Showing papers on "Displacement field published in 2003"

Solution of inverse problems in elasticity imaging using the adjoint method

Abstract: We consider the problem of determining the shear modulus of a linear-elastic, incompressible medium given boundary data and one component of the displacement field in the entire domain. The problem is derived from applications in quantitative elasticity imaging. We pose the problem as one of minimizing a functional and consider the use of gradient-based algorithms to solve it. In order to calculate the gradient efficiently we develop an algorithm based on the adjoint elasticity operator. The main cost associated with this algorithm is equivalent to solving two forward problems, independent of the number of optimization variables. We present numerical examples that demonstrate the effectiveness of the proposed approach.

A cohesive segments method for the simulation of crack growth

Abstract: A numerical method for crack growth is described in which the crack is not regarded as a single discontinuity that propagates continuously. Instead, the crack is represented by a set of overlapping cohesive segments. These cohesive segments are inserted into finite elements as discontinuities in the displacement field by exploiting the partition-of-unity property of shape functions. The cohesive segments can be incorporated at arbitrary locations and orientations and are not tied to any particular mesh direction. The evolution of decohesion of the segments is governed by a cohesive law. The independent specification of bulk and cohesive constitutive relations leads to a characteristic length being introduced into the formulation. The formulation permits both crack nucleation and discontinuous crack growth to be modelled. The implementation is outlined and some numerical examples are presented.

Deformation of a Peridynamic Bar

Abstract: The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venant's principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Green's function for general loading problems.

Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems

Abstract: Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique. The procedures yield the uniform expansions of the displacement field and the normal forms governing the slow modulations of the amplitudes and phases of the modes. The non-linear interaction coefficients appearing in the normal forms are obtained in the form of infinite series with the discretization approach or as modal projections of second-order spatial functions with the direct approach. A systematic discussion on the existence and stability of coupled/uncoupled non-linear normal modes is presented. Closed-form conditions for non-linear orthogonality of the modes, in a global and local sense, are discussed. A mechanical interpretation of these conditions in terms of virtual works is also provided.

Calibration of nonlocal damage model from size effect tests

Abstract: The calibration of nonlocal models which contain an internal length has been among the major issues conditioning the implementation of this kind of failure models. Direct calibration from uniaxial testing, where the state of strain remains homogeneous throughout the specimen, is impossible. The softening law is not directly accessible because the strains cannot remain homogeneous during the entire test. In the absence of local information on the displacement field and on micro cracking in the fracture process zone, the calibration has to rely on inverse analysis. This paper presents such a procedure based on three point bend size effect tests on notched specimens. The complete load deflection curves are used for the identification of the constitutive relations. Manual calibration is discussed first. It is emphasised that calibration on the load deflexion curve from a single experiment is not objective. We show that Bazant's size effect law, which is related to peak loads only, may serve as a helpful guide to reach the closest fit. Then, automatic calibration is described. An optimal set of model parameters can be obtained within a reasonable number of iterations.

Identification of mechanical properties by displacement field measurement: A variational approach

Abstract: We study a parameter identification problem associated with a two-dimensional mechanical problem. In the first part, the experimental technique of determining the displacement field is briefly presented. The variational method proposed herein is based on the minimization of either a separately convex functional or a convex functional that leads to the reconstruction of the elastic tensor and the stress field. These two reconstructed fields are continuous and piecewise linear on a triangulation of the two-dimensional domain. Some numerical and experimental examples are presented to test the performance of the algorithms.

A solid-like shell element allowing for arbitrary delaminations

Abstract: In this contribution a new finite element is presented for the simulation of delamination growth in thin-layered composite structures. The element is based on a solid-like shell element: a volume element that can be used for very thin applications due to a higher-order displacement field in the thickness direction. The delamination crack can occur at arbitrary locations and is incorporated in the element as a jump in the displacement field by using the partition of unity property of finite element shape functions. The kinematics of the element as well as the finite element formulation are described. The performance of the element is demonstrated by means of two examples. Copyright © 2003 John Wiley & Sons, Ltd.

Validation of a structural optimization algorithm transforming dynamic loads into equivalent static loads

Abstract: Generally, structural optimization is carried out based on external static loads. However, all forces have dynamic characteristics in the real world. Mathematical optimization with dynamic loads is almost impossible in a large-scale problem. Therefore, in engineering practice, dynamic loads are often transformed into static loads via dynamic factors, design codes, and so on. Recently, a systematic transformation of dynamic loads into equivalent static loads has been proposed in Refs. 1–3. Equivalent static loads are made to generate at each time step the same displacement field as the one generated by the dynamic loads. In this research, it is verified that the solution obtained via the algorithm of Refs. 1–3 satisfies the Karush–Kuhn–Tucker necessary conditions. Application of the algorithm is discussed.

A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

Abstract: An 8-node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions, viz., the compatibility enforcing shape functions and completeness enforcing shape functions. The former are chosen to satisfy exactly the minimum inter- as well as intra-element displacement continuity requirements, while the latter are chosen to satisfy all the (linear and higher order) completeness requirements so as to reproduce exactly a quadratic displacement field. Numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions. In this respect, the proposed 8-node unsymmetric element emerges to be better than the existing symmetric QUAD8, QUAD8/9, QUAD9, QUAD12 and QUAD16 elements, and matches the performance of the quartic element, QUAD25. For test problems involving a cubic or higher order displacement field, the proposed element yields a solution accuracy that is comparable to or better than that of QUAD8, QUAD8/9 and QUAD9 elements. Furthermore, the element maintains a good accuracy even with the reduced 2× 2 numerical integration. Copyright © 2003 John Wiley & Sons, Ltd.

Simulation of Asphalt Materials Using Finite Element Micromechanical Model with Damage Mechanics

Abstract: A theoretical and numerical study of the micromechanical behavior of asphalt concrete was undertaken. Asphalt is a heterogeneous material composed of aggregates, binder cement, and air voids. The load-carrying behavior of such a material is strongly related to the local load transfer between aggregate particles, and this is taken as the microstructural response. Numerical simulation of this material behavior was accomplished by developing a special finite element model that incorporated the mechanical load-carrying response between the aggregates. The finite element scheme incorporated a network of special frame elements, each with a stiffness matrix developed from an approximate elasticity solution of the stress and displacement field in a cementation layer between particle pairs. A damage mechanics approach was then incorporated within this solution, and this approach led to the construction of a softening model capable of predicting typical global inelastic behavior found in asphalt materials. This theory was then implemented within the ABAQUS finite element code to conduct simulations of particular laboratory specimens. A series of model simulations of indirect tension (IDT) tests were conducted to investigate the effect of variation of specimen microstructure on the sample response. Simulation results of the overall sample behavior compared favorably with experimental results. Additional comparisons were made of the evolving damage behavior within the IDT test samples, and numerical results gave reasonable predictions.

Deformation patterns of reinforced foundation sand at failure

Abstract: While the stability of foundation soils has been written about extensively, the ultimate loads on reinforced soils is a subject studied to a much lesser degree. There is convincing experimental evidence in the literature that metal strips or layers of geosynthetic reinforcement can significantly increase the failure loads on foundation soils. Laboratory tests were performed to investigate the kine- matics of the collapse of sand reinforced with a layer of flexible reinforcement. Sequential images of the deformation field under a model footing were digitally recorded. A correlation-based motion detection technique was used to arrive at an incremental displacement field under a strip footing model. Color-coded displacements are presented graphically. The mechanism retains some of the characteristic features of a classical bearing capacity pattern of failure, but the reinforcement modifies that mechanism to some extent. The strips of geotextile used as model reinforcement give rise to the formation of shear bands in a narrow layer adjacent to the geosynthetic. Reinforcement restrains the horizontal displacement of the soil and alters the collapse pattern. The mechanism of deformation identified in the tests will constitute a basis for limit analysis of reinforced foundation soils.

Stress intensity factors for an interface crack between a functionally graded coating and a homogeneous substrate

Abstract: In this paper we consider the problem of a functionally graded coating bonded to a homogeneous substrate with a partially insulated interface crack between the two materials subject to both thermal and mechanical loading. The problem is solved under the assumption of plane strain and generalized plane stress conditions. The heat conduction and the plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the temperature and the displacement fields in the medium as well as the crack tip stress intensity factors. A crack-closure algorithm recently developed by the authors is applied to handle the problem of having negative mode I stress intensity factors. The Finite Element Method was additionally used to model the crack problem and to compute the crack-tip stress intensity factors. The main objective of the paper is to study the effect of the material nonhomogeneity parameters, partial insulation of the crack surfaces and crack-closure on the crack tip stress intensity factors for the purpose of gaining better understanding of the thermo-mechanical behavior of graded coatings.

Deformations in wide, center-notched, thin panels, part I: three-dimensional shape and deformation measurements by computer vision

Abstract: The response of wide, thin, center-notched, 2024-T3 aluminum panels undergoing far-field tensile load is investigated. Three panels with a notch length to panel width of 0.33 and widths of 305, 610, and 1016 mm were subjected to far-field tensile loading. As part of the experimental program, two pairs of cameras were configured into separate stereovision systems and used to simultaneously capture both the global response of the sheet and the local response near a notch tip. Global areas, ranging in size from 250x 250 mm to 550x 550 mm, were imaged for each panel. A second stereovison system recorded images of a small area, 10×20 mm, ahead of one notch tip. Postprocessing of the stereovision measurement data from global and local systems using three-dimensional digital image correlation was used to obtain the complete displacement field at each point in the region of interest. In general, results demonstrate that the combination of stereovision and three-dimensional digital image correlation is capable of accurately measuring true, three-dimensional structural deformations in regions undergoing both large out-of-plane displacements and large displacement gradients. Furthermore, 3-D measurements on the panel specimen near the grip location are shown to provide an independent assessment of the true boundary conditions, with specimen slippage clearly noted in the 1016-mm specimen. Results from the extensive notched, wide panel experimental program demonstrate that (a) each panel has an initial shape that deviates up to 3 mm from planarity, with the greatest deviations occurring at the center of the notch, (b) the global load-displacement response is essentially linear for load levels that are well beyond the onset of large, out-of-plane displacements in the notch region, and (c) increasing the size of the notched, thin panel specimen results in distinctly different surface deformations and deformed shapes, with three separate maxima/minima in the out-of-plane component of the largest panel. The region where tensile opening strains are above 2% extends several millimeters ahead of the hole, while compressive strains parallel to the notch direction are contained within a few millimeters of the hole. The in-plane shear strains are concentrated along circular lobes at +/-45 deg from the horizontal direction, a trend which is generally consistent with plane stress conditions.

Static and dynamic 3-D human face reconstruction

Abstract: A system for three-dimensional (3-D) facial animation, including a base 3-D surface model representing a human face, and a set of displacement fields representing surface motion patterns associated with muscle movements. Each displacement field is a displacement vector varying over vertices of the base surface model and an intensity variable. Both the base surface model and displacement fields are acquired from a live subject by 3-D acquisition. The data are acquired using a surface acquisition system, capable of measuring the coordinates of a set of points on the subject's face and reconstructing the facial surface as a surface model. The base surface model is acquired by reconstructing the surface of the subject's face in the neutral pose. Each displacement field is acquired by reconstructing the surface of the subject's face in a series of poses of a muscle movement at increasing intensities. This results in a sequence of surface models exhibiting the changing shape of the face over the progress of the muscle movement. The displacement field is derived by calculating displacements at each selected intensity value and then interpolating over the known displacements.

Two-dimensional solids reinforced by thin bars using the boundary element method

Abstract: In this work, a linear boundary element formulation is presented for analyzing stiffened domains. A particular sub-region technique, in which the equilibrium is preserved along interfaces without traction approximation, is adapted to model fiber immersed in a body. The integral representation is written for a whole body, requiring only the displacement along interfaces. The sub-region is then assumed to be very thin to simulate fibers only when normal forces are taken into account. The thin sub-regions then degenerate so that they can be represented by its skeleton line. The displacement field over the fiber cross-sections is assumed to be constant. In the case of 2D problems, the degrees of freedom are reduced to two components only at each fiber node. Thus, displacement integral representations for collocations defined along the fiber skeleton are needed. The quasi-singular integrals are computed by using closed expressions or employing a numerical scheme with sub-elements. An example is solved to show that the formulation is very accurate for modeling cases of domains stiffened by fibers.

Calculation of Stress Intensity Factors and Crack Opening Displacements for Cracks Subjected to Complex Stress Fields

Abstract: Fatigue cracks in shot peened and case hardened notched machine components and high-pressure vessels are subjected to the stress fields induced by the external load and the residual stress resulting from the surface treatment or autofrettage. Both stress fields are usually nonuniform and available handbook stress intensity factor solutions are in most cases unavailable for such configurations, especially in the case of two-dimensional surface breaking cracks such as semi-elliptical and quarter-elliptical cracks at notches. The method presented in the paper makes it possible to calculate stress intensity factors for such cracks and complex stress fields by using the generalized weight function technique. It is also shown that the generalized weight functions make it possible to calculate the crack opening displacement field often used in the determination of the critical load or the critical crack size.

A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation

Abstract: A non-linear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section is developed. Non-linear in-plane and out-of-plane warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, together with the assumption that the cross-section is undeformable in its own plane, the non-linear warping is described in terms of the flexural and torsional curvatures. Due to the internal constraints, the displacement field depends on three components only, two transversal translations of the shear center and the torsional rotation. Three non-linear differential equations of motion up to the third order are derived using the Hamilton principle. Taking into account the order of magnitude of the various terms, the equations are simplified and the importance of each contribution is discussed. The effect of symmetry properties is also outlined. Finally, a discrete form of the equations is given, which is used in Part II to study dynamic coupling phenomena in conditions of internal resonance.

Dynamic shape control of beam-type structures by piezoelectric actuation and sensing

Abstract: In the present paper the concept of dynamic shape control of structures is addressed. The term “shape control”, not to be confused with automatic control, means to identify the spatial distribution (or shape) of an actuating control agency, such that a structural displacement field with a desired spatial distribution (or shape) is reached. This field aspect requires a spatially distributed control actuation, which, in the present paper, is performed by means of assigned piezoelectric eigenstrains. The goal of our proposed dynamic shape control procedure is to shape the piezoelectric actuation such as to obtain a displacement field coinciding with a dynamic displacement field induced by external forces. Equivalently, we may eliminate the force induced dynamic displacements. Bending vibrations of straight composite piezoelectric beams are studied in more detail. First, the coupled electro-mechanical field equations are developed and then the dynamic shape control problem is solved in closed form for this coupled formulation. It turns out that exact elimination of force induced vibrations is possible when the shape of the piezoelectric actuation coincides with a statically admissible bending moment distribution of the force loaded beam. Distributions characterizing non-unique solutions are discussed, and aspects of collocated sensing are addressed.

Higher correlations, universal distributions, and finite size scaling in the field theory of depinning.

Abstract: Recently we constructed a renormalizable field theory up to two loops for the quasi-static depinning of elastic manifolds in a disordered environment. Here we explore further properties of the theory. We show how higher correlation functions of the displacement field can be computed. Drastic simplifications occur, unveiling much simpler diagrammatic rules than anticipated. This is applied to the universal scaled width-distribution. The expansion in d=4-epsilon predicts that the scaled distribution coincides to the lowest orders with the one for a Gaussian theory with propagator G(q)=1/q^(d+2 \zeta), zeta being the roughness exponent. The deviations from this Gaussian result are small and involve higher correlation functions, which are computed here for different boundary conditions. Other universal quantities are defined and evaluated: We perform a general analysis of the stability of the fixed point. We find that the correction-to-scaling exponent is omega=-epsilon and not -epsilon/3 as used in the analysis of some simulations. A more detailed study of the upper critical dimension is given, where the roughness of interfaces grows as a power of a logarithm instead of a pure power.