Global Displacement Field Research Articles

Intelligent analysis method for the global vertical displacement field of foundation pits in dense karst cave areas

Prior research on excavation in dense karst cave foundation pits has primarily concentrated on evaluating the localized spatio-temporal influence and isolated geological factors. Nonetheless, this approach oversimplifies modeling conditions, thereby limiting its ability to provide a comprehensive understanding of the vertical displacement field. Consequently, this oversimplification can inflate the safety factor and increase project costs. Therefore, we propose a feedforward neural network (FNN), updated with the loop nested optimal iterative method (LNOIM), which incorporates the spatiotemporal characteristics of monitoring points and geological factors to analyze the engineering sensitivity of karst caves. Ultimately, the global foundation pit vertical displacement field was obtained. Our method has been demonstrated to be effective in a foundation pit in South China ((P value) P > 0.050, Cohen's d < 0.200). Furthermore, it has been validated in other cases ((Root Mean Square Error) RMSE = 1.576–2.916). This work provides a new perspective on the accurate reflection of the global vertical displacement state of a foundation pit. Additionally, it enhances the ability to sensitively identify caves in dense karst cave areas, thereby improving the safety of foundation pit works.

Topological characterization of rearrangements in amorphous solids.

In amorphous materials, plasticity is localized and occurs as shear transformations. It was recently shown by Wu etal. that these shear transformations can be predicted by applying topological defect concepts developed for liquid crystals to an analysis of vibrational eigenmodes [Z. W. Wu et al., Nat. Commun. 14, 2955 (2023)10.1038/s41467-023-38547-w]. This study relates the -1 topological defects to the displacement fields expected of an Eshelby inclusion, which are characterized by an orientation and the magnitude of the eigenstrain. A corresponding orientation and magnitude can be defined for each defect using the local displacement field around each defect. These parameters characterize the plastic stress relaxation associated with the local structural rearrangement and can be extracted using the fit to either the global displacement field or the local field. Both methods provide a reasonable estimation of the molecular-dynamics-measured stress drop, confirming the localized nature of the displacements that control both long-range deformation and stress relaxation.

A model for laminated composite beams based on a novel zig-zag theory

We develop a new model called C-311 for laminated composite beams based on a new zig-zag theory that is firstly proposed in this paper. The in-plane displacement field of the model includes two parts: part one is regarded as the global displacement field which is made of third-order polynomial about z, part two is the local displacement field which is the new zig-zag theory. By satisfying inter-laminar transverse stress continuity and the free surface conditions, the local displacement unknown variables can be eliminated, and the final displacement field should be expressed by the fixed number of unknown displacement variables. For testing the ability of C-311, the classical example about a simply supported laminated beam is investigated under mechanical loading in this paper. In comparison with other models of laminated composite beams, the numerical results show that the present model can give more accurate analysis, and are closer to the exact solutions. Especially the accurate transverse shear stresses are able to be directly obtained from the constitutive equations without any processing, and that is never done for other zig-zag models existing in the literature and benefit for practical engineering applications in the finite element method.

Absolute coordinate formulation and generalized component mode synthesis with rigid body coordinates

The dynamics of linearly-elastic multibody (MB) systems is conventionally modeled via the floating frame of reference formulation (FFRF); however, its equations of motion (EOMs) involve significantly more nonlinear terms and quantities than alternative formulations, such as the absolute coordinate formulation (ACF) and generalized component mode synthesis (GCMS). This large number of operations required makes computer implementations of the FFRF laborious as well as error-prone and introduces more complexity in general. These issues associated with the FFRF, and the fact that the formulations are mathematically equivalent as shown by the authors, render the ACF and its relatives appealing alternatives due to their simplistic equation structures. To make these alternatives even more appealing, this contribution proposes an improved ACF and GCMS, which (i) reduces the nonlinearity in the EOMs compared to their standard versions and (ii) eliminates the necessity to calculate the rigid body (RB) motion from the global nodal displacement field to obtain the flexible part of the degrees of freedom (DOFs) and the rotation matrix. The proposed EOMs feature a constant mass matrix, a corotated stiffness matrix in the flexible part, and a “small” nonlinear stiffness matrix in the RB rotation part. Moreover, attaching the moving reference frame to the center of mass of the underlying rigid body and employing linearized Tisserand and rotation matrix constraints eliminates coupling terms within the mass matrix and yields implementation-friendly EOMs to analyze the dynamics of linear-elastic flexible MB systems.

Water entry dynamic analysis of lightweight sandwich plates with lattice core: A semi-analytical approach and comparative study

This is the third part of water entry hydro-elastic analysis of clamped sandwich plates with lattice core (CSPLC). In the first and second parts analysis, the engineering prediction model is built to estimate the dynamic responses of CSPLC. To discuss the responses of CSPLC in more detail, the objective of the present work is to investigate the hydro-elastic responses of rectangular, clamped sandwich plates with lattice core (CSPLC) subjected to water entry via combined a novel semi-analytical approach and multi-physical numerical simulation. In the theoretical approach, the hydro-elastic characteristics are quantified by coupling a hydrodynamic water entry model with a first shear displacement model of elasticity for CSPLC. The fluid-structure interaction (FSI) effects of water entry are taken into account by dividing the total impact pressure into the rigid water entry pressure and the interacting pressure. To capture the precise responses of CSPLC, both the global displacement field and the local displacement field are considered in the analytical model. The dynamic governing equations are derived from the energy variation principle and solved by employing the modal approach. Furthermore, the 3D FSI finite element simulation is carried out to get the detailed hydro-elastic characteristics of CSPLC. This engineering semi-analytical model is found to yield accurate results for dynamic responses and offers significant savings in computational cost compared to the FSI simulations. As a comparative study, for a same geometrical stiffened panel, the FSI impact pressure and structural deflection are also given respectively.

Global and Local Deformation Effects of Dry Vacuum-Consolidated Triaxial Compression Tests on Sand Specimens: Making a Database Available for the Calibration and Development of Forward Models.

A comprehensive experimental database containing results of a series of dry vacuum-consolidated triaxial compression tests was populated. The tests were performed on sand specimens and conducted under similar experimental conditions, in which specimens’ boundary deformation was captured using a three-dimensional digital image correlation analysis (3D-DIC). The use of a standard triaxial device along with the 3D-DIC technology allowed the specimens’ global and local boundary displacement fields to be computed from start to end of the compression phase. By repeating each test under the same experimental conditions and building the specimens using the same type of sand, the boundary deformation patterns could be identified, and the statistics associated with both global and local displacement fields could be assessed. Making this experimental database available to others should serve to calibrate as well as develop new forward models to account for effects associated with the specimens’ local displacements and material heterogeneity and include statistics to represent a specimen’s random response. Moreover, this work will serve as a basis for the statistical characterization of spatio-temporal boundary localization effects used to develop stochastic models and machine-learning models, and simulate virtual triaxial tests.

Crack Propagation Simulation for Metal-Ceramic FGMs by Enriched Natural Element Method

The structural failure of functionally graded materials (FGMs) is mostly triggered by the micro-cracks so that the crack propagation is an important research subject. In this connection, the goal of this study is to develop a reliable numerical method for solving the crack propagation in 2-D inhomogeneous FGMs. To accomplish this goal, this paper introduces an improved crack propagation simulation method for 2-D FGMs by applying the enrichment technique to the natural element method (NEM). The global displacement field is interpolated using Laplace interpolation (LI) functions in NEM, and it is enhanced by the crack-tip singular displacement field. The stress intensity factors (SIFs) of FGMs having the varying Young’s modulus in space was computed by the modified interaction integral M(1,2), while the trajectories in crack propagation were predicted using the maximum principal stress (MPS) criterion and the effective SIF Kleqv at mode I. The present method was validated from the comparison with ANSYS and the unriched PG-NEM. It is found that the enriched NEM greatly enhances the simulation reliability of crack trajectory, such that it produces the crack trajectory close to one obtained by ANSYS. As well, it successfully predicted the remarkable variation of crack trajectories of FGM with exponentially varying elastic modulus.

Bioinspired cementitious-polymer composite for increased energy absorption

Preliminary results are presented on the energy absorbing characteristics of a cementitious-polymer architecture bioinspired by the organic-inorganic composite structure of nacre. The proposed bioinspired architecture consists of an open cell, platelet-shaped 3D-printed thermoplastic lattice filled with high performance cementitious paste. The hypothesis is that, similar to nacre, the platelet arrangement and differences in mechanical properties of the thermoplastic lattice and cementitious platelets would result in increased energy absorption. Initial laboratory scale investigations were performed using notched beam samples subjected to static three-point bending. Stereo-digital image correlation was used to track global strain displacement field and Hillerborg’s method was used to estimate the total fracture energy. The results indicate that this “brick-and-mortar” hierarchy can increase the energy absorbing capacity of the composite by upwards of 2490% compared with the benchmark cementitious specimen. The load-deformation behaviour and total fracture energy of the bioinspired composite were found to be influenced by the platelet arrangement and size and the lattice thickness.

A unified formulation for composite quasi-2D finite elements based on global-local superposition

A unified theory to formulate a family of quasi-2D composite finite elements that capture through-the-thickness effects is proposed. The formulation is based on the superposition of global and local displacement fields and includes thermal effects, as a response to the claim of the scientific community. First are presented the fundamentals of the unified formulation: the kinematic assumptions; the imposition of continuity of displacements and transverse stresses, and of non-homogeneous boundary conditions; and the mathematical manipulations that render its numerical efficiency. Then, the new family of elements is assessed in problems involving different boundary conditions. Results are compared to analytical and highly refined 2D finite elements solutions, providing a benchmark of the elements capabilities. Mesh convergence is assessed, together with the influence of the global and local displacement fields. The numerical efficiency of the formulation is attested and an expansion for a 3D formulation is outlined.

Isogeometric Optimal Design of Compliant Mechanisms Using Finite Deformation Curved Beam Built-Up Structures

Abstract This paper presents a configuration and sizing design optimization method for large deformation planar compliant mechanisms, using a continuum-based adjoint design sensitivity analysis (DSA) approach for built-up structures. Under the total Lagrangian formulation, the Jaumann strain formulation using the discretization of the global displacement field is employed to account for the finite deformation of arbitrarily curved Kirchhoff beams. In multipatch models, a rotational junction continuity condition is imposed using penalty and Lagrange multiplier methods. The developed adjoint DSA method can handle nonconservative loading conditions, which lead to asymmetry of tangent operator. Performance measures are displacements and rotation angles, and neutral axis configuration and cross-sectional thickness are considered as design variables. Also, analytical design sensitivity expressions for the rotation continuity condition are derived. Various compliant mechanisms including path-generators and an angular rotator are synthesized to demonstrate the applicability of the proposed method.

High-speed 3D digital image correlation for rolling deformation of a tire sidewall and measuring dynamic contact patch length.

Tires play a vital role in a vehicle's operational stability, comfort, and security. However, due to the influence of test equipment and tire operating conditions, the perception of rolling tire characteristics is still in the stage of gradual improvement, especially the analysis of sidewall rolling deformation and dynamic contact peculiarities of tires/road interactions, which has restricted the analysis of rolling energy dissipation and the accurate observation of tire forces. In this paper, the high-speed stereo-vision system was created by high-speed cameras, and the relative rigid and flexible displacement, strain, and trajectory trend of marker points and the real-time global displacement field of the sidewall during the tire-rolling cycle were acquired utilizing the improved digital image correlation algorithm under different rolling velocities. Meanwhile, the periodic dissipation of the strain concentration region was observed, and the phenomenon of strain resonance appeared at the overlap of the periods. The relative flexibility strain and shear strain of the marker point were obtained on the plane element, which resulted in the relative flexibility major principle strain of the plane element, and then it achieved recognition of the leading and trailing points of the contact patch and the accurate measurement of dynamic contact patch length.

Model-Based Digital Image Correlation for Noncontact Deformation Measurement of Strain Field and Mechanical Property

This paper presents a new method that combines a physics-based model and finite-element (FE) based digital image correlation (DIC) with an interfacial continuity condition for simultaneous deformation measurements of strain field and material mechanical properties. The dual-domain global DIC method incorporates a model-based kernel that offers useful physical insights into the complex deformation and constrains the solutions to the iterative minimization of the DIC criterion, which leads to a faster convergence. Unlike local subset-based and global FE-based methods where the DIC is formulated to solve for the nodal displacements, the model-based DIC that accounts for complex deformation in the specific region is formulated to identify system parameters. Both domains, along with their interface and the continuity condition enforced on the global displacement field, are discretized and solved simultaneously in a single calculation. The dual-domain global DIC method has been investigated by comparing reconstructed strain fields and estimated parameters against targets simulated using the commercial FEA software COMSOL and published experimental data. Unlike FE-DIC which is noise sensitive, the model-based DIC is robust against the Gaussian noise effects.

Boundary layer effect at the edge of fibrous composites using homogenization theory

Many studies in the theory of composite materials are based on the homogenization approach, which consists of the substitution of the heterogeneous medium by a homogeneous one with certain effective properties. These effective properties are obtained from internal asymptotics which excludes boundary layer effects. However, at the boundary of the individual ply the periodic solution is disrupted. This loss of periodicity of the interior asymptotic solution needs to be accounted for. Here, the boundary layer solution is a correction in the micro solution which accounts for the applied boundary condition. This error gives rise to a sharp boundary layer which decays very quickly when it travels to the interior of the domain, leaving behind a tail which has to be added to the interior solutions to get the correct representation of the global displacement field. It is shown that this correction is significant, local and affects local stress and strain predictions, which are missed by meso or classical micro mechanical solutions. This important outcome of the study proves the theoretical estimates available in the literature (via error estimates in L2 and H1 - norms) emphasizing significant improvement in the local solution at the boundary. It is shown, through numerical examples, that the corrected solution can significantly alter damage initiation predictions at the boundary.

Kirigami Mechanics as Stress Relief by Elastic Charges.

We develop a geometric approach to understand the mechanics of perforated thin elastic sheets, using the method of strain-dependent image elastic charges. This technique recognizes the buckling response of a hole under an external load as a geometrically tuned mechanism of stress relief. We use a diagonally pulled square paper frame as a model system to quantitatively test and validate our approach. Specifically, we compare nonlinear force-extension curves and global displacement fields in theory and experiment. We find a strong softening of the force response accompanied by curvature localization at the inner corners of the buckled frame. Counterintuitively, though in complete agreement with our theory, for a range of intermediate hole sizes, wider frames are found to buckle more easily than narrower ones. Upon extending these ideas to many holes, we demonstrate that interacting elastic image charges can provide a useful kirigami design principle to selectively relax stresses in elastic materials.

Structural intensity on shell structures via a Finite-Element-Method approximation

The study of energy transmission on plate-like structures is widely documented in literature. Its analysis requires the computation of the spatial derivatives of the out-of-plane displacement field of the studied sample and a priori knowledge of its material properties. However, if the structural intensity is to be assessed on irregular shells, such a study requires a more elaborate data processing. In addition to in-plane displacements, also the spatial derivatives along the sample’s local coordinates are needed. For this purpose, a method was developed to approximate both the experimental displacement data and the spatial coordinates of a given arbitrary shell using a Finite-Element-Method model. After measuring the global displacement fields and their corresponding spatial coordinates for a given sample, the data was transferred to a shell model whose basis functions were properly defined in accordance to the application’s demands and whose individual elements were assumed to behave in accordance with the Kirchhoff plate theory. The proposed method was able to process the experimental displacement and shape data of shells structures and proved to be a reliable tool to assess energy transmission.

Efficient 2D thermo-mechanical analysis of composites and sandwich laminates

ABSTRACTThe static behavior of composites and sandwich plates in thermo-mechanical environment is investigated by a two dimensional (2D) FE model. An efficient higher-order zig-zag theory (HOZT) considering actual through-thickness temperature profile and a least square error (LSE) method to accurately predict the inter-laminar stresses is implemented in this model. The in-plane displacement field is obtained by superposing a cubically varying global displacement field on a zig-zag displacement field having different slopes at each layer. This plate theory represents parabolic through thickness variation of transverse shear stresses, which satisfy the inter-laminar continuity at the layer interfaces and zero transverse shear stress conditions at the top and bottom of the plate. In the present 2D finite element (FE) model, the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C1 continuity associated with the above plate theory (HOZT). The accurate through-thickness distribution of temperature is obtained by using a linear zig-zag thermal lamination theory proposed by the authors by using the thermal conduction properties of different constituent layers in the thickness direction. The LSE method is applied at the postprocessing stage to accurately calculate the inter-laminar stresses by the 3D equilibrium equations of the plate problem, after in-plane stresses are calculated. The proposed combined FE model (HOZT+LSE) is implemented to analyze the static behavior of laminated composites and sandwich plates subjected to thermo-mechanical loadings. Many new results are also presented that should be useful for the future reference.

Quasi-static tensile properties of the Cranial Cruciate Ligament (CrCL) in adult cattle: towards the design of a prosthetic CrCL.

Mechanical properties of the Cranial Cruciate Ligament (CrCL) in adult cattle are not well documented and protocols used in the literature focus on testing a full femur-CrCL-tibia complex rather than an isolated CrCL. The aim of this study was to assess a wider range of tensile properties of the CrCL along its anatomic axis with experimental measurements of the global elongation, displacement and strain fields, in order to provide guidelines for the design of CrCL prosthetic surrogates. Fourteen bovine CrCL were harvested from seven mature cows (5.1 ± 1.3 years) weighing 631 ± 90kg. The mean CrCL length was 41.4 ± 1.5mm and its mean cross-section was 103.9 ± 23.8mm2. Pre-conditioning was achieved with 30 cycles of loading from 30 to 200N at a strain rate of 0.02s-1. Specimens were then loaded to failure at the same strain rate. The following results were obtained: the mean ultimate tensile load (UTL) 4372 ± 1485N and the median [quartiles] maximal global elongation 19.3 [17.8; 21.4] %. At first physical signs of tearing, the mean load was 3315 ± 1336N and mean elongation 13.5 ± 4.9%. The mean absorbed energy at failure was 5.23 ± 2.08 MJ.mm-3 and the mean stiffness at various levels of elongation was: 220 ± 195N.%-1 (5%), 285 ± 162N.%-1 (10%), 239 ± 200N.%-1 (15%), 146 ± 59N.%-1 (20%), 153 ± 136N.%-1 (25%). None of these properties were related to the bovine weight, age and side of the body (p > 0.05). An ideal prosthetic surrogate should then follow these sets of properties and the experimental data suggest that the in-vivo maximal elongation is below 13.5%.

Multiscale structural analysis of textile composites using mechanics of structure genome

Mechanics of structure genome (MSG) is used to develop an approach for multiscale structural analysis of textile structures. First, MSG is used to predict the properties of yarns having realistic geometry. Then, beam and plate stiffness matrices are predicted based on the yarn and matrix properties using MSG. These beam and plate stiffness matrices are used to perform structural analysis of textile beams and plates. The computed global responses are used to conduct dehomogenization to obtain local stress fields. The MSG-based global displacement and local stress fields are compared with those predicted by direct numerical simulation (DNS) to compare accuracy and computational efficiency. An excellent agreement was observed for both global displacement and local stress field results, while the computational cost and modeling effort of MSG-based analysis are significantly lower than those of DNS.

A composite beam element with through the thickness capabilities based on global-local superposition

A beam element is proposed that captures through the thickness effects in composite laminated beams, namely, transverse shear and normal stresses and strains. Stress continuity along the thickness is inherently enforced leading to a system of algebraic equations that is solved in the element level, permitting independence between the number of layers and the number of degrees of freedom, with all of them possessing a clear physical significance. Global-local superposition is performed in the thickness direction, where a cubic global displacement field, that guarantees imposition of the boundary conditions at the top and bottom surfaces of the beam, is combined with a layerwise linear local displacement distribution that assures zig-zag behavior of the stresses and displacements. The element behavior for different length-to-thickness ratios is assessed and compared to the analytical elasticity solution, as well as a commercial finite element alternative.

Structural shape reconstruction of fiber Bragg grating flexible plate based on strain modes using finite element method

Structural shape reconstruction plays an important role in structural health motoring, for example, in the fields of aerospace and civil engineering. A reconstruction algorithm is indispensable for structural shape reconstruction using discrete physical information. This article shows how to implement structural shape reconstruction using a small number of stain data measured by fiber Bragg grating sensors. First, the basic theory of structural shape reconstruction is introduced based on strain modes. A transformation from the measured discrete strain data to global displacement field is done by modal coordinate, which is the same for strain mode shape superposition and displacement mode shape superposition. Second, sensor layout optimization is carried out. Optimization analyses for the number of sensors and sensor position are implemented, and results show a better reconstruction effect for the optimized sensor layout. Meanwhile, effective parameters of the sensor layout are discussed, such as the excitation frequency, excitation amplitude, and number of mode shapes. Finally, structural shape reconstruction algorithm based on strain modes is applied in experiments. Static deformation and dynamic deformation tests are conducted to verify the proposed approach. Their results show that the reconstructed displacements match well with those measured by a laser displacement sensor.