Constrained Boundary Conditions Research Articles

Uniaxial compression performance of anti-tetrachiral structures considering the effects of cell size and boundary conditions

Anti-tetrachiral structures (AS) are typical metamaterials known for their negative Poisson's ratio. The existing analysis of the mechanical properties of AS is conducted by applying the energy method to a unit cell with periodic boundary conditions (PBC). In available works, the shear force at the structure's boundaries is neglected. But is it permissible to disregard the impact of shear forces at the boundaries on the structure's equivalent mechanical properties? By examining the deformation relationship between the ribs and the nodal rings, we developed a uniaxial compression model for AS under both free and constrained boundary conditions, which accurately predicts the mechanical properties of the AS structure. This model was validated through numerical simulations and experiments. The findings reveal that the equivalent mechanical properties of AS exhibit a size dependence related to the cell size. For example, for AS with equivalent density and identical overall dimensions, the equivalent Young's modulus of an AS with 2×2 cells will be twice that of an AS with 4×4 cells. Furthermore, the size effect of the structure can be neglected when the number of cells larger than 8×8. Moreover, it is found that the present model considering boundary conditions exhibits an equivalent Young's modulus 25% higher than the model neglecting boundary conditions. The study's findings indicate that the presence of boundary conditions can disrupt PBC, leading to significant discrepancies between theoretical derivations and practical applications.

A Study of Deployable Structures Based on Nature Inspired Curved-Crease Folding.

Fascinating 3D shapes arise when a thin planar sheet is folded without stretching, tearing or cutting. The elegance amplifies when the fold/crease is changed from a straight line to a curve, due to the association of plastic deformation via folding and elastic deformation via bending. This results in the curved crease working as a hinge support providing deployability to the surface which is of significant interest in industrial engineering and architectural design. Consequently, finding a stable form of curved crease becomes pivotal in the development of deployable structures. This work proposes a novel way to evaluate such curves by taking inspiration from biomimicry. For this purpose, growth mechanism in plants was observed and an analogous model was developed to create a discrete curve of fold. A parametric model was developed for digital construction of the folded models. Test cases were formulated to compare the behavior of different folded models under various loading conditions. A simplified way to visualize the obtained results is proposed using visual programming tools. The models were further translated into physical prototypes with the aid of 3D printing, hybrid and cured-composite systems, where different mechanisms were adopted to achieve the folds. The prototypes were further tested under constrained boundary and compressive loading conditions, with results validating the analytical model.

Geometry correction factor polynomials for corner crack test pieces accounting for constrained boundary conditions

Geometry correction factor polynomials for corner crack test pieces accounting for constrained boundary conditions

Experimental investigation on vortex-induced vibration characteristics of a segmented free-hanging flexible riser

Different from the traditional marine riser, the vertical lifting pipeline in the deep-sea mining system can be regarded as a free-hanging flexible riser with an unconstrained bottom end. Likewise, problems in terms of vortex-induced vibration (VIV) and flexible deformations can arise during operation. This paper presents experimental tests on a segmented free-hanging marine riser used for deep-sea mining, focusing on the vortex-induced vibration. The riser was formed by connecting pipe sections using flanges, with the top fixed and a lumped mass block added at the bottom to simulate the lifting pump and buffer station, resulting in weakly constrained boundary conditions. By employing the modal superposition principle and the wavelet time-frequency method, we comprehensively investigated the response characteristics of the typical pipe sections and the coupled oscillation of the riser's vibration modes. During the experiment, two important phenomena that differed from previous VIV tests were observed. Firstly, at low reduced velocities (Ur ≤ 7.77), the dominant frequency no longer conformed to the doubling principle in both directions but remained consistent. Secondly, the weak constraint effect of the concentration of mass at the bottom disrupts the formation of vortex shedding, thereby reducing the bending response of the CF direction of the bottom pipe sections. Meanwhile, the drag force enhances the bending response of the IL direction. Additionally, the middle and bottom pipe sections of the riser exhibited obvious mode competition, with the dominant mode often followed by several subordinate modes with considerable vibration energy.

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

AbstractConstruction of absorbing boundary conditions (ABCs) for nonlocal models is generally challenging, primarily due to the fact that nonlocal operators are commonly associated with volume constrained boundary conditions. Moreover, application of Fourier and Laplace transforms, which are essential for the majority of available methods for ABCs, to nonlocal models is complicated. In this paper, we propose a simple method to construct accurate ABCs for peridynamic scalar wave‐type problems in viscous media. The proposed ABCs are constructed in the time and space domains and are of Dirichlet type. Consequently, their implementation is relatively simple, since no derivatives of the wave field are required. The proposed ABCs are derived at the continuum level, from a semi‐analytical solution of the exterior domain using harmonic exponential basis functions in space and time (plane‐wave modes). The numerical implementation is done using a meshfree collocation approach employed within a boundary layer adjacent to the interior domain boundary. The modes satisfy the peridynamic numerical dispersion relation, resulting in a compatible solution of the interior region (near‐field) with that of the exterior region (far‐field). The accuracy and stability of the proposed ABCs are demonstrated with several numerical examples in two‐dimensional unbounded domains.

A variational RVE-based multiscale poromechanical formulation applied to soft biological tissues under large deformations

Investigations on micromechanics of soft biological tissues may lead to significant contributions within the fields of mechanobiology and tissue engineering. Multiscale models can be interesting numerical tools, since they may provide important insights on the micromechanics of these tissues. Due to the several multiphysics phenomena observed in soft biological tissues, the coupling between the fluid transport throughout the solid phases is one of major interest. In this regard, poromechanical models are widely used to represent the fluid diffusion through a deformable porous media. Motivated by these facts, this work presents an extension of the variational single-field multiscale formulation considering poromechanical phases at finite strain aiming at applications to soft biological tissues. To verify the suitability of the proposed formulation, numerical examples are compared with direct numerical simulations. In this case, two states are considered: an undrained and a drained condition. The results point out that the minimally constrained boundary condition for the fluid flow seems to properly represent only an undrained condition. However, for the drained one, proper multiscale boundary conditions must be developed.

Space-charge-limited current density for nonplanar diodes with monoenergetic emission using Lie-point symmetries.

Understanding space-charge-limited current density (SCLCD) is fundamentally and practically important for characterizing many high-power and high-current vacuum devices. Despite this, no analytic equations for SCLCD with nonzero monoenergetic initial velocity have been derived for nonplanar diodes from first principles. Obtaining analytic equations for SCLCD for nonplanar geometries is often complicated by the nonlinearity of the problem and over constrained boundary conditions. In this Letter, we use the canonical coordinates obtained by identifying Lie-point symmetries to linearize the governing differential equations to derive SCLCD for any orthogonal diode. Using this method, we derive exact analytic equations for SCLCD with a monoenergetic injection velocity for one-dimensional cylindrical, spherical, tip-to-tip (t-t), and tip-to-plate (t-p) diodes. We specifically demonstrate that the correction factor from zero initial velocity to monoenergetic emission depends only on the initial kinetic and electric potential energies and not on the diode geometry and that SCLCD is universal when plotted as a function of the canonical gap size. We also show that SCLCD for a t-p diode is a factor of four larger than a t-t diode independent of injection velocity. The results reduce to previously derived results for zero initial velocity using variational calculus and conformal mapping.

Thermo-mechanical buckling analysis of thick beams reinforced with agglomerated CNTs with temperature-dependent thermo-mechanical properties under a nonuniform thermal loading

In this article, the thermo-mechanical buckling analysis of composite beams reinforced with carbon nanotubes (CNTs) subjected to a nonuniform thermal loading is studied. To increase the accuracy of results, the beam is modeled based on the quasi-3D sinusoidal shear deformation beam theory which considers shear deformation and thickness stretching. Temperature-dependent thermo-mechanical properties care considered for both polymeric matrix and CNTs. The CNTs are orientated randomly and distributed along thickness direction based on various symmetric and nonsymmetric patterns and agglomeration of CNTs is incorporated. The heat conduction equation is solved analytically to obtain the temperature profile along the thickness direction. The governing equations and associated boundary conditions regarding the thermo-mechanical analysis of the beam are derived using Hamilton’s principle and solved numerically using the differential quadrature method. Convergence and accuracy of the presented solution are confirmed, and the influences of various parameters on the thermo-mechanical stability regions are investigated, such as boundary conditions, thickness-to-length ratio, mass fraction and distribution pattern of the CNTs, and the CNTs agglomeration parameters. Numerical results reveal that to achieve the most expanded thermo-mechanical stability regions, it is more helpful to distribute the CNTs as far as away from the middle axis of the beam, especially near the surface with lower temperature. It is observed that neglecting the temperature-dependency of thermo-mechanical properties results in an overestimate of the critical temperature, and this overestimate is more noticeable for the beams with more constrained boundary conditions.

Experimental Study of Modal Characteristics for Heated Composite Structures

Aircrafts with high Mach speeds are exposed to severe thermal load, which affect the performance of aircraft components. Although several experimental studies have been performed on the mechanical behavior of aircraft structures subjected to high temperatures, the experimental research on dynamic characteristics of structures under thermal load is limited. In this work, a series of laboratory modal tests have been carried out on carbon fiber reinforced silicon carbide matrix (C/SiC) composite plates under various radiant-heating conditions. High temperatures are achieved using quartz lamps, and their dynamic responses are measured using a scanning laser vibrometer. The modal parameters are evaluated by PolyMAX at several static temperature distributions. The temperatures are acquired by both full-field noncontacting infrared thermography and contacting thermocouples. The flat plates are with both free and constrained boundary conditions and the high temperature modal survey are performed for two kinds of thickness. The stiffened plate with free boundary condition is further studied as a more practical structure. Both the changes of the modal frequency and modal shape as temperature rises are presented. The effects of the thermal loads on the modal characteristics are investigated by test. The results show that the thermal stresses have a much greater effect on the modal characteristics than the change of the material properties due to heating. The test results could provide technical material for the limited database of high-temperature modal testing results.

Memory efficient constrained optimization of scanning-beam lithography.

This article describes a memory efficient method for solving large-scale optimization problems that arise when planning scanning-beam lithography processes. These processes require the identification of an exposure pattern that minimizes the difference between a desired and predicted output image, subject to constraints. The number of free variables is equal to the number of pixels, which can be on the order of millions or billions in practical applications. The proposed method splits the problem domain into a number of smaller overlapping subdomains with constrained boundary conditions, which are then solved sequentially using a constrained gradient search method (L-BFGS-B). Computational time is reduced by exploiting natural sparsity in the problem and employing the fast Fourier transform for efficient gradient calculation. When it comes to the trade-off between memory usage and computational time we can make a different trade-off compared to previous methods, where the required memory is reduced by approximately the number of subdomains at the cost of more computations. In an example problem with 30 million variables, the proposed method reduces memory requirements by 67% but increases computation time by 27%. Variations of the proposed method are expected to find applications in the planning of processes such as scanning laser lithography, scanning electron beam lithography, and focused ion beam deposition, for example.

Leaf Spring of a Lightweight Commercial Vehicle Suspension System: Structural Analysis

Mathematical and computer modeling have been playing an increasingly important role in the computer aided engineering (CAE) process of many products since last 60 years. The present work consists of a leaf spring involving two supporting leafs with the main Leaf. Main leaf and the centre leaf are having a uniform cross section throughout the length of the blade, while the bottom leaf having a thickness of around 1.5 times of that of the other two. Moreover a central bolt and two supporting U-Bolts are provided to hold leafs together. The load is applied axially downwards at the eye ends of the main leaf and a double magnitude load is applied at the bottom of the bolt. Here in the present model we have taken the steel material for the analysis purpose and the nut is firmly fixed by giving a fully constrained boundary condition on it. An octree 10 node modified tetrahedron mesh element is used for the discretization of the spring model. For the whole work i.e., from the modeling to the result generation CATIA V5R12 software is used and the obtained results are finally compared with the numerically obtained result of stress and deflection to validate the simulation results. Vehicle whose, leaf spring dimension are taken is light commercial vehicle (Ashok Leland Dost).

Uncertainty analysis and stochastic characterization of carbon nanotube-based mass sensor with multiple deposited nanoparticles

Input uncertainties on the location, mass, and geometry of multiple deposited nanoparticles for carbon nanotube-based mass sensor are investigated. The carbon nanotube (CNT) is modeled as an Euler-Bernoulli beam with end constrained boundary conditions and Eringen’s nonlocal elasticity theory to account for size-dependent phenomena. Additionally, the displacement field of the CNT is broken up into components to the left and right of each deposited particle. Utilizing the Hamilton’s principle and the discretized displacement fields, the nonlocal governing equations of motion, boundary conditions, and continuity conditions at the deposited particle locations are derived. The effects of the particle location, mass, and geometry on the inherent frequency shifts before and after deposition are studied. Following these parametric studies, sensitivity analysis and uncertainty quantification methods are applied for two different nominal configurations of the mass sensor. The importance of the input parameters is deeply discussed, focusing especially on the location effects on the first and second natural frequency shifts. It is shown that increasing the mass and size of the particle increases the overall frequency shift, while varying the location of the particles does not strictly lead to increased frequency shifts. Finally, it is shown that due to the high number of input parameters, there are multiple particle configurations that lead to identical frequency shifts. These findings may be used by other researchers to extend the usability, usefulness, and characterization of mass detection devices.

Geometric modeling of complex knitting stitches using a bicontinuous surface and its offsets

Helicoids have been utilized as a scaffold on which to define the topology and geometry of yarns in a weft-knitted fabric. The centerline of a yarn in the fabric is specified as a geodesic path, with constrained boundary conditions, running along a helicoid at a fixed distance. The properties and constraints of the yarn are formulated into a single “energy” function, which is then minimized to produce the desired resulting models. We present improvements to this approach that address the deficiencies of the original work and extend its capabilities to more complex stitches, such as transfer, tuck and miss. A single bicontinuous surface is described, which replaces discrete helicoids and produces higher quality, continuous yarn models. A new computational method is employed that significantly speeds up the optimization computations. Including offset surfaces with the scaffold, as well as removing sections of the scaffold, allow for the modeling of complex stitches. The improved approach produces superior geometric results, consisting of complex knitting stitches, at a fraction of the computational cost of the previous method.

CRACK PROPAGATION BEHAVIOR OF LATERALLY CONSTRAINED POLYMERS USED AS DIELECTRIC ELASTOMERS

ABSTRACT Dielectric elastomer-based transducers are rapidly gaining importance with the syntheses of new polymers that can potentially be used as dielectric materials. However, these materials are always prone to fracture in the presence of cracks and flaws. Failures originate from flaws (or notches), and a complete fracture may take place due to the propagation of cracks. The present work investigates the crack propagation behavior of two popular polymers, VHB 4910 and Ecoflex, that are widely used as dielectric elastomers. In this case, tensile loadings in laterally constrained boundary conditions are considered. The average crack propagation speed for Ecoflex is higher than that for VHB, implying that Ecoflex will fail earlier than that of VHB under similar conditions. Moreover, with increasing notch lengths at a fixed strain rate, the average crack propagation speed decreases appreciably but becomes constant for comparatively larger notches. The results also conclude that the average crack propagation speed and normalized crack tip diameter remain higher for VHB than for Ecoflex for larger normalized notch lengths. It is observed that the average crack propagation speed increases with strain rates, whereas the normalized crack tip diameter is independent of strain rates. Experimental results obtained here will provide a useful comparative insight to understand the failure behavior of two polymers widely used as dielectric elastomers.

Vibration characteristics analysis of an elastically restrained cylindrical shell with arbitrary thickness variation

A novel and unified solution is established to investigate the vibration characteristics of circular cylindrical shells with arbitrary variable thickness and general boundary conditions. Energy equation based on Donnell–Mushtari shell theory is formulated, with a modified Fourier series utilized for the construction of radial, circumferential and longitudinal displacements of shell structure, in which the auxiliary terms are introduced to tackle with differential discontinuity associated with elastic boundary restraints. Arbitrary thickness function is invariantly expanded into Fourier series to compress thickness variation information into Fourier coefficients accordingly. The proposed model is validated through the comparisons with those in the literature or finite element method. Excellent convergence can be observed with several truncated terms of modified Fourier series. Natural frequencies of a circular cylindrical shell with linearly varying thickness and constant mass decrease for constrained boundary conditions, and the distribution of modal displacement moves to the thinner side as the slope ratio increases. Axial thickness variation has more significant influence on natural frequencies of a step-wise cylindrical shell compared with that in circumferential direction.

Experimental study and numerical simulation of temperature gradient effect for steel-concrete composite bridge deck

The temperature distribution of the bridge and its thermal effect has always been an important issue for researchers. To investigate the temperature distribution and thermal stress in the steel-concrete composite bridge deck, a 1:4 ratio temperature gradient effect experimental study was carried out in this paper. First, a set of experimental equipment for laboratory temperature gradient loading was designed based on the principle of temperature gradient caused by solar radiation, the temperature gradient obtained from the measurements were compared with the specifications and verified by the FE method. Next, the loading of the steel-concrete composite deck at different temperatures was performed. The thermal stress response and change trend of the simply supported and continuously constrained boundary conditions under different temperature loads were analyzed. The experimental results show that the vertical temperature of steel-concrete composite bridge deck is nonlinear, which is consistent with the temperature gradient trend of specifications. The vertical temperature gradient has a great influence on the steel-concrete composite bridge deck under different constraints, and the extreme stress of concrete slab and steel beam is almost linear with the temperature gradient. Finally, some suggestions for steel-concrete composite deck design were provided based on the research results.

Thermo-mechanical characterisation of NiTi-based shape memory alloy wires for civil engineering applications

Shape Memory Alloys (SMAs) exhibit a complex material behaviour due to thermo-mechanical coupling. At present, the understanding of their material behaviour under different thermo-mechanical loading conditions relevant to civil engineering applications is lacking even though they have been widely used in the past decade or so, for example mechanical behaviour at constant stress but variable temperature loading has not received much attention in civil engineering. In this study a comparative analysis of the mechanical behaviour of various NiTi-based SMAs is carried out to investigate: (1) stress-elongation and stress-temperature response under direct tensile loading; (2) thermo-mechanical behaviour under constant stress condition but under variable temperature loading (to investigate their ability to exhibit shape memory effect, if any, while carrying load); and (3) the ability to develop and retain recovery stress at room and sub-zero temperatures. Essentially, all these tests aim at investigating the potential/limitations of the SMAs in civil engineering in particular re-centring and heat-activated prestressing applications. The results from this study showed that at a constant stress of 600 MPa a large forward transformation strain, in the range of 5%–11%, is produced in NiTi-based SMAs when the temperature is lowered to or below the room temperature. This material behaviour of NiTi-based SMAs could be detrimental in many civil engineering applications of NiTi-based SMAs. However, all NiTi-based SMAs used in this study were found to have a good potential for re-centring applications, but only austenitic NiTi-based SMAs were found suitable for long-term prestressing applications. The maximum recovery stress developed on heating with constrained boundary conditions ranged between 230–750 MPa.

Low-noise hierarchical phase unwrapping method for dual-wavelength digital holography using two synthetical wavelengths

Dual-wavelength digital holography can expand the unambiguous measurement depth in phase unwrapping by using a differential synthetic wavelength which is longer than the single illumination wavelength. However, the phase noise is significantly amplified due to the magnification of the differential synthetic wavelength, resulting in a lower measurement accuracy. On the other hand, a lower noise level can be achieved by using additive synthetic-wavelength which is shorter than the single illumination wavelength. However, the corresponding unambiguous measurement depth is greatly reduced due to the phase ambiguity. In this case, combining the merits of the differential synthetic-wavelength and the additive synthetic-wavelength, different low noise phase unwrapping algorithms have been developed in recent years. However, these algorithms are complex and time consuming because they need to calculate multiple intermediate variables or search for the constrained boundary conditions in two-dimensional space. Therefore, in this paper, we develop a hierarchical phase unwrapping algorithm by using the two synthetic wavelengths for dual-wavelength digital holography to realize low noise and fast unambiguous measurement with large depth. In this algorithm, the unwrapped phase difference obtained by the differential synthetic wavelength is used to guide the wrapped phase of one single wavelength to realize phase unwrapping, and then the optical path difference obtained by the single-wavelength unwrapped phase is employed to guide the wrapped phase sum, and thus realizing phase unwrapping. As a result, the phase noise is attenuated and the depth sensitivity is preserved for dual-wavelength phase unwrapping. After theoretical analysis, a series of simulation experiments is carried out on the reconstructed quality, anti-noise characteristics and speed through comparing with state-of-the-art dual-wavelength phase unwrapping algorithms, including the conventional algorithm, the linear programming algorithm and the direct linear programming algorithm. In this case, a flipping dual-wavelength common-path digital holography with orthogonal carrier is built to acquire multiplexed off-axis hologram in one shot and illustrate the operation of the algorithm with circular step target, and stability test of the setup. Both the simulation and experimental results show that the proposed method can be simplified and deterministic, resulting in a lower noise phase unwrapping in a time of 20.5 ms for a phase map of one megapixel. We expect that the proposed method can have practical applications in measurement that requires high accuracy, fast speed, and large depth.

Generalized Finite Difference Method for Plate Bending Analysis of Functionally Graded Materials

In this paper, an easy-to-implement domain-type meshless method—the generalized finite difference method (GFDM)—is applied to simulate the bending behavior of functionally graded (FG) plates. Based on the first-order shear deformation theory (FSDT) and Hamilton’s principle, the governing equations and constrained boundary conditions of functionally graded plates are derived. Based on the multivariate Taylor series and the weighted moving least-squares technique, the partial derivative of the underdetermined displacement at a certain node can be represented by a linear combination of the displacements at its adjacent nodes in the GFDM implementation. A certain node of the local support domain is formed according to the rule of “the shortest distance”. The proposed GFDM provides the sparse resultant matrix, which overcomes the highly ill-conditioned resultant matrix issue encountered in most of the meshless collocation methods. In addition, the studies show that irregular distribution of structural nodes has hardly any impact on the numerical performance of the generalized finite difference method for FG plate bending behavior. The method is a truly meshless approach. The numerical accuracy and efficiency of the GFDM are firstly verified through some benchmark examples, with different shapes and constrained boundary conditions. Then, the effects of material parameters and thickness on FG plate bending behavior are numerically investigated.

Analytical and Numerical Analysis of Additional Stress in Foundation of Bridge Approach Embankment

A good estimation of additional stress of a bridge approach foundation overlying embankment is of great importance in order to accurately calculate the differential settlements of foundation due to the axisymmetric distribution of the additional stress. Current design method commonly adopted in road and railway engineering in China is derived from the classical elastic theory and simply considers the embankment load by its self-weight (γh) without taking into account the influence of the width-to-height ratio. This may lead to an overestimation of additional stress and consequently an uneconomical overdesign. This study attempts to mitigate the gap by developing a new analytical solution. Based on the Boussinesq’s elastic theory, the proposed solution considers a constrained boundary condition associated with bridge abutment and incorporates a new concept of ground reaction, which takes into account the embankment load including the influence of the embankment width-to-height ratio. The proposed solution was finally validated by numerical analysis and shows a better performance in terms of additional stress estimation over the traditional calculation method. Overall, the proposed method proved to be a better alternative tool for the design of bridge approach embankment.