Journal of Engineering Mechanics

Modeling of structural nonlinear dynamic behavior is a central challenge in civil and mechanical engineering communities. The phase space embedding of response time series has been demonstrated to be an efficient coordinate basis for data-driven approximation of the modern Koopman operator, which can fully capture the global evolution of nonlinear dynamics by a linear representation. This study demonstrates that linear and nonlinear structural dynamic vibrations can be represented by a universal forced linear model in a finite dimension space projected by time-delay coordinates. Compared with the existing methods, the proposed approach improves the performance of finite linear representation of nonlinear structural dynamics on two essential issues including the robustness to measurement noise and applicability to multidegree-of-freedom (MDOF) systems. For linear structures, the dynamic mode shapes and the corresponding natural frequencies can be accurately identified by using the time-delay dynamic mode decomposition (DMD) algorithm with acceleration response data experimentally measured from an 8-story shear-type linear steel frame. Modal parameters extracted from the time-delay DMD matched well with those identified from traditional modal identification methods, such as frequency domain decomposition (FDD) and complex mode indicator function (CMIF). In addition, numerical and experimental studies on nonlinear structures are conducted to demonstrate that the finite-dimensional DMD based on the discrete Hankel singular value decomposition (SVD) coordinate is highly symmetrically structured, and is able to accurately obtain a linear representation of structural nonlinear vibration. The resulting linearized data-driven equation-free model can be used to accurately predict the responses of nonlinear systems with limited training data sets.

Abstract

A viscoelastic constitutive model constructed with fractional derivatives can accurately describe the mechanical phenomena caused by materials with time-dependent and memory properties. Using this model, the vibration characteristics and stability of a functionally graded fractional derivative viscoelastic fluid–conveying pipe with initial geometric defects are investigated in this study. First, the differential equation of motion for a functionally graded fractional derivative viscoelastic fluid–conveying pipe with initial geometric defects is derived by applying the Hamilton principle. Dimensionless differential equations and boundary conditions are discretized using a differential quadrature method, and numerical solutions of the equations under three boundary conditions are obtained. Further, the effects of geometric defect types, fractional order, power law exponent, and other factors on the complex frequency of this fluid-conveying pipe are discussed. The findings revealed that the geometric defects of the fluid-conveying pipe can significantly affect the complex frequency of the system. Moreover, the pipe’s damping characteristics vary with the fractional order of the viscoelastic material. In other words, the higher the vibration mode of the pipe, the more susceptible it is to the fractional order.

Practical Applications

Because the fractional derivative relates to values near the current moment and overall history, it can effectively describe the temporal effect of materials, such as polymers, synthetic rubber, and coatings. Many naturally occurring materials (including geological materials such as soil, oil, and minerals and biological materials such as muscle, bone, and blood) exhibit time-dependent properties. The deformation of such materials relates to the current stress level and their overall history. Because the fractional derivative constitutive relation can describe the dynamic properties of a structure in a wide frequency range with only a few parameters and terms, it represents a potential candidate for addressing the mechanical problems of viscoelastic materials and structures. In this study, functionally graded fractional derivative viscoelastic materials were applied to a fluid-conveying pipe with initial geometric curvature, as common in practical engineering. The effects of the initial geometric defect type, dimensionless delay time, power law exponent, and other parameters on the stability of the fluid-conveying pipe were discussed, providing some theoretical reference for the design of such fluid-conveying pipes.

Abstract

Recently, the unusual macromechanical behavior of a soft–hard-blend polymer composite was discovered, which is rubber-like at a low strain rate and glass-like at a high strain rate. Such strain rate–dependent mechanical behavior contributes to an outstanding impact resistant property, and consequently raises scientific interest in developing a mechanism-based modeling. In this work, a new mechanism-based modeling that involves the thermoelastoviscoplastic issues was developed. It considers the polymeric physics that accounts for the activation of the segmented molecular chains motion upon an applied stress and the strain rate dependency. The interactions of molecular chains among the hard and soft segments are captured through several relaxation spectrums. Each relaxation spectrum has a unique definition of material parameters and the activation range against the applied loading rate. In addition, the ductility originating from the large straining of soft parts that can cause a resistance to molecular chains alignment is modeled individually. Temperature rise also is addressed by a thermomechanical coupling relation which was developed by specifically targeting the related thermal parameters. The transportation of heat to the surroundings is considered on the basis of adiabatic thermodynamic flow, and the net equivalent temperature rise is calculated. The results from the modeling simulations were found to well match the experimental data. This work developed a thermomechanical model for illustrating the strain rate–dependent transition of mechanical behavior of a toughened polymer composite, which is isotropic and appliable to both small strains and large strains.

Practical Applications

This study presents a general numerical framework for developing a mechanism-based constitutive model with thermomechanical coupling considerations. It can illustrate the rate-dependent mechanical behavior of isotropic polymer composites, and can specifically target the thermal parameters based on adiabatic thermodynamic characteristics. Herein, the modeling mechanical features include the initial linear elasticity to nonlinear transition to yielding, followed by slight softening, and then plastic flow at a plateau stress with a large strain and final densification. The rate dependency involves the strain rate from $0.0001/\mathrm{s}$ to $\mathrm{7,455}/\mathrm{s}$. In the modeling, the motion activation of the segmented molecular chains upon an applied stress and the strain rate dependency are addressed. Herein, the unique definition of material parameters for several relaxation spectrums that can capture the interactions of molecular chains among the hard and soft segments is given. Each relaxation spectrum has its own activation range of the applied loading rate. In addition, the ductility, that is, from the large straining of soft segments, is addressed that has a capability of causing the resistance to the alignment of molecular chains. This model produces a well-matched illustration of stress–strain relations of polymer composites at various strain rates. This work offers an insightful understanding of the rate-dependent mechanical behavior of polymer composites for developing the high impact–resistant polymers and guiding their applications in impact events.

Stress and deformation analysis of a cavity in an infinite/finite medium is a fundamental applied mechanics problem of interest in multiple physics and engineering disciplines. This paper develops a complete semianalytical solution for the cylindrical cavity expansion in nonassociated Mohr–Coulomb materials, by using the graphical approach and Lagrangian formulation of the cavity boundary value problem (through tracing the responses of a single material point at the cavity wall). The novelty of the new solution framework lies not only in the relaxation of the stringent intermediacy assumption for the vertical stress as usually adopted in the previous analyses, but also in the comprehensive consideration of nonhydrostatic initial stress conditions via arbitrary values of $K_0$ (the coefficient of earth pressure at rest defined as the ratio between the horizontal and vertical initial stresses). The essence of the so-called graphical method, i.e., the unique geometrical analysis and tracking of the deviatoric stress trajectory, is fulfilled by leveraging the deformation requirement that during cavity expansion the progressive development of the radial and tangential strains must maintain to be compressive and tensile, respectively. With the incorporation of the radial equilibrium condition, the problem is formulated to solve a single first-order differential equation for the internal cavity pressure with respect to a pivotal auxiliary variable, for all the distinct scenarios of $K_0$ being covered. Some selected results are presented for the calculated cavity pressure-expansion curve and limit cavity pressure through an example analysis. The definitive semianalytical solution proposed will be not only substantially advancing the current state of knowledge on the fundamental cavity expansion theory, but also able to serve as a unique benchmark for truly verifying the correctness and capability of the classical cornered Mohr–Coulomb constitutive model built in commercial finite element programs.

This study proposes an efficient reliability updating method to quickly estimate the low-level failure probability, $P_f$, in structural/geotechnical systems, along with the identification of the most probable failure point (MPP). In this method, Sobol’s quasirandom low-discrepancy sequences (LDS) are employed to generate the random samples, and the system reliability updating is conducted based on a weighted low-discrepancy samplings (WLDS) technique. In particular, the updated MPP can be directly captured in $\mathbf{x}$-space according to the probabilistic weighting strategy. Coupled with the copula theory, the implementation framework of the proposed reliability updating method incorporating correlated random variables is thoroughly elaborated. The primary superiority of the proposed reliability updating method with the WLDS technique lies in that the repeated evaluation of costly performance functions is avoided during the updating process. Two illustrative examples show that 2,000 to 20,000 samples are sufficient for estimating and/or updating the probabilistic results of low-level failure probability, $P_f$, and associated MPP. The proposed method is particularly useful for conducting reliability updating of computationally costly engineering systems in conditions of changing uncertainties such as soil properties, model parameters, and applied structural loadings.

Fiber-reinforced elastomeric isolators (FREIs) have been proposed as a cost-effective solution for expanding the use of seismic isolation to normal-importance structures. By using lightweight fiber reinforcement and eliminating the attachment plates, FREIs reduce cost while improving the isolation efficiency and reducing tensile stresses in the rubber. However, the flexural flexibility of the fiber allows cross-sectional distortions (i.e., warping) to occur, which significantly impacts the stability of these devices. This paper evaluates the buckling of rectangular, circular, and annular FREIs, taking into account shear warping effects. A planar buckling theory previously proposed by the authors is adapted for the three-dimensional problem, and effective warping rigidities and warping-related areas are derived for the above bearing geometries, accounting for rubber compressibility. To assess the adequacy of the proposed buckling theory and derived warping properties in predicting the buckling of FREIs, a parametric finite element study is conducted. The critical load predictions of the proposed analytical formulation are found to be in excellent agreement with those of the numerical simulations. It is shown that traditional estimations of the buckling load that neglect warping are significantly unconservative. Finally, design recommendations and resources are provided for practice-oriented applications.

The critical-state theory (CST) serves as the foundation of modern soil mechanics and has gained widespread acceptance for explaining the behavior of granular materials. The classical CST does not address the fabric anisotropy of granular materials, leading to questions about the uniqueness of the critical state when considering such anisotropy. Recently, the development of anisotropic critical-state theory (ACST) highlights that fabric anisotropy of granular materials should reach a unique normalized value at the critical state. However, ACST does not specify which fabric tensor is used to characterize the microstructure of soils. In this study, we present a comprehensive characterization of various soil fabrics, including contact-based fabric tensor, void-based fabric tensor, and particle–void fabric, respectively. We employ three-dimensional clumped particles to approximate different shapes of Toyoura sand particles, which makes the characterization of soil fabric more accurate. Discrete element method simulations of conventional triaxial compression tests under both drained and undrained conditions are conducted on soil samples with different initial confining pressures and void ratios. The evolution of various fabrics toward the critical state is compared and discussed. Moreover, we establish relationships among contact-based fabric, void-based fabric, particle–void fabric, mean effective stress, and void ratio at the critical state. The study provides new insights into understanding the relations between the critical state and the microstructure of granular soils.

Existing research indicates that free water has a considerable influence on the dynamic mechanical properties of concrete, which is not considered in commonly used concrete models. This paper presents a high-fidelity constitutive model according to the Karagozian&Case model (K&C model) to capture the free water effect on the high dynamic properties of concrete structures. In this study, the new failure surfaces, equation of state and dynamic increase factor, which are in good agreement with experimental data, are put forward to consider the free water effect on concrete properties. On the other hand, a modified damage accumulation is introduced to better predict the soften stage of the stress-strain curve. The model is implemented into the livermore software technology corporation—dynamic nonlinear analysis (LS-DYNA) hydrocode through the user-defined materials subroutine. Numerical single element tests are first carried out to calibrate the parameters of the proposed model, where the numerical predictions are in good consistency with the test data. The perforation and penetration tests on concrete targets with different saturation are then simulated to verify the accuracy of the proposed model. Additionally, the influence of free water on the ballistic performance of concrete is investigated, and it is found that the presence of free water reduces the penetration resistance of concrete target.

Vortex-induced vibration (VIV) is a critical problem for high-rise buildings. Thus, accurate prediction of aerodynamic damping becomes a key problem in the wind-resistant design of high-rise buildings. A common feature of the existing aerodynamic damping models is that most of their model parameters are functions of wind velocity. Such a feature leads to the inconvenience of users because the parameters need to change from case to case. This paper proposes an aerodynamic damping model that is a function of structure responses only. That means it does not need to adjust its parameters in different wind speeds. A single-degree-of-freedom aeroelastic square section high-rise building model with different structural damping ratios is tested in a wind tunnel for the analytical modeling. According to the identified aerodynamic damping coefficients and its relation with response amplitudes, the nonlinear aerodynamic damping model is proposed. It is a polynomial-type function of the instantaneous velocity and displacement of the structure. The results show that predicted responses, by using one set of model parameters, are in good agreement with the experiment data in the whole tested wind speed range. These results suggest that the proposed aerodynamic model can be a generic model for prediction of wind-induced vibration.

Posttensioned rocking systems have gained attention, as they limit damage to the primary structure and simplify repairs after extreme events. The literature on posttensioned rocking walls is extensive; however, much of it is focused on numerical modeling and exploration of complex system level behavior. The present paper focuses on an analytical study of the amplitude-dependent frequencies of posttensioned flexible rocking walls. A simplified analytical model is introduced to investigate the interplay between rocking wall deformation and rocking. The study borrows analysis tools from the study of nonlinear normal modes (NNMs) and includes unforced and forced dynamics. A finite element model is also used for comparison with the static and dynamic response from the analytical model. The work shows that the unforced dynamics are able to predict peak response of forced vibration and furthermore illustrate the transition in behavior from fixed-base to flexible rocking walls. In addition, coexisting steady-state solutions were observed from the forced response. The NNMs of flexible rocking walls may aid earthquake/wind engineers in tailoring rocking systems to improve performance in the same way linear modal analysis is used in practice.