Journal of Engineering Mechanics

Cyclic peptide nanotubes (CPNs) have unique chemical and mechanical features that squarely position them to tackle persistent challenges in sensor technologies, tissue scaffolds, templates for organic and hybrid electronics, and ultrasmall electromechanical systems. These self-assembled hierarchical nanostructures are highly organized at the nanoscale and feature exceptional thermodynamical stability arising from the collective action of secondary interactions, in particular intersubunit hydrogen-bond networks. Understanding the elasticity and fracture behavior of CPNs through a multiscale analysis is crucially important for developing science-based approaches for designing the molecular subunits and hierarchical assemblies of these materials. In pursuit of addressing this need, a methodology is proposed for linking atomistic simulation results into coarser descriptions of these self-assembling soft nanostructures. This approach involves estimation of the free-energy landscape of the system along the deformation reaction coordinate from atomistic simulation trajectories using nonequilibrium statistical thermodynamics formulations, which enables bridging scales through mapping to coarse-grain or continuum descriptions. In this study, a basic multiscale approach was demonstrated for investigating the mechanics of CPNs, mapping out the elastic range of intersubunit interactions along with the large deformation and fracture regimes. This work illustrates the potential of atomistically informed methods for predicting elastic as well as large deformation behavior of high-aspect-ratio self-assembling nanostructures.

This work investigates the application of topological sensitivity (TS) toward reconstructing and identifying tissue anomalies from magnetic resonance elastography (MRE) measurements. The basic idea of MRE imaging is to apply time–harmonic vibration to a tissue inside the magnetic resonance (MR) scanner and to capture the induced three-dimensional motion response (i.e., wave patterns) via suitable sequencing of the phase-coordinated, freeze-frame MR scans. To account for the facts that (1) the displacement of a material point captured by the MRE signifies the volume average over a reference voxel size, and (2) any given measurement voxel may contain, at least partially, the lesion of interest, the concept of TS is extended to allow for (1) averaged volumetric measurements and (2) overlapping interrogation and measurement domains. To circumvent the difficulties involved in the numerical modeling of an entire organ, the latter is subdivided into an array of cubic subdomains, each consisting of $N^3$ voxels and playing the role of the reference body for TS imaging. To accomplish the sought uncoupling, the triaxial displacements along the boundary of each subdomain (captured by the MRE) are deployed as the Dirichlet data, thus specifying a separate boundary value problem, whereas the internal measurements within the subdomain are taken as the observations for lesion reconstruction. The performance of the proposed methodology for lesion reconstruction and material characterization is examined via both numerical examples and a preliminary application to in vivo MRE data. The sensitivity of the technique to measurement errors is investigated by introducing random noise to synthetic observations. The results highlight the potential of the TS method toward elevating the resolution of tissue (visco)elasticity reconstruction by MR imaging.

In this paper, the development of a multiscale model of collagen fibril is described using a hierarchical approach by combining molecular dynamics simulations and FEM. Steered molecular dynamics was used to investigate the mechanical response of collagen under various conditions that mimic the organization of collagen molecules and mineral inside the collagen fibril. The steered molecular dynamics simulations showed that the elastic properties of collagen molecules are significantly higher in the proximity of mineral. The properties of collagen and interfaces at the molecular scale were carried over to the continuum model of collagen fibril. The model of collagen fibril (measuring approximately $1\hspace{0.17em}\mu \mathrm{m}$ in length and 50 nm in diameter) was constructed using FEM. Results showed that the deformation properties of collagen fibril are significantly influenced by interactions between collagen and mineral at the molecular scale, significantly affecting the elastic properties of fibril. Here, a model of fibril is presented that, for the first time, incorporates the effects of interactions occurring between the mineral and the collagen at the molecular scale.

Some civil engineering structures, such as RC frames, exhibit hysteretic behavior when subject to dynamic loads, such as earthquakes. The identification of hysteretic systems with stiffness and strength degradations is a practical but challenging problem encountered in the engineering field. A recently developed technique, referred to as the adaptive quadratic sum-square error with unknown inputs (AQSSE-UI), is capable of identifying time-dependent parameters of hysteretic structures. In this paper, the AQSSE-UI technique is applied to the parametric identification of hysteretic RC frames with stiffness and strength degradations and with the pinching effect. The performance of the AQSSE-UI technique will be demonstrated by the experimental data. A $\mathrm{1/3}$-scaled 2-story RC frame has been tested experimentally on the shake table at the National Center for Research on Earthquake Engineering (NCREE), Taiwan. This 2-story RC frame was subject to different levels of ground excitations back to back. The structure is first considered as an equivalent linear model with time-varying stiffness parameters, and the tracking of the degradation of the stiffness parameters is carried out using the AQSSE-UI technique. The same RC frame is then considered as a hysteretic model with plastic hinges following the generalized Bouc-Wen model, and the time-varying nonlinear parameters are identified. The equations of motion of the 2-story RC frame have been formulated based on the finite-element model, and the reduced-order system has been constructed to be used for the damage detection. Experimental results demonstrate that the AQSSE-UI technique is quite effective for the tracking of (1) the stiffness degradation of linear structures, and (2) the hysteretic parameters with stiffness and strength degradations and with the pinching effect.

The universal size-shape effect law is a law that describes the dependence of nominal strength of specimen or structure on both its size and the crack (or notch) length, over the entire range of interest, and exhibits the correct small-size and large-size asymptotic properties as required by the cohesive crack model (or crack band model). The main difficulty has been the transition of crack length from 0, in which case the size effect is Type 1, to deep cracks (or notches), in which case the size effect is Type 2 and is fundamentally different from Type 1, with different asymptotes. In this transition, the problem is not linearizable because the notch is not much larger than the fracture process zone. The previously proposed universal law could not be verified experimentally for the Type 1-Type 2 transition because sufficient test data were lacking. The current study is based on recently obtained comprehensive fracture test data for three-point bend beams cast from one batch of the same concrete and cured and tested under identical conditions. The test data reveal that the Type 1-Type 2 transition in the previous universal law has insufficient accuracy and cannot be captured by Taylor series expansion of the energy release rate function of linear elastic fracture mechanics. Instead, the size effect for a zero notch and for the transitional range is now characterized in terms of the strain gradient at the specimen surface, which is the main variable determining the degree of stress redistribution by the boundary layer of cracking. The new universal law is shown to fit the comprehensive data quite well, with a coefficient of variation of only 2.3%.

The boundary effect model (BEM) for concrete fracture and the effects of specimens size and crack length has previously been criticized on theoretical grounds, but the experimental evidence found in the literature, when taken alone, has been too limited to judge the validity of BEM conclusively. New, separately published, comprehensive fracture experiments, which were made on specimens cast from one and the same batch concrete and featured a broad ranges of both the size and the crack length (including a zero crack length), change the situation. The optimum fit of the data by Hu and Duan’s model shows major deviations from these new test results. On the other hand, the Type 1 and 2 size effect laws (SELs) and their amalgamation in the universal size effect law are found to give a far better fit of the test results. Thus, regardless of the previously expounded theoretical objections, the comparison with experimental evidence alone suffices to conclude that Hu and Duan’s model is not realistic.

This study proposes a set of simple equations for the mean value and variance of the mobilized shear strength for spatially variable soil masses subjected to uniform stress states. These equations are fairly effective in explaining the complicated behaviors for the mobilized shear strengths, regardless of stress states (e.g., compression or shear), spatial variability patterns (e.g., isotropic or anisotropic), and inherent mean and variance of the random field. Two mechanisms that affect the behaviors of the mobilized shear strength are identified: (1) the averaging effect along the potential slip curves, and (2) the emergent feature of a critical slip curve. The emergence is associated with the slip curve with the minimum averaged strength. In any realization of the random field, it is not possible to know a priori the location of the minimum average; hence, it would not coincide with a prescribed average. It is shown that the well-known phenomenon of critical scale of fluctuation is the result of the trade-off between these two mechanisms.

Research efforts on nonlinear guided wave propagation have increased dramatically in the last few decades because of the high sensitivity of nonlinear waves to structural conditions (defects, quasi-static loads, instability conditions, and so on). However, the mathematical framework governing the nonlinear guided wave phenomena becomes extremely challenging in waveguides that are complex in either materials (damping, anisotropy, heterogeneous, etc.) or geometry (multilayers, geometric periodicity, etc.). The present work develops predictions of nonlinear second-harmonic generation in complex waveguides by implementing a semianalytical finite-element formulation that accounts for material nonlinearities into a highly flexible, yet very powerful, commercial finite-element code. Once formulated correctly, the proposed analysis can easily take into account damping effects, anisotropic multilayered properties, periodic geometries, and other complex waveguide properties in a computational efficient and accurate manner. Results are presented for the following cases: a railroad track, a viscoelastic plate, a composite quasi-isotropic laminate, and a RC slab. In these cases, favorable combinations of primary wave modes and resonant double-harmonic nonlinear wave modes are identified. Knowledge of such combinations is critical to the implementation of structural monitoring systems for these structures based on higher harmonic wave generation.

This paper presents an incremental approach based on the theories of continuum damage and anisotropic plasticity to model the forces that develop during timber cutting. The model used is based on the thermodynamics of irreversible processes with state variables and takes into account the coupling among orthotropic elasticity, anisotropic plasticity, and damage. The aim is to contribute to the development of simulation tools needed to optimize industrial cutting processes. Validation of the model, implemented using existing finite-element (FE) software, is carried out by simulating the forces that develop during the cutting of wooden planks and tree trunks of different timber species. It was found that there is good agreement between the predicted and measured cutting forces.

A refined three-dimensional zigzag plate model with hierarchic representation of displacements across the thickness is developed that accurately captures interlaminar stresses directly from constitutive equations with low computational effort. Thanks to the incorporation of high-order terms, the representation can be different from point to point across the thickness; thus, it can be adapted to variations of solutions. However, as in classical models, the functional degrees of freedom are the midplane displacements and the shear rotations because high-order contributions are determined without adding new unknowns. They are computed by enforcing equilibrium conditions at discrete points across the thickness and stress boundary conditions. The model fulfills a priori the interfacial stress continuity conditions at the interfaces thanks to the incorporation of appropriate continuity functions. A new approach is adopted that overcomes the drawbacks of algebraic manipulations because the expressions of high-order terms and continuity functions are derived separately only once in closed form for all conditions with a symbolic calculus tool. As shown by the numerical tests, the model accurately predicts the stress field of thick laminated and sandwich plates with abruptly changing material properties with a lower overall processing time than previously developed models by the authors.

An analytical solution for the dynamic response of an infinite beam resting on a viscoelastic foundation and subjected to arbitrary dynamic loads is developed in this paper. Fourier and Laplace transforms are utilized to simplify the governing equation of the beam to an algebraic equation, so that the solution can be conveniently obtained in the frequency domain. The convolution theorem is employed to convert the solution into the time domain. Final solutions of beam responses investigated are deflection, velocity, acceleration, bending moment, and shear force. The validation of the proposed solution is verified by considering the solutions of several special dynamic loads and comparing the degraded solution to the known results. Further complicated dynamic loads, such as impulsive loads and time-lag loads, are also discussed and analytical solutions are presented. These relationships can be an effective tool for practitioners.

Large-amplitude vibration analysis for a shear deformable cross-ply laminated composite cylindrical shell of finite length in thermal environments is presented. The material of each layer of the shell is assumed to be linearly elastic and fiber reinforced. The motion equations are based on Reddy’s higher-order shear deformation shell theory with a von Kármán-Donnell–type of kinematic nonlinearity. The thermal effects and initial imperfections of the shell are both taken into account. A two-step perturbation technique is employed to determine the linear and nonlinear frequency of the laminated cylindrical shells. The numerical illustrations concern the nonlinear vibration behavior of laminated composite cylindrical shells with different values of geometric parameters and different cases of thermal environmental conditions. The results show that the shell has relatively lower natural frequencies when the temperature-dependent properties are taken into account. The results reveal that the temperature changes, initial imperfections of the shell, and the shell geometric parameter have significant effects on the nonlinear vibration behavior of laminated composite cylindrical shells.

Solving the buckling problem for a battened column as a statically indeterminate structure yields a Euler critical load that is significantly higher than the buckling load being obtained on the basis of Engesser’s assumption. The stability problem is reduced to numerically solving a two-point boundary value problem for a system of recurrence relations between deformation parameters of adjacent joint cross sections of the column. The expression of deformation parameters for each further joint cross section of the column as a linear function of deformation parameters for the preceding joint cross section is possible using the initial-value method. For columns with any degree of static indeterminacy, the critical force is determined as the smallest eigenvalue of the fourth-order system of homogeneous linear algebraic equations. A computer realization of the present method requires implementing reorthogonalization of the vectors of particular solutions for the system of recurrence relations. Plots of the buckling load for columns with any number of panels can be constructed as a function of the batten rigidity parameter. Applying these plots can be useful in selecting a cross section of the columns being designed.

The application of meshless local integral equations based on the meshless local Petrov-Galerkin (MLPG) method for the two-dimensional (2D) dynamic stress analysis of a 2D functionally graded (FG) cylinder with two-directional grading patterns is developed. It is assumed that the inner bounding surface of the cylinder is excited by mechanical shock loading. The problem is studied in the frequency domain using the Laplace transformation, and then the stresses are transferred to the time domain using Talbot techniques. The mechanical properties of the FG materials (FGMs) are simulated using a nonlinear model with radial and axial volume fractions. The 2D propagation of stresses is tracked through the radial and axial directions in a 2D domain for various grading patterns at various time instants. By using the presented meshless technique, the maximum values of the stresses are calculated. The effects of various grading patterns on the maximum values of the stresses are studied in detail. The presented formulations based on the MLPG method furnish the groundwork for 2D stress-wave propagation in FGMs with two-directional grading patterns.

A pressure impulse (PI) diagram is a tool typically used for the preliminary design of structural members against external blasts. Using single-degree-of-freedom (SDOF) theory, numerous studies have been conducted on PI curves for structural members subjected to idealized pulse loads typically associated with free field blasts. However, owing to the complex nature of confined blasts, characterized by multiple peaks and long-lasting gas pressures, PI curves derived for such pulse loads cannot be applied to confined blasts. As the confinement magnifies the effects of a blast, resulting in more serious damage to surrounding structural members, such effects should be taken into consideration when using analysis and design tools. Therefore, a new approach for deriving PI curves was applied to confined blasts. First, the approach was used to derive PI curves for pulse loads associated with vented confined blasts. The PI curves derived using the new approach were compared to those using traditional techniques. Then, to determine the response of structural members to unvented confined blasts, in which regular PI curves cannot be used, a new technique was provided. The technique utilizes the concept of the effective pulse load to determine effective PI curves.

The biaxial tensile strength of concrete (and ceramics) can be easily tested by flexure of unreinforced circular disks. A recent experimental study demonstrated that, similar to plain concrete beams, the flexural strength of disks suffers from a significant size effect. However, the experiments did not suffice to determine the size effect type conclusively. The purpose of this study is to use three-dimensional stochastic finite-element analysis to determine the size effect type and shed more light on the fracture behavior. A finite-element code using the microplane constitutive Model M7 is verified and calibrated by fitting the previously measured load-deflections curves and fracture patterns of disks of thicknesses 30, 48, and 75 mm, similar in three dimensions, and on flexure tests on four-point loaded beams. It is found that the deformability of the supports and their lifting and sliding has a large effect on the simulations, especially on the fracture pattern, and the strength and Young’s modulus of concrete must be treated as autocorrelated random fields. The calibrated model is then used to analyze the size effect over a much broader range of disk thicknesses ranging from 20 to 192 mm. The disks are shown to exhibit the typical energetic size effect of Type I, that is, the disks fail (under load control) as soon as the macrofracture initiates from the smooth bottom surface. The curve of nominal strength versus size has a positive curvature and its deterministic part terminates with a horizontal asymptote. The fact that material randomness had to be introduced to fit the fracture patterns confirms that the Type 1 size effect must terminate at very large sizes with a Weibull statistical asymptote, although the disks analyzed are not large enough to discern it.

It is important to generalize constitutive models from the triaxial compression state to the three-dimensional stress state for soils. However, there are some deficiencies in the existing methods, e.g., the using failure criteria method, the generalized shear stress ratio method, and the shape function approach using Lode’s angle or reduced Lode’s angle, for generalizing either isotropic or anisotropic constitutive models. For example, the critical state curve may be misshapen and the yield surface will appear to be curvature discontinuous if the constitutive model is generalized with the shape function approach using reduced Lode’s angle. The transformed stress (TS) space has been proposed in the authors’ former work based on the spatially mobilized plane (SMP), the Lade’s, and the generalized nonlinear strength criterion. In this paper, the TS method based on an arbitrary failure criterion is proposed. It is a simple approach to generalize constitutive models without introducing any extra parameters. Another important work in this paper is using the TS method to generalize anisotropic models. It was found that (1) the processes for generalizing anisotropic models and isotropic models are the same, i.e., the proposed method can be directly used to generalize anisotropic models without any modification; (2) the deficiencies of other methods for generalizing constitutive models can be eliminated with the TS method (for instance, the yield surfaces appear to be curvature continuous after generalization, and the critical state curves of the generalized models are consistent with the adopted failure criteria); and (3) it is demonstrated by the comparison between the predictions and test results that the TS method is better than the shape function approach using Lode’s angle in describing the soil behaviors in the general stress state. In this paper, the modified Cam clay model is taken as an example for the discussion.

A straightforward analytical method is proposed to calculate the reinforcement loads of vertical-reinforced soil composites before the strength of soil is fully mobilized. The method assumes compatibility of soil and reinforcement deformations along the potential failure surface. It makes use of the nonlinear stress-strain relationship of soil in a plane-strain condition and Rowe’s stress-dilatancy relationship. It has the advantage of taking into account the soil dilatancy before failure and can properly estimate the reinforcement load with small or medium soil deformation. The effect of soil compaction on the reinforcement load is taken into account through an elastic unloading/reloading approach. The method was first validated against the results of a calibrated numerical analysis. It was then used to predict the reinforcement loads of two large-scale tests. The proposed method has the potential to fully develop into an analytical method for reinforced soil retaining walls, provided that effects of facing restriction and compaction can be quantified.

A closed-form solution is presented for a cantilevered sectorial plate subjected to a tip bending moment. Deflections from this solution are compared with those from a finite-element analysis and are found to be in excellent agreement. Closed-form normalized deflections and slopes at the fixed support resulting from an approximate enforcement of the boundary conditions there deviated from zero by less than 0.04%.

Many wind design codes provide recommended values of the drag coefficient for generic lattice frame geometries based on the solidity ratio. The solidity ratio of a lattice structure often varies along its length or height, resulting in a piecewise treatment in design. By definition, the solidity ratio is only dependent on the amount of solid area to the total enclosed area of a frame, and therefore does not account for member spacing or the number of members within the framework; this has implications for some nonuniform sections used in lattice tower design. Drag coefficients for many geometric configurations were measured in a wind tunnel with the focus placed on configurations having an identical solidity ratio but different member spacing ratios. It was observed that the drag coefficient varied with the member spacing ratio as well as with the number of members in the frame. The results suggest that the solidity ratio alone may not be sufficient for describing the drag coefficient of a lattice frame having a nonuniform solid area distribution. Recommendations are made for future research in this area, which may lead to better estimate of the drag coefficient for nonuniform lattice sections.