Smoothing Measured Displacements and Computing Strains Utilising Finite Element Method

Abstract: A method for smoothing measured displacements and computing strains utilising finite element and least‐squares methods is proposed. Nodal displacement values of a finite element model are determined by fitting the interpolation functions of elements to measured displacement values using the method of least‐squares. The displacements in the region where the measurement values are not obtained or unreliable are determined by solving finite element equations. Then, strains are obtained using a displacement‐strain relationship. The validity is demonstrated by applying the proposed method to the displacement distributions of a plate with a hole obtained using finite element method and those around a crack tip obtained using digital image correlation. Results show that the displacements and the strains can be determined accurately by the proposed method. Furthermore, the strains near free boundaries and strain concentration region can be computed. As strains can be evaluated easily and accurately, the proposed method can be used as one of the data processing methods for optical methods.     

Simulating the propagation of displacement discontinuities in a regularized strain‐softening medium

A numerical model is developed which allows the inclusion of displacement discontinuities in a strain‐softening medium, independent of the finite element mesh structure. Inelastic deformations develop in the continuum and, when a critical threshold of inelastic deformation is reached, a displacement discontinuity is inserted. Discontinuities are introduced using the partition of unity concept which allows discontinuous functions to be added to the standard finite element basis. It is shown that the introduction of displacement discontinuities at the later stages of the failure process can lead to a failure mode that is fundamentally different than that using a continuum model only. This combined continuum‐discontinuous model is better able to describe the entire failure process than a continuum or a discrete model alone and treats mode‐I and mode‐II failure in a unified fashion. Copyright © 2001 John Wiley & Sons, Ltd.     

Enhanced mixed interpolation XFEM formulations for discontinuous Timoshenko beam and Mindlin‐Reissner plate

Shear locking is a major issue emerging in the computational formulation of beam and plate finite elements of minimal number of degrees of freedom as it leads to artificial overstiffening. In this paper, discontinuous Timoshenko beam and Mindlin‐Reissner plate elements are developed by adopting the Hellinger‐Reissner functional with the displacements and through‐thickness shear strains as degrees of freedom. Heterogeneous beams and plates with weak discontinuity are considered, and the mixed formulation has been combined with the extended finite element method (FEM); thus, mixed enrichment functions are used. Both the displacement and the shear strain fields are enriched as opposed to the traditional extended FEM where only the displacement functions are enriched. The enrichment type is restricted to extrinsic mesh‐based topological local enrichment. The results from the proposed formulation correlate well with analytical solution in the case of the beam and in the case of the Mindlin‐Reissner plate with those of a finite element package (ABAQUS) and classical FEM and show higher rates of convergence. In all cases, the proposed method captures strain discontinuity accurately. Thus, the proposed method provides an accurate and a computationally more efficient way for the formulation of beam and plate finite elements of minimal number of degrees of freedom.     

On the strong discontinuity approach in finite deformation settings

Taking the strong discontinuity approach as a framework for modelling displacement discontinuities and strain localization phenomena, this work extends previous results in infinitesimal strain settings to finite deformation scenarios. By means of the strong discontinuity analysis, and taking isotropic damage models as target continuum (stress–strain) constitutive equation, projected discrete (tractions–displacement jumps) constitutive models are derived, together with the strong discontinuity conditions that restrict the stress states at the discontinuous regime. A variable bandwidth model, to automatically induce those strong discontinuity conditions, and a discontinuous bifurcation procedure, to determine the initiation and propagation of the discontinuity, are briefly sketched. The large strain counterpart of a non‐symmetric finite element with embedded discontinuities, frequently considered in the strong discontinuity approach for infinitesimal strains, is then presented. Finally, some numerical experiments display the theoretical issues, and emphasize the role of the large strain kinematics in the obtained results. Copyright © 2003 John Wiley & Sons, Ltd.     

Application of Digital Image Correlation to Address Complex Motions in Thermoelastic Stress Analysis

A motion compensation method for thermoelastic stress analysis (TSA) is described that uses digital image correlation (DIC) to capture the displacement field on the surface of the specimen. The displacement field is used to correct the infrared (IR) images to remove the effect of the motion of the specimen from the TSA. As the DIC displacements are obtained with a relatively high spatial resolution, sharp displacement gradients and discontinuities can be corrected. The feasibility of the motion compensation method for TSA is investigated firstly by validating the approach using data obtained from an aluminium alloy plate with a central circular hole loaded in tension and comparing the results with a finite element model. It is shown that the motion compensation approach significantly improves the accuracy of TSA, particularly when high magnification optics are used. Next, the feasibility of simultaneous capture of IR and white light images is investigated. It is shown that by using the correct combination of paints, a speckle pattern can be applied to the surface to provide contrast in the white light spectrum for the DIC but have a uniform emissivity in the IR spectrum so that there is no effect on the TSA. Thus, it is possible for the motion compensation to be conducted on data collected during fatigue tests. Finally, it is demonstrated that the motion compensation technique can be applied to discontinuous motion produced by face sheet debonding in a foam cored sandwich structure loaded in a double cantilever beam (DCB) configuration. It is shown that the motion compensation technique is capable of correcting the complex and non‐uniform motion for TSA in the DCB test, thereby enabling detailed thermoelastic data to be obtained from the vicinity of the crack tip.     

Crack tip fields in ductile crystals

Results on the asymptotic analysis of crack tip fields in elastic-plastic single crystals are presented and some preliminary results of finite element solutions for cracked solids of this type are summarized. In the cases studied, involving plane strain tensile and anti-plane shear cracks in ideally plastic f c c and b c c crystals, analyzed within conventional small displacement gradient assumptions, the asymptotic analyses reveal striking discontinuous fields at the crack tip.For the stationary crack the stress state is found to be locally uniform in each of a family of angular sectors at the crack tip, but to jump discontinuously at sector boundaries, which are also the surfaces of shear discontinuities in the displacement field. For the quasi-statically growing crack the stress state is fully continuous from one near-tip angular sector to the next, but now some of the sectors involve elastic unloading from, and reloading to, a yielded state, and shear discontinuities of the velocity field develop at sector boundaries. In an anti-plane case studied, inclusion of inertial terms for (dynamically) growing cracks restores a discontinuous stress field at the tip which moves through the material as an elastic-plastic shock wave. For high symmetry crack orientations relative to the crystal, the discontinuity surfaces are sometimes coincident with the active crystal slip planes, but as often lie perpendicular to the family of active slip planes so that the discontinuities correspond to a kinking mode of shear.The finite element studies so far attempted, simulating the ideally plastic material model in a small displacement gradient type program, appear to be consistent with the asymptotic analyses. Small scale yielding solutions confirm the expected discontinuities, within limits of mesh resolution, of displacement for a stationary crack and of velocity for quasi-static growth. Further, the discontinuities apparently extend well into the near-tip plastic zone. A finite element formulation suitable for arbitrary deformation has been used to solve for the plane strain tension of a Taylor-hardening crystal panel containing, a center crack with an initially rounded tip. This shows effects due to lattice rotation, which distinguishes the regular versus kinking shear modes of crack tip relaxation. and holds promise for exploring the mechanics of crack opening at the tip.     

Hybrid and enhanced finite element methods for problems of soil consolidation

Hybrid and enhanced finite element methods with bi‐linear interpolations for both the solid displacements and the pore fluid pressures are derived based on mixed variational principles for problems of elastic soil consolidation. Both plane strain and axisymmetric problems are studied. It is found that by choosing appropriate interpolation of enhanced strains in the enhanced method, and by choosing appropriate interpolations of strains, effective stresses and enhanced strains in the hybrid method, the oscillations of nodal pore pressures can be eliminated. Several numerical examples demonstrating the capability and performance of the enhanced and hybrid finite element methods are presented. It is also shown that for some situations, such as problems involving high Poisson's ratio and in other related problems where bending effects are evident, the performance of the enhanced and hybrid methods are superior to that of the conventional displacement‐based method. The results from the hybrid method are better than those from the enhanced method for some situations, such as problems in which soil permeability is variable or discontinuous within elements. Since all the element parameters except the nodal displacements and nodal pore pressures are assumed in the element level and can be eliminated by static condensation, the implementations of the enhanced method and the hybrid method are basically the same as the conventional displacement‐based finite element method. The present enhanced method and hybrid method can be easily extended to non‐linear consolidation problems. Copyright © 2006 John Wiley & Sons, Ltd.     

An enriched element‐failure method (REFM) for delamination analysis of composite structures

This paper develops an enriched element‐failure method for delamination analysis of composite structures. This method combines discontinuous enrichments in the extended finite element method and element‐failure concepts in the element‐failure method within the finite element framework. An improved discontinuous enrichment function is presented to effectively model the kinked discontinuities; and, based on fracture mechanics, a general near‐tip enrichment function is also derived from the asymptotic displacement fields to represent the discontinuity and local stress intensification around the crack‐tip. The delamination is treated as a crack problem that is represented by the discontinuous enrichment functions and then the enrichments are transformed to external nodal forces applied to nodes around the crack. The crack and its propagation are modeled by the ‘failed elements’ that are applied to the external nodal forces. Delamination and crack kinking problems can be solved simultaneously without remeshing the model or re‐assembling the stiffness matrix with this method. Examples are used to demonstrate the application of the proposed method to delamination analysis. The validity of the proposed method is verified and the simulation results show that both interlaminar delamination and crack kinking (intralaminar crack) occur in the cross‐ply laminated plate, which is observed in the experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

A consistent geometrically non‐linear approach for delamination

A new model is presented for the simulation of delamination in laminated composite materials. A key feature is that the material structure and the finite element mesh are uncoupled. The displacement discontinuities that arise during the delamination process are described mathematically using discontinuous functions. This leads naturally to a set of coupled equations for the continuous and the discontinuous parts of the response. Discontinuities can pass through solid finite elements arbitrarily, with the displacement jump continuous across element boundaries. The performance of the model is demonstrated for several problems of delamination and geometric instability. Copyright © 2002 John Wiley & Sons, Ltd.     

Determination of effective stress intensity factors under mixed-mode from digital image correlation fields in presence of contact stresses and plasticity

Digital image correlation (DIC) is more and more popular to monitor fatigue crack growth and to determine the stress intensity factors. However, the posttreatment of the recorded displacement fields becomes tricky when the crack faces are not stress-free and when crack tip plasticity becomes significant. Several posttreatment methods to locate the crack tip and measure the effective stress intensity factors in such cases are compared, using finite element method-computed displacement fields, and then used on real DIC fields. An approach coupling DIC and finite element method is proposed to estimate the contact stresses along the crack.     

Embedded discontinuity finite element method for modeling of localized failure in heterogeneous materials with structured mesh: an alternative to extended finite element method

In this work we discuss the finite element model using the embedded discontinuity of the strain and displacement field, for dealing with a problem of localized failure in heterogeneous materials by using a structured finite element mesh. On the chosen 1D model problem we develop all the pertinent details of such a finite element approximation. We demonstrate the presented model capabilities for representing not only failure states typical of a slender structure, with crack-induced failure in an elastic structure, but also the failure state of a massive structure, with combined diffuse (process zone) and localized cracking. A robust operator split solution procedure is developed for the present model taking into account the subtle difference between the types of discontinuities, where the strain discontinuity iteration is handled within global loop for computing the nodal displacement, while the displacement discontinuity iteration is carried out within a local, element-wise computation, carried out in parallel with the Gauss-point computations of the plastic strains and hardening variables. The robust performance of the proposed solution procedure is illustrated by a couple of numerical examples. Concluding remarks are stated regarding the class of problems where embedded discontinuity finite element method (ED-FEM) can be used as a favorite choice with respect to extended FEM (X-FEM).     

Thermoelastic stress field in a piecewise homogeneous domain under non‐uniform temperature using a coupled boundary and finite element method

This study concerns the development of a coupled finite element–boundary element analysis method for the solution of thermoelastic stresses in a domain composed of dissimilar materials with geometric discontinuities. The continuity of displacement and traction components is enforced directly along the interfaces between different material regions of the domain. The presence of material and geometric discontinuities are included in the formulation explicitly. The unknown interface traction components are expressed in terms of unknown interface displacement components by using the boundary element method for each material region of the domain. Enforcing the continuity conditions leads to a final system of equations containing unknown interface displacement components only. With the solution of interface displacement components, each region has a complete set of boundary conditions, thus leading to the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, stresses from specific BEM regions are first expressed in terms of interface displacements, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of FEM regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

A study of the displacement field around embedded fibre optic sensors

Deformation fields around optical fibres embedded in carbon fibre/epoxy composite specimens have been measured using moiré interferometry. The inclusion of the optical fibre resulted in large strain gradients. Calculated displacements from finite element analysis were compared to the experimental results. The numerical analysis showed that the displacement field on the specimen surface is smoothed out through the moiré grating thickness, an effect which is most pronounced at the material interfaces. With this influence taken into consideration a reasonable good quantitative agreement between the experiments and the finite element analysis was obtained. The finite element analysis also showed that the grating stiffness did not affect the measured displacements as long as the grating had a lower stiffness than the specimen.     

Application of digital image correlation method for analysing crack variation of reinforced concrete beams

The Digital Image Correlation (DIC) method is a fast-growing emerging technology that provides a low-cost method for measuring the strain of an object. In this study, the feasibility of using this method to observe cracks developed in reinforced concrete beams will be explored so that a practical application can be proposed. The DIC method has been applied for analysing the field of surface displacement and strain; it is not applicable for measuring non-continuous field of displacement. However, if a singular point (i.e., crack points) can be considered as the area of concentrated strain by imitating the treatment of micro-cracks using the finite element method, the region of concentrated strain field based on analyses of digital images can be applied for determining the locations of cracks. Laboratory results show that cracks developed in reinforced cement beams can be observed with a good precision using the von Mises strain field, and that smaller grids lead to clearer crack images. In addition to identifying visible cracks, the DIC image analysis will enable researchers to identify minute cracks that are not visible to naked eyes. Additionally, the DIC method has more accuracy and precision than visual observation for analysing crack loadings so that earlier warnings can be realized before cracks develop in the specimen.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

Strain fields at an interface crack in a sandwich composite

Interface damage characterization and interlaminar failure of sandwich specimens with an initial interlaminar delamination in between the face sheet and the core is done by using the digital image correlation method (DIC). Virtual strain gages are emulated at the interface and beneath it, in the polyurethane core and opening strains are measured. Peel tests reveal interesting particularities on damage localization and strain variation while damage is completed. After analyzing the experimentally obtained results, four characteristic domains which describe the opening strain variations in the core are established together with the corresponding variation of the opening strain at the interface. The critical strain at damage initiation, as well as the critical displacement when damage is finalized, is established by DIC in the cohesive zone. After the calibration of the finite element model through experimental results, the use of the cohesive elements together with a linear softening law give in the core a pattern of strain variation similar to the one obtained experimentally.     

Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution

This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.     

Study on the element with the hole and crack

A new element is proposed for describing a discontinuous medium, such as holes and cracks, inside the region of the element. The underlying idea is to construct numerically the base functions of the discontinuous region by capturing the results calculated by fine finite elements in small-scale and then to construct the element in macro-scale with the crack and hole based on the theories of the multi-scale finite element method and the extended finite element method. Some numerical analysis is performed. The results show that the proposed element can well describe the field of displacement, strain, and stress intensity of the discontinuous region inside the element and can significantly decrease the number of elements and nodes of the calculated porous structure. The precision of the proposed element is also acceptable.     

Three‐dimensional finite elements with embedded strong discontinuities to model material failure in the infinitesimal range

This paper presents new three‐dimensional finite elements with embedded strong discontinuities in the small strain infinitesimal range. The goal is to model localized surfaces of failure in solids, such as cracks at fracture, through enhancements of the finite elements that capture the propagating discontinuities of the displacement field in the element interiors. In this way, such surfaces of discontinuity can be sharply resolved in general meshes not necessarily related to the detailed geometry of the surface, unknown a priori. An important issue is also the consideration of general finite element formulations in the developments (e.g., basic displacement‐based, mixed or enhanced assumed strain finite element formulations), as needed to optimally resolve the continuum problem in the bulk. The actual modeling of the discontinuity effects, including the incorporation of the cohesive law defining the discontinuity constitutive response, is carried out at the element level with the proper enhancement of the discrete strain field of the element. The added elemental degrees of freedom approximate the displacement jumps associated with the discontinuity and are defined independently from element to element, thus allowing their static condensation at the element level without affecting the global mechanical problem in terms of the number and topology of the global degrees of freedom. In fact, this global‐local structure of the finite element methods developed in this work arises naturally from a multi‐scale characterization of these localized solutions, with the discontinuities understood to appear in the small scales, thus leading directly to these computationally efficient numerical methods for their numerical resolution, easily incorporated to an existing finite element code. The focus in this paper is on the development of finite elements incorporating a linear interpolation of the displacement jumps in the general three‐dimensional setting. These interpolations are shown to be necessary for hexahedral elements to avoid the so‐called stress locking that occurs with simpler constant approximations of the jumps (namely, a spurious transfer of stresses across the discontinuity not allowing its full release and, hence, resulting in an overstiff or locked numerical solution). The design of the new finite elements is accomplished in this work by a direct identification of the separation modes to be incorporated in the discrete strain field of the element, rather than from an assumed discontinuous interpolation of the displacements, assuring with this approach their locking‐free response by design. An additional issue addressed in the paper is the geometric characterization and propagation of the discontinuity surfaces in the general three‐dimensional setting of interest here. The paper includes a series of numerical simulations illustrating and evaluating the properties of the new finite elements. Copyright © 2012 John Wiley & Sons, Ltd.