Three-dimensional Displacement Field Research Articles

Meshless analysis of metallic and composite beam structures by advanced hierarchical models with layer-wise capabilities

This paper proposes radial basis function (RBF)-based meshless solutions for static and dynamic analyses of metallic and laminated beam-like structures with various boundary conditions. Making use of the Carrera Unified Formulation (CUF), the three-dimensional displacement field can be reduced to one-dimensional displacements related to the axial direction multiplied by cross-sectional kinematics. Locally supported Wendland’s C6 RBF is employed to interpolate the displacements in the axial domain and Hierarchical Legendre Expansion (HLE) is used to expand the kinematic unknowns over the cross-section domain, being endowed with Layer-Wise (LW) ability. The principle of virtual displacements (PVD) is adopted to derive the governing equation in a strong form, accompanied by the advent of fundamental nuclei, which are independent of the transverse assumptions in the cross section domain. Different numerical assessments on metallic and laminated structures are addressed to show the performance of RBF-based HLE models in terms of displacements, stresses and vibration modes. 3D accuracy of the obtained results is demonstrated by comparison with 3D FEM solutions provided by well-known commercial softwares (Ansys and Abaqus).

Mechanical behavior of CAD/CAM occlusal ceramic reconstruction assessed by digital color holography

ObjectivesCAD/CAM ceramic occlusal veneers are increasingly used as therapeutic options. However, little is known about their mechanical behavior under stress, as the response of the prepared tooth that supports it. The aim of this article is to use for the first time 3D color holography to evaluate the behavior of a molar occlusal veneer under stress and the response of the prepared tooth. MethodsThe occlusal surface of a lower molar is prepared to receive a specific monolithic ceramic reconstruction manufactured with a chairside CAD/CAM system. Longitudinally cut samples are used to get a planar object observation and to “look inside” the tooth. A digital holographic set-up permits to obtain the contact-less and one-shot measurement of the three-dimensional displacement field at the surface of the tooth sample; stain fields are evaluated with low noise-sensitive computation. ResultsFigures show the strain fields with micro-strain units and highlight the behavior of the ROI (region of interest) in the three directions of space. The ROI are: the ceramic, the glue junction, the dentin enamel junction, dentin and enamel. The results show an excellent behavior of the restored tooth without areas of excessive stress concentrations, but also a significant involvement of the dentin enamel junction. SignificanceThe ceramic occlusal veneer seems to behave in accordance with the biomechanical concepts ensuring the longevity of the reconstituted tooth. 3D holography is a highly recommended method for studying dental biomechanics.

Theoretical approach of characterizing the crack-tip constraint effects associated with material’s fracture toughness

The conversion problem of plane strain fracture toughness ( $$K_\mathrm{IC}$$ ) which is necessarily measured according to ASTM standards, to lower constraint applications generally existing in engineering structures, has led to extensive material and labor costs. The present paper explores and quantifies the crack-tip constraint effects by the crack-tip plastic zone to serve the prediction of material’s fracture toughness. Firstly, the approximate three-dimensional crack front displacement fields are obtained by using the variable separation method. The three-dimensional stress field is then used to predict the shape and size of the crack-tip plastic zone. Secondly, the two-dimensional in-plane and the three-dimensional out-of-plane geometric constraint effects are quantified separately, and two constraint factors, i.e., $$\alpha _\mathrm{in}$$ and $$\alpha _\mathrm{out}$$ are proposed. A series of configurations of the two-dimensional and three-dimensional crack-tip plastic zones that vary with the specimen’s geometric size (plate width, thickness, etc.) are presented, which will facilitate a better understanding of the crack-tip constraint effect. Finally, the present method is applied to elucidate the significant “thickness effect” of the X70 pipeline steel’s fracture toughness by using the out-of-plane constraint factor $$\alpha _\mathrm{out}$$ . The corresponding results are compared with the experimentally measured and FEM-predicted fracture toughness $$K_\mathrm{C}$$ . It is concluded that the parameters $$\alpha _\mathrm{out}$$ and the fracture toughness $$K_\mathrm{C}$$ can be correlated with each other, which will be beneficial to the prediction of material’s fracture toughness and the avoidance of experimental costs.

Analysis and improvement of motion encoding in magnetic resonance elastography.

Magnetic resonance elastography (MRE) utilizes phase contrast magnetic resonance imaging (MRI), which is phase locked to externally generated mechanical vibrations, to measure the three‐dimensional wave displacement field. At least four measurements with linear‐independent encoding directions are necessary to correct for spurious phase contributions if effects from imaging gradients are non‐negligible. In MRE, three encoding schemes have been used: unbalanced four‐ and six‐point and balanced four‐point (‘tetrahedral’) encoding. The first two sensitize to motion with orthogonal gradients, with the four‐point method acquiring a single reference scan without motion sensitization, whereas three additional scans with inverted gradients are used with six‐point encoding, leading to two‐fold higher displacement‐to‐noise ratio (DNR) and 50% longer scan duration. Balanced four‐point (tetrahedral) encoding encodes along the four diagonals of a cube, with one direction serving as a reference for the other three encoding directions, similar to four‐point encoding. The objective of this work is to introduce a theoretical framework to compare different motion sensitization strategies with respect to their motion encoding efficiency in two fundamental encoding limits, the gradient strength limit and the dynamic range limit, which are both placed in relation to conventional gradient recalled echo (GRE)‐ and spin echo (SE)‐based MRE sequences. We apply the framework to the three aforementioned schemes and show that the motion encoding efficiency of unbalanced four‐ and six‐point encoding schemes in the gradient‐limited regime can be increased by a factor of 1.5 when using all physical gradient channels concurrently. Furthermore, it is demonstrated that reversing the direction of the reference in balanced four‐point (tetrahedral) encoding results in the Hadamard encoding scheme, which leads to increased DNR by 2 compared with balanced four‐point encoding and 2.8 compared with unbalanced four‐point encoding. As an example, we show that optimal encoding can be utilized to reduce the acquisition time of standard liver MRE in vivo from four to two breath holds.

Micromechanical Progressive Failure Analysis of Fiber-Reinforced Composite Using Refined Beam Models

An efficient and novel micromechanical computational platform for progressive failure analysis of fiber-reinforced composites is presented. The numerical framework is based on a recently developed micromechanical platform built using a class of refined beam models called Carrera unified formulation (CUF), a generalized hierarchical formulation which yields a refined structural theory via variable kinematic description. The crack band theory is implemented in the framework to capture the damage propagation within the constituents of composite materials. The initiation and orientation of the crack band in the matrix are determined using the maximum principal stress state and the traction-separation law governing the crack band growth is related to the fracture toughness of the matrix. A representative volume element (RVE) containing randomly distributed fibers is modeled using the component-wise (CW) approach, an extension of CUF beam model based on Lagrange type polynomials. The efficiency of the proposed numerical framework is achieved through the ability of the CUF models to provide accurate three-dimensional (3D) displacement and stress fields at a reduced computational cost. The numerical results are compared against experimental data available in the literature and an analogous 3D finite element model with the same constitutive crack band model. The applicability of CUF beam models as a novel micromechanical platform for progressive failure analysis as well as the multifold efficiency of CUF models in terms of CPU time are highlighted.

SAR-revealed slip partitioning on a bending fault plane for the 2014 Northern Nagano earthquake at the northern Itoigawa–Shizuoka tectonic line

By applying conventional cross-track synthetic aperture radar interferometry (InSAR) and multiple aperture InSAR techniques to ALOS-2 data acquired before and after the 2014 Northern Nagano, central Japan, earthquake, a three-dimensional ground displacement field has been successfully mapped. Crustal deformation is concentrated in and around the northern part of the Kamishiro Fault, which is the northernmost section of the Itoigawa-Shizuoka tectonic line. The full picture of the displacement field shows contraction in the northwest–southeast direction, but northeastward movement along the fault strike direction is prevalent in the northeast portion of the fault, which suggests that a strike-slip component is a significant part of the activity of this fault, in addition to a reverse faulting. Clear displacement discontinuities are recognized in the southern part of the source region, which falls just on the previously known Kamishiro Fault trace. We inverted the SAR and GNSS data to construct a slip distribution model; the preferred model of distributed slip on a two-plane fault surface shows a combination of reverse and left-lateral fault motions on a bending east-dipping fault surface with a dip of 30° in the shallow part and 50° in the deeper part. The hypocenter falls just on the estimated deeper fault plane where a left-lateral slip is inferred, whereas in the shallow part, a reverse slip is predominant, which causes surface ruptures on the ground. The slip partitioning may be accounted for by shear stress resulting from a reverse fault slip with left-lateral component at depth, for which a left-lateral slip is suppressed in the shallow part where the reverse slip is inferred. The slip distribution model with a bending fault surface, instead of a single fault plane, produces moment tensor solution with a non-double couple component, which is consistent with the seismically estimated mechanism.

A method of monitoring three-dimensional ground displacement in mining areas by integrating multiple InSAR methods

ABSTRACTUnderground mining often causes large-gradient vertical and horizontal displacement in ground surface, resulting in the losses of coherence and difficulties in phase unwrapping. This article presents an approach to determine three-dimensional ground displacements in mining areas with the large gradient or phase decorrelation by integrating multiple interferometric synthetic aperture radar (InSAR) methods. The core of the proposed method is that the offset-tracking method is employed to solve for the displacement with the large gradient or phase decorrelation. First, the displacements in the radar line-of-sight directions are obtained from two interferometric pairs with different viewing geometries by integrating the measurements of differential InSAR and offset tracking. Then, the displacements in the azimuthal directions are obtained from two interferometric pairs with different viewing geometries by integrating the measurements of multiple aperture interferometry and offset tracking. Finally, the three-dimensional ground displacement fields are inferred from these four independent, one-dimensional displacements using the least squares method and Helmert variance component estimation. We apply this method to obtain the three-dimensional ground displacement field in the Dongtan mine region. We compare the results with those of levelling and global positioning system surveys, and the root mean square errors of the results were 24 and 43 mm in the vertical direction and horizontal directions, respectively. The experimental results indicate that the proposed method can be used to estimate three-dimensional ground displacement fields in mining areas with large-gradient displacement and phase decorrelation.

Precision of Digital Volume Correlation Approaches for Strain Analysis in Bone Imaged with Micro-Computed Tomography at Different Dimensional Levels

Accurate measurement of local strain in heterogeneous and anisotropic bone tissue is fundamental to understand the pathophysiology of musculoskeletal diseases, to evaluate the effect of interventions from preclinical studies, and to optimize the design and delivery of biomaterials. Digital volume correlation (DVC) can be used to measure the three-dimensional displacement and strain fields from micro-Computed Tomography (µCT) images of loaded specimens. However, this approach is affected by the quality of the input images, by the morphology and density of the tissue under investigation, by the correlation scheme, and by the operational parameters used in the computation. Therefore, for each application the precision of the method should be evaluated. In this paper we present the results collected from datasets analyzed in previous studies as well as new data from a recent experimental campaign for characterizing the relationship between the precision of two different DVC approaches and the spatial resolution of the outputs. Different bone structures scanned with laboratory source µCT or Synchrotron light µCT (SRµCT) were processed in zero-strain tests to evaluate the precision of the DVC methods as a function of the subvolume size that ranged from 8 to 2500 micrometers. The results confirmed that for every microstructure the precision of DVC improves for larger subvolume size, following power laws. However, for the first time large differences in the precision of both local and global DVC approaches have been highlighted when SRµCT or in vivo µCT images were used instead of conventional ex vivo µCT. These findings suggest that in situ mechanical testing protocols applied in SRµCT facilities should be optimized in order to allow DVC analyses of localized strain measurements. Moreover, for in vivo µCT applications DVC analyses should be performed only with relatively course spatial resolution for achieving a reasonable precision of the method. In conclusion, we have extensively shown that the precision of both tested DVC approaches is affected by different bone structures, different input image resolution and different subvolume sizes. Before each specific application DVC users should always apply a similar approach to find the best compromise between precision and spatial resolution of the measurements.

Finite element models with node-dependent kinematics for the analysis of composite beam structures

This paper presents refined one-dimensional models with node-dependent kinematics. The three-dimensional displacement field is discretized into two domains, namely cross-section domain and axis domain. The mechanical behaviors of the beam can be firstly captured by the cross-section functions then interpolated by the nodal shape functions of the beam element. Such a feature makes it possible to adopt different types of cross-section functions on each element node, obtaining node-dependent kinematic finite element models. Such models can integrate Taylor-based and Lagrange-type nodal kinematics on element level, bridging a less-refined model to a more refined model without using special coupling methods. FE governing equations of node-dependent models are derived by applying the Carrera Unified Formulation. Some numerical cases on metallic and composite beam-like structures are studied to demonstrate the effectiveness of node-dependent models in bridging a locally refined model to a global model when local effects should be accounted for.

Exact solutions for static analysis of laminated, box and sandwich beams by refined layer-wise theory

Abstract In the present work, a close-form solution based on a unified one-dimensional model is proposed and then applied to static response analyses of cross-ply laminated and sandwich beams subjected to simply supported boundary conditions. The hierarchical beam model is derived within the framework of the Carrera Unified Formulation (CUF), which makes use of Lagrange polynomials to express the three-dimensional (3D) displacement field via arbitrary order approximation of pure displacement variables at each layer over the cross section, in a Layer-Wise (LW) sense. The governing equations are derived via the principle of virtual work and a Navier-type close-form solution is employed to solve the resulting boundary value problem. Four benchmark numerical examples are carried out to demonstrate the efficiency of this novel method, including compact multi-layered cross-ply laminated beams, a thin-walled composite box beam and a composite sandwich-box beam. The results show that accurate displacement and stress components can be obtained as the order of the expansion increases, accompanied by a significant reduction in computational costs in comparison with the 3D finite element solutions. Besides, numerical cases in this research may be taken as benchmarks for future assessments in this field.

Retrieving Precise Three-Dimensional Deformation on the 2014 M6.0 South Napa Earthquake by Joint Inversion of Multi-Sensor SAR

We reconstructed the three-dimensional (3D) surface displacement field of the 24 August 2014 M6.0 South Napa earthquake using SAR data from the Italian Space Agency’s COSMO-SkyMed and the European Space Agency’s Sentinel-1A satellites. Along-track and cross-track displacements produced with conventional SAR interferometry (InSAR) and multiple-aperture SAR interferometry (MAI) techniques were integrated to retrieve the east, north, and up components of surface deformation. The resulting 3D displacement maps clearly delineated the right-lateral shear motion of the fault rupture with a maximum surface displacement of approximately 45 cm along the fault’s strike, showing the east and north components of the trace particularly clearly. These maps also suggested a better-constrained model for the South Napa earthquake. We determined a strike of approximately 338° and dip of 85° by applying the Okada dislocation model considering a single patch with a homogeneous slip motion. Using the distributed slip model obtained by a linear solution, we estimated that a peak slip of approximately 1.7 m occurred around 4 km depth from the surface. 3D modelling using the retrieved 3D maps helps clarify the fault’s nature and thus characterize its behaviour.

An Algorithm for Identifying a Crack Within a Measured Displacement Field

This paper discusses an algorithm for the detection of crack damage in plate structures based on the governing differential equation (GDE) of in-plane displacement of a plate. A state-of-the-art 3D scanning laser Doppler vibrometer, which is able to provide very accurate measurements of a three-dimensional displacement field, was utilized to implement the algorithm. To evaluate the GDE from the measured displacements, a 2D Savitzky–Golay differentiating filter was used. The potential of this algorithm for detecting a crack was compared against another two algorithms which are based on displacement measurements. The first is an algorithm based on the error in smoothed to measured displacements and the other uses surface strains normalized by their mean. To investigate the efficiency of the algorithms in detecting crack damage, a number of PMMA plate specimens were fabricated with a range of crack lengths extending from a V-notch and tested under a cyclic, quasi-static, uni-axial load. For each algorithm, the crack location could be positively identified. However, the two algorithms based on the error in smoothed to measured displacements and surface strains normalized by their mean were found to be dependent on the direction of the applied load, whereas, theoretically the algorithm based on the GDE of in-plane displacement would work regardless of the direction of the applied load.

An Equivalent Layer-Wise Approach for the Free Vibration Analysis of Thick and Thin Laminated and Sandwich Shells

The main purpose of the paper is to present an innovative higher-order structural theory to accurately evaluate the natural frequencies of laminated composite shells. A new kinematic model is developed starting from the theoretical framework given by a unified formulation. The kinematic expansion is taken as a free parameter, and the three-dimensional displacement field is described by using alternatively the Legendre or Lagrange polynomials, following the key points of the most typical Layer-wise (LW) approaches. The structure is considered as a unique body and all the geometric and mechanical properties are evaluated on the shell middle surface, following the idea of the well-known Equivalent Single Layer (ESL) models. For this purpose, the name Equivalent Layer-Wise (ELW) is introduced to define the present approach. The governing equations are solved numerically by means of the Generalized Differential Quadrature (GDQ) method and the solutions are compared with the results available in the literature or obtained through a commercial finite element program. Due to the generality of the current method, several boundary conditions and various mechanical and geometric configurations are considered. Finally, it should be underlined that different doubly-curved surfaces may be considered following the mathematical framework given by the differential geometry.

Two-Dimensional Phononic Crystals: Disorder Matters.

The design and fabrication of phononic crystals (PnCs) hold the key to control the propagation of heat and sound at the nanoscale. However, there is a lack of experimental studies addressing the impact of order/disorder on the phononic properties of PnCs. Here, we present a comparative investigation of the influence of disorder on the hypersonic and thermal properties of two-dimensional PnCs. PnCs of ordered and disordered lattices are fabricated of circular holes with equal filling fractions in free-standing Si membranes. Ultrafast pump and probe spectroscopy (asynchronous optical sampling) and Raman thermometry based on a novel two-laser approach are used to study the phononic properties in the gigahertz (GHz) and terahertz (THz) regime, respectively. Finite element method simulations of the phonon dispersion relation and three-dimensional displacement fields furthermore enable the unique identification of the different hypersonic vibrations. The increase of surface roughness and the introduction of short-range disorder are shown to modify the phonon dispersion and phonon coherence in the hypersonic (GHz) range without affecting the room-temperature thermal conductivity. On the basis of these findings, we suggest a criteria for predicting phonon coherence as a function of roughness and disorder.

Three-Dimensional Surface Displacement Field Associated with the 25 April 2015 Gorkha, Nepal, Earthquake: Solution from Integrated InSAR and GPS Measurements with an Extended SISTEM Approach

Three-dimensional surface displacement field associated with the 25 April 2015 Gorkha, Nepal earthquake is derived from an integration of Interferometric Synthetic Aperture Radar (InSAR) and Global Positioning System (GPS) measurements, with an extended SISTEM (Simultaneous and Integrated Strain Tensor Estimation From Geodetic and Satellite Deformation Measurements) approach (ESISTEM) proposed in this study. In ESISTEM approach, both surrounding InSAR and GPS measurements can be used as constraints in deriving surface displacements; while only surrounding GPS measurements are used in SISTEM approach. Besides the constraints from surrounding GPS measurements, the ESISTEM approach makes surrounding InSAR measurements available for constraining the derived deformations based on surface elastic theory for the first time. From the north to the south, derived surface displacement field shows prevailing southward horizontal deformations, and gradually varied vertical deformations ranging from −0.95 to 1.40 m within 120 km to the north of Kathmandu. This reveals that ruptures of Main Himalayan thrust (MHT) system were confined in subsurface and did not propagate to the Main Frontal Thrust (MFT) fault, in accordance with field investigation as well as geodetic and seismic studies. Relation between vertical deformations and earthquake-induced landslides is briefly discussed.

A one-dimensional higher-order theory with cubic distortional modes for static and dynamic analyses of thin-walled structures with rectangular hollow sections

In this paper, a one-dimensional higher-order theory is presented for both static and dynamic analyses of thin-walled structures with rectangular hollow sections. The formulation has been enriched with four cubic distortional modes, capable of reproducing three-dimensional behavior of thin-walled structures in a very accurate way. Toward this end, a systematic procedure is developed to determine higher-order distortional modes for a rectangular hollow cross section. Then, involved with the application of the modal superposition method, the two- and three-dimensional displacement fields are reduced to one dimension with 11 degrees of freedom. Next, the principle of minimum potential energy and Hamilton’s principle are applied to formulate the static and dynamic governing equations of the thin-walled structure, respectively. At last, the finite element implementation is carried out by employing the \(C_{0}\) interpolation function. Numerical examples are also presented to validate the new theory.

Hierarchical Beam Finite Elements Based Upon a Variables Separation Method

A family of hierarchical one-dimensional beam finite elements developed within a variables separation framework is presented. A Proper Generalized Decomposition (PGD) is used to divide the global three-dimensional problem into two coupled ones: one defined on the cross-section space (beam modeling kinematic approximation) and one belonging to the axis space (finite element solution). The displacements over the cross-section are approximated via a Unified Formulation (UF). A Lagrangian approximation is used along the beam axis. The resulting problems size is smaller than that of the classical equivalent finite element solution. The approach is, then, particularly attractive for higher-order beam models and refined axial meshes. The numerical investigations show that the proposed method yields accurate yet computationally affordable three-dimensional displacement and stress fields solutions.

An efficient modelling approach based on a rigorous cross-sectional analysis for analysing box girder bridge superstructures

A novel analysis technique is introduced for efficient modelling of box girder bridge decks. The general three-dimensional equations used to accurately define the deformation of these complex beam-like slender structures are decoupled into a two-dimensional cross-sectional problem and a one-dimensional beam problem through decomposition of the three-dimensional strain field. The two-dimensional cross-sectional problem is solved by a two-dimensional finite element analysis considering in-plane as well as out-of-plane warping displacements of the beam section. This gives an accurate constitutive relationship of the one-dimensional beam problem without making any major assumptions as is often done in usual beam theories. The one-dimensional beam problem is solved by a one-dimensional beam finite analysis and the results obtained are used to recover three-dimensional stress, strain and displacement fields accurately. Numerical examples of box girder bridge deck systems having thin-walled sections are solved by the proposed approach to show its performance.

A Dynamic Inflation Test for Soft Materials

We developed a dynamic inflation experiment to measure the elastodynamic behavior of soft materials. In the experiments, a shock tube was used to apply dynamic pressurization to thin polydimethylsiloxane specimens. Two high-speed cameras were used to image the deforming specimen and three-dimensional digital image correlation was used to determine the three-dimensional displacement field of the specimen surface. We applied dynamic Kirchhoff plate bending theory and concepts from structural dynamics to derive a mathematical expression for the dynamic Young’s modulus. The phase velocity of the initial transverse wave propagation response and the vibration frequency of the long-time response were captured during our experiments and were applied in the calculation of the dynamic Young’s modulus.

Synchronous triple-optical-path digital speckle pattern interferometry with fast discrete curvelet transform for measuring three-dimensional displacements

Digital speckle pattern interferometry (DSPI) is a well-established and widely used optical measurement technique for obtaining qualitative as well as quantitative measurements of objects deformation. The simultaneous measurement of an object's surface displacements in three dimensions using DSPI is of great interest. This paper presents a triple-optical-path DSPI based method for the simultaneous and independent measurement of three-dimensional (3D) displacement fields. In the proposed method, in-plane speckle interferometers with dual-observation geometry and an out-of-plane interferometer are optimally combined to construct an integrated triple-optical-path DSPI system employing the phase shift technique, which uses only a single laser source and three cameras. These cameras are placed along a single line to synchronously capture real-time visible speckle fringe patterns in three dimensions. In addition, a pre-filtering method based on the fast discrete curvelet transform (FDCT) is utilized for denoising the obtained wrapped phase patterns to improve measurement accuracy. Finally, the simultaneous measurement of the 3D displacement fields of a simple beam and a composite laminated plate respectively subjected to three-point and single-point bend loading are investigated to validate the feasibility and effectiveness of the proposed method.