On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion

Standard measures of local deformation such as deformation gradient, strain, elastic deformation, and plastic deformation are dimensionless. However, many macroscopic and submacroscopic geometrical changes observed in continuous bodies result in the formation of zones across whose boundaries significant changes in geometry can occur. In order to predict the sizes of such zones and their influence on material response, theories of elasticity and plasticity have been employed in which second gradients of deformation, gradients of strain, as well as gradients of elastic or of plastic deformation are taken into account. The theory of structured deformations provides additive decompositions of first deformation gradient and of second deformation gradient, valid for large deformations of any material, in which each term has a multiscale geometrical interpretation corresponding to the presence or absence of submacroscopic disarrangements (non-smooth geometrical changes such as slips and void formation). This article provides a field theory that broadens the earlier field theory, elasticity with disarrangements, by including energetic contributions from submacroscopic “gradient-disarrangements” (limits of averages of jumps in gradients of approximating deformations) and by treating particular kinematical conditions as internal constraints. An explicit formula is obtained showing the manner in which submacroscopic gradient-disarrangements determine a defectiveness density analogous to the dislocation density in theories of plasticity. A version of the new field theory incorporates this defectiveness density to obtain a counterpart of strain-gradient plasticity, while another instance of elasticity with gradient-disarrangements recovers an instance of strain-gradient elasticity with symmetric Cauchy stress. All versions of the new theory included here are compatible with the Second Law of Thermodynamics.     

Invertible structured deformations and the geometry of multiple slip in single crystals

Invertible structured deformations are employed to derive the basic kinematical relations of crystalline plasticity without the use of an intermediate configuration. All of the quantities appearing in these relations have definite geometrical interpretations in terms of either smooth or non-smooth geometrical changes (disarrangements) occuring at macroscopic or sub-macroscopic length scales. For f.c.c. crystals, the kinematical relations are valid for each family of invertible structured deformations. For other single crystals, an appropriate collection of invertible structured deformations is identified and the validity of the kinematical relations is established within this (possibly smaller) collection.     

Moving interfaces that separate loose and compact phases of elastic aggregates: a mechanism for drastic reduction or increase in macroscopic deformation

Granular materials such as sand may be viewed as continuous bodies composed of much smaller elastic bodies. The multiscale geometry of structured deformations captures the contribution at the macrolevel of the smooth deformation of each small body in the aggregate (deformation without disarrangements) as well as the contribution at the macrolevel of the non-smooth deformations such as slips and separations between the small bodies in the aggregate (deformation due to disarrangements). When the free energy response of the aggregate depends only upon the deformation without disarrangements, is isotropic, and possesses standard growth and semi-convexity properties, we establish (i) the existence of a compact phase in which every small elastic body deforms in the same way as the aggregate and, when the volume change of macroscopic deformation is sufficiently large, (ii) the existence of a loose phase in which every small elastic body expands and rotates to achieve a stress-free state with accompanying disarrangements in the aggregate. We show that a broad class of elastic aggregates can admit moving surfaces that transform material in the compact phase into the loose phase and vice versa and that such transformations entail drastic changes in the level of deformation of transforming material points.     

Digital Image Correlation under Scanning Electron Microscopy: Methodology and Validation

The recent combination of scanning electron microscopy and digital image correlation (SEM-DIC) enables the experimental investigation of full-field deformations at much smaller length scales than is possible using optical digital image correlation methods. However, the high spatial resolution of SEM-DIC comes at the cost of complex image distortions, long image scan times that can capture gradients from stress relaxation, and a high noise sensitivity to SEM parameters. In this paper, it is shown that these sources of error can significantly impact the quality of the results and must be accounted for in order to perform accurate SEM-DIC experiments. An existing framework for distortion corrections is adapted to improve accuracy and the procedures are described in detail. As the results demonstrate, time varying drift distortion is a larger problem at high magnification while spatial distortion is more problematic at low magnification. Additionally, the new use of sample-independent calibration and a method to eliminate the detrimental effects of stress relaxation in the displacement fields prior to distortion correction are introduced. The impact of SEM settings on image noise is quantified and noise minimization schemes are examined. Finally, a uniaxial tension test on coarse-grained 1100-O aluminum is used to demonstrate these techniques, where active slip planes are identified and strain localization is examined in relation to the underlying microstructure.     

Mode I and mixed mode crack-tip fields in strain gradient plasticity

Strain gradients develop near the crack-tip of Mode I or mixed mode cracks. A finite strain version of the phenomenological strain gradient plasticity theory of Fleck-Hutchinson (2001) is used here to quantify the effect of the material length scales on the crack-tip stress field for a sharp stationary crack under Mode I and mixed mode loading. It is found that for material length scales much smaller than the scale of the deformation gradients, the predictions converge to conventional elastic-plastic solutions. For length scales sufficiently large, the predictions converge to elastic solutions. Thus, the range of length scales over which a strain gradient plasticity model is necessary is identified. The role of each of the three material length scales, incorporated in the multiple length scale theory, in altering the near-tip stress field is systematically studied in order to quantify their effect.     

Regularized variational theories of fracture: A unified approach

The fracture pattern in stressed bodies is defined through the minimization of a two-field pseudo-spatial-dependent functional, with a structure similar to that proposed by Bourdin-Francfort-Marigo (2000) as a regularized approximation of a parent free-discontinuity problem, but now considered as an autonomous model per se. Here, this formulation is altered by combining it with structured deformation theory, to model that when the material microstructure is loosened and damaged, peculiar inelastic (structured) deformations may occur in the representative volume element at the price of surface energy consumption. This approach unifies various theories of failure because, by simply varying the form of the class for admissible structured deformations, different-in-type responses can be captured, incorporating the idea of cleavage, deviatoric, combined cleavage-deviatoric and masonry-like fractures. Remarkably, this latter formulation rigorously avoid material overlapping in the cracked zones. The model is numerically implemented using a standard finite-element discretization and adopts an alternate minimization algorithm, adding an inequality constraint to impose crack irreversibility (fixed crack model). Numerical experiments for some paradigmatic examples are presented and compared for various possible versions of the model.     

Suppressed plastic deformation at blunt crack-tips due to strain gradient effects

Large deformation gradients occur near a crack-tip and strain gradient dependent crack-tip deformation and stress fields are expected. Nevertheless, for material length scales much smaller than the scale of the deformation gradients, a conventional elastic–plastic solution is obtained. On the other hand, for significant large material length scales, a conventional elastic solution is obtained. This transition in behaviour is investigated based on a finite strain version of the Fleck–Hutchinson strain gradient plasticity model from 2001. The predictions show that for a wide range of material parameters, the transition from the conventional elastic–plastic to the elastic solution occurs for length scales ranging from 0.001 times the size of the plastic zone to a length scale of the same order of magnitude as the plastic zone.     

Toward a Field Theory for Elastic Bodies Undergoing Disarrangements

Structured deformations are used to refine the basic ingredients of continuum field theories and to derive a system of field equations for elastic bodies undergoing submacroscopically smooth geometrical changes as well as submacroscopically non-smooth geometrical changes (disarrangements). The constitutive assumptions employed in this derivation permit the body to store energy as well as to dissipate energy in smooth dynamical processes. Only one non-classical field G, the deformation without disarrangements, appears in the field equations, and a consistency relation based on a decomposition of the Piola-Kirchhoff stress circumvents the use of additional balance laws or phenomenological evolution laws to restrict G. The field equations are applied to an elastic body whose free energy depends only upon the volume fraction for the structured deformation. Existence is established of two universal phases, a spherical phase and an elongated phase, whose volume fractions are (1?γ₀)³ and (1?γ₀) respectively, with γ₀:=( $ \sqrt 5 $ ?1)/2 the “golden mean”.     

On the formulations of higher-order strain gradient crystal plasticity models

Recently, several higher-order extensions to the crystal plasticity theory have been proposed to incorporate effects of material length scales that were missing links in the conventional continuum mechanics. The extended theories are classified into work-conjugate and non-work-conjugate types. A common feature of the former is that existence of higher-order stresses work-conjugate to gradients of plastic strain is presumed and an extended principle of virtual work involving such an additional virtual work contribution is formulated. Meanwhile, in the latter type, the higher-order stress quantities do not appear explicitly. Instead, rates of crystallographic slip are influenced by back stresses that arise in response to spatial gradients of the geometrically necessary dislocation densities. The work-conjugate type and the non-work-conjugate type of theories have different theoretical backgrounds and very unlike mathematical representations. Nevertheless, both types of theories predict the same kind of material length scale effects. We have recently shown that there exists some equivalency between the two approaches in the special situation of two-dimensional single slip under small deformation. In this paper, the discussion is extended to a more general situation, i.e. the context of multiple and three-dimensional slip deformations.     

Phenomenological modeling of viscous electrostrictive polymers

A common usage for electroactive polymers (EAPs) is in different types of actuators, where advantage is taken of the deformation of the polymer due to an electric field. It turns out that time-dependent effects are present in these applications. One of these effects is the viscoelastic behavior of the polymer material. In view of the modeling and simulation of applications for EAP within a continuum mechanics setting, a phenomenological framework for an electro-viscoelastic material model is elaborated in this work. The different specific models are fitted to experimental data available in the literature. While the experimental data used for inherent electrostriction is restricted to small strains, a large strain setting is used for the model in order to account for possible applications where the polymers undergo large deformations, such as in pre-strained actuators.     

Hydraulic jump structures in the presence of an ice sheet

Jumps of the bore type arising in a fluid layer with an ice sheet are investigated. These jump structures are considered for a determining mechanism in the form of dispersion due to the presence of an ice sheet. For this purpose a generalized Korteweg-de Vires equation [1] is used. The structure of these jumps consists of a wave zone that expand with time. On the boundary of the wave zone there are transitions between uniform and periodic states which can be locally considered as jumps. Among them are jumps which can be regarded as steady in the coordinate system moving with the boundary of the wave zone. These are jumps between a sequence of solitons and a uniform state (jumps of soliton type) on the boundary of the wave zone and jumps between periodic and uniform states (jumps with radiation). In addition, there are jumps which are unsteady even from the standpoint of a local analysis. In order to investigate the effect of dissipation processes on the jumps considered a system of generalized Boussinesq equations is derived with allowance for bottom slope and bottom and ice friction. The jump damping process is investigated numerically. This system of equations also makes it possible to investigate undamped jumps of the floodwater wave type. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 139–146. July–August, 2000.     

Nonhomogeneous Initial-Boundary Value Problems for Coercive and Self-Controlling Models of Monotone Type

Based on the idea of partial Yosida approximation we prove global in time existence of nonhomogeneous initial-boundary value problems for the class of coercive models of monotone type from the theory of inelastic deformations of metals. Then using this result and the coercive limits idea introduced in [8], we approximate self-controlling problems with nonhomogeneous boundary data by a sequence of coercive models and prove a convergence result. Received March 30, 2000     

数字图像相关法测量局域变形场中形函数和模板尺寸的影响

数字图像相关法测量均匀变形场已被普遍接受, 其测量结果可与应变片测量结果比较. 然而, 在工程测量中, 针对局域变形场(应变高度集中, 如波特文-勒夏特利埃带、试件缺口附近和裂纹尖端等), 应变片受限于其尺寸, 其测量结果是接触面内的平均应变值. 此时, 采用数字图像相关法能够测量这些局域变形场. 但形函数和模板尺寸等计算参数对计算结果影响很大, 这也导致使用者很难判断计算结果的可靠性. 论文通过对合金拉伸实验获得的不同应变梯度的波特文-勒夏特利埃带和模拟生成的带的计算分析, 发掘了形函数和模板尺寸作用于计算结果的深层机制, 证明了二阶形函数比一阶形函数更适用于高度非均匀的局域变形场. 提出了在局域应变场测量中, 当一阶和二阶形函数计算结果的相对误差小于10% 时, 二阶形函数的结果是可靠的判据.      

Deformation measurements by digital image correlation: Implementation of a second-order displacement gradient

This paper outlines the procedure for refining the digital image correlation (DIC) method by implementing a second-order approximation of the displacement gradients. The second-order approximation allows the DIC method to directly measure both the first- and second-order displacement gradients resulting from nonlinear deformation. Thirteen unknown parameters, consisting of the components of displacement, the first- and second-order displacement gradients and the gray-scale value offset, are determined through optimization of a correlation coefficient. The previous DIC method assumes that the local deformation in a subset of pixels is represented by a first-order Taylor series approximation for the displacement gradient terms, so actual deformations consisting of higher order displacement gradients tend to distort the infinitesimal strain measurements. By refining the method to measure both the first- and second-order displacement gradients, more accurate strain measurements can be achieved in large-deformation situations where second-order deformations are also present. In most cases, the new refinements allow the DIC method to maintain an accuracy of ±0.0002 for the first-order displacement gradients and to reach ±0.0002 per pixel for the second-order displacement gradients.     

A computational homogenization approach for the yield design of periodic thin plates. Part II : Upper bound yield design calculation of the homogenized structure

In the first part of this work (Bleyer and de Buhan, 2014), the determination of the macroscopic strength criterion of periodic thin plates has been addressed by means of the yield design homogenization theory and its associated numerical procedures. The present paper aims at using such numerically computed homogenized strength criteria in order to evaluate limit load estimates of global plate structures. The yield line method being a common kinematic approach for the yield design of plates, which enables to obtain upper bound estimates quite efficiently, it is first shown that its extension to the case of complex strength criteria as those calculated from the homogenization method, necessitates the computation of a function depending on one single parameter. A simple analytical example on a reinforced rectangular plate illustrates the simplicity of the method. The case of numerical yield line method being also rapidly mentioned, a more refined finite element-based upper bound approach is also proposed, taking dissipation through curvature as well as angular jumps into account. In this case, an approximation procedure is proposed to treat the curvature term, based upon an algorithm approximating the original macroscopic strength criterion by a convex hull of ellipsoids. Numerical examples are presented to assess the efficiency of the different methods.     

Numerical analysis of the stress concentration near holes originating in previously loaded viscoelastic bodies at finite strains

The state of stress and strain of previously loaded viscoelastic bodies with holes originating in them, successively or simultaneously, is analyzed under finite plane deformations. The problem statement and solution are based on the theory of repeatedly superimposed large deformations. The material mechanical properties are described using integral relations of the convolution type over time with a weakly singular kernel. The problem solving is based on the finite-element method. To calculate the integral of the convolution type, a recurrence formula is used that can be obtained by approximating the initial kernel with a linear combination of exponential functions (the truncated Prony’s series). The nonlinear effects and the effect of the interaction between holes on the stress concentration are analyzed. For the dynamic problems, the results for incompressible and weakly compressible materials are compared.     

An approach for choosing discretization schemes and grid size based on the convection-diffusion equation

A new approach for selecting proper discretization schemes and grid size is presented. This method is based on the convection-diffusion equation and can provide insight for the Navier-Stokes equation. The approach mainly addresses two aspects, i.e., the practical accuracy of diffusion term discretization and the behavior of high wavenumber disturbances. Two criteria are included in this approach. First, numerical diffusion should not affect the theoretical diffusion accuracy near the length scales of interest. This is achieved by requiring numerical diffusion to be smaller than the diffusion discretization error. Second, high wavenumber modes that are much smaller than the length scales of interest should not be amplified. These two criteria provide a range of suitable scheme combinations for convective flux and diffusive flux and an ideal interval for grid spacing. The effects of time discretization on these criteria are briefly discussed.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations

In three recent papers [6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of a second-order linear differential operator D², which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude, the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered in Part II [16]. This revised version was published online in August 2006 with corrections to the Cover Date.     

Active slip-band separation and the energetics of slip in single crystals

This research supports recent efforts to provide an energetic approach to the prediction of stress–strain relations for single crystals undergoing single slip and to give precise formulations of experimentally observed connections between hardening of single crystals and separation of active slip-bands. Non–classical, structured deformations in the form of two-level shears permit the formulation of new measures of the active slip-band separation and of the number of lattice cells traversed during slip. A formula is obtained for the Helmholtz free energy per unit volume as a function of the shear without slip, the shear due to slip, and the relative separation of active slip-bands in a single crystal. This formula is the basis for a model, under preparation by the authors, of hardening of single crystals in single slip that is consistent with the Portevin-Le Chatelier effect and the existence of a critical resolved shear stress.