Nonlinear Mathematical Models of Metamaterials Defined as “Mass-in-Mass” and “Damper-in-Mass” Chains

To describe dynamic processes in an acoustic (mechanical) metamaterial, there are proposed models that are a one-dimensional chain containing the same masses connected by linearly elastic (or nonlinearly elastic) elements (springs) with the same stiffness. In this case, it is assumed that each mass contains inside itself a series connection of another mass and an elastic element or viscous element (damper).

Influence of Impulse Moments On Single-Frequency Torsional Oscillations of Nonlinearly Elastic Bodies

Analytical study of the impulse moment influences on the nonlinear torsional oscillations in the homogeneous constant cross-section of a body under classical boundary conditions of the first, second, and third types has been developed. For the case when the elastic material properties meet the body close to the power law of elasticity, mathematical models of the process are obtained. They are the boundary value problems for an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of pulse momentum on the oscillatory process. The peculiarities of resonant oscillations are established. Relative torsional oscillations of a nonlinear elastic body that rotates around the axis with a constant portable angular velocity are considered, taking into account the periodic action of pulse momentum acting in a fixed cross-section. The reliability of the obtained calculation formulas is confirmed.

Identification of the Model of Nonlinear Elasticity in Dynamic Experiments

Dynamic methods for identifying a model of a nonlinearly elastic deformable body are considered. By the effective phase velocities of longitudinal and transverse waves propagating along and across the axis of the compressed bar, it is possible to determine five elastic constants of the second and third orders included in the model relations. Calculation formulae are obtained and an example of determining the dependence of phase velocities on the preliminary deformation for polyamide 6 is given. The influence of preliminary deformations on polar diagrams of wave velocities is investigated.

Mathematical modeling and numerical simulation of nonlinearly elastic yarn in ring spinning

In this paper, yarn dynamic behavior in the ring spinning system has been studied. A new model has been proposed by considering nonlinear elastic yarn. Equations of motion were derived by Newton’s second law and resolved by the finite difference method. Some results were given and relationships among models of inextensible yarn, linear elastic yarn, and nonlinearly elastic yarn were discussed. Experiments were conducted to evaluate the accuracy of the proposed model in terms of yarn tension and balloon profile and a good agreement has been made between the predicted data and experimental results.

Nonlinearly Elastic Constitutive Relation of Anisotropic Aggregate of Cube Crystallites

Under Voigt model, Barsch and Johnson gave the formula of nonliearly elastic constitutive relations for isotropic aggregates of cubic crystallites and orthorhambic aggregates of cubic crystallies, respectively. In this paper, a nonlinear elastic constitutive relation based on Voigt model, which is more general than Barsch's and Johnson's results, is derived for the set of anisotropic cubic grains. The anisotropy of metals is described by the texture coefficient.