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Research Papers

J. Appl. Mech. 2018;85(7):071001-071001-11. doi:10.1115/1.4039622.

Understanding the buckling and post-buckling behavior of rods confined in a finite space is of both scientific and engineering significance. Under uniaxial compression, an initially straight and slender rod confined in a tube may buckle into a sinusoidal shape and subsequently evolve into a complicated, three-dimensional (3D) helical shape. In this paper, we combine theoretical and numerical methods to investigate the post-buckling behavior of confined rods. Two theoretical models, which are based on the inextensible and extensible rod theories, respectively, are proposed to derive the analytical expressions for the axial compressive stiffness in the sinusoidal post-buckling stage. The former is concise in formulation and can be easily applied in engineering, while the latter works well in a broader scope of post-buckling analysis. Both methods can give a good approximation to the sinusoidal post-buckling stiffness and the former model is proved to be a zeroth-order approximation of the latter. The flexible multibody dynamics method based on the Timoshenko's geometrically exact beam theory is used to examine the accuracy of the two models. The methods presented in this work can be used in, for example, drilling engineering in oil and gas industries.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071002-071002-9. doi:10.1115/1.4039815.

Soft network materials constructed with horseshoe microstructures represent a class of bio-inspired synthetic materials that can be tailored precisely to match the nonlinear, J-shaped, stress–strain curves of human skins. Under a large level of stretching, the nonlinear deformations associated with the drastic changes of microstructure geometries can lead to an evident mechanical anisotropy, even for honeycomb and triangular lattices with a sixfold rotational symmetry. Such anisotropic mechanical responses are essential for certain targeted applications of these synthetic materials. By introducing appropriate periodic boundary conditions that apply to large deformations, this work presents an efficient computational model of soft network materials based on the analyses of representative unit cells. This model is validated through comparison of predicted deformed configurations with full-scale finite element analyses (FEA) for different loading angles and loading strains. Based on this model, the anisotropic mechanical responses, including the nonlinear stress–strain curves and Poisson's ratios, are systematically analyzed for three representative lattice topologies (square, triangular and honeycomb). An analytic solution of the geometry-based critical strain was found to show a good correspondence to the critical transition point of the calculated J-shaped stress–strain curve for different network geometries and loading angles. Furthermore, the nonlinear Poisson's ratio, which can be either negative or positive, was shown to depend highly on both the loading angle and the loading strain.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071003-071003-8. doi:10.1115/1.4039672.

Viscoelasticity plays an important role in the instability and performance of soft transducers. Wrinkling, an instability phenomenon commonly observed on soft materials, has been studied extensively. In this paper, we theoretically investigate the viscoelastic effect on the wrinkle formation of a dielectric-elastomer (DE) balloon subjected to combined electromechanical loads. Results show that the critical voltage for the wrinkle formation of a DE balloon gradually decreases as the material undergoes viscoelastic relaxation and finally reaches a stable value. The wrinkles in the lateral direction always have critical voltages equal to or lower than those in the longitudinal direction. What is more, the nucleation sites of wrinkles always move from the apex to the rim of DE balloon with the viscoelastic relaxation of DE. It takes less time for the DE balloon with higher pressure to reach the stable state. Higher pressure also leads to the stable wrinkle nucleation site moving closer to the fixed edge of the DE balloon. An experiment is conducted to illustrate the effect of viscoelasticity on the wrinkle propagation of a DE balloon, and the results agree well with the model predictions. This study provides a guide in the wrinkling control of a DE balloon and may help the future design of DE transducers.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071004-071004-11. doi:10.1115/1.4039951.

In this work, the surface wrinkle modulation mechanism of the three-dimensional (3D) film/substrate system caused by biaxial eigenstrains in the films is studied. A theoretical model based on the energy minimization of the 3D wrinkled film/substrate system is proposed which shows that the change of the surface wrinkle amplitude is determined by four dimensionless parameters, i.e., the eigenstrain in the film, plane strain modulus ratio between the film and substrate, film thickness to wrinkle wavelength ratio, and initial wrinkle amplitude to wavelength ratio. The surface wrinkle amplitude decreases (even almost flat) upon contraction eigenstrain in the film, while increases for that of expansion eigenstrain. Parallel finite element method (FEM) simulations are carried out which have good agreements with the theoretical predictions, and experimental verifications are also presented to verify the findings. Besides, different patterns of 3D surface wrinkles are studied and the similar surface wrinkle modulation is also observed. The findings presented herein may shed useful insights for the design of complex stretchable electronics, cosmetic products, soft devices and the fabrication of 3D complex structures.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071005-071005-10. doi:10.1115/1.4039881.

Brittleness in paper is one of the primary reasons library books are removed from circulation, digitized, or have their access limited. Yet, paper brittleness is difficult to characterize as it has multiple definitions and no single measurable physical or chemical property associated with it. This study reevaluates the cantilever test as applied to aged papers. In this nondestructive test, the deflection of a strip of paper held horizontally is measured across its length. The deflection data are then fit to nonlinear bending theories assuming large deflection of a cantilever beam under a combined uniform and concentrated load. Fitting the shape of the deflection profiles provides bending and elastic moduli, the bending length, and confirms that the paper sheets respond linearly. The results are compared to those calculated from a simplified single point measurement of the maximum deflection of the cantilevered sample. Young's modulus measured by the cantilever test is lower for paper-based materials than that measured by tensile testing, and the bending modulus was found to correlate with the destructive Massachusetts Institute of Technology (MIT) fold endurance test.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071006-071006-6. doi:10.1115/1.4039757.

In a bilayer structure consisting of a stiff film bonded to a soft substrate, the stress in the film is much larger when the rigidity of the film is much higher than that of the substrate so that film cracking is a common phenomenon in bilayer structures such as flexible electronics and biological tissues. In this paper, a theoretical model is developed to analyze the normal stress distribution in the structure to explain the mechanism of the formation of periodic crack patterns. The effects of geometrical and material parameters are systematically discussed. The analytical result agrees well with finite element analysis, and the prediction of spacing between cracks agrees with experiments from the literature.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071007-071007-7. doi:10.1115/1.4039964.

The use of cellular substrates for stretchable electronics minimizes not only disruptions to the natural diffusive or convective flow of bio-fluids, but also the constraints on the natural motion of the skin. The existing analytic constitutive models for the equivalent medium of the cellular substrate under finite stretching are only applicable for stretching along the cell walls. This paper aims at establishing an analytic constitutive model for the anisotropic equivalent medium of the cellular substrate under finite stretching along any direction. The model gives the nonlinear stress–strain curves of the cellular substrate that agree very well with the finite element analysis (FEA) without any parameter fitting. For the applied strain <10%, the stress–strain curves are the same for different directions of stretching, but their differences become significant as the applied strain increases, displaying the deformation-induced anisotropy. Comparison of the results for linear and nonlinear elastic cell walls clearly suggests that the nonlinear stress–strain curves of the cellular substrate mainly result from the finite rotation of cell walls.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071008-071008-14. doi:10.1115/1.4040079.

Analytical displacement and stress fields with stress concentration factors (SCFs) are derived for linearly elastic annular regions subject to inhomogeneous boundary conditions: an infinite class of the mth order polynomial antiplane tractions or displacements. The solution of the Laplace equation governing the out-of-plane problem covers both rigid and void circular inclusions forming the core of the annulus. The results show first that the SCF and the loading order are inversely proportional. In particular, the SCF approaches value 2 when either the outer boundary of the annulus tends to infinity or the order of the polynomial loading increases. Second, the number of peculiar points on the inner contour having null stress increases with the increasing loading order. The analytical solution is confirmed and extended to noncircular enclosures via finite element analysis by exploiting the heat-stress analogy. The results show that the closed-form solution for a circular annulus can be used as an accurate approximation for noncircular enclosures. Altogether, the results shown can be exploited for analyzing complex loading conditions and/or multiple rigid or void inclusions for enhancing the design of hollow and reinforced composites materials.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071009-071009-10. doi:10.1115/1.4039950.

In this paper, a new simplified indirect measuring method (SIMM) is proposed for the notch stress of a circumferentially notched thin cylindrical shell by measuring the stresses away from the notch with the conventional strain gauges. The explicit relationships between the measurable stresses and notch-root stress in both the elastic and plastic stages are derived. A refined finite element modeling indicates that the developed measuring method for notch stress is feasible, and the measuring accuracy is satisfactory. A series of quasi-static tensile experiments were conducted, with both the strain gauges and advanced optical measuring method applied. Good agreement with the optical measuring results further confirms the validity and accuracy of the present method. Our method has the advantages of low cost, easy implementation, and independence of the environmental disturbance such that it has potential for wide applicability in both laboratory and in situ notch stress measurements, which is of great significance for the design of some important aerospace structures such as pyrotechnic separation devices.

Topics: Stress , Pipes
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):071010-071010-9. doi:10.1115/1.4039898.

In this paper, inerter-based dynamic vibration absorbers (IDVAs) are applied in elastic metamaterials to broaden low-frequency band gaps. A discrete mass-spring lattice system and a distributed metamaterial beam carrying a periodic array of IDVAs are, respectively, considered. The IDVA consists of a spring and an inerter connected to a traditional mass-spring resonator. Compared to the traditional resonators, the special designed IDVAs generate two local-resonance (LR) band gaps for the discrete lattice system, a narrow low-frequency band gap and a wider high-frequency one. For the distributed IDVA-based metamaterial beam, in addition to the generated two separated LR band gaps, the Bragg band gap can also be significantly broadened and the three band gaps are very close to each other. Being able to amplify inertia, the IDVAs can be relatively light even operated for opening up low-frequency band gaps. When further introducing a dissipative damping mechanism into the IDVA-based metamaterials, the two close-split LR band gaps in the lattice system are merged into one wide band gap. As for the metamaterial beam with the dissipative IDVAs, an even wider band gap can be acquired due to the overlap of the adjacent LR and Bragg-scattering band gaps.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Appl. Mech. 2018;85(7):074501-074501-7. doi:10.1115/1.4039880.

A necessary and sufficient condition in terms of explicit algebraic inequalities on its five on-axis material constants and a similarly formulated sufficient condition on its entire set of nine material constants are given for the first time to guarantee a calibrated Gotoh's fourth-order yield function to be convex. When considering the Gotoh's yield function to model a sheet metal with planar isotropy, a single algebraic inequality has also been obtained on the admissible upper and lower bound values of the ratio of uniaxial tensile yield stress over equal-biaxial tensile yield stress at a given plastic thinning ratio. The convexity domain of yield stress ratio and plastic thinning ratio defined by these two bounds may be used to quickly assess the applicability of Gotoh's yield function for a particular sheet metal. The algebraic convexity conditions presented in this study for Gotoh's nonquadratic yield function complement the convexity certification based on a fully numerical minimization algorithm and should facilitate its wider acceptance in modeling sheet metal anisotropic plasticity.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):074502-074502-5. doi:10.1115/1.4039952.

The structural symmetry of a material can be manifested at a multitude of length scales such as spatial arrangement of atoms in a crystal structure, preferred orientation of grains in a polycrystalline material, alignment of reinforcing particles/fibers in composites or the micro-architecture of members in cellular solids. This paper proofs, in a simple yet rigorous manner, that six axes of fivefold structural symmetry is necessary and sufficient for isotropy of the elastic moduli tensor in the three-dimensional (3D) context.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(7):074503-074503-4. doi:10.1115/1.4040018.

Rubber bearings, used for seismic isolation of structures, undergo large shear deformations during earthquakes as a result of the horizontal motion of the ground. However, the bearings are also compressed by the weight of the structure and possible traffic on it. Hence, failure analysis of rubber bearings should combine compression and shear. Such combination is considered in the present communication. In order to analyze failure, the strain energy density is enhanced with a limiter, which describes rubber damage. The inception of material instability and the onset of damage are marked by the violation of the condition of strong ellipticity, which is studied in the present work. Results of the studies suggest that horizontal cracks should appear because of the dominant shear deformation in accordance with the experimental observations. It is remarkable that compression delays failure in terms of the critical stretches.

Commentary by Dr. Valentin Fuster

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