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 Generalized Formulation of Elastodynamics: Small on Rigid

The conventional theory of linearized elastodynamics addresses the case of motions that have small displacement gradients with respect to a reference configuration of the elastic body that is unstressed and at rest. Here, we develop a theory of much wider applicability in which the linearization is with respect to a reference configuration that is in rigid motion. More specifically, with an eye toward application of the theory to analysis of the motions of relatively inflexible machine parts, we view the motion as being composed of a rigid motion, which corresponds to the applied loads and initial conditions for the body under consideration, and an infinitesimal motion, in which the displacement from the rigid motion has a small gradient.     

水荷载引起混凝土重力坝位移的理论研究

基于多孔连续介质模型,从理论上探讨了作用在地基上的水荷载作为渗流体荷载时引起混凝土重力坝的位移,导出了均质各向同性地基在渗流体荷载作用下的应力解答和位移解答.通过理论分析得到:(1)由于渗流体荷载引起上游地基下沉,下游地基上抬,从而使地基转动,导致坝体向上游位移.(2)作用在地基上的水荷载按面荷载分析的位移大于按渗流体荷载分析的位移,但它们都引起坝体向上游位移.     

One-dimensional problems in the stability of thin shells

Summary Algebraic conditions, sufficient for the infinitestimal stability of elastic shells in the class of one-dimensional perturbations from homogeneous ground states, are considered. A necessary and sufficient condition for superstability is also deduced.The shells are modelled according to the director theory. Unlike three-dimensional elastic continua, the strain energy depends on the displacement components and not only on their gradients. This plays a central role in the analysis of these problems.     

On the velocity of ghost particles and the bias errors in Tomographic-PIV

The paper discusses bias errors introduced in Tomographic-PIV velocity measurements by the coherent motion of ghost particles under some circumstances. It occurs when a ghost particle is formed from the same set of actual particles in both reconstructed volumes used in the cross-correlation analysis. The displacement of the resulting ghost particle pair is approximately the average displacement of the set of associated actual particles. The effect is further quantified in a theoretical analysis and in numerical simulations and illustrated in an actual experiment. It is shown that the bias error does not significantly affect the measured flow topology as deduced in an evaluation of the local velocity gradients. Instead, it leads to a systematic underestimation of the measured particle displacement gradient magnitude. This phenomenon is alleviated when the difference between particles displacement along the volume depth is increased beyond a particle image diameter, or when the reconstruction quality is increased or when the accuracy of the tomographic reconstruction is improved. Furthermore, guidelines to detect and avoid such bias errors are proposed.     

Full-field representation of discretely sampled surface deformation for displacement and strain analysis

A detailed evaluation of the feasibility of determining displacements and displacement gradients from measured surface displacement fields is presented. An improved methodology for both the estimation and elimination of noise is proposed. The methodology is used to analyze the gradients for three tests: (1) uniform rotation, (2) uniform strain, and (3) crack-tip displacement fields. Results of the study indicate that the proposed methodology can be used to extract the underlying two-dimensional displacements and their corresponding gradients from the noisy data with reasonable accuracy. Specifically, it is shown that (a) the digital correlation method for acquiring displacement fields has an error in strain of approximately 150 strain at each point, (b) the average strain in a region of uniform strain has much less error, typically on the order of 20 strain, (c) the displacement nolse present in digital correlation is very small, approximately 0.01 pixels, (d) the proposed methodology for reducing noise in the data is essential to the accurate evaluation of displacement gradients and (e) the successful evaluation of displacement and displacement gradients for all three cases indicates that the proposed methodology can be used both to quantify the displacement fields and to reasonably estimate the overall gradient trends.     

A nonlinear theory for doubly curved anisotropic sandwich shells with transversely compressible core

In the present paper, an advanced geometrically nonlinear shell theory of doubly curved structural sandwich panels with transversely compressible core is presented. The model is based on the adoption of the Kirchhoff theory for the face sheets and a second/third order power series expansion for the core displacements. The theory accounts for dynamic effects as well as for initial geometric imperfections. In the v. Kármán sense, large displacement theory is employed with respect to the transverse direction while the displacement gradients with respect to the tangential directions are assumed to be small. The equations of motion are derived by means of Hamilton’s principle and hold valid for all types of elastic and elastic–plastic material models. The theory is illustrated by an analysis of the elastic buckling and postbuckling behavior of flat and curved sandwich panels using an extended Galerkin scheme. Owing to the assumed transverse flexibility of the core, both the global and the local (face wrinkling) instability modes can be addressed.     

Interlaminar hygrothermal stresses in laminated plates

Within the elasticity formulation the most general displacement field for hygrothermal problems of long laminated composite plates is presented. The equivalent single-layer theories are then employed to determine the global deformation parameters appearing in the displacement fields of general cross-ply, symmetric, and antisymmetric angle-ply laminates under thermal and hygroscopic loadings. Reddy’s layerwise theory is subsequently used to determine the local deformation parameters of various displacement fields. An elasticity solution is also developed in order to validate the efficiency and accuracy of the layerwise theory in predicting the interlaminar normal and shear stress distributions. Finally, various numerical results are presented for edge-effect problems of several cross-ply, symmetric, and antisymmetric angle-ply laminates subjected to uniform hygrothermal loads. All results indicate high stress gradients of interlaminar normal and shear stresses near the edges of laminates.     

A Robust Finite Element-based Filter for Digital Image and Volume Correlation Displacement Data

Background

Digital Image and Volume Correlation (DIC and DVC) are non-contact measurement techniques that are used during mechanical testing for quantitative mapping of full-field displacements. The relatively high noise floor of DIC and DVC, which is exasperated when differentiated to obtain strain fields, often requires some form of filtering. Techniques such as median filters or least-squares fitting perform poorly over high displacement gradients, such as the strain localisation near a crack tip, discontinuities across crack flanks or large pores. As such, filtering does not always effectively remove outliers in the displacement field.

Objective

This work proposes a robust finite element-based filter that detects and replaces outliers in the displacement data using a finite element method-based approximation.

Methods

A method is formulated for surface (2D and Stereo DIC) and volumetric (DVC) measurements. Its validity is demonstrated using analytical and experimental displacement data around cracks, obtained from surface and full volume measurements.

Results

It is shown that the displacement data can be filtered in such a way that outliers are identified and replaced. Moreover, data can be smoothed whilst maintaining the nature of the underlying displacement field such as steep displacement gradients or discontinuities.

Conclusions

The method can be used as a post-processing tool for DIC and DVC data and will support the use of the finite element method as an experimental–numerical technique.

     

Constitutive equations for multiphase mixtures of fluids

Constitutive equations for a multiphase mixture of fluids are presented. The mixture is assumed to consist of a single non-uniform temperature and no change is allowed. The theory is based on the conservation and balance equations of multiphase mixtures proposed by Dobran, and the constitutive assumption allows for the effects of temperature gradient, density gradients, velocity gradients, velocities and accelerations. A linearized form of the constitutive equations is presented for an arbitrary number of phases, and restrictions on the constitutive assumption are investigated by the second law of thermodynamics. The theory yielded a significant number of results and they are compared with previous investigations.     

正交各向异性弹性问题的规则化边界元法

本文致力于平面正交各向异性弹性问题的规则化边界元法研究,提出了新的规则化边界元法的理论和方法。对问题的基本解的特性进行了研究,确立基本解的积分恒等式,提出一种基本解的分解技术,在此基础上,结合转化域积分方程为边界积分方程的极限定理,建立了新颖的规则化边界积分方程。和现有方法比,本文不必将问题变换为各向同性的去处理,从而不含反演运算,也有别于Galerkin方法,无需计算重积分,因此所提方法不仅效率高,而且程序设计简单。特别是,所建方程可计算任何边界位移梯度,进而可计算任意边界应力,而不仅限于面力。数值实施时,采用二次单元和椭圆弧精确单元来描述边界几何,使用不连续插值逼近边界函数。数值算例表明,本文算法稳定、效率高,所取得的边界量数值结果与精确解相当接近。     

On the Use of Implicit Integration Methods and the Absolute Nodal Coordinate Formulation in the Analysis of Elasto-Plastic Deformation Problems

In the general theory of continuum mechanics, the state of rotation and deformation of material points can be uniquely defined from the displacement field by using the nine independent components of the displacement gradients. For this reason, the use of the absolute rotation parameters as nodal coordinates, without relating them to the displacement gradients, leads to coordinate redundancy that leads to numerical and fundamental problems in many existing large rotation finite element formulations. Because of this fundamental problem, special measures that require modifications of the numerical integration methods were proposed in the literature in order to satisfy the principle of work and energy. As demonstrated in this paper, no such measures need to be taken when the finite element absolute nodal coordinate formulation is used since the principle of work and energy are automatically satisfied. This formulation does not suffer from the problem of coordinate redundancy and ensures the continuity of stresses and strains at the nodal points. In this study, the use of the implicit integration methods with the consistent Lagrangian elasto-plastic tangent moduli is examined when the absolute nodal coordinate formulation is used. The performance of different numerical integration methods in the dynamic analysis of large elasto-plastic deformation problems is investigated. It is shown that all these methods, in the case of convergence, yield a solution that satisfies the principle of work and energy without the need of taking any special measures. Semi-implicit integration methods, however, can lead to numerical difficulties in the case of very stiff problems due to the linearization made in these methods in order to avoid the iterative Newton--Raphson procedure. It is also demonstrated that the use of the consistent Lagrangian-plastic tangent moduli derived in this investigation using the absolute nodal coordinate formulation leads to better convergence of the iterative Newton--Raphson procedure used in the implicit integration methods.     

Absolute Nodal Coordinates in Digital Image Correlation

A great deal of progress has been made in recent years in the field of global digital image correlation (DIC), where higher-order, element-based approaches were proposed to improve the interpolation performance and to better capture the displacement fields. In this research, another higher-order, element-based DIC procedure is introduced. Instead of the displacements, the elements’ global nodal positions and nodal position-vector gradients, defined according to the absolute nodal coordinate formulation, are used as the searched parameters of the Newton–Raphson iterative procedure. For the finite elements, the planar isoparametric plates with 24 nodal degrees of freedom are employed to ensure the gradients’ continuity among the elements. As such, the presented procedure imposes no linearization on the strain measure, and therefore indicates a natural consistency with the nonlinear continuum theory. To verify the new procedure and to show its advantages, a real large deformation experiment and several numerical tests on the computer-generated images are studied for the standard, low-order, element-based digital image correlation and the presented procedure. The results show that the proposed procedure proves to be accurate and reliable for describing the rigid-body movement and simple deformations, as well as for determining the continuous finite strain field of a real specimen.     

A consideration of curved interfaces in stress-assisted martensite formation

A purely mechanical, sharp interface model is developed to consider curved interfaces that have been observed between martensite phase variants. The approach is based on a theory of small strains as distinct from small displacement gradients. It admits a realistic characterization of each phase with standard elasticity tensors and allows for inhomogeneous states of strain within each phase including inhomogeneous, finite rotations. The model indicates that any signficant interface curvature must be due to material rotation because interfaces cannot be finitely curved with respect to the material lattice. It is also found that the interface driving traction is not influenced by local lattice rotations unless inertia affects the reaction.     

Turbulent diffusion in a medium subjected to uniform deformation

Turbulent diffusion in media subjected to uniform deformation caused by the presence of average-velocity gradients which are constant throughout the space is an idealization of real processes, in particular, of processes such as diffusion in channels of variable cross-sectional area [1], in the lowest layers of the atmosphere [2], etc. In this article we formulate the problem of the connection between the statistical characteristics of the transfer of a passive substance in turbulent diffusion in deformed media with the statistical characteristics of the turbulence. The statistical transfer characteristics generally used are the first two moments of the vector of random displacement of a liquid particle under the action of turbulent pulsations in velocity: the average displacement and the components of the dispersion tensor of the displacement of a liquid particle. We obtain connecting relations for the dispersion tensor of a liquid particle in turbulent diffusion of a passive substance in a uniform turbulent medium subjected to uniform deformation caused by average-velocity gradients which are constant throughout the space. These relations are a generalization of known expressions for undeformed media [2, 3]. We investigate the case of rapid deformation when the turbulent characteristics of the medium vary in accordance with the linear theory [4].     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Micro-macro homogenization of gradient-enhanced Cosserat media

Based on the Hill’s lemma for classical Cauchy continuum, a generalized Hill’s lemma for micro-macro homogenization modeling of heterogeneous gradient-enhanced Cosserat continuum is presented in the frame of the average-field theory. In this context not only the strain and stress tensors defined in classical Cosserat continuum but also their gradients at each macroscopic sampling point are attributed to associated microstructural representative volume element (RVE). The admissible boundary conditions required to prescribe on the RVE for the modeling are extracted as a corollary of the presented generalized Hill’s lemma and discussed to ensure the satisfaction of the enhanced Hill–Mandel energy condition and the average-field theory.     

Coupled Solvent and Heat Transport of a Mixture of Swelling Porous Particles and Fluids: Single Time-Scale Problem

A three-spatial scale, single time-scale model for both moisture and heat transport is developed for an unsaturated swelling porous media from first principles within a mixture theoretic framework. On the smallest (micro) scale, the system consists of macromolecules (clay particles, polymers, etc.) and a solvating liquid (vicinal fluid), each of which are viewed as individual phases or nonoverlapping continua occupying distinct regions of space and satisfying the classical field equations. These equations are homogenized forming overlaying continua on the intermediate (meso) scale via hybrid mixture theory (HMT). On the mesoscale the homogenized swelling particles consisting of the homogenized vicinal fluid and colloid are then mixed with two bulk phase fluids: the bulk solvent and its vapor. At this scale, there exists three nonoverlapping continua occupying distinct regions of space. On the largest (macro) scale the saturated homogenized particles, bulk liquid and vapor solvent, are again homogenized forming four overlaying continua: doubly homogenized vicinal fluid, doubly homogenized macromolecules, and singly homogenized bulk liquid and vapor phases. Two constitutive theories are developed, one at the mesoscale and the other at the macroscale. Both are developed via the Coleman and Noll method of exploiting the entropy inequality coupled with linearization about equilibrium. The macroscale constitutive theory does not rely upon the mesoscale theory as is common in other upscaling methods. The energy equation on either the mesoscale or macroscale generalizes de Vries classical theory of heat and moisture transport. The momentum balance allows for flow of fluid via volume fraction gradients, pressure gradients, external force fields, and temperature gradients.     

Time resolved visualization of structures of velocity gradients measured with near Kolmogorov-scale resolution in turbulent free-shear flows

A three-dimensional visualization of velocity gradients at the finest scales in a turbulent free-shear flow is obtained using a time-resolved scanning PIV system with near Kolmogorov-scale resolution. A minimal observation volume capable of capturing single velocity gradient structures is used. The structures are identified by iso-surfaces revealing four basic shapes in the makeup of the velocity gradients: sheets, tubes, square ribbons and spherical blobs. Their size is also obtained. The 3D local acceleration structure is visualized showing strong anti-alignment with the convective acceleration. The statistics of the velocity gradient, and higher order moments (skewness and kurtosis) are calculated and compared favorably with published experimental work and with homogeneous isotropic theory.