Geometric Nonlinearity Research Articles

Symmetric and asymmetric vibration and buckling of a thermal rotating annular-plate with Pasternak foundation

This paper studies the symmetric and asymmetric vibration and buckling modes of a rotating thermal annular plate resting on a Pasternak foundation. With considering the von Kármán geometric nonlinearity and the first-order shear deformation theory, the governing equations of the system are established via Hamilton's principle, and then solved by using the differential quadrature method. With both inner and outer boundaries constrained, the modes of the rotating system vary differently depending on the mode symmetry. The symmetric modes are mainly affected by the generated centrifugal force, while the asymmetric ones are affected by both centrifugal and Coriolis forces. The additional Coriolis, with mainly two directions, strengthens the forward travelling parts of the asymmetric modes, but makes the backward counterparts buckled more easily. Also, the thermal effect, as well as boundary elasticity and foundation coefficients, are also investigated from the viewpoint of mode symmetry. Results show that the temperature rise can shift the fundamental vibration mode from the symmetric one to the asymmetric one and backward travelling waves for rotation, which also adds to the complexity of mode variation with joint parameters. For the joint effect of the temperature rise and rotating, the annular plate tends to buckle more readily while the effects of the two factors are linearly superimposed.

A Foundational Study on Rational Optimization of Damping Ratio for Accurate Dynamic Simulation with Ultra Large Displacement

The integration of dynamic simulation analysis has become widespread in general-purpose software, providing enhanced capabilities. However, accurately tracking deformations based on complete equilibrium solutions remains a significant challenge in problems characterized by strong geometric nonlinearity. This study examines the accuracy of the combined Newmark β method and Tangent stiffness method in dynamic analysis with ultra large displacements and evaluates the utility of Rayleigh proportional damping in numerical simulations compared to experimental models. An experimental model of a slender steel plate undergoing free vibrations after being released from a deformed state was created. Video footage capturing the deformation histories was compared to computational simulations to verify accuracy. The study also examines the appropriate values of the damping ratio (ζ) and the Newmark β value in the simulations. The results indicate that adjusting various damping ratio and a β value of 1/2 yield more realistic simulations with longer conservation of mechanical energy. The findings suggest that incorporating numerical damping into actual damping settings can achieve a more realistic simulation of dynamic behaviour with ultra-large displacements. Furthermore, experiments and analyses were performed to correct natural frequency by changing Young’s modulus, which is a key factor influencing natural frequencies, with observed correlations to plate thickness, setting the stage for further research under varied conditions to develop a more rational methodology for structural analysis.

Role of inertial nonlinearity and coupling stiffness on a series of coupled harvesters

This study investigates the effect of coupling configuration on a state-of-the-art harvesting system consisting of a series of spring-coupled piezoelectric nonlinear monostable tip-loaded cantilever-based vibration energy harvesters (VEHs). The investigation finds favorable coupling configurations that exploit complex nonlinear interaction for improving system performance. The system model incorporates the beam's geometric nonlinearities in the governing equation using inertial and stiffness nonlinear terms. The study first analytically investigates the effect of grounding stiffness on a single harvester's performance. Then, it numerically investigates the effect of increasing array size and changing stiffness distribution. The study discovers a novel hardening/softening transition phenomenon at a critical grounding stiffness value induced by the inertial nonlinear effect. This critical event also maximizes peak output power near the largest eigenfrequency at an optimum grounding stiffness value. At larger arrays, the peak power shows a complex bifurcation behavior, hinting at the presence of multiple attractors. The numerical study of different stiffness configurations and linearized eigenanalysis reveals the necessity of a multilevel optimization strategy focused on both the stiffness ratio between the coupling springs' and their overall stiffness level. This study bridges the gap between previous studies on spring-coupled linear and bistable arrays, offering new insights into coupled nonlinear interactions and proposing novel optimization strategies.

Size-dependent nonlinear free vibration of magneto-electro-elastic nanobeams by incorporating modified couple stress and nonlocal elasticity theory

Magneto-Electro-Elastic (MEE) Composites, as an innovative functional material blend, are composed of multiple materials, boasting exceptional strength, rigidity, and an extraordinary magneto-electric interaction effect. This paper establishes a nonlocal modified couple stress (NL-MCS) magneto-electro-elastic nanobeam dynamic model. To accurately capture the intricate influences of scale effects on nanostructures, This model meticulously examines scale effects from two distinct perspectives: leveraging nonlocal elasticity theory to elucidate the softening phenomena in nanostructures stemming from long-range particle interactions, and employing modified couple stress theory to reveal the hardening effects attributed to the rotational behavior of particles within the structure. By incorporating Von Karman geometric nonlinearity, Reddy’s third-order shear deformation theory and Maxwell’s equations, the governing equations for the nonlinear free vibration of MEE nanobeams are derived using Hamilton’s principle. Finally, a two-step perturbation method is employed to solve these equations. Two-step perturbation method disintegrates the solution process into two stages, iteratively approximating and refining the solution, thereby progressively unraveling the intricate details and enhancing the precision of the solution in a systematic manner. Finally, the nonlinear free vibration behavior of MEE nanobeams is explored under the coupled magnetic-electric-elastic fields, with a focus on the effects of various factors that including length scale parameters, nonlocal parameters, Winkler-Pasternak coefficients, span-to-thickness ratios, applied voltages and magnetic potentials.

Numerical Simulation for Strength and Stability of RC Tapered Columns

This study investigates the strength and stability of Reinforced Concrete (RC) linearly tapered square columns. Evaluating the slenderness ratio of an RC column requires the radius of gyration of its cross-section, which is well-defined for a prismatic column but not for a non-prismatic one. This study primarily investigates the application of the ACI Code formulae to evaluate the slenderness ratio of RC columns with the studied geometry. Validated numerical models, using Abaqus, were employed to perform nonlinear first- and second-order analyses on the investigated columns subjected to eccentric axial loads. The concrete damaged plasticity model was employed to simulate the nonlinear behavior of concrete. The static Riks solver, available in Abaqus, was utilized for nonlinear analyses: first, with an inactivated geometric nonlinearity for a first-order analysis, and second, with an activated geometric nonlinearity to consider the effects of secondary moments (p-δ effects). The findings indicate the reliability of defining the slenderness ratio of an RC linearly tapered column based on the ACI Code formulae, using the average cross-section of the tapered column.

Study on fracture of hyperelastic Kirchhoff-Love plates and shells by phase field method

Thin walled structures such as plates and shells are widely used in many engineering fields. To Predict its fracture behavior is of great significance for integrity design and strength evaluation of engineering structures. Numerical simulation of the fracture behavior of hyperelastic plates and shells is a challenge due to complex kinematic description, hyperelastic constitutive relationship, geometric nonlinearity and the degradation on elastic parameter caused by fracture damage. Combining Kirchhoff Love (K-L) shell theory with the fracture phase field method, and numerically discretizing the first and second order partial derivatives of displacement field and phase field by using T-splines and meeting the requirements of K-L plate and shell theory for the C1 continuity of the shape function, a model for the isogeometric analysis numerical formulation of the phase field fracture in hyperelastic K-L plates and shells is established. The fracture failure behavior of hyperelastic K-L plates and shells under the uniform load and displacement load is simulated, and the effect of the Gaussian curvature on the fracture behavior of hyperelastic K-L shells is studied. The simulation results show that the present numerical scheme can effectively capture the complex crack propagation path of plates and shells under the uniform load, and the displacement field can effectively reflect the crack distribution of materials. The thin shell with negative Gaussian curvature shows the excellent fracture performance under the internal pressure, and can withstand the greater internal pressure.

Novel Curve-Shape Sandwich Composites with Flexible Cores for Rehabilitation of Buried Infrastructure: Experimental and Analytical Studies Considering Geometric Nonlinearity

Novel Curve-Shape Sandwich Composites with Flexible Cores for Rehabilitation of Buried Infrastructure: Experimental and Analytical Studies Considering Geometric Nonlinearity

Unravelling the mechanics of knitted fabrics through hierarchical geometric representation

Knitting interloops one-dimensional yarns into three-dimensional fabrics that exhibit behaviour beyond their constitutive materials. How extensibility and anisotropy emerge from the hierarchical organization of yarns into knitted fabrics has long been unresolved. We seek to unravel the mechanical roles of tensile mechanics, assembly and dynamics arising from the yarn level on fabric nonlinearity by developing a yarn-based dynamical model. This physically validated model captures the mechanical response of knitted fabrics, analogous to flexible metamaterials and biological fibre networks due to geometric nonlinearity within such hierarchical systems. Fabric anisotropy originates from observed yarn–yarn rearrangements during alignment dynamics and is topology-dependent. This yarn-based model also provides a design space of knitted fabrics to embed functionalities by varying geometric configuration and material property in instructed procedures compatible to machine manufacturing. Our hierarchical approach to build up a knitted fabric computationally modernizes an ancient craft and represents a first step towards mechanical programmability of knitted fabrics in wide engineering applications.

Mechanical Quantities Prediction of Metal Cutting by Machine Learning and Simulation Data

Metal cutting is an important process in industrial manufacturing. Using the mechanical quantities of metal cutting to optimize process design is helpful to improve productivity. However, it is expensive to obtain these quantities due to the complexity of the cutting process, including material nonlinearity, geometric nonlinearity, state nonlinearity and their interactions. In this paper, a prediction model is constructed by combining machine learning (ML) and simulation data to quickly acquire multi-difficult-to-obtain metal cutting mechanical quantities to solve this problem. First, Adaptive Smoothed Particle Hydrodynamics (ASPH) is used to generate a simulation dataset of 2000 metal cutting cases. Based on the simulation data, six machine learning (ML) methods are employed to establish two prediction models, single-task learning and multi-task learning, to predict the mechanical quantities of metal cutting. The experimental results demonstrate that the ML method can predict abundant reference data efficiently after understanding the relationship between simulation parameters and mechanical quantities from simulation data, which is expected to replace some similar and repetitive simulation work. The Multilayer Perceptron (MLP) model under the multi-task setting provides the best prediction performance, fastest prediction time efficiency, and stable model behavior. Additionally, input erasure experiments reveal that the prediction of maximum equivalent plastic strain is significantly affected by particle spacing, and cutting speed plays a vital role in predicting maximum velocity. This work highlights the promotion of the data-driven ML method in quickly obtaining abundant reference data for the metal cutting process, and provides an auxiliary means for process optimization.

Total harmonic distortion estimation in piezoelectric micro-electro-mechanical-system loudspeakers via a FEM-assisted reduced-order-model

Piezoelectric micro-electro-mechanical-system (MEMS) loudspeakers are attracting growing research interest in the last years due to the increasing interest towards miniaturization required by new wireless audio devices. Finite Element Models (FEM) and Lumped Element Models (LEM) able to accurately simulate their linear response have been recently proposed in the literature. However, a nonlinear model suitable to predict the Total Harmonic Distortion (THD) of these devices is to date still missing. In this work, we present a FEM-assisted lumped-parameters equivalent circuit for THD estimation which accounts for geometric nonlinearities and piezoelectric hysteresis. The loudspeaker nonlinear electro-mechanical domain is simulated through a Reduced Order Model (ROM) which considers as basis function the pre-stressed undamped actuated mode of the device computed via FEM, to account for loudspeaker diaphragms of arbitrarily complex geometry. Parameters of the acoustical circuit are computed through analytical formulas. The good matching between numerical predictions and experimental results, carried out on a piezoelectric MEMS loudspeaker prototype for in-ear condition, demonstrates the accuracy of the proposed tool.

Nonlinear Seismic Analysis of Concrete Dams Using ABAQUS-Based Scaled Boundary Finite Element Method

ABSTRACT The scaled boundary finite element method (SBFEM) is implemented in ABAQUS through the user subroutine interface. The accuracy of the proposed method and computer program are verified by using two benchmark examples. With the coupled SBFEM and standard finite element method (FEM), the dynamic interaction model of concrete dam – reservoir – foundation considering the geometrical nonlinearity of transverse joints is established using the viscous-spring absorbing boundary conditions. The overlaying element technique is applied to define the contact interface of transverse joints. The dynamic characteristics and linear seismic response of the Koyna gravity dam have been analyzed using polyhedral elements. Finally, nonlinear seismic analysis of the NG5 arch dam has been conducted. The calculated results are in good agreement with those of ABAQUS built-in elements. Moreover, numerical examples demonstrate that proposed method has high computational accuracy and can be applied to the dynamic analysis of complex engineering structures.

Nonlinear vibrations of variable speed rotating graphene platelets reinforced blades subjected to combined parametric and forced excitation

It is generally acknowledged that the disturbance of rotating speed will cause parametric resonance for blades, and the presence of rotor displacement also can make the blade vibrate transversely. However, the coupled resonance mechanism of blades under the combined action of rotating speed disturbance and rotor displacement is not yet clear. To answer this question, this article investigates the nonlinear coupled resonance behavior of rotating blades in two cases: typical tuned (the frequency ratio of parametric and forced excitations equals 2:1) and frequency ratio detuned. A nonlinear vibration equation for rotating blades is established based on Euler beam theory in conjunction with geometric nonlinearity. The mixing rule and modified Halpin-Tsai model are used to calculate the effective properties of GPLRMF materials. Subsequently, using the method of varying amplitudes (MVA), an approximate analytical solution of the coupled resonance response is derived, the Jacobian matrix is used to determine the stability of non-trivial solutions, and the accuracy of the analytical results is verified with the aid of numerical solutions. Finally, the steady-state response of typical tuned and detuned coupled resonance is analyzed, and the influence mechanism of factors such as excitation amplitude, excitation phase angle and other parameters on coupled resonance is studied in detail. The results indicate that the supercritical coupled resonance curve exhibits double resonance peaks and jumps. Meanwhile, the steady-state response of coupled resonance highly depends on the excitation phase angle.

Fluidic Multichambered Actuator and Multiaxis Intrinsic Force Sensing.

Soft robots have morphological characteristics that make them preferred candidates, over their traditionally rigid counterparts, for executing physical interaction tasks with the environment. Therefore, equipping them with force sensing is essential for ensuring safety, enhancing their controllability, and adding autonomy. At the same time, it is necessary to preserve their inherent flexibility when integrating sensory units. Soft-fluidic actuators (SFAs) with hydraulic actuation address some of the challenges posed by the compressibility of pneumatic actuation while maintaining system compliance. This research further investigates the feasibility of utilizing the incompressible actuation fluid as the means of actuation and of multiaxial sensing. We have developed a hyperelastic model for the actuation pressure, acting as a baseline pressure. Any disparities from the baseline have been mapped to external forces, using the principle of pressure-based fluidic soft sensor. Computed tomography imaging has been used to examine inner deformation and validate the analytically derived actuation-pressure model. The induced stresses within the SFA are examined using COMSOL simulations, contributing to the development of a calibration algorithm, which accounts for geometric and cross-sectional nonlinearities and maps pressure variations with tip forces. Two force types (concentrated and distributed) acting on our SFA under different configurations are examined, using two experimental setups described as "Point Load" and "Distributed Force." The force sensing algorithm achieves high accuracy with a maximum absolute error of 0.32N for forces with a magnitude of up to 6N.

Simultaneous effects of material and geometric nonlinearities on nonlinear vibration of nanobeam with surface energy effects

Simultaneous effects of material and geometric nonlinearities on nonlinear vibration of nanobeam with surface energy effects

Deployment Dynamic Modeling and Driving Schemes for a Ring-Truss Deployable Antenna

Mesh reflector antennas are widely used in space tasks owing to their light weight, high surface accuracy, and large folding ratio. They are stowed during launch and then fully deployed in orbit to form a mesh reflector that transmits signals. Smooth deployment is essential for duty services; therefore, accurate and efficient dynamic modeling and analysis of the deployment process are essential. One major challenge is depicting time-varying resistance of the cable network and capturing the cable-truss coupling behavior during the deployment process. This paper proposes a general dynamic analysis methodology for cable-truss coupling. Considering the topological diversity and geometric nonlinearity, the cable network’s equilibrium equation is derived, and an explicit expression of the time-varying tension of the boundary cables, which provides the main resistance in truss deployment, is obtained. The deployment dynamic model is established, which considers the coupling effect between the soft cables and deployable truss. The effects of the antenna’s driving modes and parameters on the dynamic deployment performance were investigated. A scaled prototype was manufactured, and the deployment experiment was conducted to verify the accuracy of the proposed modeling method. The proposed methodology is suitable for general cable antennas with arbitrary topologies and parameters, providing theoretical guidance for the dynamic performance evaluation of antenna driving schemes.

Carbon nanotubes as a basis of metamaterials and nanostructures: Crafting via design optimization

Nanotruss structures made of carbon nanotubes are investigated in two conceptual applications: either as building blocks of metamaterials or for nanostructural applications. The nanotrusses are optimized for different purposes, including various loadings, boundary conditions, parameterizations, objectives and constraints used to formulate optimization problems. The procedure relies on a recently developed framework consisting of molecular dynamics simulations, neural networks and finite elements. This framework is now used in the design optimization of nanostructures and the performances of different popular heuristic optimization methods are compared. Five applications of nanotrusses made of carbon nanotubes are analyzed in detail to investigate the mechanical behavior of such structures and the efficiency of the optimizations. Besides an introductory example, the design of an energy trapping carbon nanotube nanotruss, an auxetic nanotruss, a cantilever nanotruss and the maximization of the compressive strength of a metamaterial are presented. It is shown that the exceptional mechanical properties of carbon nanotubes can indeed be exploited for the development of structures and materials with extraordinary mechanical properties. Although hampered by material and geometrical nonlinearity of the problem, most of the tested optimization methods have proven to be a good choice for the design of such materials and structures.

Nonlinear vibrations of temperature-dependent FGM cylindrical panels subjected to instantaneous surface heating

An analysis is performed in this research to deal with the large amplitude vibrations caused by rapid surface heating in an FGM cylindrical panel. It is assumed that the properties of the panel are distributed through the thickness in terms of volume fraction of constituents. Following the first-order shear deformation theory, von Karman type of kinematics, and Donnell kinematic assumptions, definition of kinetic and strain energies of the shell are established. These obtained equations are discretized via the method of Ritz, where the shape functions are estimated by the Chebyshev polynomials. Temperature distribution within the body is also obtained using the one-dimensional heat conduction equation. This equation is discretized using the finite different method and solved iteratively following the Picard and Crank-Nicolson methods. Thermally induced force and moment resultants are evaluated and inserted into the matrix representation of equations of motion. The temporal evolution of displacement is obtained using the iterative Picard method and Newmark time marching method. Results of this study are compared with the available data in the open literature and after that novel results are given to explore the effects of geometrical, mechanical, and thermal parameters. It is highlighted that power law index, shallowness, thickness, edge supports, temperature dependency, geometrical nonlinearity and thermal boundary conditions are all important factors on the response of the structure under thermal shock. It is highlighted that thermally induced vibrations indeed exists especially in thin class of shells. Besides, thermally induced deflections may be controlled through using a proper power law index.

Geometrically nonlinear analysis of plates and shells by a cell-based smoothed CS-MITC18+ flat shell element with drilling degrees of freedom

A development of a 3-node triangular flat shell element to analyze geometric nonlinearity of plate and shell structures is presented in this study. The proposed flat shell element has 6 degrees of freedom per node, including the drilling rotational ones, by using the Allman-type approximations for the in-plane displacements and the bending displacement approximations enriched by a cubic shape function at the bubble node located at the element's centroid. The geometrically nonlinear behaviors are described by the von Karman's large deflection assumption. The membrane and bending strains of the presented flat shell elements are averaged over sub-triangular elements based on the cell-based smoothed (CS) technique. To attenuate the shear locking phenomenon, the transverse shear strains are re-interpolated following the mixed interpolation of tensorial components (MITC) technique designed for the 3-node triangular degenerated shell elements enriched by a cubic bubble function (MITC3+). Combined with the Newton-Raphson iterations and load increments predicted by the arc-length method, the suggested 3-node triangular flat shell elements, namely the CS-MITC18+ element, can analyze moderately geometrical nonlinearity, including the snap-thought and snap-back behaviors, of various plates and shells with the different shapes, thickness, and boundaries. The numerical investigations show that the load-displacement curves provided by the CS-MITC18+ flat shell elements well agree with other references.

Geometrically nonlinear topology and fiber orientation optimization of composite structures using membrane-embedded model

The membrane-embedded model can be efficiently used in the finite element analysis of composite structures considering geometrical nonlinearity because of its practicability. This paper extends this model to the topology and fiber orientation optimization considering geometrical nonlinearity. The discrete design variables of topology and fiber orientation are continuous by the solid isotropic material with penalization method and bi-value coding parameterization method. An optimized interpolation scheme for the membrane-embedded model is constructed for geometrical nonlinearity. The filtering function is added to improve the convergence of the fiber orientation. To reduce the computational complexity, topology and fiber orientation are independently updated at each iteration during the optimization process. Numerical examples demonstrate the effectiveness of the proposed method for membrane-embedded composite considering geometrical nonlinearity.

Directional effects on the nonlinear response of vehicle-bridge system under correlated wind and waves

Long-span railway sea-crossing bridges are subjected to strong winds and high waves. Due to variations in structural features, including stiffness and size across diverse orientations, the dynamic response of vehicles and bridges can exhibit significant differences under distinct wind and wave directions. Consequently, a novel approach for analyzing bridge and vehicle dynamic performance called the directional wind-wave-vehicle-bridge (DWWVB) model has been developed. To acquire the statistical characterization of the correlation between wind and wave conditions, a three-dimensional joint probability distribution of wind and wave characteristics derived from collected data at the bridge site is established. The wind loads are derived from wind tunnel tests and computational fluid dynamics simulations. The nonlinear wave loads are solved utilizing the hydrodynamic solver ANSYS-AQWA. Based on the finite element bridge model and a 23-degree-of-freedom mass-spring-damper vehicle model, the aerodynamic nonlinearity and initial geometric nonlinearity are considered during the iteration process. The Newmark-β method is employed to solve the dynamic response of bridges and vehicles. The calculation results reveal that the most unfavorable response occurs when the wind direction is 45° and the wave direction is 90°. The directional analysis provides valuable insights for selecting optimal bridge locations and reasonable design environmental parameters.