Modeling the thermomechanical behavior of physically nonlinear materials under monoharmonic loading

Abstract

We analyze approximate approaches to the modeling of the thermomechanical behavior of physically nonlinear materials under harmonic loading. The approaches are based on various harmonic-linearization schemes and the concept of complex moduli. Mechanical and mathematical features of various schemes are considered. Some modifications of the model are proposed to account for various aspects of material behavior under harmonic loading. The problems of vibration and dissipative heating of physically nonlinear bodies are formulated. The main thermomechanical characteristics are analyzed for some classes of problems.Introduction. Many structural elements and technological objects experience cyclic deformation during use or treatment [2,3,[30][31][32]61]. Loads are sometimes so high that the deformation process becomes nonlinear and mechanical elements may suffer low-cycle fatigue. Apart from purely mechanical fatigue failure, polymeric products may undergo thermal failure, i.e., softening or even melting due to vibrational heating, which is because of high hysteresis losses and low heat conductivity [32,59,60].In the cases mentioned, modeling the thermomechanical behavior of nonlinearly dissipative materials is one of the major tasks to be accomplished in evaluating the durability or working characteristics of cyclically deformed bodies.There are currently two approaches to solve such problems. One employs the constitutive equations valid for arbitrary or, at least, rather wide classes of loading histories. Quasistatic problems for inelastic bodies, specifically beams and plates, were addressed in [77,81,82]; and vibration problems for viscoplastic structures in [66,73,78,79].This approach was used to solve problems for thin-walled structural members (mainly beams) in a practically important formulation that allows for the elastoviscoplasticity and geometrical nonlinearity of the material. Such formulations in combination with well-tested computational schemes allow us to analyze mechanical effects of the interaction of physical and geometrical nonlinearities, specifically snap buckling of initially curved elements [92] and chaotic motions in such elements under harmonic loading [42,89].By using exact models under harmonic loading, we can study a number of effects accompanying vibrations such as snap buckling, drift of average plastic strain, dynamic buckling, chaotic motions, etc. Study of such processes is difficult because of

Vibration analysis of thin-wall structures containing piezoactive layers

“…Dependencies of the storage modulus, G', the loss modulus, G", and the real and imaginary parts of the plasticity coefficients p κ ′ and p κ ′ ′ on the stress intensity, e i , are also shown in [15]. Let us emphasise that for the simplified model adopted here, the volumetric modulus K V is a real quantity.…”

“…One employs the constitutive equations valid for arbitrary or, at least, rather wide classes of loading histories [12][13][14]. The other approach is based on simplified models for specific deformation history [1,15,16]. Within the framework of unified flow theories used in the first approach, the time dependent inelastic behaviour of crystalline solids is represented as part of a combined elastic-inelastic formulation that may not rely on a yield criterion or loading and unloading conditions.…”

“…This model is based on the concept of complex moduli, which are determined by a modified technique of equivalent linearization [1,15]. In terms of these moduli, the initial problem is reduced to a scleronomous system of equations for complex amplitudes of mechanical and electrical variablesdisplacements, stresses, total and inelastic strains, voltage and electric current.…”

“…In terms of these moduli, the initial problem is reduced to a scleronomous system of equations for complex amplitudes of mechanical and electrical variablesdisplacements, stresses, total and inelastic strains, voltage and electric current. Equations of the simplest version of the model for monophase loading were developed in [15].…”

“…The complex moduli for aluminium alloy are taken from [15] where the detailed procedure for calculating them is also described. Dependencies of the storage modulus, G', the loss modulus, G", and the real and imaginary parts of the plasticity coefficients p κ ′ and p κ ′ ′ on the stress intensity, e i , are also shown in [15].…”