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J. Appl. Mech. 2018;85(8):081001-081001-9. 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, which 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 (thermodynamic) 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.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(8):081002-081002-11. doi:10.1115/1.4040017.

A major challenge in designing a perfect invisibility cloak for elastic waves is that the mass density and elasticity tensor need to be independent functions of its radius with a linear transformation medium. The traditional cloak for out-of-plane shear waves in elastic membranes exhibits material properties with inhomogeneous and anisotropic shear moduli and densities, which yields a poor or even negative cloaking efficiency. This paper presents the design of a cylindrical cloak for elastic 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. A 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 elastic 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 the ray trajectory equation can also be used to design nonlinear cloaks for other elastic waves.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(8):081003-081003-9. doi:10.1115/1.4040117.

The elastic interaction energy between several precipitates is of interest since it may induce ordering of precipitates in many metallurgical systems. Most of the works on this subject assumed homogeneous systems, namely, the elastic constants of the matrix and the precipitates are identical. In this study, the elastic fields, and self and interaction energies of inhomogeneous anisotropic precipitates have been solved and assessed, based on a new iterative approach using the quasi-analytic Fourier transform method. This approach allows good approximation for problems of several inhomogeneous precipitates in solid matrix. We illustrate the calculation approach on γ-Ni3Ti precipitates in A-286 steel and demonstrate that the influence of elastic inhomogeneity is in some incidences only quantitative, while in others it has essential effect. Assuming homogeneous system, disk shape precipitate is associated with minimum elastic energy. Only by taking into account different elastic constants of the precipitate, the minimum self-energy is found to be associated with spherical shape, and indeed, this is the observed shape of the precipitates in A-286 steel. The elastic interaction energy between two precipitates was calculated for several configurations. Significant differences between the interactions in homogeneous and inhomogeneous were found for disk shape morphologies. Only quantitative differences (9% higher interaction between inhomogeneous precipitates) were found between two spherical precipitates, which are the actual shape.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2018;85(8):081004-081004-7. doi:10.1115/1.4040119.

The transverse ballistic impact on a two-dimensional (2D) membrane causes a truncated deformation cone to develop in the wake of tensile implosion waves. Here, the cone wave reflected from the finite boundaries of the elastic membrane has been studied analytically. A first-order linear nonhomogeneous differential equation for the ratio of the reflected cone wave front velocity to the speed of tensile waves is derived, which is further used to calculate the traveling time taken by the reflected cone wave to reach to the projectile surface. Since the reflected wave starts when the membrane is already in a deformed configuration, the speed of the reflected cone wave is a function of radius r in the cylindrical coordinates as opposed to almost constant speed of the incoming cone wave studied in the literature. The analytical results are validated with molecular dynamics (MD) simulations of the ballistic impact of projectiles onto a single layer of coarse-grained (CG) graphene. In the second part of the paper, we analyze the membrane impact problem for linear isotropic viscoelastic materials and find that the tensile wave speed for stresses and displacements is the same as that obtained in the case of a linear isotropic elastic material. We also show that only under special conditions, self-similar solutions for the cone wave are possible in viscoelastic materials modeled by Maxwell, Kelvin–Voigt, or a combination of similar models. Our findings lay some grounds on which further studies on the ballistic response of viscoelastic materials can be performed.

Commentary by Dr. Valentin Fuster

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