Basic Constitutive Equations Research Articles

Crystal Plasticity Elastic-Plastic Rate-Independent Numerical Analyses of Pollycrystalline Materials

Macroscopic analyses of plastic forming processes give only the overall description of the problem without the consideration of mechanisms of plastic deformation and the microstructure evolution. For the consideration of these processes, numerical simulations within crystal plasticity include the change of texture, anisotropy, and strain hardening of the material are used. In this paper, a crystal plasticity rate-independent model proposed by Anand and Kothari is applied for numerical analyses of polycrystalline materials. The slip was considered as the main mechanism of the plastic deformation. Basic constitutive equations of crystal plasticity for large deformation theories are presented. The selected results of elastic-plastic problems obtained using both macro- and micro- scales software for the explicit and implicit integration are featured here. The heterogeneous distribution of strain and stress in different grains are obtained, which is associated with the various crystal orientation. The crystal plasticity modelling of materials subject to plastic deformation involves not only the information about the change of a material’s shape in a macro-scale, but also describes the phenomena occurring in material in a micro-scale.

Load Ratios Carried by Each Constituent for Some Problems of a Particulate Composite Modeled as a Mixture of Two Linear Elastic Solids

ABSTRACTThe basic constitutive equations of theory of mixtures obtained for a mixture of two linear elastic solids can be used as an alternative way to describe the mechanical behavior of binary composite materials. Determining the load ratios carried by each constituent solid of a binary composite is one of challenges of this theory. In this study, the results of directly calculating the ratios of external load carried by each constituent solid for the case of perfectly bonded interface between binary mixture constituents are discussed. Thus, the effects of loading type and volume fraction of the constituent solids to the load ratios carried by each constituent solid are investigated by using three different loading cases and three different volume fractions. Finally, displacement, stress and diffusive force results of two constituent solids using the calculated load ratios are given.

Influence of Inclined Magnetic Field on Carreau Nanoliquid Thin Film Flow and Heat Transfer with Graphene Nanoparticles

The thermodynamics of a Carreau nanoliquid thin film embedded with graphene nanoparticles past a stretching sheet is studied in the presence of inclined magnetic field and non-uniform heat source/sink. Graphene is a new two-dimensional amphiphilic macromolecule which has great applications due to its electrical and mechanical properties. The basic constitutive equations of Carreau nanoliquid for velocity and temperature have been used. Similarity transformations are adopted to achieve the nonlinear coupled differential equations accompanying boundary conditions embedded with different parameters. HAM (Homotopy Analysis Method) is used to solve the transformed equations for expressions of velocity and temperature. Graphs are shown which illustrate the effects of various parameters of interest. There exists a nice agreement between the present and published results. The results are useful for the thermal conductivity and in the analysis and design of coating processes.

Exegesis of Sect. II and III.A from “Fundamentals of the Mechanics of Continua” by E. Hellinger

AbstractThis is a second exegetic essay on the fundamental review article DIE ALLGEMEINEN ANSÄTZE DER MECHANIK DER KONTINUA in the Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Bd. IV‐4, Hft. 5 (1913) by Ernst Hellinger which concerns the translation and the commentary of pp. 629–662. Within these pages, the setting of the basic constitutive equations for field theories, whose formulations are based on the principle of virtual work or on the principle of stationary action, are discussed. The interest for a contemporary reader for the herein presented subject is still substantial, as this article clearly contains some considerations and visions being still topical. However, there is also an epistemological interest in examining it from the point of view of a historian of science. Indeed, it represents an available but forgotten source of an important piece of mechanical sciences. Available, because it is still present in our libraries in its complete form, but forgotten because, being written in German by a Jewish refugee escaped to the United States, has been ignored by the main stream of the dominant groups in continuum mechanics. The ideas by Hellinger and the German school of continuum mechanics (remark that this school includes, as we have discovered reading this article, even Gauss himself!) were lost or dramatically deformed in translation. We believe that the destiny of Hellinger's paper is an evidence supporting Lucio Russo's view about history of science. Our aim is to trace the origins of current ideas of mechanical sciences to their original sources.

A basic hypoplastic constitutive model for sand

Hypoplastic constitutive models are based on nonlinear tensor functions and are characterized by simple formulation and few parameters. In its early stage, mainly basic hypoplastic constitutive equations were concerned, where the stress tensor is assumed as the only state variable. There followed some enhanced models based on the basic constitutive equation by including void ratio as an additional state variable. In this paper, we first show that the widely used hypoplastic model by Wolffersdorff is seriously flawed because the underlying basic equation does not perform properly. We proceed to develop a basic hypoplastic constitutive equation by introducing a new tensorial term, which preserves the critical state at large strain. The model performance is demonstrated by parameter study for some element tests. This simple and robust basic equation is well suited to build more sophisticated models.

Magnetohydrodynamic Nanoliquid Thin Film Sprayed on a Stretching Cylinder with Heat Transfer

The magnetohydrodynamic thin film nanofluid sprayed on a stretching cylinder with heat transfer is explored. The spray rate is a function of film size. Constant reference temperature is used for the motion past an expanding cylinder. The sundry behavior of the magnetic nano liquid thin film is carefully noticed which results in to bring changes in the flow pattern and heat transfer. Water-based nanofluids like Al 2 O 3 -H 2 O and CuO-H 2 O are investigated under the consideration of thin film. The basic constitutive equations for the motion and transfer of heat of the nanofluid with the boundary conditions have been converted to nonlinear coupled differential equations with physical conditions by employing appropriate similarity transformations. The modeled equations have been computed by using HAM (Homotopy Analysis Method) and lead to detailed expressions for the velocity profile and temperature distribution. The pressure distribution and spray rate are also calculated. The comparison of HAM solution predicts the close agreement with the numerical method solution. The residual errors show the authentication of the present work. The CuO-H 2 O nanofluid results from this study are compared with the experimental results reported in the literature showing high accuracy especially, in investigating skin friction coefficient and Nusselt number. The present work discusses the salient features of all the indispensable parameters of spray rate, velocity profile, temperature and pressure distributions which have been displayed graphically and illustrated.

Logarithmic Spin, Logarithmic Rate and Material Frame-Indifferent Generalized Plasticity

In this work, we present a new rate type formulation of large deformation generalized plasticity which is based on the consistent use of the logarithmic rate concept. For this purpose, the basic constitutive equations are initially established in a local rotationally neutralized configuration which is defined by the logarithmic spin. These are then rephrased in their spatial form, by employing some standard concepts from the tensor analysis on manifolds. Such an approach, besides being compatible with the notion of (hyper)elasticity, offers three basic advantages, namely: (i) The principle of material frame-indifference is trivially satisfied. (ii) The structure of the infinitesimal theory remains essentially unaltered. (iii) The formulation does not preclude anisotropic response. A general integration scheme for the computational implementation of generalized plasticity models which are based on the logarithmic rate is also discussed. The performance of the scheme is tested by two representative numerical examples.

Spectral stochastic finite element vibration analysis of fiber-reinforced composites with random fiber orientation

Fiber-reinforced composites exhibit random fiber orientations due to the manufacturing tolerances. The present study concerns with a numerical investigation of the vibrational behavior of the long fiber-reinforced composites under uncertainty in fiber orientations. The basic constitutive equations of laminate theory based on the first-order shear deformation are developed in stochastic form. It is assumed that each ply-orientation exhibits individual spatial and random variations from the nominal mean value. These lead to local variations in mechanical properties such as structural stiffness and mass matrices and, accordingly, in structural responses. The finite element formulation utilizes the spectral stochastic discretization in the sense that the random fiber orientations are approximated by a truncated Karhunen–Loève expansion. The expansion provides a framework to capture the spatial and random variations. The generalized polynomial chaos with unknown deterministic coefficients is employed to represent the structural responses. The coefficients are estimated using probabilistic collocation points where any deterministic in-house code or commercial FEM package can be treated as a black-box for the modeling and analysis of structure. A numerical case study is used to illustrate the features of the method, where impact of the uncertainty in fiber orientations on the natural frequencies and mode shapes of a 12 plies rectangular composite plate with nominal orientation of [0/90]s is investigated.

On the constitutive modeling of partially saturated interfaces

Generalization of soil–structure interface models from dry/saturated states to consider partially saturated states is studied in this paper. For this purpose, basic constitutive equations of a conventional elasto-plastic interface model are firstly presented. Then, consideration is given to the effect of partial saturation on definition of effective stress, location of the critical state line as well as the impact of interface state on plastic hardening modulus and dilatancy. For each concern, proper independent approaches together with associated constitutive equations are discussed to be included in the basic model as complementary ingredients. Among many different possibilities to combine complementary constitutive equations for effective stress, relocation of the critical state line with degree of saturation, and impact of the interface state on plastic hardening modulus and dilatancy, six essential cases are selected. Evaluations show that all six cases can realistically consider the impact of partial degree of saturation on the peak and residual shear strengths as well as the volume change behavior of unsaturated interfaces.

Spectral element solution for transversely isotropic elastic multi-layered structures subjected to axisymmetric loading

Spectral element method is proposed to analyze the mechanical response of transversely isotropic elastic multi-layered pavement structure subjected to axisymmetric loading. Based on the basic constitutive equations and modified Love’s function for transversely isotropic elastic media, the governing state equation of a multi-layered transversely isotropic medium was deduced. From the equation, s spectral layer element for a single layer (i.e., a stiffness matrix) was acquired. The global stiffness matrix was obtained by assembling the interrelated layer elements based on the principle of the finite element method and the boundary conditions, and the solution for the corresponding problem was obtained by solving the algebraic equations of the global stiffness matrix. The solutions in the physical domain were acquired by means of the Fourier–Bessel superposition. The validity of the proposed solution was examined by comparing with an existing exact solution and finite element method respectively. Subsequent to solution validation, a parametric study by varying n-values of unbound layer was carried to investigate the influence of the transversal isotropy on pavement response. The proposed analytical solution demonstrated the ability of solving the mechanical response of transversely isotropic elastic multi-layered pavement structures subjected to axisymmetric loading.

Nonlinear dynamic model with multi-fields coupling effects for giant magnetostrictive actuators

This paper addresses the modeling of complex hysteresis behavior for giant magnetostrictive actuator system under magnetically unbiased conditions. The hysteresis behavior is modeled by establishing a novel nonlinear dynamic model with multi-fields coupling effects, in which both the eddy current effects and the change of stress are considered. The former is included in the nonlinear transient constitutive model with magnetic-elastic-thermal coupling effect, which is employed as the basic constitutive equations of Terfenol-D. The latter is characterized through the structural dynamic behavior of actuator system itself, which is modeled by theorem of momentum. The quantitative agreements between numerical simulation results and existing experimental data indicate that nonlinear dynamic model can accurately describe the complex hysteresis behavior of the giant magnetostrictive actuator system not only under quasi-static operating conditions but also under dynamic operating conditions. The numerical simulation results also indicate that both the eddy current effects and structural dynamic behavior are the origin of frequency-dependent hysteresis behavior for giant magnetostrictive actuator system, and demonstrate the significance and necessity of simultaneously considering the eddy current effects and the change of stress in the system-level. Thus, the nonlinear dynamic model established in this paper is a system-lever coupled theoretical model, which can be directly used in the active vibration control and any other engineering application of the giant magnetostrictive actuators.

Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment

In this paper, the axisymmetric buckling analysis of circular single-layered graphene sheets is studied by decoupling the basic constitutive equations based on the nonlocal theory of Eringen. The influences of temperature change, surface parameters and nonlocality on the buckling response of single-layered graphene sheet are investigated considering size-dependent material properties. Numerical solutions for buckling loads are computed using differential quadrature method (DQM). For comparison purpose, Galerkin method is also used to solve the nonlocal governing differential equation. DQM results are successfully verified with those of Galerkin method. The comparison of present results with the available molecular dynamics simulation data from the literature shows that the present formulation with appropriate values of surface and nonlocal parameters provides more accurate results than those obtained by the classical plate model. The results of present work can be used as benchmarks to evaluate future analyses of the circular nanoplates.

Analytical layer-element solutions for a multi-layered transversely isotropic elastic medium subjected to axisymmetric loading

This paper presents an analytical layer-element method used to analyze the displacement of a multi-layered transversely isotropic elastic medium of arbitrary depth subjected to axisymmetric loading. Based on the basic constitutive equations and the HU Hai-chang’s solutions for transversely isotropic elastic media, the state vectors of a multi-layered transversely isotropic medium were deduced. From the state vectors, an analytical layer element for a single layer (i.e., a symmetric and exact stiffness matrix) was acquired in the Hankel transformed domain, which not only simplified the calculation but also improved the numerical efficiency and stability due to the absence of positive exponential functions. The global stiffness matrix was obtained by assembling the interrelated layer elements based on the principle of the finite layer method. By solving the algebraic equations of the global stiffness matrix which satisfy the boundary conditions, the solutions for multi-layered transversely isotropic media in the Hankel transformed domain were obtained. The actual solutions of this problem in the physical domain were acquired by inverting the Hankel transform. This paper presents numerical examples to verify the proposed solutions and investigate the influence of the properties of the multi-layered medium on the load-displacement response.

Viscosity sensor using ZnO and AlN thin film bulk acoustic resonators with tilted polar c-axis orientations

We present a theoretical analysis of liquid viscosity sensors using ZnO and AlN thin film bulk acoustic wave resonators (FBARs) with tilted polar c-axis orientations. Besides the thickness longitudinal mode, the tilted c-axis orientation induces thickness shear mode in the resonator, which allows resonators operated in liquid medium without significant damping for sensory application. The equation for predicting electric impedance of shear mode film bulk acoustic wave resonators (FBARs) with a viscous liquid loading was derived from the basic piezoelectric constitutive equations. The viscosity sensitivity of shear mode ZnO and AlN resonators was achieved by calculation of resonant frequency shift due to viscous liquid loading. In the simulation, it is assumed that all the resonators have 2 μm thickness and 300 μm×300 μm electrode area; three different liquids (water, acetone, and olive oil) were chosen as the liquid loadings; and different tilt c-axis angles for both ZnO FBARs and AlN FBARs have been examined. It was found that the sensitivities of shear mode resonators to the three liquid loading are very close, and do not change much with the c-axis tilt angle with a value rang from 0.91e-3 to 0.97e-3 (kg m−3 Pa·S)−0.5 for ZnO FBARs and from 1.12e−3 to 1.13 e−3 (kg m−3 Pa·S)−0.5 for AlN FBARs. When the resonator’s mechanical quality factor (Q) is changed from 50 to 10 000, viscosity sensitivities are almost same. However, Q has a great effect on resonator impedance; and if Q is too low or the viscosity of the liquid is high, the maximum phase angle of the resonator will be less than 0, which makes excitation of the oscillation difficult if an oscillator circuit is used for sensor measurement. The results can be used for design and application of ZnO or AlN FBARs to monitor liquid viscosity.

Modeling of turbulent transport term of interfacial area concentration in gas–liquid two-phase flow

A modeling of basic transport equation and constitutive equations of turbulent transport terms of interfacial area concentration was carried out. Based on the local instant formulation, interfacial area concentration was formulated in term of spatial correlation functions of characteristic function (local instant volume fraction) and its directional derivative of each phase. In the basic transport equations, interfacial area concentration is transported by time averaged interfacial velocity. In the previous models, interfacial velocity is roughly approximated by velocity of each phase. In the present model, interfacial velocity is formulated in term of spatial correlation functions of characteristic function and velocity of each phase and their directional derivatives. In this formulation, the time averaged interfacial velocity was shown to be correlation functions of fluctuation of velocity and local instant void fraction and their derivatives which reflect the transport of interfacial area concentration due to interaction between interfacial area and turbulence of each phase. Basic conservation equations of spatial correlation functions of characteristic function and velocity of each phase were also derived based on the conservation equations momentum and its fluctuation of each phase. For practical purpose, further modeling of this turbulent transport terms of interfacial area concentration was carried out. As a result, constitutive equations of turbulent diffusion and lateral migration of interfacial area concentration were obtained which can be applied to various flow regime of two-phase flow.

Mass sensitivity of thin film bulk acoustic resonator sensors based on polar c-axis tilted zinc oxide and aluminum nitride thin film

The mass sensitivity of thin film bulk acoustic wave resonator (FBAR) sensors using ZnO and AlN thin films with tilted polar c-axis has been theoretically investigated. The tilted c-axis orientation induces normal plane and in-plane polarizations, which leads to the coexistence of thickness longitudinal mode and thickness shear mode in the resonators. The equation for predicting electric impedance of FBARs with a mass loading layer was derived from the basic piezoelectric constitutive equations. The mass sensitivity of ZnO and AlN dual mode resonators was found by calculating the resonant frequency shifts of the thickness shear mode and thickness longitudinal mode due to thin film mass loading. In the calculation, ZnO and AlN thin film has a c-axis tilt angle from 0°–90°, 2 μm thickness, and 300×300 μm2 electrode area; four different materials (Al, SiO2, Au, and Pt) were considered as the thin film mass loading. It was found that both longitudinal resonance frequency and shear resonance frequency for different c-axis angle have a significant shift due to the mass loading; the mass sensitivities, defined as Δf/(fΔm), of the longitudinal and shear mode for this four mass loading materials are very close, and do not change much with c-axis tilt angle with a value rang around −900 cm2/g for ZnO FBARs and −1550 cm2/g for AlN FBARs. The results can be used in the design and applications of ultrahigh sensitive ZnO or AlN FBAR mass sensors.

Differential geometry of viscoelastic models with fractional-order derivatives

Viscoelastic materials with memory effect are studied based on the fractional rheonomic geometry. The geometric objects are regarded as basic quantities of fractional viscoelastic models, i.e. the metric tensor and torsion tensor are interpreted as the strain and the fractional strain rate, respectively. The generalized viscoelastic equations are expressed by the geometric objects. Especially, the basic constitutive equations such as Voigt and Maxwell models can be derived geometrically from the generalized equation. This leads to the fact that various viscoelastic models can be unified into one geometric expression.

Analytical study of dual-mode thin film bulk acoustic resonators (FBARs) based on ZnO and AlN films with tilted c-axis orientation

In this paper, we present the analytical study of thin film bulk acoustic wave resonators (FBARs) using ZnO and AlN films with a c-axis tilt angle (off-normal) from 0 degrees to 180 degrees. The tilted c-axis orientation induces normal plane and inplane polarizations, which leads to the coexistence of the longitudinal mode and shear mode in the resonator. The equation for predicting electric impedance of FBARs was derived from the basic piezoelectric constitutive equations. Material properties including elastic, dielectric, and piezoelectric coefficients, bulk wave properties including acoustic velocity and electromechanical coupling coefficient, and impedance of FBARs were calculated and showed strong dependence on the tilt angle. Interestingly, it was found that for ZnO FBARs, pure thickness longitudinal modes occur at 0 degrees and 65.4 degrees, and pure thickness shear modes occur at 43 degrees and 90 degrees. For AlN FBARs, pure longitudinal modes occur at 0 degrees and 67.1 degrees, and pure shear modes occur at 46.1 degrees and 90 degrees for AlN. In other words, pure thickness longitudinal and shear modes exist in ZnO and AlN FBARs at specific tilted polarization angles. In addition, two peaks of shear mode electromechanical coefficient are found at 33.3 degrees and 90 degrees for ZnO, and 34.5 degrees and 90 degrees for AlN. Therefore, ZnO and AlN films with specific tilt angles may provide options in the design and fabrication of FBARs, considering their strong shear resonance with high electromechanical coefficients. The use of dual-mode FBARs for mass sensors is also analyzed; the calculated large resonant frequency shift caused by mass loading shows that they have good prospects for use in sensor applications with high sensitivity. The simulation results agreed well with the reported experiment results, and can be used for design and application of FBARs.

Vibrations of porous piezoelectric ceramic plates

The basic constitutive equations and equations of motion are derived for anisotropic porous piezoelectric materials by making use of variational principle. These equations are first derived for three-dimensional case and then reduced to the two-dimensional case by expanding mechanical displacement and electric potential as power series. The results are obtained in particular for monoclinic (2) porous piezoelectric materials. Thickness shear correction factors are determined for high frequency case. The thickness shear resonance frequencies are obtained numerically for a particular model of PZT.

Model of FRP-Confined Concrete Cylinders in Axial Compression

This paper introduces a dilatancy-based analytical model of the response of an axially loaded concrete cylinder, confined with a fiber-reinforced polymer (FRP) composite jacket. Model construction is based on the experimentally based observation that the relation between axial secant stiffness and the lateral (dilatancy) strain is effectively unique for cylinders with the same unconfined concrete strength, although the confinement levels may differ. Model development incorporates strength degradation of the concrete with dilatancy (lateral dilation); this feature allows one to demonstrate that the performance of FRP-confined concrete is consistent with the strength envelope obtained from triaxial tests. Model validation is accomplished by comparisons with existing test database and the new results on large-scale concrete cylinders. The results of the validation reveal good agreement with key response functions and parameters. The present study illustrates basic constitutive equations to model FRP-confined...