Accepted Manuscripts

Summer Shahzad and Jarkko Niiranen
J. Appl. Mech   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 m-th 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 non-circular 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 non-circular enclosures. Altogether, the results shown can be exploited for analyzing complex loading conditions and/or multiple rigid or void incusions for enhancing the design of hollow and reinforced composites materials.
TOPICS: Stress concentration, Annulus, Boundary-value problems, Stress, Polynomials, Displacement, Laplace equations, Approximation, Design, Finite element analysis, Heat, Composite materials
Arghavan Louhghalam, Roland J.-M. Pellenq and Franz-Josef Ulm
J. Appl. Mech   doi: 10.1115/1.4040080
Structural damping, that is the presence of a velocity dependent dissipative term in the equation of motion, is rationalized as a thermalization process between a structure (here a beam) and an outside bath (understood in a broad sense as a system property). This is achieved via the introduction of the kinetic temperature of structures, and formalized by means of an extended Lagrangian formulation of a structure in contact with an outside bath at a given temperature. Using the Nosé-Hoover thermostat, the heat exchange rate between structure and bath is identified as a mass damping coefficient, which evolves in time in function of the kinetic energy/temperature history exhibited by the structure. By way of application to a simple beam structure subjected to eigen vibrations and dynamic buckling, commonality and differences of the Nosé-Hoover beam theory with constant mass damping models are shown, that permit a handshake between classical damping models and statistical mechanics--based thermalization models. The solid foundation of these thermalization models in statistical physics provides new insights into stability and instability for engineering structures. Specifically, since two systems are considered in equilibrium when they have the same temperature, we show in the case of dynamic buckling that a persistent steady-state difference in kinetic temperature between structure and bath is but indicative of the instability of the system. This shows that the kinetic temperature can serve as a structural order parameter to identify and comprehend failure of structures, possibly well beyond the elastic stability considered here.
TOPICS: Structural dynamics, Damping, Temperature, Stability, Buckling, Failure, Steady state, Euler-Bernoulli beam theory, Heat, Vibration, Statistical mechanics, Structures, Kinetic energy, Temperature controls, Equilibrium (Physics), Equations of motion
Mao Liu and Weidong Zhu
J. Appl. Mech   doi: 10.1115/1.4040017
A major challenge in designing a perfect invisibility cloak for elastic waves is that density and elasticity tensors need to be independent functions of its radius with a linear transformation medium. The traditional cloak for out-of-plane shear waves in membranes exhibits material properties with inhomogeneous and anisotropic shear moduli and densities, which yields a poor or even negative cloaking efficiency. This paper presents design of a cylindrical cloak for shear waves based on a nonlinear transformation. This excellent broadband nonlinear cloak only requires variation of its shear modulus, while the density in the cloak region remains unchanged. The nonlinear ray trajectory equation for out-of-plane shear waves is derived and a parameter to adjust the efficiency of the cylindrical cloak is introduced. Qualities of the nonlinear invisibility cloak are discussed by comparison with those of a cloak with the linear transformation. Numerical examples show that the nonlinear cloak is more effective for shielding out-of-plane shear waves from outside the cloak than the linear cloak and illustrate that the nonlinear cloak for shear waves remains highly efficient in a broad frequency range. The proposed nonlinear transformation in conjunction with ray trajectory equations can also be used to design nonlinear cloaks for other elastic waves.
TOPICS: Shear waves, Design, Trajectories (Physics), Density, Elastic waves, Shear modulus, Materials properties, Elasticity, Anisotropy, Tensors, Membranes
Technical Brief  
Pulela Mythravaruni and Konstantin Volokh
J. Appl. Mech   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 is 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.
TOPICS: Rubber, Shear (Mechanics), Bearings, Compression, Failure, Damage, Failure analysis, Shear deformation, Traffic, Delays, Earthquakes, Fracture (Materials), Density, Weight (Mass), Deformation
Feng Zhu, Hanbin Xiao, Yeguang Xue, Xue Feng, Yonggang Huang and Yinji Ma
J. Appl. Mech   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 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 suggest that the nonlinear stress-strain curves of the cellular substrate mainly result from the finite rotation of cell walls.
TOPICS: Deformation, Anisotropy, Stress-strain curves, Constitutive equations, Finite element analysis, Fittings, Skin, Electronics, Fluids, Rotation, Flow (Dynamics)
Zhangxian Yuan and Dr. George Kardomateas
J. Appl. Mech   doi: 10.1115/1.4039953
This is a series of two papers in which the nonlinear stability behavior of sandwich panels is investigated. This part presents the buckling behavior, and focuses on the critical load and the buckling mode. The buckling analysis is based on the Extended High-order Sandwich Panel Theory (EHSAPT) which takes transverse compressibility and axial rigidity of the core into account. It allows for the interaction between the faces and core. The geometric nonlinearity, i.e., large displacement with moderate rotation, is considered in both faces and core. The weak form governing equations are derived based on the EHSAPT based element. Detailed formulations and analysis procedures are provided. It presents a general approach for arbitrary buckling type without decoupling it into isolated global buckling and wrinkling. There are no additional assumptions made about the pre-buckling state and buckling mode shape, which are commonly presumed in literature. In addition, edge effects, which are also commonly neglected are included. The pre-buckling state is determined via a nonlinear static analysis. Solving an eigen-value problem yields the critical load and the corresponding eigen-vector gives the buckling mode. Sandwich panels with different lengths are studied as examples. Both global buckling and wrinkling are observed. It shows that the axial rigidity of the core has a pronounced effect on both the critical load and the buckling mode.
TOPICS: Stability, Buckling, Stress, Stiffness, Mode shapes, Displacement, Eigenvalues, Rotation, Compressibility
Zhangxian Yuan and Dr. George Kardomateas
J. Appl. Mech   doi: 10.1115/1.4039954
The nonlinear post-buckling response of sandwich panels based on the Extended High-order Sandwich Panel Theory (EHSAPT) is presented. The model includes the transverse compressibility, the axial rigidity, and the shear effect of the core. Both faces and core are considered undergoing large displacements with moderate rotations. Based on the nonlinear weak form governing equations, the post-buckling response is obtained by the arc-length continuation method together with the branch switching technique. Furthermore, the post-buckling response with imperfections is studied. The numerical examples discuss the post-buckling response corresponding to global buckling and wrinkling. It is found that due to the interaction between faces and core, localized effects may be easily initiated by imperfections after the sandwich structure has buckled globally. Furthermore, this could destabilize the post-buckling response. The post-buckling response verifies the critical load and buckling mode given by the buckling analysis in Part I. The axial rigidity of the core, although it is very small compared to that of the faces, has a significant effect on the post-buckling response.
TOPICS: Stability, Buckling, Stiffness, Sandwich structures, Compressibility, Stress, Shear (Mechanics)
Bo Wang, YunFeng Shi, Rui Li and Bin Wang
J. Appl. Mech   doi: 10.1115/1.4039950
In this paper, a new simplified indirect measuring method 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, Strain gages, Separation (Technology), Aerospace industry, Design, Finite element analysis, Modeling
Mengjie Li, Huasong Qin, Jingran Liu and Yilun Liu
J. Appl. Mech   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 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.
TOPICS: Wavelength, Manufacturing, Simulation, Finite element methods, Design, Energy conservation, Engineering simulation, Film thickness, Plane strain, Electronics
Technical Brief  
Can Ayas and Cihan Tekoglu
J. Appl. Mech   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/ fibres in composites or the micro-architecture of members in cellular solids. This paper proofs, in a simple yet rigorous manner, that six axes of five-fold structural symmetry is necessary and sufficient for isotropy of the elastic moduli tensor in the three-dimensional context.
TOPICS: Elastic moduli, Isotropy, Atoms, Crystal structure, Solids, Composite materials, Fibers, Particulate matter, Tensors
Xiang Fang, Kuochih Chuang, Xiao-ling Jin and Z. L. Huang
J. Appl. Mech   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 is respectively considered. Unlike traditional local resonators, the proposed IDVAs generate two local-resonance 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, other than the generated two separated local-resonance band gaps, the Bragg band gap can also be significantly broadened. When further introducing a dissipative damping mechanism into the IDVA-based metamaterials, the two close split local-resonance band gaps in the lattice system can be 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 local-resonance and Bragg-scattering band gaps. This work shows that the IDVA-based metamaterials can possess much wider band gaps than the traditional local resonator-based metamaterials.
TOPICS: Vibration absorbers, Energy gap, Metamaterials, Resonance, Scattering (Physics), Radiation scattering, Electromagnetic scattering, Damping, Springs
Technical Brief  
Wei Tong
J. Appl. Mech   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 non-quadratic yield function complement the convexity certification based on a fully numerical minimization method and should facilitate its wider acceptance in modeling sheet metal anisotropic plasticity.
TOPICS: Algebra, Yield stress, Sheet metal, Anisotropy, Modeling, Isotropy, Plasticity
Andrea K.I. Hall, Thomas C. O'Connor, Molly K. McGath and Patricia M. McGuiggan
J. Appl. Mech   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 non-destructive test, the deflection of a strip of paper held horizontally is measured across its length. The deflection data is then fit to non-linear 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 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. The 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 MIT fold endurance test.
TOPICS: Cantilever beams, Brittleness, Nondestructive evaluation, Stress, Chemical properties, Cantilevers, Deflection, Elastic moduli, Fittings, Shapes, Strips, Tensile testing, Young's modulus
Armanj D. Hasanyan and Anthony M. Waas
J. Appl. Mech   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 and used 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 in the continuum model is naturally introduced by modeling the composite as a micropolar continuum. A new approach is introduced to approximate the effective transversely-isotropic micropolar constitutive model of a fiber composite. Using an updated Lagrangian, nonlinear finite element code, the simulation of localized deformation in the continuum model, corresponding to fiber kinking, is demonstrated and it is found to be comparable with the micromechanics simulation results. Most importantly, the elusive kink band width is a natural outcome of the developed model.
TOPICS: Composite materials, Fibers, Failure, Deformation, Modeling, Micromechanics (Engineering), Constitutive equations, Failure mechanisms, Finite element analysis, Fiber reinforced composites, Simulation, Stress, Performance, Computation, Displacement, Simulation results, Damage
Xianhong MENG, Zihao WANG, Sandra Vinnikova and Shuodao WANG
J. Appl. Mech   doi: 10.1115/1.4039757
In a bilayer structure consists 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 (FEA), and the prediction of spacing between cracks agrees with experiments from the literature.
TOPICS: Cracking (Materials), Fracture (Process), Finite element analysis, Fracture (Materials), Biological tissues, Stress, Stress concentration, Flexible electronics, Stiffness

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