide range of striking velocities.

A direct method for solving quasi-static, mixed-mode fracture problems is presented. The element-free Galerkin method is used in order to allow for crack growth without remeshing. An expression for the normalized, critical traction is derived in terms of the fracture resistance (R-curve) and a crack-dependent function. Sample problems demonstrate the capability of this method to accurately compute the post-peak equilibrium paths for structures with growing cracks.

Element-free Galerkin (EFG) methods are presented and applied to static and dynamic fracture problems. EFG methods, which are based on moving least-square (MLS) interpolants, require only nodal data; no element connectivity is needed. The description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks. This makes the method particularly attractive for growing crack problems, since only minimal remeshing is needed to follow crack growth. In moving least-square interpolants, the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point. Numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method.

Element-free Galerkin method (EFG) is extended to dynamic problems. EFG method, which is based on moving least square interpolants (MLS), requires only nodal data; no element connectivity is needed. This makes the method particularly attractive for moving dynamic crack problems, since remeshing can be avoided. In contrast to the earlier formulation for static problems by authors, the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions. With this formulation, the stiffness matrix is symmetric and positive semi-definite, and hence the consistency, conergence and stability analyses of time integration remain the same as those in finite element method. Numerical examples are presented to illustrate the performance of this method. The relationship between the element-free Galerkin method and the smooth particle hydrodynamics (SPH) method is also discussed in this paper. Results are presented for some one-dimensional problems and two-dimensional problems with static and moving cracks.

A simple, accurate technique for recovery of displcements and derivatives, such as strains is presented. The technique is based on local interpolation of nodal displacements using a moving least square method. The strains are then recovered by taking appropriate derivatives of this interpolant. Numerical experiments in linear elasticity and heat conduction on the convergence and accuracy of the recovered derivatives show very good results and superconvergence for strains in many cases; the technique is also effective for displacement interpolation for projection methods.

Various types of error criteria are examined and it is shown that for problems involving plastic response or localization, an error criterion based on an L 2 -projection of strains is the most effective for the constant strain elements considered here. Examples of one-dimensional and two-dimensional localization (shear band formation) problems are given

UNLABELLED This study evaluated the strength of laser-welded arteriovenous shunts established using St. Jude BioPolyMeric vascular grafts. The arterial anastomoses of the biological graft were laser welded with and without the addition of soluble collagen or fibrin sealant. In four dogs, 16 arteriovenous grafts were implanted between the femoral artery and vein or the carotid artery and jugular vein using a 6 cm long, 4 mm internal diameter prosthesis. The 16 arterial anastomoses were evenly divided into four groups: sutured control, laser welded (LW), LW with soluble collagen applied immediately before and during welding, and LW with fibrin sealant applied after welding. All arterial control and venous anastomoses were sutured using continuous 6-0 polypropylene suture. All LW anastomoses were initially divided into six 5 mm long segments using six evenly spaced 6-0 polypropylene stay sutures. Each segment was laser welded using 15 to 18 5-sec pulses of the 0.5 W (7.5 W/cm 2) argon laser energy delivered via a 300 mum fiber while cooling the tissue with slow-drip saline irrigation. Blood flow was established and maintained through each anastomosis for 1 h. The vessels were then controlled, and anastomotic bursting pressure was determined with infusion of heparinized blood. RESULTS An additional hemostatic suture was required in 3 LW anastomoses (2 LW, 1 LW with collagen). Mean bursting pressures (mm Hg) of the arterial anastomoses were as follows: sutured controls 165 +/- 159, LW 144 +/- 58, LW and collagen 93 +/- 47, LW and fibrin sealant 181 +/- 45.(ABSTRACT TRUNCATED AT 250 WORDS)

Intraluminal ultrasound (ILUS) is a new catheter-based system which produces two-dimensional (2D), cross-sectional images of tubular structures. The principle of image acquisition using 5.0 Fr (30 MHz) and 8.0 Fr (20 MHz) ILUS catheters containing ultrasound transducers at the tip, is equally applicable to all tubular or hollow organs. This article illustrates the feasibility of intraluminal imaging of hollow organ structures, and describes computerized three-dimensional (3D) reconstruction of the 2D images, using a PC-based image analysis system. A set (n = 90) of longitudinally aligned, consecutive 2D images was sampled from various organs (canine bladder, urethra, jejunum, esophagus, and trachea) and computer processed to produce 3D images. By adjusting image density threshold and viewing angle, the lumen and wall morphology can be examined in any projection. Possible applications include diagnosis of lumen encroaching pathology, guidance of intraluminal instruments, and assessment of the effects of endoluminal interventions.

A particle model for brittle aggregate composite materials such as concretes, rocks, or ceramics is presented. The model is also applicable to the behavior of unidirectionally reinforced fiber composites in the transverse plane. A method of random computer generation of the particle system meeting the prescribed particle size distribution is developed. The particles are assumed to be elastic and have only axial interactions, as in a truss. The interparticle contact layers of the matrix are described by a softening stress‐strain relation corresponding to a prescribed microscopic interparticle fracture energy. Both two‐ and three‐dimensional versions of the model are easy to program, but the latter poses, at present, forbidding demands for computer time. The model is shown to simulate realistically the spread of cracking and its localization. Furthermore, the model exhibits a size effect on: (1) The nominal strength, agreeing with the previously proposed size effect law; and (2) the slope of the post‐peak l...

We measure deformations of electrostatically actuated cantilever beams adhered to a substrate and compare results to numerical simulations. Beams are peroxide treated and either supercritically dried or coated with octadecyltrichlorosilane (ODTS). In air with relative humidity (RH) of about 50%, measured deformations are consistent with numerical results when an effective insulator thickness of 14 nm is assumed. This value is a measure of the roughness of the substrate. Deformations are generally reversible for voltages up to 2 V. With RH of 75%, deformations of supercritically dried beams are dominated by capillary rather than electrostatic forces. The ODTS-coated beams exhibit less effect from the capillary forces, as expected for a hydrophobic coating.

Acute thoracic aortic dissection is a life-threatening disorder, with frequency of approximately 2,000 new cases per year in the USA. Overall mortality remains high despite advances in both operative and nonoperative management.1 Critical variables in determining treatment options and outcome are the exact anatomic location of the lesion and the premorbid general condition of the patient. Diagnostic techniques include angiography, computerized tomography, magnetic resonance imaging, and transthoracic and transesophageal echocardiography. Each of these techniques has limitations related to the method of image production and resolution.2 Recent advances in catheter-based technology have provided devices suitable for both diagnosing aortic dissection and treating the lesion without recourse to major surgery. Intravascular ultrasound is a rapidly developing method for defining luminal and transmural anatomy of vascular structures.3 It is capable of producing high-resolution images of acute aortic dissection, and may have advantages over conventional imaging techniques in identifying the origin and distal end point and relation of the false lumen to major visceral aortic branches.4 Intravascular balloon-expandable stents have been used to successfully obliterate acute aortic dissections in dogs, but require total occlusion of the aorta during deployment as the balloon is expanded to force the stent against the aortic wall. Recently available self-expanding stents have the potential advantages of higher flexibility and a deployment mechanism that does not obstruct blood flow during insertion.

A case of acute dissecting aortic aneurysm is described in which intravascular ultrasonography was used at the time of aortography to produce real-time, 360 degree cross-sectional images of the aorta. The transmural vessel morphology visualized by this new catheter-based technology allowed confirmation of the diagnosis and identification of distal extension to the aortic bifurcation. The case demonstrates the unique potential of this modality in diagnosis and possible therapy in vascular diseases. (J VASC SURG 1991;13:510-2.)

A procedure has been developed to represent the loading on a penetrator and its motion during oblique penetration into geologic media. The penetrator is modeled with the explicit dynamics, finite element computer program PRONTO 3D and the coupled pressure on the penetrator is given in a new loading option based on a separate cavity expansion (CE) solution that accounts for the pressure reduction from a nearby target free surface. The free-surface influence distance is selected in a predictive manner by considering the pressure to expand a spherical cavity in a finite radius sphere of the target material. The CE/PRONTO 3D procedure allows a detailed description of the penetrator for predicting shock environments or structural failure during the entire penetra- tion event and is sufficiently rapid to be used in design optimization. It has been evaluated by comparing its results with data from two field tests of a full-scale penetrator into frozen soil at an impact angles of 49.6 and 52.5 degrees from the horizontal. The measured penetrator rotations were 24 and 22 degrees, respectively. In the simulation, the rotation was 21 degrees and predom- inately resulted from the pressure reduction of the free surface. Good agreement was also found formore » the penetration depth and axial and lateral acceleration at two locations in the penetrator.« less

We find the exact solution of two interacting friendly directed walks (modelling polymers) on the square lattice. These walks are confined to the quarter plane by a horizontal attractive surface, to capture the effects of DNA-denaturation and adsorption. We find the solution to the model's corresponding generating function by means of the obstinate kernel method. Specifically, we apply this technique in two different instances to establish partial solutions for two simplified generating functions of the same underlying model that ignore either surface or shared contacts. We then subsequently combine our two partial solutions to find the solution for the full generating function in terms of the two simpler variants. This expression guides our analysis of the model, where we find the system exhibits four phases, and proceed to delineate the full phase diagram, showing that all observed phase transitions are second-order.

The numerical computation of dynamic crack growth is a challenging problem which is of importance to the prediction of ultimate capacities and failure of structures. However, while substantial success has been achieved in modeling rectilinear crack growth by finite elements [1, 2], the solution of problems with arbitrary crack growth has proven very difficult. We have recently demonstrated that the element-free Galerkin method is well suited to the problem of modeling or simulating arbitrary dynamic crack propagation in two dimensions [3, 4].

We solve exactly a two-dimensional partially directed walk model of a semi-flexible polymer that has one end tethered to a sticky wall, while a pulling force away from the adsorbing surface acts on the free end of the walk. This model generalizes a number of previously considered adsorption models by incorporating individual horizontal and vertical stiffness effects, in competition with a variable pulling angle. A solution to the corresponding generating function is found by means of the kernel method. While the phases and related phase transitions are similar in nature to those found previously the analysis of the model in terms of its physical variables highlights various novel structures in the shapes of the phase diagrams and related behaviour of the polymer. We review the results of previously considered sub-cases, augmenting these findings to include analysis with respect to the model?s physical variables?namely, temperature, pulling force, pulling angle away from the surface, stiffness strength and the ratio of vertical to horizontal stiffness potentials, with our subsequent analysis for the general model focusing on the effect that stiffness has on this pulling angle range. In analysing the model with stiffness we also pay special attention to the case where only vertical stiffness is included. The physical analysis of this case reveals behaviour more closely resembling that of an upward pulling force acting on a polymer than it does of a model where horizontal stiffness acts. The stiffness?temperature phase diagram exhibits re-entrance for low temperatures, previously only seen for three-dimensional or co-polymer models. For the most general model we delineate the shift in the physical behaviour as we change the ratio of vertical to horizontal stiffness between the horizontal-only and the vertical-only stiffness regimes. We find that a number of distinct physical characteristics will only be observed for a model where the vertical stiffness dominates the horizontal stiffness.

This report describes an occlusive canine iliac artery model for use in experimental angioplasty procedures. Lesions were induced by overdistending and breaking the internal elastic lamina of the artery and implanting a 2-cm long occlusive collagen plug. The collagen plug diameter was varied to fit the iliac artery and caused immediate bilateral iliac occlusions. Histology of the lesions at 9 and 14 days (n = 2) showed that the lumen was filled with a mixture of fresh thrombus and collagen pad material (more collagen than thrombus) with focal disruptions of the internal elastic lamina. At 36 days (n = 2) the lumen was obliterated with a mixture of organizing thrombus and collagen pad material (more organizing thrombus than collagen). At 60 (n = 4) and 80 (n = 8) days the collagen pad had been completely replaced with organizing thrombus consisting of collagen fibers and hemosiderin-laden macrophages with persistent focal disruptions of the internal elastic lamina. These initial data demonstrate the ability to create fibrotic arterial occlusions in as short a time as 60 days and to provide a practical model to study methods for treating localized peripheral vascular occlusive disease.

This study reports three-dimensional reconstruction of two-dimensional intravascular ultrasound images obtained along 5 cm vessel segments. Each three-dimensional image was produced by computerized "stacking" of a set of consecutive two-dimensional images (mode 90 images per set; range 32 to 256). Three-dimensional images (n = 26) were obtained from 11 human normal and atherosclerotic arteries (three in vitro and eight in vivo) and five in vivo canine studies. In vivo human examinations included three iliac, one deep, and three superficial femoral arteries and one aortic dissection. Five in vivo canine vessels (three iliac stenoses and two aortic dissections) were imaged before and after intraluminal stent deployment. Images were displayed on a gray-scale monitor, allowing examination of vessel images as complete cylinders or longitudinal hemisections in any user-defined plane. This enabled global examination of vascular segments and identified luminal shape, wall thickness, and distribution and morphology of plaques. Reconstructions of aortic dissections illustrated the extent of the dissection and produced an anatomic picture of the false lumen. Three-dimensional imaging enhanced stent deployment by identifying the site for deployment, dimensions of the vessel lumen, and changes in morphology after stent insertion. There was good correlation in vessel dimensions measured by angiography, two-dimensional intravascular ultrasonography and longitudinal gray-scale reconstructions (r = 0.74 to 0.95; p = 0.34 to 0.001) but poor correlation with measurements from three-dimensional surface-rendered images (r = 0.13 to 0.48; p = 0.42 to 0.87). We conclude that three-dimensional intravascular ultrasound imaging is an innovative new method for identifying the type, extent, and spatial configuration of arterial disease, with promising diagnostic and therapeutic applications.

Intravascular ultrasonography is developing rapidly as a method for defining the transmural anatomy of vascular structures, with diagnostic and therapeutic applications. The ultrasound technology not only has unique diagnostic capabilities by defining the distribution and character of lesions, but also provides accurate control information regarding efficacy of angioplasty methods. An exciting recent development is the three-dimensional reconstruction of two-dimensional images which permits global examination of luminal and transmural vessel morphology. This technology may enable improved guidance of intraluminal devices to enhance lesion removal without damaging adjacent normal wall structure and appropriate device selection by differentiating specific plaque characteristics.

Intravascular ultrasound (IVUS) is a new catheter-based system that produces two-dimensional (2D) images of vascular structures. Existing systems produce real-time, cross-sectional "slices" of vessels using 5.0 French (Fr) (30 MHz) and 8.0 Fr (20 MHz) IVUS catheters containing ultrasound transducers at the tip. Computerized, three-dimensional (3D) reconstruction of these 2D images using a personal computer- (PC) based image analysis system is described. A set (n = 90) of longitudinally aligned, consecutive images is sampled from a 5.0 cm vessel segment, and computerized processing creates rendered 3D images. By adjusting image density threshold and viewing angle, the morphology, location, and spatial distribution of arterial pathology can be seen. Refinements in computer hardware and software have reduced processing time and improved image resolution to the point where 3D IVUS imaging is a clinically applicable tool. Possible applications include diagnosis of complex arterial pathology, guidance of intraluminal instruments, and assessment of the effects of endovascular interventions.

This study evaluates the ability of intravascular ultrasound (IUS) to image normal and mildly diseased human ileo-femoral vessels during angioplasty or vascular bypass procedures. Five Fr. and 8Fr. rotating A scan IUS catheters were used to obtain 43 images in 4 superficial femoral arteries, and 5 iliac arteries in 8 vascular surgery patients. Luminal cross sectional (LCS) areas measured by IUS were compared to LCS areas calculated by uniplanar angiography (ANGIO) at the same location in the vessel. The correlation between the areas (IUS vs ANGIO) for all images was significant (n = 43, r = 0.90, P

The paper presents a simple new method to calculate column-interaction diagrams, which takes into account slenderness effects. The method consists of a simple incremental loading algorithm that traces the load-deflection curve at constant eccentricity of the axial load. The column failure is defined for design purposes as the peak of the diagram of axial load versus midlength bending moment at constant load eccentricity. The tangent modulus load is found to be apporximately equal to the peak load of column with load eccentricity 0.01 of the cross-sectional thickness and represents a lower bound for the maximum loads at still smaller eccentricities. Strain irreversibility at unloading can be taken into account but its effect is very small. The method is compared with the ACI moment magnification method and with the CEB Model column method based on moment-curvature relations. The agreement with the CEB method is very close, but with respect to the ACI method there are large discrepancies.

Tissue fusion by laser energy is an intriguing arxi very promising new application for laser technology. In coniparison to using high laser energy to ablate tissue as in the aricplasty application, laser tissue fusion is possible in any soft tissue by deliverir appropriate lc levels of energy to the cppose tissue surfaces. This technology is particularly appealing for vascular applications in inakir sutureless blood vessel anastornosis and for securir the endpOints of erxarterect:amies ani dissection planes. Although there have been limitel evaluations of this technology the preliminary experimental ar1 clinical data is very promisir for continu development arid application.

This paper presents a comparison of various models for strain‐softening due to damage such as cracking or void growth, as proposed recently in the literature. Continuum‐based models expressed in terms of softening stress—strain relations, and fracture‐type models expressed in terms of softening stress—displacement relations are distinguished. From one‐dimensional wave propagation calculations, it is shown that strain‐localization into regions of finite size cannot be achieved. The previously well‐documented spurious convergence is obtained with continuum models, while stress—displacement relations cannot model well smeared‐crack situations. Continuum models may, however, be used in general if a localization limiter is implemented. Gradient‐type localization limiters appear to be rather complicated; they require solving higher‐order differential equations of equilibrium with additional bourdary conditions. Non‐local localization limiters, especially the non‐local continuum with local strain, in which only the energy dissipating variables are non‐local, is found to be very effective, and also seems to be physically realistic. This formulation can correctly model the transition between homogeneous damage states and situations in which damage localizes into small regions that can be viewed as cracks. The size effect observed in the experimental and numerical response of specimens in tension or compression is shown to be a consequence of this progressive transition from continuum‐type to fracture‐type formulations.

Micromechanics of damage in brittle heterogeneous materials and composites requires analysis of a system of interacting cracks. The response path of a crack system typically exhibits bifurcations such that the states on each post-bifurcation branch can be stable yet only one branch can be reached in a stable manner. Recent results on thermodynamic criteria for stable states and stable response paths of inelastic structures are reviewed and formulated in terms of the incremental internal entropy of the system. The incremental entropy, which can be expressed in terms of second order work, is then calculated for various points on the response paths of some typical symmetric crack systems. It is shown that while the symmetric states may be stable, the path which leads to them is unstable and cannot occur in reality. Generally, nonsymmetry develops at the beginning of softening. The results show that it is insufficient to model distributed cracking only by means of crack systems and linear elastic fracture mechanics. Further aspects, such as material heterogeneity, residual stresses, and cohesive fracture zones for the microcracks, might have to be taken into account.