Hereditary Integral Research Articles

Dynamic Relaxation Nonlinear Viscoelastic Analysis of Annular Sector Composite Plate

The geometrically nonlinear laminated annular plate structure with nonlinear viscoelastic orthotropic composite is considered using the dynamic relaxation (DR) method for the first time in the present study. The DR technique in conjunction with the finite difference discretization is used to study the behavior of a Mindlin plate. The plate is made of nonlinear viscoelastic orthotropic material expressed by the Schapery single integral model. A recursive-iterative formulation is implemented for the hereditary integral of the material. Under uniform lateral pressure and clamped and hinged edged constraints, the unsymmetrical laminated sector plate deformation is predicted in different times. The present results are compared with the elastic finite element (FE) generated numerical results. The correlations are very satisfactory. The load-center deflections for linear and nonlinear viscoelastic material plate along the symmetrical radial line for elastic and viscoelastic plate are illustrated. The dimensionless deflection results for up to 480min are predicted to show the creep behavior of the material. Finally, the dimensionless stress couple results are obtained after 8 h creep time for a graphite/epoxy composite plate.

Comparison between the dynamic moduli of fully non-linear and quasilinear viscoelastic materials

This paper deals with the non-linear viscoelastic model with multiple hereditary integrals (MHI) in the frequency domain and the conditions that it reduces to single hereditary integral or the quasilinear viscoelastic (QLV) model. It is shown that when the higher order complex moduli are related to the first-order modulus as E n ( ω 1 , … , ω n ) = α n q E 1 ( ω 1 + ⋯ + ω n ) the MHI model reduces to the QLV model. The coefficients of quasilinearity α n q should be real and independent of amplitude and frequency.

On modelling of anisotropic viscoelasticity for soft tissue simulation: Numerical solution and GPU execution

Efficient and accurate techniques for simulation of soft tissue deformation are an increasingly valuable tool in many areas of medical image computing, such as biomechanically-driven image registration and interactive surgical simulation. For reasons of efficiency most analyses are based on simplified linear formulations, and previously almost all have ignored well established features of tissue mechanical response such as anisotropy and time-dependence. We address these latter issues by firstly presenting a generalised anisotropic viscoelastic constitutive framework for soft tissues, particular cases of which have previously been used to model a wide range of tissues. We then develop an efficient solution procedure for the accompanying viscoelastic hereditary integrals which allows use of such models in explicit dynamic finite element algorithms. We show that the procedure allows incorporation of both anisotropy and viscoelasticity for as little as 5.1% additional cost compared with the usual isotropic elastic models. Finally we describe the implementation of a new GPU-based finite element scheme for soft tissue simulation using the CUDA API. Even with the inclusion of more elaborate constitutive models as described the new implementation affords speed improvements compared with our recent graphics API-based implementation, and compared with CPU execution a speed up of 56.3 x is achieved. The validity of the viscoelastic solution procedure and performance of the GPU implementation are demonstrated with a series of numerical examples.

Interfacial stresses and fracture analysis of a two-phase composite under time-dependent heat flux

An analysis of a two-phase composite component under time-dependent heat flux is presented. The fundamental thermoelastic solution is obtained in terms of complex potentials via the technique of the analytical continuation in order to satisfy the continuous conditions on the interface. The hereditary integral associated with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking inverse Laplace transform. Some typical examples of interface stresses induced by various time-dependent heat flux are discussed. Finally, the solution of a crack embedded in the bi-material subjected to a uniform heat flux is also discussed.

Thermal stresses in a viscoelastic three-phase composite cylinder

A thermoviscoelastic analysis of a three-phase composite cylinder is presented in this work. The solutions of the heat conduction and thermoelastic problem for three dissimilar media are derived based on the method of analytic continuation associated with the alternation technique. A rapidly convergent series solution for both the temperature and stress field, which is expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. According to the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real time solution can then be solved numerically by taking inverse Laplace transform. Finally, some typical examples of interfacial stresses are discussed.

Thermal stresses in a viscoelastic trimaterial with a combination of a point heat source and a point heat sink

A general solution for a thermoviscoelastic trimaterial combined with a point heat source and a point heat sink is presented in this work. Based on the method of analytic continuation associated with the alternation technique, the solutions to the heat-conduction and thermoelastic problems for three dissimilar, sandwiched media are derived. A rapidly convergent series solution for both the temperature and stress field, expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties, while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking the inverse Laplace transform. A typical example concerning the interfacial stresses generated from a combined arrangement of a heat source and sink are discussed in detail. The corresponding thin-film problem is also discussed.

Analytical techniques for indentation of viscoelastic materials

Indentation of linearly viscoelastic materials is explored using elastic–viscoelastic correspondence analysis for both conical-pyramidal and spherical indentation. Boltzmann hereditary integrals are used to generate displacement–time solutions for loading at constant rate and creep following ramp loading. Experimental data for triangle- and trapezoidal-loading are examined for commercially-available polymers and compared with analytical solutions. Emphasis is given to the use of multiple experiments to test the fidelity and predictive capability of the obtained material creep function. Plastic deformation occurs in sharp indentation of glassy polymers and is found to complicate the viscoelastic analysis. A new method is proposed for estimating a material time-constant from peak displacement or hardness data obtained in pyramidal indentation tests performed at different loading rates.

A simplified methodology to predict the dynamic stiffness of carbon-black filled rubber isolators using a finite element code

A new and different approach to the inclusion of the amplitude-dependent effect, known as the Fletcher–Gent effect or Payne effect, in a linear viscoelastic rubber material model is presented to predict the dynamic stiffness of filled rubber isolators using a finite element (FE) code. The technique is based on providing a linear viscoelastic model with the adequate material data set, once the dynamic strain amplitude, to which the rubber mount is subjected, is estimated. A generalized Zener model is adopted to describe the frequency-dependent behaviour of the material through the use of hereditary integrals. The dynamic strain amplitude dependence is not modelled through any friction model or plasticity theory, as usually is in literature. It is introduced by considering the frequency-dependent properties of the compound at an adequate strain value, which enforces the estimation of an equivalent strain value. As a first approximation, a quasi-static value is used as the reference value at which material properties should be provided to the linear viscoelastic model. The technique works directly in frequency domain, the dynamic stiffness of the bushing being directly obtained. The methodology is applied to evaluate the dynamic stiffness of a real bushing in working conditions with very satisfactory results. Despite the assumptions made, especially regarding the estimation of the equivalent strain amplitude value, errors of the predictions fall within the limits usually accepted by rubber manufacturers.

A constitutive law for snow taking into account the compressibility

AbstractA constitutive law for snow derived from a complementary power potential is proposed. The total deformation of snow is divided into elastic and creep parts. A hereditary integral using Norton’s power law is employed to describe primary creep. The concept of effective stress, which takes compressibility of snow into account, is used to calculate creep deformation. The hereditary integral is approximated by a non-linear spring–dashpot model. Results from uniaxial compression experiments (stress range 15– 45 kPa) on sieved snow of density range 180–470 kgm-3 were used to determine the constants appearing in the constitutive equation. The response of snow to constant strain rate (7.4×10-6 s-1 to 2.2×10-5 s-1) under bilaterally confined conditions was found with an iterative scheme employing the proposed constitutive law. The simulated results agree well with the measured axial stresses and volumetric changes.

Fractional calculus in bioengineering, part 2.

Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner. In Part 1 of this review (Crit Rev Biomed Eng 2004; 32(1):1-104), the fundamental properties of fractional calculus were introduced. The reader should refer to that material, particularly the sections on special functions and fractional order differential equations, for background.

Fractional Calculus in Bioengineering, Part3

Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner. In Part 2 of this review (Crit Rev Biomed Eng 2004; 32(1):105-193), fractional calculus was applied to problems in nerve stimulation, dielectric relaxation, and viscoelastic materials by extending the governing differential equations to include fractional order terms. In this third and final installment, we consider distributed systems that represent shear stress in fluids, heat transfer in uniform one-dimensional media, and subthreshold nerve depolarization. Classic electrochemical analysis and impedance spectroscopy are also reviewed from the perspective of fractional calculus, and selected examples from recent studies in neuroscience, bioelectricity, and tissue biomechanics are analyzed to illustrate the vitality of the field.

Fractional calculus in bioengineering.

Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner.

The initiation J-integral for linear viscoelastic solids with constant Poisson's ratio

The present work examines the fracture response of homogeneous and linear isotropic visco-elastic solids with constant Poisson's ratio. For this type of materials it is shown that the J-integral at fixed crack size is proportional to the area under the load-vs-load point displacement record of a fracture test, and that, for given boundary conditions, the factor of proportionality depends on crack and specimen geometry – and also, perhaps, on Poisson's ratio – but is independent of material heredity. As a consequence, the factor of proportionality – or crack configuration factor – can be established by analysis of the test article, subjected to the same displacement and traction boundary conditions as the test, but using – if desired – an arbitrary linear elastic material with the constant Poisson's ratio of the viscoelastic solid. The same approach is shown valid for a class of nonlinear viscoelastic materials whose constitution involves single hereditary integrals, and whose property functions do not depend on stress. The methodology is used with the results of constant displacement-rate tests of centrally cracked specimens to construct a portion of the master initiation J-integral function for a solid propellant.

The problem of a viscoelastic cylinder rolling on a rigid half-space

The problem of a viscoelastic cylinder rolling on a rigid base, propelled by a line force acting at its centre, is solved in the noninertial approximation. The method used is based on a decomposition of hereditary integrals developed by the authors in previous work, and on the viscoelastic Kolosov-Muskhelishvili equations which are used to generate a Hilbert problem. In this formulation, the problem reduces to a nonsingular integral equation in space and time, which simplifies under steady-state conditions and for exponential decay materials, to algebraic form. There are also two subsidiary conditions. In the case of a standard linear model, explicit analytic results and numerical examples are given for the pressure function, for surface displacements, and also for hysteretic friction.

Nonlinear viscoelastic effects in oscillatory shear deformation of brain tissue

Two nonlinear constitutive models were used to describe the dynamic viscoelastic behavior of brain tissue. Small disc-shaped samples of bovine brain tissue were tested in simple shear using forced vibrations (0.5 to 200 Hz) with finite amplitudes (up to 20% Lagrangian shear strain). The samples response to simple, double, and triple harmonic inputs was determined in order to characterize the nonlinearities up to the third-order. A quasilinear viscoelastic model was proposed to describe the spatial nonlinearity. A fully nonlinear viscoelastic model with product-form multiple hereditary integrals was proposed to describe the spatial as well as the temporal nonlinearities. The fully nonlinear model demonstrated superiority at high frequencies (above 44 Hz). Under finite strains, the linear complex modulus showed nonrecoverable asymptotic strain conditioning behavior. Discrepancies observed in previously published studies and the threshold of functional failure of the neural tissue were shown to be related to this strain conditioning effect.

Indentation rheometry for glass-forming materials

Theoretical considerations are made on the time-dependent viscoelastic deformation and flow during penetration by a flat-ended, spherical, conical, or a pyramidal indenter into glass-forming materials at temperatures near the glass-transition. The concept of representative stress and strain combined with the hereditary integral leads to principal constitutive equations for linear viscoelastic indentation. An experimental study of the viscoelastic behavior of soda-lime silica glass, as a representative of glass-forming materials, is conducted with the use of a trigonal sapphire indenter (Berkovich indenter). The self-consistency of the present theoretical framework for viscoelastic indentation is demonstrated on the basis of experimental results. It is emphasized that the indentation technique provides the most efficient micro/nano-probes for examining viscoelastic behaviors of glass-forming materials at temperatures near the glass-transition; testing specimens with extremely small dimensions and thin films with no special preparation, as well as testing with compressive loading for elevated temperatures.

A spatially nonlocal model for polymer-penetrant diffusion

Diffusion of a penetrant in a polymer entanglement network cannot be described by Fick's Law alone; rather, one must incorporate other nonlocal effects. In contrast to previous viscoelastic models which have modeled these effects through hereditary integrals in time, a new model is presented exploiting the disparate lengths of the polymer in the glassy (dry) and rubbery (saturated) states. This model leads to a partial integrodifferential equation which is nonlocal in space. The system is recast as a moving boundary-value problem between sets of coupled partial differential equations. Using singular perturbation techniques, sorption in a semi-infinite polymer is studied on several time scales with varying exposed interface conditions. Though some of the results match with those from viscoelastic models, new physically relevant behaviors also appear. These include the formation of stopping fronts and overshoot in the pseudostress.

Method of calculating the hereditary integral at variable temperature

Method of calculating the hereditary integral at variable temperature

Pyramidal indentation load–depth curve of viscoelastic materials

The indentation load P versus depth h curves are examined to investigate the time-dependent surface deformation of viscoelastic materials. The viscoelastic P–h curves significantly depend on the temperature of measurement and the penetration rate of indentation. Sneddon's elastic solution of a conical indentation is extended to a viscoelastic one for a conical or a pyramidal indentation in terms of the hereditary integral. Several types of viscoelastic problems are discussed in relation to the test techniques and analyses for determining the relaxation modulus E(t) and the creep compliance D(t). The superposition rules of time–temperature, penetration depth–temperature, and penetration depth–penetration rate are examined. The viscoelastic indentation tests (constant rate penetration test and constant load creep test) of amorphous Se are conducted at temperatures from 10 to 42 °C. The theoretical considerations and the test results encourage pyramidal indentation as an efficient microprobe for the viscoelastic characterization, in particular, of extremely small-size test specimens and ceramic, metal, and polymer thin films coated on substrate.

Damage Dependent Constitutive Behavior and Energy Release Rate for a Cohesive Zone in a Thermoviscoelastic Solid

In this paper a cohesive zone is introduced ahead of a crack tip in order to avoid the singularity at the crack tip. By applying thermodynamics to the cohesive zone and the surrounding body, a fracture criterion will be established so that the inelastic energy dissipation both in the cohesive zone and the surrounding bulk material can be distinguished from the energy released by fracture, and the propagation of crack can be predicted. In addition, the cohesive zone constitutive equation is constructed utilizing the Helmholtz free energy in the form of a single hereditary integral for a nonlinear viscoelastic material. The resulting constitutive model for the cohesive zone contains an internal state variable which represents the damage state within the cohesive zone. When the cohesive zone opening displacement is known, the energy release rate is thus history dependent, which is expressed in terms of the damage state, the length of separation in the cohesive zone and the geometric configuration of the cohesive zone opening displacement. Example results contained herein demonstrate this effect.