Showing papers on "Displacement field published in 2007"

On modifications of Newton's second law and linear continuum elastodynamics

Abstract: In this paper, we suggest a new perspective, where Newton’s second law of motion is replaced by a more general law which is a better approximation for describing the motion of seemingly rigid macroscopic bodies. We confirm a finding of Willis that the density of a body at a given frequency of oscillation can be anisotropic. The relation between the force and the acceleration is non-local (but causal) in time. Conversely, for every response function satisfying these properties, and having the appropriate high-frequency limit, there is a model which realizes that response function. In many circumstances, the differences between Newton’s second law and the new law are small, but there are circumstances, such as in specially designed composite materials, where the difference is enormous. For bodies which are not seemingly rigid, the continuum equations of elastodynamics govern behaviour and also need to be modified. The modified versions of these equations presented here are a generalization of the equations proposed by Willis to describe elastodynamics in composite materials. It is argued that these new sets of equations may apply to all physical materials, not just composites. The Willis equations govern the behaviour of the average displacement field whereas one set of new equations governs the behaviour of the averageweighted displacement field, where the weighted displacement field may attach zero weight to ‘hidden’ areas in the material where the displacement may be unobservable or not defined. From knowledge of the average-weighted displacement field, one obtains an approximate formula for the ensemble averaged energy density. Two other sets of new equations govern the behaviour when the microstructure has microinertia, i.e. where there are internal spinning masses below the chosen scale of continuum modelling. In the first set, the average displacement field is assumed to be observable, while in the second set an average-weighted displacement field is assumed to be observable.

A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics

Abstract: This paper proposes a three-dimensional meshfree method for arbitrary crack initiation and propagation that ensures crack path continuity for non-linear material models and cohesive laws. The method is based on a local partition of unity. An extrinsic enrichment of the meshfree shape functions is used with discontinuous and near-front branch functions to close the crack front and improve accuracy. The crack is hereby modeled as a jump in the displacement field. The initiation and propagation of a crack is determined by the loss of hyperbolicity or the loss of material stability criterion. The method is applied to several static, quasi-static and dynamic crack problems. The numerical results very precisely replicate available experimental and analytical results.

A meshfree method based on the local partition of unity for cohesive cracks

Abstract: We will present a meshfree method based on the local partition of unity for cohesive cracks. The cracks are described by a jump in the displacement field for particles whose domain of influence is cut by the crack. Particles with partially cut domain of influence are enriched with branch functions. Crack propagation is governed by the material stability condition. Due to the smoothness and higher order continuity, the method is very accurate which is demonstrated for several quasi static and dynamic crack propagation examples.

Upper bound limit analysis using simplex strain elements and second-order cone programming

Abstract: In geomechanics, limit analysis provides a useful method for assessing the capacity of structures such as footings and retaining walls, and the stability of slopes and excavations. This paper presents a finite element implementation of the kinematic (or upper bound) theorem that is novel in two main respects. First, it is shown that conventional linear strain elements (6-node triangle, 10-node tetrahedron) are suitable for obtaining strict upper bounds even in the case of cohesive-frictional materials, provided that the element sides are straight (or the faces planar) such that the strain field varies as a simplex. This is important because until now, the only way to obtain rigorous upper bounds has been to use constant strain elements combined with a discontinuous displacement field. It is well known (and confirmed here) that the accuracy of the latter approach is highly dependent on the alignment of the discontinuities, such that it can perform poorly if an unstructured mesh is employed. Second, the optimization of the displacement field is formulated as a standard second-order cone programming (SOCP) problem. Using a state-of-the-art SOCP code developed by researchers in mathematical programming, very large example problems are solved with outstanding speed. The examples concern plane strain and the Mohr-Coulomb criterion, but the same approach can be used in 3D with the Drucker-Prager criterion, and can readily be extended to other yield criteria having a similar conic quadratic form.

Terrestrial laser scanner to detect landslide displacement fields: a new approach

Abstract: A terrestrial laser scanner (TLS) allows the generation of a detailed model of a landslide surface. In this way, when two or more georeferenced models obtained by multi-temporal scans are available, the landslide displacement field can be computed. Nevertheless, such a computation is a relatively complex task because the recognition of correspondences among the multi-temporal models is required. The Iterative Closest Point (ICP) algorithm allows the alignment of two 3D objects having a common part by iterative shape matching. A new method for the automatic calculation of a landslide displacement field is presented here. It is based on a piecewise application of the ICP algorithm and is made possible by the robustness of this algorithm against noise and small morphological modifications. After a series of numerical experimentations, this method was successfully applied to two test sites located in the North-Eastern Italian Alps affected by high-risk landslides of the slump type (Perarolo di Cadore and Lamosano) with very different observational conditions.

Extended meshfree methods without branch enrichment for cohesive cracks

Abstract: An extended meshless method for both static and dynamic cohesive cracks is proposed. This new method does not need any crack tip enrichment to guarantee that the crack closes at the tip. All cracked domains of influence are enriched by only the sign function. The domain of influence which includes a crack tip is modified so that the crack tip is always positioned at its edge. The modification is only applied for the discontinuous displacement field and the continuous field is kept unchanged. In addition to the new method, the use of Lagrange multiplier is explored to achieve the same goal. The crack is extended beyond the actual crack tip so that the domains of influence containing the crack tip are completely cut. It is enforced that the crack opening displacement vanishes along the extension of the crack. These methods are successfully applied to several well-known static and dynamic problems.

Finite elements for symmetric tensors in three dimension

Abstract: We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger--Reissner mixed formulation of the elasticty equations, when standard discontinous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex, and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.

Calibration of Stress-triaxiality Dependent Crack Formation Criteria: A New Hybrid Experimental–Numerical Method

Abstract: A new experimental technique has been developed to investigate the onset of fracture in metals at low and intermediate stress triaxialities. The gage section of a flat specimen has been designed such that cracks are most likely to initiate within the specimen center, remote from the specimen boundaries. Along with the specimen, a biaxial testing device has been built to apply a well-defined displacement field to the specimen shoulders. The stress state within the specimen is adjusted by changing the biaxial loading angle. Using this new experimental technique, the crack initiation in metals can be studied experimentally for stress triaxialities ranging from 0.0 to 0.6. The stress and strain fields within the specimen gage section are determined from finite element analysis. The reliability of the computational model of the test set-up has been verified by comparing the simulation results with laser speckle-interferometric displacement measurements during testing. Sample experiments have been performed on the Al-7Si-Mg gravity die casting alloy. A three-step hybrid experimental–numerical calibration procedure has been proposed and applied to determine a phenomenological crack formation criterion for the Al-7Si-Mg alloy.

Two-dimensional multi-level strain estimation for discontinuous tissue.

Abstract: A large number of the strain estimation methods presented in the literature are based on the assumption of tissue continuity that establishes a continuous displacement field. However, in certain locations in the body such as the arteries in vivo scanning may produce displacement fields that are discontinuous between the two walls of the artery. Many of the displacement or strain estimators fail when the displacement fields are discontinuous. In this paper, we present a new 2D multi-level motion or displacement tracking method for accurate estimation of the strain in these situations. The final high-resolution displacement estimate is obtained using two processing steps. The first step involves an estimation of a coarse displacement estimate utilizing B-mode or envelope signals. To reduce computational time, the coarse displacement estimates are obtained starting from down-sampled B-mode pre- and post-compression image pairs using a pyramidal processing approach. The coarse displacement estimate obtained from the B-mode data is used to guide the final 2D cross-correlation computations on radio-frequency (RF) data. Results from finite element simulations and in vivo experimental data demonstrate the feasibility of this approach for imaging tissue with discontinuous displacement fields.

Stress intensity factor gauging by digital image correlation: application in cyclic fatigue

Abstract: A fatigue crack in steel (CCT geometry) is studied via digital image correlation. The measurement of the stress intensity factor (SIF) change during one cycle is performed using a decomposition of the displacement field onto a tailored set of elastic fields. The same analysis is performed using two different routes, namely, the first one consists in computing the displacement field using a general correlation technique providing the displacement field projected onto finite element shape functions, and then analysing this displacement field in terms of the selected mechanically relevant fields. The second strategy, called integrated approach, directly estimates the amplitude of these elastic fields from the correlation of successive images. Both procedures give consistent results, and offer very good performances in the evaluation of the crack tip position (uncertainty of about 20 μm for a 14.5-mm crack), SIFs (uncertainty <1 MPa ) and opening properties.

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

Abstract: We consider several inverse problems motivated by elastography. Given the (possibly transient) displacement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density ρ, determine the Lame parameters λ and μ. We consider several special cases of this problem: (a) for μ known a priori, λ is determined by a single deformation field up to a constant. (b) Conversely, for λ known a priori, μ is determined by a single deformation field up to a constant. This includes as a special case that for which the term λ∇ ⋅ u ≡ 0. (c) Finally, if neither λ nor μ is known a priori, but Poisson's ratio ν is known, then μ and λ are determined by a single deformation field up to a constant. This includes as a special case plane stress deformations of an incompressible material. Exact analytical solutions valid for 2D, 3D and transient deformations are given for all cases in terms of quadratures. These are used to show that the inverse problem for μ based on the compressible elasticity equations is unstable in the limit λ → ∞. Finally, we use the exact solutions as a basis to compute non-trivial modulus distributions in a simulated example.

Upper bound limit analysis using discontinuous quadratic displacement fields

Abstract: The classical lower and upper bound theorems allow the exact limit load for a perfectly plastic structure to be bracketed in a rigorous manner. When the bound theorems are implemented numerically in combination with the finite element method, the ability to obtain tight bracketing depends not only on the efficient solution of the arising optimization problem, but also on the effectiveness of the elements employed. Elements for (strict) upper bound analysis pose a particular difficulty since the flow rule is required to hold throughout each element, yet it can only be enforced at a finite number of points. For over 30 years, the standard choice for this type of analysis has been the constant strain element combined with discontinuities in the displacement field. Here we show that, provided certain conditions are observed, conventional linear strain triangles and tetrahedra can also be used to obtain strict upper bounds for a general convex yield function, even when the displacement field is discontinuous. A specific formulation for the Mohr-Coulomb criterion in plane strain is given in terms of second-order cone programming, and example problems are solved using both continuous and discontinuous quadratic displacement fields.

A second-order homogenization procedure for multi-scale analysis based on micropolar kinematics

Abstract: The paper presents a higher order homogenization scheme based on non-linear micropolar kinematics representing the macroscopic variation within a representative volume element (RVE) of the material. On the microstructural level the micro–macro kinematical coupling is introduced as a second-order Taylor series expansion of the macro displacement field, and the microstructural displacement variation is gathered in a fluctuation term. This approach relates strongly to second gradient continuum formulations, presented by, e.g. Kouznetsova et al. (Int. J. Numer. Meth. Engng 2002; 54:1235–1260), thus establishing a link between second gradient and micropolar theories. The major difference of the present approach as compared to second gradient formulations is that an additional constraint is placed on the higher order deformation gradient in terms of the micropolar stretch. The driving vehicle for the derivation of the homogenized macroscopic stress measures is the Hill–Mandel condition, postulating the equivalence of microscopic and macroscopic (homogenized) virtual work. Thereby, the resulting homogenization procedure yields not only a stress tensor, conjugated to the micropolar stretch tensor, but also the couple stress tensor, conjugated to the micropolar curvature tensor. The paper is concluded by a couple of numerical examples demonstrating the size effects imposed by the homogenization of stresses based on the micropolar kinematics. Copyright © 2006 John Wiley & Sons, Ltd.

Full-field assessment of the damage process of laminated composite open-hole tensile specimens. Part I: Methodology

Abstract: The present paper is the first part of a study dedicated to the assessment of the damage taking place in composite open-hole tensile specimens using full-field strain measurements. It concentrates on the thorough validation of the methodology used to process the full-field data. First, the grid method used to provide the two components of the in-plane displacement field at the surface of the test specimens is described. Then, the data processing procedure to obtain the displacement maps during the test is discussed. The main part of the paper is dedicated to the derivation of strains from displacements, investigating two procedures: local differentiation and polynomial fitting. The effect of the grid is also investigated. Finally, successful experimental results are presented.

Plastic response of a two-dimensional amorphous solid to quasistatic shear: transverse particle diffusion and phenomenology of dissipative events.

Abstract: We perform extensive simulations of a two-dimensional Lennard-Jones glass subjected to quasistatic shear deformation at T=0 . We analyze the distribution of nonaffine displacements in terms of contributions of plastic, irreversible events, and elastic, reversible motions. From this, we extract information about correlations between plastic events and about the elastic nonaffine noise. Moreover, we find that nonaffine motion is essentially diffusive, with a clearly size-dependent diffusion constant. These results, supplemented by close inspection of the evolving patterns of the nonaffine tangent displacement field, lead us to propose a phenomenology of plasticity in such amorphous media. It can be schematized in terms of elastic loading and irreversible flips of small, randomly located shear transformation zones, elastically coupled via their quadrupolar fields.

Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm

Abstract: The displacement field in highly nonuniformly strained crystals is obtained by addition of constraints to an iterative phase retrieval algorithm. These constraints include direct space density uniformity and also constraints to the sign and derivatives of the different components of the displacement field. This algorithm is applied to an experimental reciprocal space map measured using high-resolution x-ray diffraction from an array of silicon lines, and the obtained component of the displacement field is in very good agreement with the one calculated using a finite element model.

Thermo elastic analysis of a functionally graded rotating disk with small and large deflections

Abstract: A functionally graded (FG) rotating disk with axisymmetric bending and steady-state thermal loading is studied. The material properties of the disk are assumed to be graded in the direction of the thickness by a power law distribution of volume fractions of the constituents. First-order shear deformation Mindlin plate and von Karman theories are employed. New set of equilibrium equations with small and large deflections are developed. Using small deflection theory an exact solution for displacement field is given. Solutions are obtained in series form in case of large deflection. Mechanical responses are compared small deflection versus large deflection as well as homogeneous versus FG disks. It is observed that for particular values of the grading index n of material properties mechanical responses in FG disk can be smaller than in a homogeneous disk. It is seen that given the non-dimensional maximum vertical displacement wmax/h close to 0.4 for a homogeneous (full-ceramic in this study) disk greater errors in the mechanical responses for FG disks would be introduced if one uses small deflection theory.

Kinematics of fault‐related folding derived from a sandbox experiment

Abstract: We analyze the kinematics of fault tip folding at the front of a fold-and-thrust wedge using a sandbox experiment The analog model consists of sand layers intercalated with low-friction glass bead layers, deposited in a glass-sided experimental device and with a total thickness h = 48 cm A computerized mobile backstop induces progressive horizontal shortening of the sand layers and therefore thrust fault propagation Active deformation at the tip of the forward propagating basal decollement is monitored along the cross section with a high-resolution CCD camera, and the displacement field between pairs of images is measured from the optical flow technique In the early stage, when cumulative shortening is less than about h/10, slip along the decollement tapers gradually to zero and the displacement gradient is absorbed by distributed deformation of the overlying medium In this stage of detachment tip folding, horizontal displacements decrease linearly with distance toward the foreland Vertical displacements reflect a nearly symmetrical mode of folding, with displacements varying linearly between relatively well defined axial surfaces When the cumulative slip on the decollement exceeds about h/10, deformation tends to localize on a few discrete shear bands at the front of the system, until shortening exceeds h/8 and deformation gets fully localized on a single emergent frontal ramp The fault geometry subsequently evolves to a sigmoid shape and the hanging wall deforms by simple shear as it overthrusts the flat ramp system As long as strain localization is not fully established, the sand layers experience a combination of thickening and horizontal shortening, which induces gradual limb rotation The observed kinematics can be reduced to simple analytical expressions that can be used to restore fault tip folds, relate finite deformation to incremental folding, and derive shortening rates from deformed geomorphic markers or growth strata

Transient landscapes at fault tips

Abstract: [1] Fault growth produces patterns of displacement and slip rate that are highly variable in both space and time. This transience is most pronounced near fault tips, where along-strike displacement gradients vary in time as the fault array lengthens. We use a set of statistical and field observations to quantify the response of catchments and their associated fans in three large normal fault arrays to transient patterns of displacement and slip rate. Catchments near the fault tips show distinct scaling of channel slope with drainage area compared with catchments near the strike center. This scaling becomes uniform beyond about ∼10 km from the fault tips and is therefore like footwall relief, largely decoupled from the fault displacement profile. The estimated catchment response times to a change in slip rate also vary between fault tips and strike center. The response times for tip catchments are much longer than the inferred time since fault activity began, indicating that they are unlikely to be in equilibrium with the current fault displacement field. This disequilibrium, combined with the decoupling of slope-area scaling from displacement, indicates that landscapes are most sensitive to fault activity near fault tips. Active faults characterized by along-strike variation in slip rate thus provide excellent opportunities to explore the transient response of landscapes to tectonic forcing.