Microstructural origins of continuum stress fields - A brief history and some unresolved issues

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Piezoelectric nanomaterials PNs are attractive for applications including sensing, actuating, energy harvesting, among others in nano-electro- mechanical -systems NEMS because of their excellent electromechanical coupling, mechanical and physical properties.

Continuum Mechanics - Ch 8 - Lecture 4 - Principal Stress Space

However, the properties of PNs do not coincide with their bulk counterparts and depend on the particular size. A large amount of efforts have been devoted to studying the size-dependent properties of PNs by using experimental characterization, atomistic simulation and continuum mechanics modeling with the consideration of the scale features of the nanomaterials.

This paper reviews the recent progresses and achievements in the research on the continuum mechanics modeling of the size-dependent mechanical and physical properties of PNs. We start from the fundamentals of the modified continuum mechanics models for PNs, including the theories of surface piezoelectricity, flexoelectricity and non-local piezoelectricity, with the introduction of the modified piezoelectric beam and plate models particularly for nanostructured piezoelectric materials with certain configurations.

Then, we give a review on the investigation of the size-dependent properties of PNs by using the modified continuum mechanics models , such as the electromechanical coupling, bending, vibration, buckling, wave propagation and dynamic characteristics. Finally, analytical modeling and analysis of nanoscale actuators and energy harvesters based on piezoelectric nanostructures are presented. A micro- mechanism -based corrosion fatigue damage model is developed for studying the high-cycle corrosion fatigue of steel from multi-scale viewpoint.

The developed physical corrosion fatigue damage model establishes micro-macro relationships between macroscopic continuum damage evolution and collective evolution behavior of microscopic pits and cracks, which can be used to describe the multi-scale corrosion fatigue process of steel. It shows that the model is effective and can be used to evaluate the continuum macroscopic corrosion fatigue damage and study microscopic corrosion fatigue mechanisms of steel. In this work, the fiber kinking phenomenon, which is known as the failure mechanism that takes place when a fiber reinforced polymer is loaded under longitudinal compression, is studied.

A computational micromechanics model is employed to interrogate the assumptions of a recently developed mesoscale continuum damage mechanics CDM model for fiber kinking based on the deformation gradient decomposition DGD and the LaRC04 failure criteria. A continuum mechanics -based musculo- mechanical model for esophageal transport. In this work, we extend our previous esophageal transport model using an immersed boundary IB method with discrete fiber-based structural model , to one using a continuum mechanics -based model that is approximated based on finite elements IB-FE.

To deal with the leakage of flow when the Lagrangian mesh becomes coarser than the fluid mesh, we employ adaptive interaction quadrature points to deal with Lagrangian-Eulerian interaction equations based on a previous work Griffith and Luo [1].


In particular, we introduce a new anisotropic adaptive interaction quadrature rule. The new rule permits us to vary the interaction quadrature points not only at each time-step and element but also at different orientations per element. This helps to avoid the leakage issue without sacrificing the computational efficiency and accuracy in dealing with the interaction equations. For the material model , we extend our previous fiber-based model to a continuum -based model. We present formulations for general fiber-reinforced material models in the IB-FE framework.

The new material model can handle non-linear elasticity and fiber-matrix interactions, and thus permits us to consider more realistic material behavior of biological tissues. To validate our method, we first study a case in which a three-dimensional short tube is dilated. Results on the pressure-displacement relationship and the stress distribution matches very well with those obtained from the implicit FE method. We remark that in our IB-FE case, the three-dimensional tube undergoes a very large deformation and the Lagrangian mesh-size becomes about 6 times of Eulerian mesh-size in the circumferential orientation.

To validate the performance of the method in handling fiber-matrix material models , we perform a second study on dilating a long fiber-reinforced tube. Errors are small when we compare numerical solutions with analytical solutions. The technique is then applied to the problem of esophageal transport. We use two. Ranges of applicability for the continuum beam model in the mechanics of carbon nanotubes and nanorods. Limitations in the validity of the continuum beam model for carbon nanotubes NTs and nanorods are examined. Applicability of all assumptions used in the model is restricted by the two criteria for geometric parameters that characterize the structure of NTs.

The key non-dimensional parameters that control the NT buckling behavior are derived via dimensional analysis of the nanomechanical problem. A mechanical law of geometric similitude for NT buckling is extended from continuum mechanics for different molecular structures. A model applicability map, where two classes of beam-like NTs are identified, is constructed for distinct ranges of non-dimensional parameters. Expressions for the critical buckling loads and strains are tailored for two classes of NTs and compared with the data provided by the molecular dynamics simulations.

All rights reserved. Time dependent reliability model incorporating continuum damage mechanics for high-temperature ceramics. Presently there are many opportunities for the application of ceramic materials at elevated temperatures. In the near future ceramic materials are expected to supplant high temperature metal alloys in a number of applications. It thus becomes essential to develop a capability to predict the time-dependent response of these materials. The creep rupture phenomenon is discussed, and a time-dependent reliability model is outlined that integrates continuum damage mechanics principles and Weibull analysis.

Several features of the model are presented in a qualitative fashion, including predictions of both reliability and hazard rate. In addition, a comparison of the continuum and the microstructural kinetic equations highlights a strong resemblance in the two approaches. Continuum mechanical model for cross-linked actin networks with contractile bundles. In the context of a mechanical approach to cell biology, there is a close relationship between cellular function and mechanical properties.

In recent years, an increasing amount of attention has been given to the coupling between biochemical and mechanical signals by means of constitutive models. In particular, on the active contractility of the actin cytoskeleton. Given the importance of the actin contraction on the physiological functions, this study propose a constitutive model to describe how the filamentous network controls its mechanics actively. Embedded in a soft isotropic ground substance, the network behaves as a viscous mechanical continuum , comprised of isotropically distributed cross-linked actin filaments and actomyosin bundles.

The performance of a state-of-the-art continuum damage mechanics model for interlaminar damage, coupled with a cohesive zone model for delamination is examined for failure prediction of quasi-isotropic open-hole tension laminates. Limitations of continuum representations of intra-ply damage and the effect of mesh orientation on the analysis predictions are discussed.

It is shown that accurate prediction of matrix crack paths and stress redistribution after cracking requires a mesh aligned with the fiber orientation. Based on these results, an aligned mesh is proposed for analysis of the open-hole tension specimens consisting of different meshes within the individual plies, such that the element edges are aligned with the ply fiber direction.

The modeling approach is assessed by comparison of analysis predictions to experimental data for specimen configurations in which failure is dominated by complex interactions between matrix cracks and delaminations. It is shown that the different failure mechanisms observed in the tests are well predicted. In addition, the modeling approach is demonstrated to predict proper trends in the effect of scaling on strength and failure mechanisms of quasi-isotropic open-hole tension laminates. This paper evaluates the ability of progressive damage analysis PDA finite element FE models to predict transverse matrix cracks in unidirectional composites.

The results of the analyses are compared to closed-form linear elastic fracture mechanics LEFM solutions. Matrix cracks in fiber-reinforced composite materials subjected to mode I and mode II loading are studied using continuum damage mechanics and zero-thickness cohesive zone modeling approaches.

The FE models used in this study are built parametrically so as to investigate several model input variables and the limits associated with matching the upper-bound LEFM solutions. Specifically, the sensitivity of the PDA FE model results to changes in strength and element size are investigated. A new model is proposed that represents the kinematics of kink-band formation and propagation within the framework of a mesoscale continuum damage mechanics CDM model. The model uses the recently proposed deformation gradient decomposition approach to represent a kink band as a displacement jump via a cohesive interface that is embedded in an elastic bulk material.

The model is capable of representing the combination of matrix failure in the frame of a misaligned fiber and instability due to shear nonlinearity. In contrast to conventional linear or bilinear strain softening laws used in most mesoscale CDM models for longitudinal compression, the constitutive response of the proposed model includes features predicted by detailed micromechanical models.

These features include: 1 the rotational kinematics of the kink band, 2 an instability when the peak load is reached, and 3 a nonzero plateau stress under large strains. The Z-vector method is modified to include induced dipoles and induced surface charges to determine the MP2 response density matrix, which can be used to evaluate MP2 properties. In particular, analytic nuclear gradient is derived and implemented for this method.

Differential continuum damage mechanics models for creep and fatigue of unidirectional metal matrix composites. Each model is phenomenological and stress based, with varying degrees of complexity to accurately predict the initiation and propagation of intergranular and transgranular defects over a wide range of loading conditions.

The development of these models is founded on the definition of an initially transversely isotropic fatigue limit surface, static fracture surface, normalized stress amplitude function and isochronous creep damage failure surface, from which both fatigue and creep damage evolutionary laws can be obtained. The anisotropy of each model is defined through physically meaningful invariants reflecting the local stress and material orientation. All three transversely isotropic models have been shown, when taken to their isotropic limit, to directly simplify to previously developed and validated creep and fatigue continuum damage theories.

Results of a nondimensional parametric study illustrate 1 the flexibility of the present formulation when attempting to characterize a large class of composite materials, and 2 its ability to predict anticipated qualitative trends in the fatigue behavior of unidirectional metal matrix composites. Additionally, the potential for the inclusion of various micromechanical effects e. In this paper, a continuum damage mechanics CDM -based creep model was proposed to study the creep behavior of T91 and T92 steels at high temperatures. Long-time creep tests were performed for both steels under different conditions.

The creep rupture data and creep curves obtained from creep tests were captured well by theoretical calculation based on the CDM model over a long creep time. It is shown that the developed model is able to predict creep data for the two ferritic steels accurately up to tens of thousands of hours. Fracture simulation of restored teeth using a continuum damage mechanics failure model.

The aim of this paper is to validate the use of a finite-element FE based continuum damage mechanics CDM failure model to simulate the debonding and fracture of restored teeth. Fracture testing of plastic model teeth, with or without a standard Class-II MOD mesial-occusal-distal restoration, was carried out to investigate their fracture behavior.

The material parameters needed for the CDM model to simulate fracture are obtained through separate mechanical tests. The predicted results are then compared with the experimental data of the fracture tests to validate the failure model. The failure processes of the intact and restored model teeth are successfully reproduced by the simulation.

However, the fracture parameters obtained from testing small specimens need to be adjusted to account for the size effect. The results indicate that the CDM model is a viable model for the prediction of debonding and fracture in dental restorations. Published by Elsevier Ltd. Nonlocal continuum -based modeling of mechanical characteristics of nanoscopic structures. Insight into the mechanical characteristics of nanoscopic structures is of fundamental interest and indeed poses a great challenge to the research communities around the world.

These structures are ultra fine in size and consequently performing standard experiments to measure their various properties is an extremely difficult and expensive endeavor. Hence, to predict the mechanical characteristics of the nanoscopic structures, different theoretical models , numerical modeling techniques, and computer-based simulation methods have been developed. Among several proposed approaches, the nonlocal continuum -based modeling is of particular significance because the results obtained from this modeling for different nanoscopic structures are in very good agreement with the data obtained from both experimental and atomistic-based studies.

A review of the essentials of this model together with its applications is presented here. Our paper is a self contained presentation of the nonlocal elasticity theory and contains the analysis of the recent works employing this model within the field of nanoscopic structures. In this review, the concepts from both the classical local and the nonlocal elasticity theories are presented and their applications to static and dynamic behavior of nanoscopic structures with various morphologies are discussed. We first introduce the various nanoscopic structures, both carbon-based and non carbon-based types, and then after a brief review of the definitions and concepts from classical elasticity theory, and the basic assumptions underlying size-dependent continuum theories, the mathematical details of the nonlocal elasticity theory are presented.

A comprehensive discussion on the nonlocal version of the beam, the plate and the shell theories that are employed in modeling of the mechanical properties and behavior of nanoscopic structures is then provided. Next, an overview of the current literature discussing the application of the nonlocal models. Dexterous continuum manipulators DCMs have been widely adopted for minimally- and less-invasive surgery. During the operation, these DCMs interact with surrounding anatomy actively or passively.

The interaction force will inevitably affect the tip position and shape of DCMs, leading to potentially inaccurate control near critical anatomy. In this paper, we demonstrated a 2D mechanical model for a tendon actuated, notched DCM with compliant joints. The model predicted deformation of the DCM accurately in the presence of tendon force, friction force, and external force. A partition approach was proposed to describe the DCM as a series of interconnected rigid and flexible links. Beam mechanics , taking into consideration tendon interaction and external force on the tip and the body, was applied to obtain the deformation of each flexible link of the DCM.

The model results were compared with experiments for free bending as well as bending in the presence of external forces acting at either the tip or body of the DCM. The overall mean error of tip position between model predictions and all of the experimental results was 0. The results suggest that the proposed model can effectively predict the shape of the DCM. We study the stone-impact resistance of a monolithic glass ply using a combined experimental and computational approach. Instrumented stone impact tests were first carried out in controlled environment.

Explicit finite element analyses were then used to simulate the interactions of the indentor and the glass layer during the impact event, and a continuum damage mechanics CDM model was used to describe the constitutive behavior of glass. The experimentally measured strain histories for low velocity impact served as validation of the modeling procedures.

The purpose of this study is to establish the modeling procedures and the CDM critical stress parameters under impact loading conditions. The modeling procedures and the CDM model will be used in our future studies to predict through-thickness damage evolution patterns for different laminated windshield designs in automotive applications. An enhanced version of a bone-remodelling model based on the continuum damage mechanics theory.

In their paper, they stated that the evolution of the internal variables of the bone microstructure, and its incidence on the modification of the elastic constitutive parameters, may be formulated following the principles of CDM, although no actual damage was considered. The resorption and apposition criteria similar to the damage criterion were expressed in terms of a mechanical stimulus.

However, the resorption criterion is lacking a dimensional consistency with the remodelling rate. We propose here an enhancement to this resorption criterion, insuring the dimensional consistency while retaining the physical properties of the original remodelling model. We then analyse the change in the resorption criterion hypersurface in the stress space for a two-dimensional 2D analysis. We finally apply the new formulation to analyse the structural evolution of a 2D femur. This analysis gives results consistent with the original model but with a faster and more stable convergence rate.

Continuum modeling of the mechanical and thermal behavior of discrete large structures. In the present paper we introduce a rather straightforward construction procedure in order to derive continuum equivalence of discrete truss-like repetitive structures. Once the actual structure is specified, the construction procedure can be outlined by the following three steps: a all sets of parallel members are identified, b unidirectional 'effective continuum ' properties are derived for each of these sets and c orthogonal transformations are finally used to determine the contribution of each set to the 'overall effective continuum ' properties of the structure.

Here the properties includes mechanical stiffnesses , thermal coefficients of thermal expansions and material densities. Once expanded descriptions of the steps b and c are done, the construction procedure will be applied to a wide variety of discrete structures and the results will be compared with those of other existing methods.

The fatigue behavior of a cellular composite with an epoxy matrix and glass foam granules is analyzed and modeled by means of continuum damage mechanics. The investigated cellular composite is a particular type of composite foam, and is very similar to syntactic foams. In contrast to conventional syntactic foams constituted by hollow spherical particles balloons , cellular glass, mineral, or metal place holders are combined with the matrix material metal or polymer in the case of cellular composites. A microstructural investigation of the damage behavior is performed using scanning electron microscopy.

For the modeling of the fatigue behavior, the damage is separated into pure static and pure cyclic damage and described in terms of the stiffness loss of the material using damage models for cyclic and creep damage. Both models incorporate nonlinear accumulation and interaction of damage. A cycle jumping procedure is developed, which allows for a fast and accurate calculation of the damage evolution for constant load frequencies.

The damage model is applied to examine the mean stress effect for cyclic fatigue and to investigate the frequency effect and the influence of the signal form in the case of static and cyclic damage interaction. The calculated lifetimes are in very good agreement with experimental results. Combined binary collision and continuum mechanics model applied to focused ion beam milling of a silicon membrane. Many experiments indicate the importance of stress and stress relaxation upon ion implantation. In this paper, a model is proposed that is capable of describing ballistic effects as well as stress relaxation by viscous flow.

It combines atomistic binary collision simulation with continuum mechanics. The only parameters that enter the continuum model are the bulk modulus and the radiation-induced viscosity. The shear modulus can also be considered but shows only minor effects. A boundary-fitted grid is proposed that is usable both during the binary collision simulation and for the spatial discretization of the force balance equations.

As an application, the milling of a slit into an amorphous silicon membrane with a 30 keV focused Ga beam is studied, which demonstrates the relevance of the new model compared to a more heuristic approach used in previous work. Linear response and variational treatment are formulated for Hartree-Fock HF and Kohn-Sham density functional theory DFT methods and combined discrete- continuum solvation models that incorporate self-consistently induced dipoles and charges. Due to the variational treatment, analytic nuclear gradients can be evaluated efficiently for these discrete and continuum solvation models.

The forces and torques on the induced point dipoles and point charges can be evaluated using simple electrostatic formulas as for permanent point dipoles and point charges, in accordance with the electrostatic nature of these methods. Implementation and tests using the effective fragment potential EFP, a polarizable force field method and the conductorlike polarizable continuum model CPCM show that the nuclear gradients are as accurate as those in the gas phase HF and DFT methods.

Nonlinear Continuum Mechanics. This report summarizes the key continuum mechanics concepts required for the systematic prescription and numerical solution of finite deformation solid mechanics problems. Topics surveyed include measures of deformation appropriate for media undergoing large deformations, stress measures appropriate for such problems, balance laws and their role in nonlinear continuum mechanics , the role of frame indifference in description of large deformation response, and the extension of these theories to encompass two dimensional idealizations, structural idealizations, and rigid body behavior.

There are three companion reports that describe the problem formulation, constitutive modeling , and finite element technology for nonlinear continuum mechanics systems. A viscoelastic model for dielectric elastomers based on a continuum mechanical formulation and its finite element implementation. Smart materials are active and multifunctional materials, which play an important part for sensor and actuator applications.

These materials have the potential to transform passive structures into adaptive systems. However, a prerequisite for the design and the optimization of these materials is, that reliable models exist, which incorporate the interaction between the different combinations of thermal, electrical, magnetic, optical and mechanical effects.

Polymeric electroelastic materials, so-called electroactive polymer EAP , own the characteristic to deform if an electric field is applied. EAP's possesses the benefit that they share the characteristic of polymers, these are lightweight, inexpensive, fracture tolerant, elastic, and the chemical and physical structure is well understood. However, the description "electroactive polymer" is a generic term for many kinds of different microscopic mechanisms and polymeric materials.

Based on the laws of electromagnetism and elasticity, a visco-electroelastic model is developed and implemented into the finite element method FEM. The presented three-dimensional solid element has eight nodes and trilinear interpolation functions for the displacement and the electric potential. The continuum mechanics model contains finite deformations, the time dependency and the nearly incompressible behavior of the material. To describe the possible, large time dependent deformations, a finite viscoelastic model with a split of the deformation gradient is used.

Thereby the time dependent characteristic of polymeric materials is incorporated through the free energy function. The electromechanical interactions are considered by the electrostatic forces and inside the energy function. A physically-based continuum damage mechanics model for numerical prediction of damage growth in laminated composite plates. Rapid growth in use of composite materials in structural applications drives the need for a more detailed understanding of damage tolerant and damage resistant design.

Current analytical techniques provide sufficient understanding and predictive capabilities for application in preliminary design, but current numerical models applicable to composites are few and far between and their development into well tested, rigorous material models is currently one of the most challenging fields in composite materials. The present work focuses on the development, implementation, and verification of a plane-stress continuum damage mechanics based model for composite materials. A physical treatment of damage growth based on the extensive body of experimental literature on the subject is combined with the mathematical rigour of a continuum damage mechanics description to form the foundation of the model.

The model has been implemented in the LS-DYNA3D commercial finite element hydrocode and the results of the application of the model are shown to be physically meaningful and accurate. Furthermore it is demonstrated that the material characterization parameters can be extracted from the results of standard test methodologies for which a large body of published data already exists for many materials. Two case studies are undertaken to verify the model by comparison with measured experimental data.

The predicted force-time and force-displacement response of the panels compare well with experimental measurements. To further demonstrate the physical nature of the model , a IM6. This paper introduces a graduate course, continuum mechanics , which is designed for and taught to graduate students in a Mechanical Engineering ME program.

The significance of continuum mechanics in engineering education is demonstrated and the course structure is described. Methods used in teaching this course such as topics, class…. Experimental verification of a progressive damage model for composite laminates based on continuum damage mechanics. Thesis Final Report.

Progressive failure is a crucial concern when using laminated composites in structural design. Therefore the ability to model damage and predict the life of laminated composites is vital. The purpose of this research was to experimentally verify the application of the continuum damage model , a progressive failure theory utilizing continuum damage mechanics , to a toughened material system. Crack density and delamination surface area were used to calculate matrix cracking and delamination internal state variables, respectively, to predict stiffness loss.

A damage dependent finite element code qualitatively predicted trends in transverse matrix cracking, axial splits and local stress-strain distributions for notched quasi-isotropic laminates. The predictions were similar to the experimental data and it was concluded that the continuum damage model provided a good prediction of stiffness loss while qualitatively predicting damage growth in notched laminates.

Fundamentals of continuum mechanics — classical approaches and new trends. Continuum mechanics is a branch of mechanics that deals with the analysis of the mechanical behavior of materials modeled as a continuous manifold. Continuum mechanics models begin mostly by introducing of three-dimensional Euclidean space. The points within this region are defined as material points with prescribed properties.

Each material point is characterized by a position vector which is continuous in time. Thus, the body changes in a way which is realistic, globally invertible at all times and orientation-preserving, so that the body cannot intersect itself and as transformations which produce mirror reflections are not possible in nature. For the mathematical formulation of the model it is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.

If the physical fields are non-smooth jump conditions must be taken into account. The basic equations of continuum mechanics are presented following a short introduction.

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Additionally, some examples of solid deformable continua will be discussed within the presentation. Finally, advanced models of continuum mechanics will be introduced. A continuum model of transcriptional bursting. Transcription occurs in stochastic bursts. Early models based upon RNA hybridisation studies suggest bursting dynamics arise from alternating inactive and permissive states.

Here we investigate bursting mechanism in live cells by quantitative imaging of actin gene transcription, combined with molecular genetics, stochastic simulation and probabilistic modelling. In contrast to early models , our data indicate a continuum of transcriptional states, with a slowly fluctuating initiation rate converting the gene between different levels of activity, interspersed with extended periods of inactivity.

We place an upper limit of 40 s on the lifetime of fluctuations in elongation rate, with initiation rate variations persisting an order of magnitude longer. TATA mutations reduce the accessibility of high activity states, leaving the lifetime of on- and off-states unchanged. A continuum or spectrum of gene states potentially enables a wide dynamic range for cell responses to stimuli.

A simple continuum damage mechanics CDM based 3D progressive damage analysis PDA tool for laminated composites was developed and implemented as a user defined material subroutine to link with a commercially available explicit finite element code. This PDA tool uses linear lamina properties from standard tests, predicts damage initiation with an easy-to-implement Hashin-Rotem failure criteria, and in the damage evolution phase, evaluates the degradation of material properties based on the crack band theory and traction-separation cohesive laws. It follows Matzenmiller et al. Since nonlinear shear and matrix stress-strain relations are not implemented, correction factors are used for slowing the reduction of the damaged shear stiffness terms to reflect the effect of these nonlinearities on the laminate strength predictions.

Strength predictions obtained, using this VUMAT, are correlated with test data for a set of notched specimens under tension and compression loads. A method has been proposed for developing structure-property relationships of nano-structured materials. This method serves as a link between computational chemistry and solid mechanics by substituting discrete molecular structures with equivalent- continuum models.

It has been shown that this substitution may be accomplished by equating the vibrational potential energy of a nano-structured material with the strain energy of representative truss and continuum models. As important examples with direct application to the development and characterization of single-walled carbon nanotubes and the design of nanotube-based devices, the modeling technique has been applied to determine the effective- continuum geometry and bending rigidity of a graphene sheet.

A representative volume element of the chemical structure of graphene has been substituted with equivalent-truss and equivalent continuum models. As a result, an effective thickness of the continuum model has been determined. This effective thickness has been shown to be significantly larger than the interatomic spacing of graphite.

The effective thickness has been shown to be significantly larger than the inter-planar spacing of graphite. The effective bending rigidity of the equivalent- continuum model of a graphene sheet was determined by equating the vibrational potential energy of the molecular model of a graphene sheet subjected to cylindrical bending with the strain energy of an equivalent continuum plate subjected to cylindrical bending. A computational continuum model of poroelastic beds.

Despite the ubiquity of fluid flows interacting with porous and elastic materials, we lack a validated non-empirical macroscale method for characterizing the flow over and through a poroelastic medium. We propose a computational tool to describe such configurations by deriving and validating a continuum model for the poroelastic bed and its interface with the above free fluid. We show that, using stress continuity condition and slip velocity condition at the interface, the effective model captures the effects of small changes in the microstructure anisotropy correctly and predicts the overall behaviour in a physically consistent and controllable manner.

Moreover, we show that the performance of the effective model is accurate by validating with fully microscopic resolved simulations. The proposed computational tool can be used in investigations in a wide range of fields, including mechanical engineering, bio-engineering and geophysics. A method has been developed for modeling structure-property relationships of nano-structured materials. In support of this, theories of biological network decomposition and regulatory motif detection need to be developed so that analysis of pieces of the system can proceed without explicitly including every interaction in the cell.

All the while, these analyses must drive experiment and be validated in detail by the experimental data. Here we describe some of our approaches to all these different problems and demonstrate them on particular bacterial and eukaryotic systems. Progress on this tool and its theory will also be discussed. Traditional algorithms design assumes that the problem is described by a single objective function.

One of the main current trends of work focuses on approximation algorithm, where computing the exact optimum is too hard. However, there is an additional difficulty in a number of settings. It is natural to consider algorithmic questions where multiple agents each pursue their own selfish interests.

We will discuss problems and results that arise from this perspective. With both computational cost and coding effort for RBF approximations independent of the number of spatial dimensions, it is not surprising that RBFs have since found use in many applications. Their use as basis functions for the numerical solution of PDEs is however surprisingly novel. In this Colloquium, we will discuss RBF approximations from the perspective of someone interested in pseudospectral spectral collocation methods primarily for wave-type equations. Reflection seismograms contain enormous amounts of information about the Earth's structure, obscure by complex reflection and refraction effects.

Modern mathematical understanding of wave propagation in heterogeneous materials has aided in the unraveling of this complexity. The speaker will outline some advances in the theory of oscillatory integrals which have had immediate practical application in seismology. The genomes of more than fifty other organisms have also been completed.

Thus we have in these genomes the information that encodes the description of not only the majority of the protein and nucleic acid components essential for their operation but also the details of when and where they are to be produced. A major challenge of the next decade will be the determination of how the control elements associated with the structural genes function.

We must develop methods that allow these elements to be recognized in the genomic sequence. This has been done for only a small number of genes at this time. Comparative genomics provides a powerful approach to this problem. Comparison of the genomic sequence of organisms that are evolutionarily separated by 30 to million years provides way of identifying regions that contain conserved regulatory signals. In addition this type of comparison also often allows identification previously unrecognized genomic features.

I will present some examples of this approach. Algorithms and Software for Dynamic Optimization with Application to Chemical Vapor Deposition Processes Linda Petzold University of California at Santa Barbara Date: November 1, Location: UBC Abstract In recent years, as computers and algorithms for simulation have become more efficient and reliable, an increasing amount of attention has focused on the more computationally intensive tasks of sensitivity analysis and optimal control.

In this lecture we describe algorithms and software for sensitivity analysis and optimal control of large-scale differential-algebraic systems, focusing on the computational challenges. An application from the chemical vapor deposition growth of a thin film YBCO high-temperature superconductor will be described. Unusual comparison properties of capillary surfaces Robert Finn Stanford University Date: May 8, Location: UBC Abstract This talk will address a question that was raised about 30 years ago by Mario Miranda, as to whether a given cylindrical capillary tube always raises liquid higher over its section than does a cylinder whose section strictly contains the given one.

Depending on the specific shapes, the answer can take unanticipated forms exhibiting nonuniformity and discontinuous reversal in behavior, even in geometrically simple configurations. The presentation will be for the most part complete and self-contained, and is intended to be accessible for a broad mathematical audience. One is the nonlinear median filter, that chooses the median of the sample values in the sliding window. This deals effectively with "outliers" that are beyond the correct sample range, and will never be chosen as the median.

A straightforward implementation of the filter is expensive for large windows, particularly in two dimensions for images. The second filter is linear, and known as "Savitzky-Golay". It is frequently used in spectroscopy, to locate positions and peaks and widths of spectral lines. This filter is based on a least-squares fit of the samples in the sliding window to a polynomial of relatively low degree.

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The filter coefficients are unlike the equiripple filter that is optimal in the maximum norm, and the "maxflat" filters that are central in wavelet constructions. Should they be better known? We will discuss the analysis and the implementation of both filters. Guessing Secrets Ron Graham University of California, San Diego Date: May 24, Location: UBC Abstract We will describe a variant of the familiar "20 questions" problem in which one tries to discover the identity of some unknown "secret" by asking binary questions. In this variation, there is now a set of two or more secrets.

For each question asked, an adversary gets to choose which of the secrets to use in supplying the answer, which in any case must be truthful. We will discuss a number of algorithms for dealing with this problem, although we are still far from a complete understanding of the situation.

Problems of this type have recently arisen in connection with certain Internet traffic routing applications, although it turns out that such problems have in fact occurred in the literature at least 40 years ago. Fingerprint Matching Anil K. Because of the need to enhance security e. Among the various characteristics e. Inspite of over 50 years of research and development, fingerprint matching and classification continue to be challenging research problems.

This talk will present methods for fingerprint representation, matching and classification. Current research related to multimodal biometrics e. I will describe the sources of data and give an overview of the problems being addressed. I will go into some detail on a particular problem-forecasting travel times over a network of freeways. Although the underlying system is very complex and tempting to model, a simple model is surprisingly effective at forecasting. Modeling stock returns and calculating portfolio risk is almost invariably accomplished by fitting a linear model, called a "factor" model in the finance community, using the sanctified method of ordinary least squares OLS.

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However, it is well-known that stock returns are often non-Gaussian by virtue of containing outliers, and that OLS estimates are not robust toward outliers. Modern robust regression methods are now available that are not for stock returns using firm size and book-to-market as the factors, where we show that OLS gives a misleading result. Then we show how Trellis graphics displays can be used to obtain quick, penetrating visualization of stock returns factor model data, and to obtain convenient comparisons of OLS and robust factor model fits.

Last but not least, we point out that robust factor model fits and Trellis graphics displays are in effect powerful "data mining tools" for better understanding of financial data. In contrast to the standard modelling of jumps for asset returns, the jump component of our process can display finite or infinite activity, and finite or infinite variation. Empirical investigations of time series indicate that index dynamics are essentially devoid of a diffusion component, while this component may be present in the dynamics of individual stocks.

This result leads to the conjecture that the risk-neutral process should be free of a diffusion component for both indices and individual stocks. Empirical investigation of options data tends to confirm this conjecture. We conclude that the statistical and risk-neutral processes for indices and stocks tend to be pure jump processes of infinite activity and finite variation.

Arithmetic progressions of primes. Mahler measure, factors of Markov shifts and symbolic representations of group automorphisms Klaus Schmidt University of Vienna Date: May 8, Location: UBC Abstract The topic of this lecture is a construction of symbolic representations of hyperbolic toral automorphisms originally due to Vershik at least in a special case and later developed further by Vershik, Sidorov, Kenyon, Einsiedler, Lind and myself. Skein theory in knot theory and beyond Vaughan F.

It was extended somewhat in the 's when new knot polynomials were discovered. More recently it has proved useful in the theory of subfactors as a tool to construct and analyse certain exotic structures. These structures remain related to low dimensional topology as they have topological quantum field theories attached. The unifying language is that of planar algebra. Care will be taken to explain the economic framework and the tools of differential geometry. Arnold Institute for Mathematics and its Applications Date: May 16, Location: UBC Abstract An ineluctable, though subtle, consequence of Einstein's theory of general relativity is that relatively accelerating masses generate tiny ripples on the curved surface of spacetime which propagate through the universe at the speed of light.

Although such gravity waves have not yet been detected, it is believed that technology is reaching the point where detection is possible, and a massive effort to construct worldwide network of interferometer gravity wave observatories is well underway.

They promise to be our first window to the universe outside the electromagnetic spectrum and so, to astrophysicists and others trying to fathom a universe composed primarily of electromagnetically dark matter, the potential payoff is enormous. If gravitational wave detectors are to succeed as observatories, we must learn to interpret the wave forms which are detected. This requires the numerical simulation of the violent cosmic events, such as black hole collisions, which are the most likely sources of detectable radiation, via the numerical solution of the Einstein field equations.

The Einstein equations form a system of ten second order nonlinear partial differential equations in four-dimensional spacetime which, while having a very elegant and fundamental geometric character, are extremely complex. Their numerical solution presents an enormous computational challenge which will require the application of state-of-the-art numerical methods from other areas of computational physics together with new ideas. This talk aims to introduce some of the scientific, mathematical, and computational problems involved in the burgeoning field of numerical relativity, discuss some recent progress, and suggest directions of future research.

Ratiu will present Hamiltonian systems whose configuration space is a Lie group. The systems discussed are all geodesic flows of certain metrics with a number of common characteristics. He will begin with the classical example introduced by Euler: a free rigid body whose configuration space is a proper rotation group. Then he will discuss a homogeneous incompressible fluid flow whose configuration space is a group of volume preserving diffeomorphisms of a smooth manifold.

The Arnold-Ebin-Marsden program for the analysis of the equations of motion will be also presented. Another example is the Korteweg-de Vries equation whose configuration space is the Bott-Virasoro group. Generalization of this is the Camassa-Holm equation, and Prof. Ratiu will discuss its possible choices of configuration spaces. The averaged incompressible Euler flow and recent results about it will be presented.

Finally, Prof. It will be shown how one can determine the dynamics of the corresponding quadratic maps by visualizing tiny regions in the Mandelbrot set as well as how the size and location of the bulbs in the Mandelbrot set is governed by Farey arithmetic. The lecture is about some aspects of that conjecture and of the analogue for idempotent matrices. There are results for various classes of groups, mainly appearing in geometry or topology. One uses suitable trace concepts, and methods such as the Kaplansky Theorem, the Bass Conjecture, and the Gromov type geometry of groups.

This means that the infinite system is replaced by the sequence of finite sections of the original system and it is expected that the solutions of the finite systems converge to the solution of the infinite system. This method has a rich history of years and many distinguished mathematicians have made important contributions in this area.

The talk will present the early history and recent important achievements. Unexpected examples and computational experiments will motivate and illustrate the main results. Special attention will be paid to the case of Toeplitz matrices with continuous and discontinuous symbols. The talk is planned for a wide audience. The aim is to study these sequences. The orbits are studied using dynamical systems and algebraic geometry. Following Lyness, Ladas showed that f leaves invariant a family of nested closed curves filling the positive quadrant.

We show that on each curve f, is conjugate to a rigid rotation, and so the orbits on that curve are either all periodic or all dense in that curve. The extension to higher dimensions leads to further theorems and conjectures. Define an a , b -code as a set of words of weight greater than or equal to b called code words , such that no word of weight greater than or equal to a occurs as a consecutive substring of two different code words or more than once as a substring of the same code word.

We show how to construct a , b -codes of near-maximum cardinality. The problem arises in molecular biology.

  1. Creating Neighbourhoods and Places in the Built Environment (Built Environment Series of Textbooks)!
  2. HS1.2 – Cross-cutting hydrological sessions!
  3. From discontinuous models towards a continuum description?
  4. HS1.2 – Cross-cutting hydrological sessions.
  5. Introduction!
  6. Collected Writings of Gordon Daniels (Collected Writings of Modern Western Scholars on Japan).
  7. Winning the Race for Value: Strategies to Create Competitive Advantage in the Emerging.

The Watson-Crick complement of a molecule is the molecule obtained by replacing each nucleotide by its complement, where A and T are complementary and C and G are complementary. Complementary or near-complementary sequences tend to hybridize stick together. For various molecular recognition tasks we seek a set of molecules such that each molecule in the set hybridizes strongly to its own complement, but does not hybridize strongly to the complement of any other molecule in the set.

Other definitions lead to alternate formulations that we also discuss. Modelling, analysis and computation of crystalline microstructures Mitchell Luskin University of Minnesota Date: October 29, Location: UBC Abstract Microstructure is a feature of crystals with multiple symmetry-related energy-minimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetry-related states on a fine scale to minimize energy.

We have developed an approximation theory for crystal microstructure which gives an analysis of the stability of macroscopic variables with respect to small energy perturbations. This theory has been applied to analyze several numerical methods for the approximation of martensitic microstructure and ferromagnetic microstructure. Some natural computational tasks are infeasible e. Probabilistic algorithms can be much more efficient than deterministic ones. As it does with other notions e. One major achievement of this study is the following surprising?

I plan to explain the sequence of important ideas, definitions, and techniques developed in the last 20 years that enable a formal statement and proof of such theorems. Many of them, such as the formal notions of "pseudo-random generator", and "computational indistinguishability" are of fundamental interest beyond derandomization; they have far reaching implications on our ability to build efficient cryptographic systems, as well as our inability to efficiently learn natural concepts and effectively prove natural mathematical conjectures such as 1 above.

Phase-contrast computed tomography CT is one such technique that exploits differences in the real part of the refractive index distribution of an object to form an image using a spatially coherent light source. Of particular interest is the ability of phase-contrast CT to produce useful images of objects that have very similar or identical absorption properties. In applications such as microtomography, it is imperative to reconstruct an image with high resolution.

Experimentally, the demand of increased resolution can be achieved by highly collimating the incident light beam and using a microscope optic to focus the transmitted image, formed on a scintillator screen, onto the detector. When the object is larger than the field-of-view FOV of the imaging system, the measured phase-contrast projections are necessarily truncated and one is faced with the so-called local CT reconstruction problem.

To circumvent the non-local nature of conventional CT, local CT algorithms have been developed that aim to to reconstruct a filtered image that contains detailed information regarding the location of discontinuities in the imaged object. Such information is sufficient for determining the structural composition of an object, which is the primary task in many biological and materials science imaging applications. Theory of Local Phase-Contrast Tomography: We have recently demonstrated that the mathematical theory of local CT, which was originally developed for absorption CT, can be applied naturally for understanding the problem of reconstructing the location of image boundaries i.

Our analysis suggested the use of a simple backprojection-only algorithm for reconstructing object discontinuities from truncated phase-contrast projection data that is simpler and more theoretically appropriate than use of the FBP algorithm or use of the exact reconstruction algorithm for phase-contrast CT that was recently proposed by Bronnikov [1]. We demonstrated that the reason why this simple backprojection-only procedure represents an effective local reconstruction algorithm for phase-contrast CT is that the filtering operation that needs to be explicitly applied to the truncated projection data in conventional absorption CT is implicitly applied to the phase-contrast projection data before they are measured by the act of paraxial wavefield propagation in the near-field.

In this talk, we review the application of local CT reconstruction theory to the phase-contrast imaging problem. Using concepts from microlocal analysis, we describe the features of an object that can be reliably reconstructed from incomplete phase-contrast projection data. In many applications, the magnitude of the refractive index jump across an interface may provide useful information about the object of interest. For the first time, we demonstrate that detailed information regarding the magnitude of refractive index discontinuities can be extracted from the phase-contrast projections.

Moreover, we show that these magnitudes can be reliable reconstructed using adaptations of algorithms that were originally developed for absorption local CT. Numerical Results: We will present extensive numerical results to corroborate our theoretical assertions. We will compare the ability of the available approximate and exact reconstruction algorithms to provide images that contain accurate information regarding the location and magnitude of refractive index discontinuities. The stability of the algorithms to data noise and inconsistencies will be reported. In Fig.

SUMMARY In this talk, we address the important problem of reconstructing the location and magnitude of refractive index discontinuities in phase-contrast tomography. We theoretically investigate existing and novel reconstruction algorithms for reconstructing such information from truncated phase-contrast tomographic projections and numerically corroborate our findings using simulation and experimental data. Bronnikov, "Theory of quantitative phase-contrast computed tomography," Journal of the Optical Society of America A , vol.

Reconstruction Methods in Optical Tomography and Applications to Brain Imaging Simon Arridge University College London Date: August 7, Location: UBC Abstract In the first part of this talk I will discuss methods for reconstruction of spatially-varying optical absorbtion and scattering images from measurements of transmitted light through highly scattering media. The problem is posed in terms of non-linear optimisation, based on a forward model of diffusive light propgation, and the principle method is linearisation using the adjoint field method.

In the second part I will discuss the particular difficulties involved in imaing the brain. These include: - Accounting for non or weakly scattering regions that do not satisfy the diffusion approximation the void problem - Accounting for anisotropic scattering regions - Constructing realistic 3D models of the head shape - Dynamic imaging incorporating temporal regularisation. Computational Anatomy: Quantifying Shape and Size of Anatomical Objects via Metrics on Flows of Diffeomorphisms Mirza Faisal Beg Johns Hopkins University Date: August 7, Location: UBC Abstract Development of mathematical models and computational tools that can quantify shape and size of anatomical structures in normal and diseased states and during growth and aging is an important and challenging task in advancing the field of medical image understanding.

I will present an overview of the Metric Pattern Theory of Miller, Trouve and Younes in which anatomical objects as modelled by images or landmarked representations of these images are an orbit of a template object under the group of diffeomorphisms. Metric distances between objects images or landmarked representations of images come from the geodesic distance between points on the manifold of diffeomorphisms that register these objects and are calculated by the variational optimization of a cost associated to a path of deformation on the manifold of diffeomorphisms.

I will present the Euler-Lagrange equations to estimate diffeomorphisms for registering images and sets of landmarks and issues in numerical and parallel implementation of these equations. I will also present applications in quantifying shape and size changes in neurodegenerative diseases like Huntington's, Alzheimer's and Schizophrenia. Fast Hierarchical Algorithms for Tomography Yoram Bresler University of Illinois at Urbana-Champaign Date: August 8, Location: UBC Abstract The reconstruction problem in practical tomographic imaging systems is recovery from samples of either the x-ray transform set of the line-integral projections or the Radon transform set of integrals on hyperplanes of an unknown object density distribution.

The method of choice for tomographic reconstruction is filtered backprojection FBP , which uses a backprojection step. These algorithms employ a divide-and-conquer strategy in the image domain, and rely on properties of the harmonic decomposition of the Radon transform. For image sizes typical in medical applications or airport baggage security, this results in speedups by a factor of 50 or greater. Such speedups are critical for next-generation real-time imaging systems. For that evaluation, contrast resolution, of highest importance for modality selection in most cases, is defined as the signal to background for the desired biochemical or physiological parameter.

But a particular modality which has exquisite biological potential e. MRI and SPECT for atherosclerosis characterization might not be deployed in medical science because appropriate algorithms are not available to deal with problems of blurring, variable point spread function, background scatter, detection sensitivity, attenuation and refraction.

Trade-offs in technique selection frequently pit contrast resolution against intrinsic instrument resolution temporal and spatial and depth or size of the object. For example, imaging vulnerable carotid plaques using a molecular beacon with signal to background and with 7 mm resolution in the human neck can be argued as superior to imaging tissue characteristics with signal to background at 0. Another example is the use of the multidetector CT helical due to its relative speed instead of MRI to characterize coronary plaques even though MRI has much better intrinsic contrast mechanisms.

The superior speed of modern CT argues for its preferred use. Some old examples of how mathematics of the inverse problem have enabled medical science advances include incorporation of attenuation compensation in SPECT imaging which brought SPECT to a quantitative technique, light transmission and fluorescence emission tomography, iterative reconstruction algorithm for all methods, and incorporation of phase encoding for MRI reconstruction.

Current work on new mathematical approaches includes endeavors to improve resolution, improve sampling speed, decrease background and achieve reliable quantitation. Examples are rf exposure reduction in MRI by selective radio frequency pulses requiring low peak power, dose reduction by iterative reconstruction schemes in X-Ray CT, implementation of coded aperture models for emission tomography, 3D and time reversal ultrasound, a multitude of transmission and stimulated emission methods for light wavelength of nm to 3 cm, and electrical potential and electric source imaging.

Many of these subjects will be discussed at this workshop and all rely on innovations in mathematics applied to the inverse problem. Firstly, we study the fan-beam Radon transform of symmetric solenoidal tensor fields of arbitrary rank in a unit disc. The inversion formula for a fan-beam transform follows from its singular value decomposition SVD. For that we first build polynomial solenoidal orthonormal basis tensor fields with the use of Zernike polynomials.

As a result, SVD of the fan-beam transform is a double Fourier series of trigonometric functions. This algorithm allows to investigate the velocity distribution in a flow or the material stress distribution in metals by differential ultrasonic time-of-flight measurements. We provide numerical evaluation of the given algorithms for recovering scalar and vector fields from uniform and non-uniform discrete fan-beam projections. Then we consider the problem of two-dimensional single photon emission computerized tomography SPECT in the case of a fan-beam geometry. The problem of recovering emission map from SPECT data arises in different industrial and medical applications.

This problem is mathematically investigated using the attenuated Radon transform mostly within the framework of either a parallel-beam geometry or a fan-beam geometry. The first explicit inversion formula in the case of a fan-beam geometry was obtained by A. Later on, another explicit inversion formulas for reconstructing the emission map in the case of a parallel-beam geometry were derived by R.

We also give an implementation of the inversion formula for recovering the emission map provided that the attenuation map is known. New Multiscale Thoughts on Limited-Angle Tomography Emmanuel Candes California Institute of Technology Date: August 4, Location: UBC Abstract This talk is concerned with the problem of reconstructing an object from noisy limited-angle tomographic dataa problem which arises in many important medical applications. Here, a central question is to describe which features can be reconstructed accurately from such data and how well, and which features cannot be recovered.

We argue that curvelets, a recently developed multiscale system, may have a great potential in this setting. Conceptually, curvelets are multiscale elements with a useful microlocal structure which makes them especially adapted to limited-angle tomography. We develop a theory of optimal rates of convergence which quantifies that features which are microlocally in the "good" direction can be recovered accurately and which shows that adapted curvelet-biorthogonal decompositions with thresholding can achieve quantitatively optimal rates of convergence.

We hope to report on early numerical results. Scott Carney University of Illinois at Urbana-Champaign Date: August 7, Location: UBC Abstract Near-field optics provides a means to observe the electromagnetic field intensity in close proximity to a scattering of radiating sample. Modalities such as near-field scanning optical microscopy NSOM and photon scanning tunneling microscopy PSTM accomplish these measurements by placing a small probe close to the object in the "near-zone" and then precision controlling the position.

The data are usually plotted as a function probe position and the resulting figure is called an image. These modalities provide a means to circumvent the classical Rayleigh-Abbe resolution limits, providing resolution on scales of a small fraction of a wavelength. There are a number of problems associated with the interpretation of near-field images. If the probe is slightly displaced from the surface of the object, the image quality degrades dramatically. If the sample is thick, the subsurface features are obscured.

The quantitative connection between the measurements and the optical properties of the sample is unknown. To resolve all these problems it is desirable to solve the inverse scattering problem ISP for near-field optics. The solution of the ISP provides a means to tomographically image thick samples and assign quantitative meaning to the images. Furthermore, data taken at distances up to one wavelength from the sample may be processed to obtain a focused, or reconstructed image of the sample at subwavelength scales.

Applications of sampling theory in tomography include the identification of efficient sampling schemes; a qualitative understanding of some artifacts; numerical analysis of reconstruction methods; and efficient interpolation schemes for non-equidistant sampling. In this talk we present an application of periodic sampling theorems in three-dimensional multisclice helical tomography shedding light on the question of preferred pitches.

These are measured by transducers, and the problem is to recover the density of emitters. This may be modelled as the recovery of the initial value of the time derivative of the solution of the wave equation from knowledge of the solution on part of the boundary of the domain. This talk, in conjunction with the talk by Sarah Patch, will report on recent work by the author, S. Patch and Rakesh on uniqueness and stability and an inversion formula, in odd dimensions, for the special case when measurements are taken on an entire sphere surrounding the object.

The well-known relation between spherical means and solutions of the wave equation then implies results on recovery of a function from its spherical means. It explains the great interest for developing quantitative imaging of the shear modulus distribution map. This can be achieved by observing with NMR or with ultrasound the propagation of low frequency shear waves between 50 Hz and Hz in the body.

We have developed an ultra high-rate ultrasonic scanner that can give With such a high frame-rate we can follow in real time the propagation of transient shear waves, and from the spatio-temporal evolution of the displacement fields, we can use inversion algorithm to recover the shear modulus map. New inversion algorithm can be used that are no more limited by diffraction limits. In order to obtain unbiased shear elasticity map, different configurations of shear sources induced by radiation pressure of focused transducer arrays are used. A very interesting configuration that induces quasi plane shear waves will be described.

It used a sonic shear source that moves at supersonic velocities, and that is created by using a very peculiar beam forming in the transmit mode. In vitro and in vivo results will be presented that demonstrate the interest of this new transient elastographic technique. Finally, the anisotropy of strain hardening induced by the emergence of internal stress fields will be reviewed.

The direc- tionality of the sharp yield point in strain-aged steels and the occurrence of a Baushinger effect after a sequence of forward-reverse straining will receive interpretation within the framework of a field dislocation theory coupling the evolution of statistical and polar dislocation densities with that of point de- fects due to strain-aging [27].

By considering internal stresses due to dislocation - dislocation interactions, alternative modeling approaches such as statistical mechanics [4], phase field [28] and discrete dislocation dynamics methods [22, 29, 30] reproduce the scale-invariance of plastic activity. They have a potential for retrieving length- scale dependence of material properties, but usually consider periodic bound- ary conditions over small domains. Further, both phase field and statistical mechanics methods have not been shown to retrieve the propagative features related with dislocation transport.

In discrete dislocation dynamics simula- tions, transport of dislocation densities is present, but fully resolved into the motion of individual dislocations. As a rule of thumb, using present day computing facilities, dislocation dynamics codes are able to handle a tenfold increase of the initial number of dislocations [31]. Thus, if not for the treatment of boundary con- ditions, geometric and elastic nonlinearity, and inertia, the chances to tackle large-scale engineering problems in the future with discrete dislocation dy- namics simulations are slim, and field dislocation theories seem to be more fitted for real scale boundary value problems.

The Chapter is organized as follows. In Section 2, we provide an overview of the current field dislocation dynamics theories, augmented with recent devel- opments in macroscopic polycrystal response. Section 3 deals with the effects of sample size on mechanical response, and is illustrated with the example of ice single crystals submitted to torsion creep where robust size effects are observed.

Section 4 is devoted to scale-invariance and transport effects in the intermittency of crystal plasticity. Examples include the behavior of copper single crystals in tension. Section 5 deals with the anisotropy in mechanical properties induced by complex strain paths, with the exemple of the direc- tionality of the sharp yield point and the occurrence of a Baushinger effect in strain-aged polycrystalline steels.

The concluding section provides insights into the flexibility of the theory regarding the scale of resolution and its ability to deal with fine - scale vs. When the size of S, i. In intermediate cases, the net Burgers vector b is non-zero, but part of the dislocations threading S may remain unresolved. In this model of dislocation mechanics, the total displacement field, u, does not represent the actual physical motion of atoms involving topological changes but only a consistent shape change and hence is not required to be discontinuous. Applying this operator to Eqs.

Initial and boundary conditions for are important from the physical model- ing point of view [15], particularly in the context of triggering inhomogeneity under boundary conditions corresponding to homogeneous deformation in conventional plasticity theory [17]. Through the curl of the total plastic distortion rate tensor U it couples the polar and statistical dislocation densities for the nucleation of polar dislocations. Complementing the above equations with a constitutive relation for the average dislocation velocity V as a function of stress and dis- location orientation, and with phenomenological evolution equations for the statistical densities involved in the conventional velocity gradient Lp , one obtains a closed theory in the sense that it contains enough statements to derive uniquely the dynamics of stress and dislocation densities in a bounded domain from boundary and initial conditions.

In the sit- uation when the dislocation density may not be expressed as an elementary dyad formed from a Burgers vector direction and a line direction, the defi- nition 26 arises as a sufficient condition for pressure independence of the polar dislocation plastic strain rate and ensuring positive dissipation. Thus d represents the fact that mixed or edge dislocations cannot climb or cross-slip whereas screw dis- locations are unrestricted in their motion. It is perhaps insightful to evoke analogies between dislocation dynam- ics and eddy dynamics in turbulent flow [34].

Turbulent flow is characterized by eddies at all scales. Averaging in space and time the Navier-Stokes equations provides equations for large resolved eddies, while unresolved ones are dealt with using additional sub- grid-scale variables. Closure of the theory is obtained through sub-grid phe- nomenological models featuring scaling character [35]. In dislocation dynam- ics, averaging in space secures equations for polar dislocations while providing the link with conventional plasticity: closure for the unresolved variables Lp derives from well-established models for the viscoplasticity of crystalline ma- terials, i.

Also in contrast to turbulence, scaling behavior is associated with grid scale level, not sub-grid scale, as we show in Section 4. Two types of solutions are offered in what follows, in order to provide various insights. First we conduct full 3-D numerical solutions of Eqs. Large n values reflect abruptness of dislocation unpinning from obstacles. The velocity V of polar dislocations is taken as the average of the statistical slip velocity absolute values Vs over all slip systems.

Hence, the same physics applies to both dislocation species. We shall also consider simplified situations with dislocations pertaining to, and gliding in a single slip plane with no out-of-plane motion. The latter simulations provide for representative behavior of some portion of a slip plane in a single crystal experiment. When investigating the plastic response of materials, this gradient is commonly viewed as a drawback of torsion testing. It becomes beneficial when the material behavior involves internal length scales associated with emerging dislocation microstructures.

The inhomogeneity of the boundary conditions then generates polar dislocations, which give rise to long-range elastic stress fields. Hence torsion is a challenging case for theories of plastic- ity with internal length scales. Size effects were reported, and the trend is that the greater is the imposed gradient, the greater is the degree of hardening. However, the large strains achieved, the polycrystalline character of the material, the texture evolution and varying grain size of the samples may have complicated the interpretation.

In the present Chapter, the creep response of ice single crystals in torsion, a much simpler material and experimental configuration, is described with focus on the effects of the sample dimensions on this response. As an hcp material with a strong anisotropy of plasticity, ice is a choice material in this respect.

It deforms plastically by the activity of basal slip systems almost exclusively [36] and it is characterized by a low Peierls stress [37]. In [19], the torsion creep tests were carried out on cylindrical samples ma- chined from laboratory grown single crystals. In each sample, height and diameter are equal, in order to avoid any bias due to end effects.

Courtesy of Juliette Chevy [38]. A forward and reverse creep curve, with the torque sign changed at reversal, is also shown below in Fig. Conversely, distinct curves in this plot are evidence for an effect of size on the plastic response. Dispersion in the curves is existing, but limited. It results mainly from uncontrollable fluc- tuations in the initial dislocation microstructure, which lead to uneven initial creep strain rates.

Hard X-ray diffraction analyzes performed on slices extracted from the strained samples show that plasticity is almost exclusively due to polar dislo- cations of screw character gliding in basal planes, with very few mobile statis- tical dislocations [39]. Dislocation mediated continuum plasticity 13 In addition, the analyses reveal a scale invariant arrangement of polar dislo- cations along the torsion axis suggesting propagation of slip in this direction. The latter can be explained by the occurrence of double cross-slip of screw dislocations through prismatic planes [40].

Interpretation in terms of dislocation dynamics of these observations is now provided on the basis of the model described in Section 2. We begin with a simplified 1-D model designed for twofold purpose: to illustrate the critical aspects of the theory; to allow for effective parametric study of size ef- fects. In this idealization for deformation under a gradient of simple shear, we consider screw dislocation density of infinite extent in the x1 , x3 tangential and axial directions, line and Burgers vector along the tangential direction x1 and transport in the radial direction x2.

It represents the transport of screw dislocations along the radius with a source term due to gradients in statistical dislocation mobility. Account of the physics of dislocation velocity and of the history of straining is now made through phenomenological statements. They are identified from the experimental data [36, 37]. Relation 35 is similar to the Armstrong- Frederick law for kinematic hardening [41], but here the back-stress builds up from polar dislocation mobility only.

It increases due to dislocation sources associated with edge jogs in prismatic planes [42]. Its nucleation rate is supposed to be proportional to the shear strain rate, with coefficient C1. Saturation of mobile dislocations results from their mutual annihilation, with coefficient C2. Their presence in the phenomenology of the 1-D formulation through Eqs.

The figure highlights the development of stress due to the multipli- cation of polar dislocations, from the effectively elastic solution with low mobile density. Plot A also shows end effects in the distribution of stress. Model parameters and initial conditions are given in Table I. The blue dashed line shows the forward creep curve for a sample with halved radius and height.

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It is seen that the acceleration of creep increases when the sample size is reduced. The green continuous line, obtained for conventional elasto-viscoplastic EVP treatment, shows that the latter is unable to retrieve the acceleration of creep. Note that the 3-D stress distribution supports the assumption made in the 1-D idealization. As seen in Fig. The reverse torsion behavior is also shown in the figure. At torque reversal, an increase in the creep rate absolute value is observed in the experiments. It is fully retrieved by both the 1-D and 3-D models. This asymmetry of slopes at the reversal point can be attributed to the positive screw dislocation pile-ups built in forward torsion.

Screw dislocations of neg- ative sign are nucleated in reverse loading, which progressively annihilate with the positive screw pile-ups created in forward loading. Hence, the total polar screw density decreases, with the consequence that the positive screw pile-ups are dismantled and that the creep rate absolute value decreases. The minimum creep rate value is reached at the inflexion of the creep curve. At this point, creep is mostly accommodated by statistical dislocations. In the rest of the reverse creep curve, creep keeps accelerating while negative screw pile-ups are created, in a fashion similar to forward loading, though obviously with reversed sign.

Hence, the anisotropy of creep behavior derives from the nucleation, transport and annihilation of inhomogeneous polar screw density distributions. Sample size effects on the creep response are obtained. By reducing the sample diameter, screw nucleation is promoted and acceleration of the creep rate increases, in close agreement with experimental data see Figs.

Thus, the greater is the imposed gradient, the greater is the degree of soft- ening, not hardening as would have been expected if polar screw dislocations had been contributing to isotropic statistical hardening in a way similar to statistical dislocations. Rather, the polar screw dislocations induce softening due to larger rates of plastic distortion.

Several other effects of sample size on mechanical response were predicted by the model and observed in the exper- iments. We report here on two such effects, observed in reverse torsion and shown in Fig. Firstly, the larger is the imposed gradient the smaller is the sample diameter , the larger is the increase in the creep rate absolute value at torque reversal.

Secondly, the larger is the imposed gradient, the larger is the decrease in the creep rate, from torque reversal to inflexion of the creep curve. Note that the latter is a hardening effect on the creep response. As mentioned above, the asymmetry of the creep curve at torsion reversal re- flects screw dislocation pile-ups and internal stresses built up during forward torsion. Hence, the larger is the imposed gradient, the larger are the internal stress level and creep rate acceleration at reversal.

The interpretation sug- gested by the model for the second effect is as follows. Since the inflexion point corresponds to the instant when the polar screw density is the closer to zero, a larger deceleration in the creep rate is representative of a larger nucle- ation of polar screw density, which in turn is due to a larger imposed gradient.

In pure Cu, experimental evidence of sample size effects on mechanical response is still controversial. The work by Fleck et al. However, data on Cu single crystals loaded in tension rather indicate strong decrease of hardening in the easy glide region when crystal radius is decreased [43]. In this reference, the reduction in the density of polar screw dislocations in the center region of the sample is seen as the origin of hardening. As its plastic distortion is reduced near the axis, the metal behaves more like an elastic solid and, as such, it becomes harder.

The difference in behavior with ice single crystals might then be attributed to the difference in the elastic constants values the elastic shear modulus in ice is of the order of 3GP a, much smaller than the 40GP a Cu value. On the basis of the above simulations, dislocation transport and long-range internal stress build-up appear as the controlling mechanisms for the rarefaction of polar screw dislocations in the center of the sample, through polar screw pile-up formation. For example, it serves as a cornerstone in the theory of fluid dynam- ics.

When envisioned on length scales over which areal densities of dislocations may be envisaged, dislocation motion is amenable to the transport of these densities. The fundamental equation for dislocation transport 7 has been known for half a century [12, 13], mostly as a curiosity, and only recently has it been effectively used for dislocation dynamics predictions [9]. Similarly, the relevant length scale for the observation in dislocation dynamics of the propagative features associated with transport has remained elusive, and only recently has experimental evidence been provided [26], although observation of strain waves [45] could perhaps have given a clue earlier.

An inherent connection between dislocation transport and the intermittency of plastic activity is revealed in [26] by applying high resolution extensometry to Cu single crystals in tension. When oriented for multislip, Cu single crys- tals represent the truly emblematic situation where material instability can be ruled out and homogeneous straining in a traditional mechanical sense expected at small strains see loading curve in Fig. Yet, the inhomo- geneous dislocation microstructure and the intermittent dislocation activity at a microscopic scale may well induce intermittency and inhomogeneity in dislocation transport at some intermediate scale.

Note that the maximum size of the fluctuations 2. The extensometry method is based on a digital image correlation technique in one-dimensional setting. The sample surface is painted with al- ternated black and white strips, which perfectly reflect the material displace- ment underneath. A high resolution CCD camera with recording frequency Hz and pixel size 1. Despite smoothness of the loading curve in Fig. The figure suggests that the intermittency of dislocation motion at the microscopic scale shows up at a somewhat larger scale.

The prob- ability density for the size of jerks in Fig. This exponent is consistent with the Fig. Such scaling is evidence for self-organization of the observed fluctuations, which is also demonstrated by Fig. The figure features a space-time diagram for local fluctuations about the driving strain-rate during the elasto-plastic transition in the test shown in Fig. It displays spots of intense activity dotted along straight lines, suggesting wave propagation at constant average speed. It is of the order of the average velocity of dislocation ensembles, which suggests that the observed waves reflect the underlying collective motion of dislocations.

In this interpretation, the dotted pattern of spots along the characteristic lines is manifestation of the intermittency of collective dislocation motion. It is also proof to the wavy structure of plastic activity, when envisioned at ap- propriate length scale. At larger strains, this wavy pattern is seen on shorter time and length scales, because the dislocation mean free path decreases in relation with the multiplication of forest obstacles. A 3-D generic simulation of the above experiments using field dislocation dynamics is now briefly outlined.

The reader is referred to [26] and [46] for further details. A flat Cu whisker is clamped to the left end, while the right end has constant velocity. Dotted characteristic lines run from the left and right of the gauge length, reflecting intermittency and transport. Fluctuations can be as high as 2. C0 , C1 and C2 are material parameters ac- counting for the interaction between polar and forest dislocations, the mobile dislocation generation and loss, respectively.

The material parameters are listed in Table 2.

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  • Note that there is no inhomogeneity introduced in either the material parameters or the initial conditions. Table 2 Material parameters used in the 3D Cu whisker simulation. Thus, inhomogeneity of plastic straining clearly stems from the inhomogeneity of the boundary conditions. This prediction of a yield drop is in full agreement with experimental data on Cu whiskers [48, 49, 50].

    Propagation does not occur in this flat sample if transport is turned off in the equations. For further details on the propagation of slip along the plateau, the reader is referred to [46]. Here, we focus on the intermittency of plastic activity during the eventual linear hardening period. Bursts in stress - Fig. The highlighted portion corresponds to the strain rate plots shown in Fig. One particular sequence, highlighted in Fig. In this figure, intermittent events and transport are clearly seen, with a general progression of plastic activity from left to right of the sample.

    In view of these results, 2D simulations of course more tractable than 3D were carried out in order to check for scaling behavior at a smaller scale and for possible variation of the scaling exponent under diverse material and ex- perimental conditions. In these simulations, a rectangle subjected to constant shear rate v1,3 at boundaries is considered in a glide plane of a Cu single crys- tal. Only dislocations pertaining to, and gliding in this plane are considered. Out of plane motion by cross-slip and climb is not considered, and single slip activity is assumed.

    As all gradients normal to the slip plane are ignored, out-of-plane features of lattice incompatibility and internal stresses are lost in this simplified description. The sequence highlights a single plastic burst shown in Fig In the initial configuration, polar dislocations are absent and the statistical mobile density is chosen at random about an average value. Since the boundary conditions are homogeneous in contrast to the above 3D simulation , the incompatibility arising from the distribution of statistical dislocations is initially the only source for polar dislocations. The information on material parameters, initial and boundary conditions is summed up in Table 3.

    In qual- Fig. The figure shows the evolution in time of the strain rate profile seen along the x2 direction. This pattern follows naturally from the development of polar dislocation density, by virtue of dislocation transport and internal stress. The velocity obtained from the slopes in Fig. This result confirms that the fluc- tuations in Fig. With unchanged geometry and loading conditions, possible influence of material behavior was also investigated by switching from a thermally activated law in Cu Eq.

    Thus, the scaling behavior of intermittency obtained from the model features a rather universal scaling exponent. The event size is defined as the maximum strain rate value during the event. The model implies that both dislocation transport and long-range interactions play a role in the emergence of the scale-invariant be- havior of intermittency. As dislocation transport involves such mechanisms as double cross-slip of screw dislocations by-passing short-range obstacles, it follows from this remark that short-range interactions play a significant role in the intermittency of plastic activity.

    Such a conclusion is fully consistent with the observations of dislocation avalanches arrested on obstacles made in dislocation dynamics simulations [30]. The plastically strained area then spreads along the sample. A clear cut front separates this area from the unstrained one, into which it propagates, until the sample is uniformly stretched.