Integral Constitutive Equation Research Articles

Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models

This paper presented an efficient finite element formulation for analysis of the buckling and free vibration of the axial functionally graded (AFG) Euler-Bernoulli nanobeam based on minimum total potential energy principle with linear constraints. The stress- and strain-driven two-phase local/nonlocal integral models are utilized to address small scale effects. The material properties of the AFG nanobeam vary exponentially along the axial direction. The nonlocal integral constitutive equation can be equivalently transformed into a differential equation with two constitutive boundary conditions. The governing equation and standard boundary conditions are derived via Hamilton's principle. The weak form associated with total potential energy is derived and the constitutive boundary conditions can be converted to external forces at the boundaries. By employing a numerical solution methodology on the basis of the finite element method (FEM) together with Lagrangian multiplier method (LMM), a unified finite element formulation based on stress- and strain-driven two-phase local/nonlocal elasticity is constructed to obtain the critical buckling load and vibration frequency. Meanwhile, the numerical results obtained through generalized differential quadrature method (GDQM) validate the efficiency and accuracy of the present results. The influence of gradient index, nonlocal parameter and local phase parameter on buckling and free vibration of AFG Euler-Bernoulli nanobeam is investigated in detail under different nonlocal models and boundary conditions. It is demonstrated that the gradient index has more effect on mode shapes than the buckling load and vibration frequency between the local stress- and strain-driven two-phase local/nonlocal models.

60 years of the Kaye–Bernstein, Kearsley, Zapas rheological constitutive law for polymers

The K–BKZ (Kaye–Bernstein, Kearsley, Zapas) rheological constitutive model is now 60 years old. The paper reviews the connections of the model and its variants with continuum mechanics and experimental evidence in polymer melt flow, presenting an up-to-date recap of research and major findings in the open literature. In the Introduction, an historical perspective is given on developments in the last 60 years of the K–BKZ model. Then, a section on mathematical modeling of polymer flows follows, including the governing equations of flow, the rheological constitutive equations (with emphasis on the viscoelastic integral constitutive equations of the K–BKZ type), dimensionless numbers controlling the flow, and relevant boundary conditions. The “Method of Solution” section reviews the major developments of numerical techniques for particle tracking and integral evaluation for the viscoelastic stresses. Finally, selected examples of successful application of the K–BKZ model in polymer flows are presented including considerations of wall slip and non-isothermal flows.

Numerical Simulation of Three-Dimensional Free Surface Flows Using the K-BKZ-PSM Integral Constitutive Equation.

This work introduces a novel numerical method designed to address three-dimensional unsteady free surface flows incorporating integral viscoelastic constitutive equations, specifically the K-BKZ-PSM (Kaye-Bernstein, Kearsley, Zapas-Papanastasiou, Scriven, Macosko) model. The new proposed methodology employs a second-order finite difference approach along with the deformation fields method to solve the integral constitutive equation and the marker particle method (known as marker-and-cell) to accurately capture the evolution of the fluid's free surface. The newly developed numerical method has proven its effectiveness in handling complex fluid flow scenarios, including confined flows and extrudate swell simulations of Boger fluids. Furthermore, a new semi-analytical solution for velocity and stress fields is derived, considering fully developed flows of a K-BKZ-PSM fluid in a pipe.

Theoretical thermal damping vibration analysis of functionally graded viscoelastic Timoshenko microbeam with integral nonlocal strain gradient model

In this work, nonlocal strain gradient integral model is applied to investigate the free damping vibration analysis of functionally graded (FG) viscoelastic Timoshenko microbeam with immovable boundary conditions in thermal environment. The microbeam is assumed to be viscoelastic and modeled by Kelvin–Voigt model. The differential governing equations and corresponding boundary conditions are derived with the Hamilton’s principle. Combining nonlocal strain gradient integral model and Kelvin–Voigt viscoelastic model, the integral constitutive equations of nonlocal stress with thermal effect are derived and then converted into differential form plus constitutive constraints. The size-dependent axial force due to thermal expansion is explicitly derived through the immovable boundary conditions. The bending deflection and moment, cross-sectional rotation as well as shear force are explicitly derived with Laplace transformation for linear thermo-elastic vibration. Taking into account the boundary conditions as well as constitutive constraints, one gets a nonlinear equation with complex coefficients, from which one can determine the complex characteristic frequency. A two-step numerical method is proposed to solve the elastic vibration frequency and damping ratio. The influence of length scale parameters, viscos coefficient, FG index, thermal effect, vibration order and ratio between beam length and thickness on the vibration frequencies and damping ratio is investigated numerically for FG viscoelastic Timoshenko microbeams under different immovable boundary conditions.

Numerical Simulation on Structural Integrity of Grain with Large Increased Area Ratio for Solid Rocket Motor

To evaluate the structural integrity of a type of solid rocket motor grain with a largely increased area ratio using free-loading modified double-base grain, the Schapery integral nonlinear viscoelastic constitutive equation with damage is derived, and the numerical simulation model of solid rocket motor grain is constructed. According to the mission profile of the grain during the equipment period, the numerical simulation research on the mechanical response characteristics of the grain under three typical working conditions, namely, platform vibration, underwater explosion impact, and ignition pressurization during the launch is carried out. The simulation results show that under the three typical working conditions, the grain will not produce structural damage and defects, and the mechanical response of the grain under the platform vibration and explosion impact load is far less than that under the launch ignition pressurization. The ignition pressurization should be considered in the grain structural integrity design of this type or similar structure solid rocket motor.

Fiber Spinning with Molecular Models

Abstract The isothermal fiber spinning of ciscoelastic liquid is analyzed successfully by means of the Curtiss-Bird integral constitutive equation derived from molecular theory, Glaerkin/finite element method, and Newton iteration. The analysis accounts for shear and elongational prehistories, and specified draw ratio or drawing force. Solutions are obtained at high values of elasticity, draw ratio and drawing force. The predictions agree with the analytic solution for Newtonian liquid in the limit of zero relaxation time and with experimental data from the literature on spinning of polystyrene and polypropylene melts.

Local/nonlocal mixture integral models with bi-Helmholtz kernel for free vibration of Euler-Bernoulli beams under thermal effect

We present predictive models of the free vibration of Euler-Bernoulli beams subjected to a uniformly thermal environment using two-phase local/nonlocal mixture theory of strain- and stress-driven types. The equation of motion and standard boundary conditions are derived via Hamilton's principle and the constitutive equation is expressed in local/nonlocal mixed integral form with bi-Helmholtz kernel. The temperature effect is taken as equivalent to a thermal load taking into account the two-phase nonlocal effects. After transforming the local/nonlocal mixture integral constitutive equation to differential form with four constitutive boundary conditions, the problem is solved numerically via a generalized differential quadrature method (GDQM). We provide a series of numerical examples in which we present predictions of the size-effects on beams with various boundary edges using different local/nonlocal mixture models. The influence of the two-phase thermal load on the natural frequencies of the beam is also examined. Furthermore, it can be observed that the normalized frequency value varies consistently with the increase of the order of the vibration mode.

Spectral method for time-strain separable integral constitutive models in oscillatory shear

The time-strain separable Kaye–Bernstein–Kearsley–Zappas model (tssKBKZM) is a popular integral constitutive equation that is used to model the nonlinear response of time-strain separable materials using only their linear viscoelastic properties and damping function. In oscillatory shear, numerical evaluation of tssKBKZM is complicated by the infinite domain of integration, and the oscillatory nature of the integrand. To avoid these problems, a spectrally accurate method is proposed. It approximates the oscillatory portion of the integrand using a discrete Fourier series, which enables analytical evaluation of the resulting integrals for the Maxwell model. The spectral method is generalized for arbitrary discrete and continuous relaxation spectra. Upper bounds for quadrature error, which can often be driven to machine precision, are presented. The Doi–Edwards model with independent-alignment approximation (DE-IA) is a special case of tssKBKZM; for DE-IA, the spectral method is compared with trapezoidal rule to highlight its accuracy and efficiency. The superiority of the proposed method is particularly evident at large strain amplitude and frequency. For continuous relaxation spectra, the spectral method transforms the double integral corresponding to the tssKBKZM to a single integral. Solutions computed to a specified level of accuracy using standard numerical libraries show that the spectral method is typically two to three orders of magnitude faster. Extensions to fractional rheological models, materials with nonzero equilibrium modulus, stretched exponential models, etc., are also discussed.

A bi-Helmholtz type of two-phase nonlocal integral model for buckling of Bernoulli-Euler beams under non-uniform temperature

It is well-acknowledged by the scientific community that Eringen’s nonlocal integral theory is not applicable to nanostructures of engineering interest due to conflict between equilibrium and constitutive requirements. In this paper, a well-posed two-phase nonlocal integral elasticity with the bi-Helmholtz kernel is developed to study the size-dependent buckling response of Bernoulli-Euler beams under non-uniform temperatures. The governing equation is derived by invoking the variational principle of virtual work, and the temperature effect is equivalent to the thermal load along the axial direction, which is determined by nonlocal heat conduction. The two-phase nonlocal integral constitutive equation is transformed into a differential one equipped with four constitutive boundary conditions, and then exact solutions for the buckling loads of the beam with various boundary edges are obtained. Numerical results are validated by comparing them with those from local elasticity. Moreover, the effects of parameters related to the two-phase nonlocal elastic model and the nonlocal heat conductive model are investigated.

A constitutive equation for Rouse model modified for variations of spring stiffness, bead friction, and Brownian force intensity under flow

We derived a constitutive equation for the Rouse model (the most frequently utilized bead-spring model) with its spring constant κ, bead friction coefficient ζ, and the (squared) Brownian force intensity B being allowed to change under flow. Specifically, we modified the Langevin equation of the original Rouse model by introducing time (t)-dependent κ, ζ, and B (of arbitrary t dependence), which corresponded to the decoupling and preaveraging approximations often made in bead-spring models. From this modified Langevin equation, we calculated time evolution of second-moment averages of the Rouse eigenmode amplitudes and further converted this evolution into a constitutive equation. It turned out that the equation has a functional form, σ(t)=∫−∞tdt′{κ(t)/κ(t′)}M(t,t′)C−1(t,t′), where σ(t) and C−1(t,t′) are the stress and Finger strain tensors, and M(t,t′) is the memory function depending on κ(t′), ζ(t′), and B(t′) defined under flow. This equation, serving as a basis for analysis of nonlinear rheological behavior of unentangled melts, reproduces previous theoretical results under specific conditions, the Lodge–Wu constitutive equation for the case of t-independent κ, ζ, and B [A. S. Lodge and Y. Wu, “Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm,” Rheol. Acta 10, 539 (1971)], the finite extensible nonlinear elastic (FENE)-Peterlin mean-Rouse formulation for the case of t-dependent changes of the only κ reported by Wedgewood and co-workers [L. E. Wedgewood et al., “A finitely extensible bead-spring chain model for dilute polymer solutions,” J. Non-Newtonian Fluid Mech. 40, 119 (1991)], and analytical expression of steady state properties for arbitrary κ(t), ζ(t), and B(t) reported by ourselves [H. Watanabe et al., “Revisiting nonlinear flow behavior of Rouse chain: Roles of FENE, friction reduction, and Brownian force intensity variation,” Macromolecules 54, 3700 (2021)]. It is to be added that a constitutive equation reported by Narimissa and Wagner [E. Narimissa and M. H. Wagner, “Modeling nonlinear rheology of unentangled polymer melts based on a single integral constitutive equation,” J. Rheol. 64, 129 (2020)] has a significantly different functional form and cannot be derived from the Rouse model with any simple modification of the Rouse–Langevin equation.

Numerical Simulation of KBKZ Integral Constitutive Equations in Hierarchical Grids

In this work, we present the implementation and verification of HiGTree-HiGFlow solver (see for numerical simulation of the KBKZ integral constitutive equation. The numerical method proposed herein is a finite difference technique using tree-based grids. The advantage of using hierarchical grids is that they allow us to achieve great accuracy in local mesh refinements. A moving least squares (MLS) interpolation technique is used to adapt the discretization stencil near the interfaces between grid elements of different sizes. The momentum and mass conservation equations are solved by an implicit method and the Chorin projection method is used for decoupling the velocity and pressure. The Finger tensor is calculated using the deformation fields method and a three-node quadrature formula is used to derive an expression for the integral tensor. The results of velocity and stress fields in channel and contraction-flow problems obtained in our simulations show good agreement with numerical and experimental results found in the literature.

Bending and Buckling Analysis of Functionally Graded Euler–Bernoulli Beam Using Stress-Driven Nonlocal Integral Model with Bi-Helmholtz Kernel

Static bending and elastic buckling of Euler–Bernoulli beam made of functionally graded (FG) materials along thickness direction is studied theoretically using stress-driven integral model with bi-Helmholtz kernel, where the relation between nonlocal stress and strain is expressed as Fredholm type integral equation of the first kind. The differential governing equation and corresponding boundary conditions are derived with the principle of minimum potential energy. Several nominal variables are introduced to simplify differential governing equation, integral constitutive equation and boundary conditions. Laplace transform technique is applied directly to solve integro-differential equations, and the nominal bending deflection and moment are expressed with six unknown constants. The explicit expression for nominal deflection for static bending and nonlinear characteristic equation for the bucking load can be determined with two constitutive constraints and four boundary conditions. The results from this study are validated with those from the existing literature when two nonlocal parameters have same value. The influence of nonlocal parameters on the bending deflection and buckling loads for Euler–Bernoulli beam is investigated numerically. A consistent toughening effect is obtained for stress-driven nonlocal integral model with bi-Helmholtz kernel.

Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model

A torsional static and free vibration analysis of the functionally graded nanotube (FGNT) composed of two materials varying continuously according to the power-law along the radial direction is performed using the bi-Helmholtz kernel based stress-driven nonlocal integral model. The differential governing equation and boundary conditions are deduced on the basis of Hamilton’s principle, and the constitutive relationship is expressed as an integral equation with the bi-Helmholtz kernel. Several nominal variables are introduced to simplify the differential governing equation, integral constitutive equation, and boundary conditions. Rather than transforming the constitutive equation from integral to differential forms, the Laplace transformation is used directly to solve the integro-differential equations. The explicit expression for nominal torsional rotation and torque contains four unknown constants, which can be determined with the help of two boundary conditions and two extra constraints from the integral constitutive relation. A few benchmarked examples are solved to illustrate the nonlocal influence on the static torsion of a clamped-clamped (CC) FGNT under torsional constraints and a clamped-free (CF) FGNT under concentrated and uniformly distributed torques as well as the torsional free vibration of an FGNT under different boundary conditions.

A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

Prediction of Deformation Processes of Polymeric Materials Taking into Account Plastic Irreversible Deformation

The need to develop new methods for predicting the properties of polymeric materials is also justified by the goal of designing new innovative materials with the required functional properties and increased competitiveness. The classical methods for predicting deformation processes of polymeric materials are based on the numerical solution of integral constitutive equations for polymer viscoelasticity of the Boltzmann-Volterra type, which do not consider corrections for the irreversibility of the plastic component of deformation and therefore can lead to significant prediction errors. To improve the accuracy of predicting the deformation processes of polymer materials it is proposed to introduce a physically justified correction with account for the irreversibility of the plastic component of deformation. The introduction of this correction significantly increases the reliability and accuracy of predicting the functional and operational properties of polymer materials. The article suggests demonstrating the methods for predicting deformation processes with the example of the polyester textile yarn made of polyester fibers. Unlike many other synthetic fibers, the polyester ones have such important properties as structural stability, softness along with high strength, elasticity, resilience, tensile strength, crease and pilling resistance, temperature regulation, shape retention, etc. The polyester fiber has a hollow structure and its single components have the form of spiral springs which give the effect of a springy base when intertwined.

Fractional Nonlocal Elasticity and Solutions for Straight Screw and Edge Dislocations

Nonlocal elasticity assumes integral stress constitutive equation, takes into account interatomic long-range forces, reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit. In this paper, we propose the nonlocal elasticity theory with the kernel (the weight function in the integral stress constitutive equation) which is the Green function of the Cauchy problem for the fractional diffusion equation and is expressed in terms of the Mainardi function and Wright function being the generalizations of the exponential function. The solutions for the straight screw and edge dislocations in an infinite solid are obtained in the framework of the proposed theory.

Exact solutions for size-dependent bending of Timoshenko curved beams based on a modified nonlocal strain gradient model

Size-dependent bending analysis of Timoshenko curved beams is performed with a modified nonlocal strain gradient integral model, in which the integral constitutive equation is transformed into an equivalent differential form equipped with two constitutive boundary equations. The governing equations and boundary conditions are derived via the minimum total potential energy principle and solved analytically using the Laplace transformation technique and its inverse version. In numerical examples, the inconsistency of the nonlocal strain gradient model is examined extendedly under different boundary and loading conditions, while consistent softening and stiffening responses can be observed via the modified nonlocal strain gradient integral model. In addition, within the modified nonlocal strain gradient model, numerical examples also show that the increase of the opening angle can affect the total size effects of the combination of the two scale parameters (i.e., nonlocal and gradient), and these effects are inconsistent for different beam boundaries. Finally, by comparing with the results of the Euler–Bernoulli theory, an interesting finding is that as the nonlocal (or gradient length-scale) parameter increases, the shear deformations of simply supported-simply supported beams (or clamped-clamped/simply supported beams) become more significant.

Universality of steady shear flow of Rouse melts

The data set of steady and transient shear data reported by Santangelo and Roland Journal of Rheology 45: 583–594, (2001) in the nonlinear range of shear rates of an unentangled polystyrene melt PS13K with a molar mass of 13.7 kDa is analysed by using the single integral constitutive equation approach developed by Narimissa and Wagner Journal of Rheology 64:129–140, (2020) for elongational and shear flow of Rouse melts. We compare model predictions with the steady-state, stress growth, and stress relaxation data after start-up shear flows. In characterising the linear-viscoelastic relaxation behaviour, we consider that in the vicinity of the glass transition temperature, Rouse modes and glassy modes are inseparable, and we model the terminal regime of PS13K by effective Rouse modes. Excellent agreement is achieved between model predictions and shear viscosity data, and good agreement with first normal stress coefficient data. In particular, the shear viscosity data of PS13K as well as of two polystyrene melts with M = 10.5 kDa and M = 9.8 kDa investigated by Stratton Macromolecules 5 (3): 304–310, (1972) agree quantitatively with the universal mastercurve predicted by Narimissa and Wagner for unentangled melts, and approach a scaling of Wi−1/2at sufficiently high Weissenberg numbers Wi. Some deviations between model predictions and data are seen for stress growth and stress relaxation of shear stress and first normal stress difference, which may be attributed to limitations of the experimental data, and may also indicate limitations of the model due to the complex interactions of Rouse modes and glassy modes in the vicinity of the glass transition temperature.Graphical abstract

Buckling analysis of curved sandwich microbeams made of functionally graded materials via the stress-driven nonlocal integral model

Size-dependent buckling analysis for slightly curved sandwich microbeams made of functionally graded (FG) materials is performed via a stress-driven nonlocal model. The Fredholm integral constitutive equations are transformed into the Volterra type of the first kind and then solved analytically using the Laplace transformation and its inversion under different boundary conditions. The exact solutions are validated against those existing results. The effect of the nonlocal parameter, thickness ratio of core-to-skin layers, FG power-law index, and length-height ratio on the buckling loads, as well as on the ratio of the result predicted by two common beam-theories is investigated.

The consistency of the nonlocal strain gradient integral model in size-dependent bending analysis of beam structures

Despite the extensive applications of the nonlocal strain gradient theory to predict the size-effect of beam-like structures, the inconsistencies of its differential form have been uncovered recently. For bounded beam structures, there are additional constitutive boundary conditions should be taken into account to ensure the strict equivalence between the differential constitutive relation and the originally integral ones. Such an improvement would lead the problems to have more mandatory boundary conditions than those required, and therefore, there must be a trade-off between these boundary conditions. A strategy by omitting the higher-order (non-standard) boundary conditions has been presented recently and from which consistently size-effects can be obtained. However, it was pointed out that this treatment no longer fully meets the requirements of the energy-variational principle. In this paper, an alternative methodology by directly using the originally integral model is proposed for the first time. The advantages of such a strategy are pertinently evidenced by investigating Euler-Bernoulli curved beams made of functionally graded materials. The Laplace transform technique is applied to solve the integral constitutive equations and differential governing equations, which is an original contribution of this paper. In contrast to the differential forms, the originally integral model can satisfy energy conservation well. Meanwhile, adopting specific higher-order boundary conditions, numerical results show consistently stiffening and softening effects for all boundary edges and loading conditions.