Boltzmann Superposition Integral Research Articles

Stochastic analysis of small-scale beams with internal and external damping

Random flexural vibrations of small-scale Bernoulli–Euler beams with internal and external damping are investigated in the paper. Such a kind of problem occurs in design and optimization of structural components of smart miniaturized electromechanical systems. Internal damping is related to the viscoelastic stress–strain law describing the beam time-dependent behavior while the external damping represents the interaction between the structure and the surrounding viscous environment. In the proposed formulation, Boltzmann superposition integral and fractional-order viscoelasticity are exploited to capture the above mentioned non-conventional phenomena. Size effects caused by long range interactions arising at small-scale are also taken into account and modeled by means of an integral nonlocal formulation based on the stress-driven approach. Moreover, in order to reproduce stochastic vibrations of small-size devices, such as resonators and sensors, the external loading are modeled by taking into account their random nature. The relevant dynamic problem is thus described by a fractional-order stochastic partial differential equation in space and time equipped with standard and non-classical boundary conditions. A proper differential eigenanalysis for the nonlocal viscoelastic bending problem is performed to find the modal shapes and closed-form expressions of power spectral density and stationary variance of displacements are provided. The proposed methodology, accounting for size-effects, hereditariness behaviors, viscous properties of surrounding environment and randomness of external loadings, is suitable to model and capture the effective behavior of miniaturized devices. The presented theoretical findings and numerical outcomes show the influence of size-effects and viscoelastic parameters on the structural response.

Free and forced wave propagation in beam lattice metamaterials with viscoelastic resonators

Beam lattice materials are characterized by a periodic microstructure realizing a geometrically regular pattern of elementary cells. The dispersion properties governing the free dynamic propagation of elastic waves can be studied by formulating parametric discrete models of the cellular microstructure and applying the Floquet-Bloch theory. Within this framework, governing the wave propagation by means of spectral design techniques and/or energy dissipation mechanisms is a major issue of theoretical and applied interest. Specifically, the wave propagation can be inhibited by purposely designing the microstructural parameters to open stop bands in the material spectrum at target center frequencies. Based on these motivations, a general dynamic formulation is presented for determining the dispersion properties of mechanical metamaterials, modeled as locally resonant beam lattices with generic coordination number. The mechanism of local resonance is realized by tuning periodic auxiliary oscillators, viscoelastically coupled with the beam lattice microstructure. As peculiar aspect, the viscoelastic coupling is derived by a mechanical formulation based on the Boltzmann superposition integral, whose kernel is approximated by a Prony series. Consequently, the free propagation of damped waves is governed by a linear homogeneous system of integro-differential equations of motion. Therefore, differential equations of motion with frequency-dependent viscoelastic coefficients are obtained by applying the in-space Z-transform and in-time bilateral Laplace transform. The complex-valued branches characterizing the dispersion spectrum are determined and parametrically analyzed for the beam lattice characterized by quadrilateral periodic cell. The spectral branches may exceed the model dimension, due to the occurrence of pure-damping spectral components. Particularly, the spectra corresponding to Laurent series approximations of the viscoelastic coefficients are investigated and the solution admissibility and convergence for increasing order series is analyzed. The standard dynamic equations with linear viscous damping are recovered at the first-order approximation. Low-order approximations are found to underestimate the real and imaginary parts of the spectrum, as well as the low-frequency stop bandwidth. Finally, the forced response to a harmonic mono-frequent external point excitation is investigated. The metamaterial responses to non-resonant, resonant and quasi-resonant external forces are compared and discussed from a qualitative and quantitative viewpoint.

Free damped vibration analysis of Mindlin plates with hybrid material-foundation viscoelasticity

Suppressed vibration of shear deformable plates with hybrid viscoelastic damping of material and foundation is investigated. Viscoelasticity of foundation is established by adopting Kelvin-Voigt model while viscoelasticity of materials is expressed by Boltzmann superposition integral, which uses dynamic mechanical analysis (DMA) results in terms of Prony series. Application of Laplace transformation and Mindlin plate theory with Hamilton principle led to a system of coupled integro-PDE of motions. Weighted residual method, numerical iterative algorithm and Fourier transform are implemented to achieve complex frequencies and transient responses of plates. Assessment of accuracy is carried out in the both elastic and viscoelastic domains. In the earlier, the frequencies are compared with counterparts obtained by p-version Ritz method and in the latter, dynamic characteristics of Mindlin and classic plates are examined that represent acceptable correlation. The influences of hybrid material-foundation damping, edge conditions, foundation characteristics and geometrical parameters on dynamic characteristics of plates are studied via a set of parametric study.

Hybrid material and foundation damping of Timoshenko beams

In this paper, vibration suppression of shear deformable beams with hybrid material-foundation viscoelastic damping is studied. The precise behavior of viscoelastic polymeric structures is taken into consideration by Boltzmann superposition integral and utilizing dynamic mechanical analysis (DMA) results while foundation viscoelasticity is adapted by Kelvin-Voigt model. The integro-partial differential equations of motion are derived based on the Timoshenko theory via Hamilton principle. Assessment of the solution procedure is carried out by comparison with elastic Timoshenko and viscoelastic Euler-Bernoulli cases. Transient response, natural frequency and modal loss factor are attained by Fourier transform, weighted residual method and numerical iterative algorithm. Influences of hybrid viscoelastic damping, foundation properties, geometrical parameters and boundary conditions on vibration characteristics and dynamic response are investigated via comprehensive parametric study. Due to the absence of similar results in the literature, this paper is likely to fill a gap in the state of the art of this problem.

An analytical study on peeling of an adhesively bonded joint based on a viscoelastic Bernoulli–Euler beam model

Peeling of a thin film adhesively bonded to a rigid substrate is analytically studied using a Bernoulli–Euler beam theory for viscoelastic materials. The film (adherend) is modeled as a viscoelastic Bernoulli–Euler beam, and the normal and shear stresses on the film-adhesive interface are assumed to satisfy constant traction laws. Closed-form solutions are derived for the following two cases: (i) only the interfacial normal stress is present (mode I loading) and (ii) the interfacial shear stress is acting alone (mode II loading). The Boltzmann superposition integral is used to obtain the constitutive relations for the viscoelastic beam, and the methods of separation of variables and Laplace transforms are employed in the formulation. To illustrate the newly derived analytical solutions, sample cases are quantitatively studied. The three-parameter Kohlrausch–Williams–Watts model is adopted to compute the compliance. The numerical results show that both the vertical displacement under mode I loading and the horizontal displacement under mode II loading increase with time and/or temperature.

Efficient higher-order zig-zag theory for viscoelastic laminated composite plates

An efficient higher-order plate theory for viscoelastic materials is developed to predict the time-dependent mechanical behaviors of composite laminates. In-plane displacement fields are constructed by superimposing a cubic varying displacement field on a linear zig-zag varying field. Time-dependent relaxation moduli have the form of Prony series, which can be determined by the master curve based on experimental data. The constitutive equation of linear viscoelastic materials in the form of the Boltzmann superposition integral is simplified by the convolution theorem of the Laplace transform to avoid direct integration as well as to improve both computational accuracy and efficiency. By using the equivalent linear elastic stress–strain relationship in the corresponding Laplace domain, the transverse shear stress-free conditions at the top and bottom surfaces and the transverse shear stress continuity conditions at the interfaces between layers are satisfied conveniently. Finally, the viscoelastic responses in the time domain are obtained through various numerical inverse Laplace transforms. To validate the present theory, the numerical results for graphite/epoxy GY70/339 material are obtained and compared with the solutions of elastic composite laminated plates.

Accurate damping analysis of viscoelastic composite beams and plates on suppressive foundation

An accurate study for dynamic behavior of viscoelastic thin laminated composite beams and plates resting on viscoelastic foundations is presented. Boltzmann superposition integral based on the dynamic mechanical analysis results is adapted to accurately predict viscoelastic behavior of polymeric fiber-reinforced composite structures. Also, foundation viscoelasticity is described based on the Kelvin–Voigt model. Integro-differential governing equations of motion are derived based on classical lamination theory via Hamilton principle. Galerkin weighted residual method, iterative QZ algorithm, and Fourier transform are applied to obtain natural frequency, loss factor, and transient response of structures resting on suppressive foundations. Influence of foundation and geometrical parameters and also boundary conditions on the dynamic behavior is put into evidence via a parametric study, and pertinent conclusions are outlined. Due to the absence of similar results in the literature, this paper is likely to fill a gap in the state of the art of this problem.

Stress-sensitive tissue regeneration in viscoelastic biomaterials subjected to modulated tensile strain

This research contribution addresses the mechanochemistry of intra-tissue mass transfer for nutrients, oxygen, growth factors, and other essential ingredients that anchorage-dependent cells require for successful proliferation on biocompatible surfaces. The unsteady state reaction–diffusion equation (i.e., modified diffusion equation) is solved according to the von Kármán–Pohlhausen integral method of boundary layer analysis when nutrient consumption and tissue regeneration are stimulated by harmonically imposed stress. The mass balance with diffusion and stress-sensitive kinetics represents a rare example where the Damköhler and Deborah numbers appear together in an effort to simulate the development of mass transfer boundary layers in porous viscoelastic biomaterials. The Boltzmann superposition integral is employed to calculate time-dependent strain in terms of the real and imaginary components of dynamic compliance for viscoelastic solids that transmit harmonic excitation to anchorage-dependent cells. Rates of nutrient consumption under stress-free conditions are described by third-order kinetics which include local mass densities of nutrients, oxygen, and attached cells that maintain dynamic equilibrium with active protein sites in the porous matrix. Thinner nutrient mass transfer boundary layers are stabilized at shorter dimensionless diffusion times when the stress-free intra-tissue Damköhler number increases above its initial-condition-sensitive critical value. The critical stress-sensitive intra-tissue Damköhler number, above which it is necessary to consider the effect of harmonic strain on nutrient consumption and tissue regeneration, is proportional to the Deborah number and corresponds to a larger fraction of the stress-free intra-tissue Damköhler number in rigid biomaterials.

Nonlinear viscoelasticity of a vulcanized elastomer

AbstractThe material nonlinearities in the viscoelastic behavior of rubbers at modest‐to‐large deformations have been studied. A vulcanized network based on the styrene–butadiene elastomer has been tested in stress‐relaxation experiments at room temperature for a wide range of deformations. After time–strain separabality was confirmed, the damping function could be calculated and introduced into the Boltzmann superposition integral. The equilibrium stress–strain behavior of the system could then be described, while comparison with other theoretical equations has also been made. © 1994 John Wiley & Sons, Inc.

Analysis of the tensile stress‐strain behavior of elastomers at constant strain rates. I. Criteria for separability of the time and strain effects

AbstractConsidering the case where the relaxation time spectrum is preserved at finite deformations, a theoretical analysis of the tensile stress‐strain relation of elastomers at constant strain rates has been carried out. The finite strain effect is taken into account by replacing the Cauchy strain by a general strain function, ƒ(ϵ), in the Boltzmann superposition integral. The analysis shows that there are two cases where the time and strain effects are separable when: (1) the segment of the stress relaxation modulus which coincides with the experimental time of stretching can be represented by a single power law; and (2) the general strain function, ƒ(ϵ), is linearly proportional to the Cauchy strain. Separability of the time and strain effects, therefore, can be achieved by adjusting the stretching time (or strain) and temperature, if the relaxation time spectrum remains unchanged by the deformation. The tensile stress‐strain relations derived from the theoretical analysis were applied to analyze data on a crosslinked styrene butadiene rubber obtained in the temperature range −40 to 60°C. Γ(ϵ), which describes the strain dependence of tensile stress, Bϵ, the ratio of isochronal stresses at different strains, and ai, slope of a segment of the relaxation modulus Ei(t) on log t plot, were obtained directly from the experiment. Values of Γ(ϵ), Bϵ and ai obtained at −40°C are quite different from those obtained at −30°C or higher. Results obtained from our analysis are generally in agreement with those obtained by an empirical method for analyzing the experimental data.

Phenomenological aspects of the deformation of elastomeric networks

The application of finite linear viscoelastic theory to elastomeric networks is discussed. It is shown that the time dependent behaviour of rubber-like materials in moderately large deformations can be described successfully by introducing a suitable chosen non-linear strain measure into the Boltzmann superposition integral. A new two-term elastic potential is proposed. The first term is the neo-Hookean potential predicted by the statistical theory of rubber elasticity for a ‘perfect’ network. The second accounts for contributions arising from the topological constraints which are generally present in real networks. The resulting two-network theory is verified on hand of published data.

Uniaxial pulse propagation in a viscoelastic medium governed by a Boltzmann superposition integral

The problem of pulse propagation through a linear viscoelastic medium is formulated by using the Boltzmann superposition integral. The relaxation times and coefficients which describe the modulus of a real material are therefore used directly in the formulation. The basic equation governing the wave motion is a partial integro-differential equation. The resulting initial value problem is solved via Laplace transforms and the asymptotic behavior of the solution is discussed. Pulse shapes for a real material (monodispersive polystrene) are calculated by using the numerical procedure of Dubner and Abate to evaluate the inverse laplace transforms.

Study of the viscoelastic behavior of uncrosslinked (gum) rubbers in moderately large deformations

AbstractTwo models, one for materials possessing a well‐defined reference state (solid‐like behavior, model S), the other for materials lacking such reference (liquid‐like behavior, model L) were tested with the aid of experimental data obtained in various strain histories in simple tension on an uncrosslinked styrene–butadiene rubber (SBR) gum rubber over the temperature range from −20 to 23°C.The models resulted from the introduction of the nonlinear n‐measure of strain into the Boltzmann superposition integral. Both gave excellent predictions of the response of the SBR to step and to ramp functions of strain. Published data on the response of polyisobutylene to an exponential stretch ratio excitation were in similarly good agreement with both models. Sensitive tests in which small deformations were superposed on a finite stretch allowed a distinction to be made in favor of model S.In uncrosslinked materials the strain parameter n was found to be small (typically about 0.5) and sensibly independent of temperature in the range investigated. By contrast, in crosslinked rubbers, n is typically between 1 and 2 at room temperature and generally decreases with temperature. The dependence of n on the nature of the material, crosslink density, temperature, and crosslinking conditions was examined.

Time-dependent response of soft polymers in moderately large deformations

We present a theory that successfully describes the time-dependent mechanical behavior of soft incompressible isotropic polymers in moderately large deformations. The theory is based on the introduction of a generalized measure of strain into the Boltzmann superposition integral.

Interrelationships among the viscoelastic functions for anisotropic solids: Application to calcified tissues and related systems

The Boltzmann superposition integral is applied to the anisotropic solid in order to derive relationships between various viscoelastic functions for orthotropic solids with special emphasis on describing biological systems such as calcified and connective tissues. These relations show that the results of uniaxial tension or compression experiments are rather complicated functions of the stiffness modulus tensor elements; as such the data could not be readily treated in a theoretical manner. On the other hand, torsional type experiments can be used to directly yield the appropriate shear elements of the tensor. Applying these derived relationships, the experimental results on cortical bone in uniaxial compression were transformed by means of computer techniques into a common representation (the complex dynamic modulus). The lack of agreement among these transformed results, based on the data from several different experimenters, as well as the failure of data from individual experiments to satisfy internal consistency tests, lead to the conclusion that the Boltzmann integral fails to describe properly the viscoelastic properties of cortical bone in compression. This implies that linear viscoelasticity theory, invariably assumed for bone, does not apply to bone in compression over the entire frequency range studied. For the purpose of comparison, dynamic moduli for synthetic (PMMA and high density linear polyethylene) and other natural (dentin and cancellous bone) materials are also presented. Fourier analytic techniques similar to those used for the above have been applied to the time-dependence of knee-joint forces obtained from gaiting studies.

Biaxial stress relaxation in glassy polymers: polymethylmethacrylate

Biaxial stress relaxation studies were performed on glassy polymethylmethacrylate in combined torsion-tension strain fields using a specially designed apparatus with exceptionally high stiffness and low cross talk between the torsional and tensile load measuring transducers. It was found that at low strain levels uniaxial tension relaxation is slower than pure torsion relaxation; tensile-component relaxation rates are unaffected by the level of torsional strain; torsional-component relaxation rates decrease as tensile strain is increased; uniaxial tension relaxation rates approach the pure torsion rates at higher strains (∼2%). A phenomenological treatment is presented which shows that relaxation rates can be coupled to the strain fields in which they are observed and yet be consistent with the concepts of linear viscoelasticity and the Boltzmann superposition integral. It is concluded that the mean normal strain, and not the octahedral shear strain, is the principal factor that governs both the effect of strain field on relaxation kinetics and the onset of nonlinear viscoelastic behavior. A mechanistic interpretation based on long-term relaxation processes in glassy polymers is offered to explain the effects of mean normal strain on relaxation kinetics.