Showing papers on "Displacement field published in 2009"

Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review

Abstract: As a practical and effective tool for quantitative in-plane deformation measurement of a planar object surface, two-dimensional digital image correlation (2D DIC) is now widely accepted and commonly used in the field of experimental mechanics. It directly provides full-field displacements to sub-pixel accuracy and full-field strains by comparing the digital images of a test object surface acquired before and after deformation. In this review, methodologies of the 2D DIC technique for displacement field measurement and strain field estimation are systematically reviewed and discussed. Detailed analyses of the measurement accuracy considering the influences of both experimental conditions and algorithm details are provided. Measures for achieving high accuracy deformation measurement using the 2D DIC technique are also recommended. Since microscale and nanoscale deformation measurement can easily be realized by combining the 2D DIC technique with high-spatial-resolution microscopes, the 2D DIC technique should find more applications in broad areas.

A synthetic Schlieren method for the measurement of the topography of a liquid interface

Abstract: An optical method for the measurement of the instantaneous topography of the interface between two transparent fluids, named free-surface synthetic Schlieren (FS-SS), is characterised. This method is based on the analysis of the refracted image of a random dot pattern visualized through the interface. The apparent displacement field between the refracted image and a reference image obtained when the surface is flat is determined using a digital image correlation (DIC) algorithm. A numerical integration of this displacement field, based on a least square inversion of the gradient operator, is used for the reconstruction of the instantaneous surface height, allowing for an excellent spatial resolution with a low computational cost. The main limitation of the method, namely the ray crossing (caustics) due to strong curvature and/or large surface-pattern distance, is discussed. Validation experiments using a transparent solid model with a wavy surface or plane waves at a water–air interface are presented, and some additional time-resolved measurements of circular waves generated by a water drop impact are discussed.

A phantom node formulation with mixed mode cohesive law for splitting in laminates

Abstract: A phantom node method with mixed mode cohesive law is proposed for the simulation of splitting in laminates. With this method, a discontinuity in the displacement field can be modeled at arbitrary locations. The micromechanical phenomenon that splitting cracks grow parallel to the fiber, is incorporated on the mesolevel, i.e., in the homogenized ply, by setting the direction of the crack propagation equal to the fiber direction. A new mixed mode cohesive law is introduced for increased robustness of the incremental-iterative solution procedure. The model is validated with mixed mode bending tests, and its utility is illustrated with examples for a single ply and for a laminate.

Eigenfracture: An Eigendeformation Approach to Variational Fracture

Abstract: We propose an approximation scheme for a variational theory of brittle fracture. In this scheme, the energy functional is approximated by a family of functionals depending on a small parameter and on two fields: the displacement field and an eigendeformation field that describes the fractures that occur in the body. Specifically, the eigendeformations allow the displacement field to develop jumps that cost no local elastic energy. However, this local relaxation requires the expenditure of a certain amount of fracture energy. We provide a construction, based on the con- sideration of e-neighborhoods of the support of the eigendeformation field, for calculating the right amount of fracture energy associated with the eigendeformation field. We prove the Γ-convergence of the eigendeformation functional sequence, and of finite element approximations of the eigende- formation functionals, to the Griffith-type energy functional introduced in Francfort and Marigo (J. Mech. Phys. Solids, 46 (1998), pp. 1319-1342). This type of convergence ensures the convergence of eigendeformation solutions, and of finite element approximations thereof, to brittle-fracture solu- tions. Numerical examples concerned with quasi-static mixed-mode crack propagation illustrate the versatility and robustness of the approach and its ability to predict crack-growth patterns in brittle solids.

Measuring the mechanical properties of human skin in vivo using digital image correlation and finite element modelling

Abstract: The mechanical properties of the skin are important in many applications, but are not well understood. This paper presents a method for measuring the mechanical properties of human skin in vivo using digital image correlation, with a finite element model that was used to optimize the material properties to obtain the best match with the model data. The skin was modelled as an Ogden hyperelastic membrane, with a tension field wrinkling model and an initial stretch identified as an additional material parameter, and the boundary conditions were the measured load and the displacements around the edge of the region of interest. Fast, reliable convergence was obtained using a Hager–Zhang non-linear conjugate gradient solver. A stochastic optimization procedure was used to identify the material parameters. Good estimates of the material parameters could be obtained from the displacement field at a single time point. Typical material parameters were μ = 10 Pa, α = 26, and an initial strain of 0.2. These parameters were not unique; the stochastic optimization procedure gave good global convergence and an indication of the overall uncertainty in the identification of the results. It is argued that the use of the DIC technique, which generates very large amounts of data, also gave a clearer picture of the overall uncertainty.

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

Abstract: In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.

An extended and integrated digital image correlation technique applied to the analysis of fractured samples The equilibrium gap method as a mechanical filter

Abstract: To reduce the measurement uncertainty, a measurement technique is proposed for es- timating full displacement fields by complementing digital image correlation with an additional penalization on the distance between the estimated displacement field and its projection onto the space of elastic solutions. The extended finite element method is used for inserting disconti- nuities independently of the underlying mesh. An application to the brittle fracture of a silicon carbide specimen is used to illustrate the application. To complete the analysis, the crack tip location and the stress intensity factors are estimated. This allows for a characterization of the measurement and identification procedure in terms of uncertainty.

Numerical simulation of dynamic fracture using finite elements with embedded discontinuities

Abstract: This paper presents the extension of some finite elements with embedded strong discontinuities to the fully transient range with the focus on dynamic fracture. Cracks and shear bands are modeled in this setting as discontinuities of the displacement field, the so-called strong discontinuities, propagating through the continuum. These discontinuities are embedded into the finite elements through the proper enhancement of the discrete strain field of the element. General elements, like displacement or assumed strain based elements, can be considered in this framework, capturing sharply the kinematics of the discontinuity for all these cases. The local character of the enhancement (local in the sense of defined at the element level, independently for each element) allows the static condensation of the different local parameters considered in the definition of the displacement jumps. All these features lead to an efficient formulation for the modeling of fracture in solids, very easily incorporated in an existing general finite element code due to its modularity. We investigate in this paper the use of this finite element formulation for the special challenges that the dynamic range leads to. Specifically, we consider the modeling of failure mode transitions in ductile materials and crack branching in brittle solids. To illustrate the performance of the proposed formulation, we present a series of numerical simulations of these cases with detailed comparisons with experimental and other numerical results reported in the literature. We conclude that these finite element methods handle well these dynamic problems, still maintaining the aforementioned features of computational efficiency and modularity.

Displacement field estimation for a two-dimensional structure using fiber Bragg grating sensors

Abstract: The structure shape itself is of great interest for many aerospace applications. For example, the stability of the surface shape of large, high precision or space reflectors is essential for the communication performance. The knowledge of static and dynamic displacements of these structures would provide the possibility to enhance their performance by appropriate countermeasures. During operation, however, the direct measurement of displacements of the whole structure is often difficult. This study investigates whole displacement field estimation using strain measurement and a displacement–strain-transformation approach. The use of fiber Bragg gratings (FBGs) as strain sensors for this application offers the possible implementation of an integrated sensor network including many measurement points within only a few optical fibers. This paper discusses many issues related to the displacement field estimation of a dynamically excited plate using a transformation matrix based on a modal approach. In order to reduce systematic displacement estimation errors due to aliasing, a parametric study was performed and the sensor locations were optimized. Experimental validation was also conducted using a cantilever plate equipped with 16 FBG sensors in an optimized configuration. The estimated displacements showed good agreements with those measured directly from laser displacement sensors.

A three‐dimensional C1 finite element for gradient elasticity

Abstract: In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C1 continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C1 finite elements and their perceived computational cost. In this context, the lack of three-dimensional C1 elements is of particular concern. In this paper we present a new C1 hexahedral element which, to the best of our knowledge, is the first three-dimensional C1 element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C1 elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good-quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.

Quantitative three-dimensional elasticity imaging from quasi-static deformation: a phantom study

Abstract: We present a methodology to image and quantify the shear elastic modulus of three-dimensional (3D) breast tissue volumes held in compression under conditions similar to those of a clinical mammography system. Tissue phantoms are made to mimic the ultrasonic and mechanical properties of breast tissue. Stiff lesions are created in these phantoms with size and modulus contrast values, relative to the background, that are within the range of values of clinical interest. A two-dimensional ultrasound system, scanned elevationally, is used to acquire 3D images of these phantoms as they are held in compression. From two 3D ultrasound images, acquired at different compressed states, a three-dimensional displacement vector field is measured. The measured displacement field is then used to solve an inverse problem, assuming the phantom material to be an incompressible, linear elastic solid, to recover the shear modulus distribution within the imaged volume. The reconstructed values are then compared to values measured independently by direct mechanical testing.

Limit analysis of plates using the EFG method and second‐order cone programming

Abstract: The meshless element-free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation that involves approximating the displacement field using the moving least-squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations using compatible elements. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non-smooth minimization problem can be transformed into a form suitable for solution using second-order cone programming. The procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper-bound solutions for benchmark problems.

Traction patterns of tumor cells

Abstract: The traction exerted by a cell on a planar deformable substrate can be indirectly obtained on the basis of the displacement field of the underlying layer. The usual methodology used to address this inverse problem is based on the exploitation of the Green tensor of the linear elasticity problem in a half space (Boussinesq problem), coupled with a minimization algorithm under force penalization. A possible alternative strategy is to exploit an adjoint equation, obtained on the basis of a suitable minimization requirement. The resulting system of coupled elliptic partial differential equations is applied here to determine the force field per unit surface generated by T24 tumor cells on a polyacrylamide substrate. The shear stress obtained by numerical integration provides quantitative insight of the traction field and is a promising tool to investigate the spatial pattern of force per unit surface generated in cell motion, particularly in the case of such cancer cells.

Measurement of Fracture Parameters for a Mixed-Mode Crack Driven by Stress Waves using Image Correlation Technique and High-Speed Digital Photography

Abstract: Measurement of fracture parameters for a rapidly growing crack in syntactic foam sheets using image correlation technique and high-speed photography is presented. The performance of a rotating mirror-type multi-channel high-speed digital camera to measure transient deformations is assessed by conducting benchmark tests on image intensity variability, rigid translation and rigid rotation. Edge-cracked foam samples are subjected to eccentric impact loading relative to the initial crack plane to produce mixed-mode loading conditions in a three-point bend configuration. High-speed photography is used to record decorated random speckles in the vicinity of the crack tip at a rate of 200 000 frames per second. Two sets of images are recorded, the first set before impact and the second after impact. Using image correlation methodology, crack-tip displacement field histories and dominant strains from the time of impact up to complete fracture are mapped. Over-deterministic least-squares analyses of crack-tip radial and tangential displacements are used to obtain mixed-mode fracture parameters. The measurements are compared with complementary finite element results. The fracture parameters determined from radial displacements seem more robust even when fewer number of higher order terms in the crack-tip asymptotic expansion are used.