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### Research Papers

J. Appl. Mech. 2018;85(9):091001-091001-15. doi:10.1115/1.4039754.

Micromechanics models of fiber kinking provide insight into the compressive failure mechanism of fiber reinforced composites, but are computationally inefficient in capturing the progressive damage and failure of the material. A homogenized model is desirable for this purpose. Yet, if a proper length scale is not incorporated into the continuum, the resulting implementation becomes mesh dependent when a numerical approach is used for computation. In this paper, a micropolar continuum is discussed to characterize the compressive failure of fiber composites dominated by kinking. Kink banding is an instability associated with a snap-back behavior in the load–displacement response, leading to the formation of a finite region of localized deformation. The challenge in modeling this mode of failure is the inherent geometric and matrix material nonlinearity that must be considered. To overcome the mesh dependency of numerical results, a length scale is naturally introduced when modeling the composite as a micropolar continuum. A new approach is presented to approximate the effective transversely isotropic micropolar constitutive relation of a fiber composite. Using an updated Lagrangian, nonlinear finite element code, previously developed for incorporating the additional rotational degrees-of-freedom (DOFs) of micropolar theory, the simulation of localized deformation in a continuum model, corresponding to fiber kinking, is demonstrated and is found to be comparable with the micromechanics simulation results. Most importantly, the elusive kink band width is a natural outcome of the continuum model.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091002-091002-12. doi:10.1115/1.4040332.

The ring-core method is often used in residual stress analysis. It is applied to macro- and microscale stress analysis and has a unique advantage of releasing the residual stress across the core instead of at a single point, which makes it possible to measure an uneven residual stress field within a limited area, especially when the area is too small to be measured by other techniques. We developed a new layer-by-layer stress analysis method based on the ring-core method to retrieve the uneven in-plane stress, in which a nonuniform load that surrounds the core is approximated by discrete loading and then used to reveal the stress distribution within the core. The displacement–stress relationship is calibrated through finite element simulation. Because of the difficulty of preparing a standard specimen with an accurate high-gradient in-plane field stress in an area of several micrometers, the performance of the method was tested by a finite element simulation experiment. Good matches were achieved when comparing the calculated stress fields and the stress fields in the simulation experiments, including the biaxial, uniaxial and high-gradient cases. The method was applied to a piece of superconductor stand with a highly nonuniform stress by using a 4 μm-diameter-area ring core cut with focused ion beam.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091003-091003-10. doi:10.1115/1.4040331.

Based on the KGD scheme, this paper investigates, with both analytical and numerical approaches, the propagation of a hydraulic fracture with a fluid lag in permeable rock. On the analytical aspect, the general form of normalized governing equations is first formulated to take into account both fluid lag and leak-off during the process of hydraulic fracturing. Then a new self-similar solution corresponding to the limiting case of zero dimensionless confining stress ($T=0$) and infinite dimensionless leak-off coefficient ($L=∞$) is obtained. A dimensionless parameter $R$ is proposed to indicate the propagation regimes of hydraulic fracture in more general cases, where $R$ is defined as the ratio of the two time-scales related to the dimensionless confining stress $T$ and the dimensionless leak-off coefficient $L$. In addition, a robust finite element-based KGD model has been developed to simulate the transient process from $L=0$ to $L=∞$ under $T=0$, and the numerical solutions converge and agree well with the self-similar solution at $T=0$ and $L=∞$. More general processes from $T=0$ and $L=0$ to $T=∞$ and $L=∞$ for three different values of $R$ are also simulated, which proves the effectiveness of the proposed dimensionless parameter $R$ for indicating fracture regimes.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091004-091004-9. doi:10.1115/1.4040336.

The whole peeling behavior of thin films on substrates attract lots of research interests due to the wide application of film-substrate systems, which was well modeled theoretically by introducing Lennard–Jones (L-J) potential to describe the interface in Peng and Chen (2015, Effect of Bending Stiffness on the Peeling Behavior of an Elastic Thin Film on a Rigid Substrate,” Phys. Rev. E, 91(4), p. 042401). However, it is difficult for real applications because the parameters in the L-J potential are difficult to determine experimentally. In this paper, with the help of the peeling test and combining the constitutive relation of a cohesive zone model (CZM) with the L-J potential, we establish a new method to find the parameters in the L-J potential. The whole peeling process can then be analyzed quantitatively. Both the theoretical prediction and the experimental result agree well with each other. Finite element simulations of the whole peeling process are carried out subsequently. Quantitative agreements among the theoretical prediction, numerical calculation, and the experiment measurement further demonstrate the feasibility of the method. Effects of not only the interface strength but also the interface toughness on the whole peeling behavior are analyzed. It is found that the peeling force at a peeling angle of 90 deg during the steady-state stage is affected only by the interface toughness, while the peeling force before the steady-state stage would be influenced significantly by the interface toughness, interface strength, and bending stiffness of the film. All the present results should be helpful for deep understanding and theoretical prediction of the interface behavior of film-substrate systems in real applications.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091005-091005-8. doi:10.1115/1.4040330.

Most tough hydrogels suffer accumulated damages under cyclic loads. The damages may stem from breakage of covalent bonds, unzipping of ionic crosslinks, or desorption of polymer chains from nanoparticle surfaces. Recent experiments report that when a tough hydrogel is subject to cyclic loads, the stress–stretch curves of tough hydrogels change cycle by cycle and approach a steady-state after thousands of cycles, denoted as the shakedown phenomenon. In this paper, we develop a phenomenological model to describe the shakedown of tough hydrogels under prolonged cyclic loads for the first time. We specify a new evolution law of damage variable in multiple cycles, motivated by the experimental observations. We synthesize nanocomposite hydrogels and conduct the cyclic tests. Our model fits the experimental data remarkably well, including the features of Mullins effect, residual stretch and shakedown. Our model is capable of predicting the stress–stretch behavior of subsequent thousands of cycles by using the fitting parameters from the first and second cycle. We further apply the model to polyacrylamide (PAAM)/poly(2-acrylanmido-2-methyl-1-propanesulfonic acid) (PAMPS) and PAAM/alginate double-network hydrogels. Good agreement between theoretical prediction and experimental data is also achieved. The model is hoped to serve as a tool to probe the complex nature of tough hydrogels, through cyclic loads.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091006-091006-11. doi:10.1115/1.4040335.

In this study, the governing equation of motion for a general arbitrary higher-order theory of rods and tubes is presented for a general material response. The impetus for the study, in contrast to the classical Cosserat rod theories, comes from the need to study bulging and other deformation of tubes (such as arterial walls). While Cosserat rods are useful for rods whose centerline motion is of primary focus, here we consider cases where the lateral boundaries also undergo significant deformation. To tackle these problems, a generalized curvilinear cylindrical coordinate (CCC) system is introduced in the reference configuration of the rod. Furthermore, we show that this results in a new generalized frame that contains the well-known orthonormal moving frames of Frenet and Bishop (a hybrid frame) as special cases. Such a coordinate system can continuously map the geometry of any general curved three-dimensional (3D) structure with a reference curve (including general closed curves) having continuous tangent, and hence, the present formulation can be used for analyzing any general rod or pipe-like 3D structures with variable cross section (e.g., artery or vein). A key feature of the approach presented herein is that we utilize a non-coordinate “Cartan moving frame” or orthonormal basis vectors, to obtain the kinematic quantities, like displacement gradient, using the tools of exterior calculus. This dramatically simplifies the calculations. By the way of this paper, we also seek to highlight the elegance of the exterior calculus as a means for obtaining the various kinematic relations in terms of orthonormal bases and to advocate for its wider use in the applied mechanics community. Finally, the displacement field of the cross section of the structure is approximated by general basis functions in the polar coordinates in the normal plane which enables this rod theory to analyze the response to any general loading condition applied to the curved structure. The governing equation is obtained using the virtual work principle for a general material response, and presented in terms of generalized displacement variables and generalized moments over the cross section of the 3D structure. This results in a system of ordinary differential equations for quantities that are integrated across the cross section (as is to be expected for any rod theory).

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091007-091007-10. doi:10.1115/1.4040480.

In this paper, we present an approach for characterizing the interfacial region using the molecular dynamics (MD) simulations and the shear deformation model (SDM). The bulk-level mechanical properties of graphene-reinforced nanocomposites strongly depend on the interfacial region between the graphene and epoxy matrix, whose thickness is about 6.8–10.0 Å. Because it is a challenge to experimentally investigate mechanical properties of this thin region, computational MD simulations have been widely employed. By pulling out graphene from the graphene/epoxy system, pull-out force and atomic displacement of the interfacial region are calculated to characterize the interfacial shear modulus. The same processes are applied to 3% grafted hydroxyl and carboxyl functionalized graphene (OH-FG and COOH-FG)/epoxy (diglycidyl ether of bisphenol F (DGEBF)/triethylenetetramine (TETA)) systems, and influences of the functionalization on the mechanical properties of the interfacial region are studied. Our key finding is that, by functionalizing graphene, the pull-out force moderately increases and the interfacial shear modulus considerably decreases. We demonstrate our results by comparing them with literature values and findings from experimental papers.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091008-091008-11. doi:10.1115/1.4040453.

The paper addresses the problem of isochronous beams, namely those that oscillate with a frequency that is independent of the amplitude also in the nonlinear regime. The mechanism adopted to obtain this goal is that of having, as a boundary condition, a roller that can slide on a given path. A geometrically exact Euler–Bernoulli formulation is considered, and the nonlinear analysis is done by the multiple time scale method, that is applied directly to the partial differential equations governing the motion without an a priori spatial reduction. The analytical expression of the backbone curve is obtained, up to the third-order, and its dependence on the roller path is addressed. Conditions for having a straight backbone curve, i.e., the isochronous beam, are determined explicitly. As a by-product of the main result, the free and forced nonlinear oscillations of a beam with an inclined support sliding on an arbitrary path have been investigated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091009-091009-11. doi:10.1115/1.4040454.

This paper presents B-splines and nonuniform rational B-splines (NURBS)-based finite element method for self-consistent solution of the Schrödinger wave equation (SWE). The new equilibrium position of the atoms is determined as a function of evolving stretching of the underlying primitive lattice vectors and it gets reflected via the evolving effective potential that is employed in the SWE. The nonlinear SWE is solved in a self-consistent fashion (SCF) wherein a Poisson problem that models the Hartree and local potentials is solved as a function of the electron charge density. The complex-valued generalized eigenvalue problem arising from SWE yields evolving band gaps that result in changing electronic properties of the semiconductor materials. The method is applied to indium, silicon, and germanium that are commonly used semiconductor materials. It is then applied to the material system comprised of silicon layer on silicon–germanium buffer to show the range of application of the method.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(9):091010-091010-14. doi:10.1115/1.4040455.

This paper investigates issues that have arisen in recent efforts to revise long-standing knockdown factors for elastic shell buckling, which are widely regarded as being overly conservative for well-constructed shells. In particular, this paper focuses on cylindrical shells under axial compression with emphasis on the role of local geometric dimple imperfections and the use of lateral force probes as surrogate imperfections. Local and global buckling loads are identified and related for the two kinds of imperfections. Buckling loads are computed for four sets of relevant boundary conditions revealing a strong dependence of the global buckling load on overall end-rotation constraint when local buckling precedes global buckling. A reasonably complete picture emerges, which should be useful for informing decisions on establishing knockdown factors. Experiments are performed using a lateral probe to study the stability landscape for a cylindrical shell with overall end rotation constrained in the first set of tests and then unconstrained in the second set of tests. The nonlinear buckling behavior of spherical shells under external pressure is also examined for both types of imperfections. The buckling behavior of spherical shells is different in a number of important respects from that of the cylindrical shells, particularly regarding the interplay between local and global buckling and the post-buckling load-carrying capacity. These behavioral differences have bearing on efforts to revise buckling design rules. The present study raises questions about the perspicacity of using probe force imperfections as surrogates for geometric dimple imperfections.

Commentary by Dr. Valentin Fuster