Additional Material Constants Research Articles

Non-classical theory of electro-thermo-elasticity incorporating local mass displacement and nonlocal heat conduction

A non-classical local gradient theory of nonferromagnetic thermoelastic dielectrics is presented, incorporating both the local mass-displacement process and the heat-flux gradient effect. The process of local mass displacement is related to the changes in material microstructure. The nonlocal heat conduction law is also addressed in the model. Thus, the generalized relationship between the higher-grade heat and entropy fluxes is adopted. The gradient-type constitutive relations and governing equations are derived using the fundamental principles of continuum mechanics, non-equilibrium thermodynamics, and electrodynamics. Due to the contribution of higher-grade flux, the nonlocal law of heat conduction is obtained. The constitutive relations for isotropic materials with the corresponding additional material constants are derived. To illustrate the local gradient theory and to show the electro-thermo-mechanical coupling effect in isotropic materials, a straightforward problem is analytically solved for a layered non-piezoelectric structure under non-uniform temperature distribution. The analytical results reveal that the thermal polarization effect can also be pronounced in isotropic materials. To illustrate the model considering the effect of nonlocal heat conduction, the propagation of spherical thermoelastic harmonic waves in a homogeneous and isotropic elastic medium with non-classical heat conduction law is studied.

A Miner's rule based fatigue life prediction model for combined high and low cycle fatigue considering loading interaction effect

AbstractThe key components in service, such as aero‐engine turbine blades, are generally bearing complex interactions of high and low cycle fatigue loading. It is essential to make accurate fatigue life predictions for these components. In this paper, based on experimental analysis and the Miner's rule, a combined cycle fatigue life prediction model is proposed. The effect of the loading interaction of the high and low cycle fatigue is considered through the coupled damage, which is proposed based on the investigation of the fatigue test results and does not require additional material constants except for the S‐N curve parameters. The applicability of the proposed model is validated by the experimental data of three materials and the turbine blade. Compared with the T‐K model and Miner's rule, the results demonstrate that the proposed model provides high precision accuracy.

Bending, Buckling and Vibration Analysis of Complete Microstructure-Dependent Functionally Graded Material Microbeams

A new functionally graded non-classical Timoshenko microbeam model is developed by using a variational formulation. The new model incorporates strain gradient, couple stress (rotation gradient) and velocity gradient effects and explains a power-law variation of two-phase materials through the thickness direction. The new model contains three additional material constants to account for the relative three gradient effects. The static bending, buckling and free vibration problems of the newly developed functionally graded material (FGM) beam are then analytically solved. Numerical results show that the differences between the prediction results of the current model and the classic model are significant when FGM beam thickness is very small, but they are diminishing as the thickness increases.

Threshold damage-based fatigue life prediction of turbine blades under combined high and low cycle fatigue

Life prediction based on the damage mechanics of aero-engine turbine blades is crucial for performing their strength design and ensuring the operational reliability under combined high and low cycle fatigue (CCF) loadings. In view of this, a simple and efficient life prediction method is developed to consider the interaction of high cycle fatigue (HCF) and low cycle fatigue (LCF) without any additional material constants, and as a basis of that a new nonlinear damage accumulation model is proposed by introducing threshold damage of the component related to material type and current HCF damage. In order to validate the presented methodology, model predictions were compared with the experimental data sets of turbine blades and its alloy materials. The predicted results indicate that the simple life prediction model is capable of estimating the fatigue life quickly with an acceptable accuracy in contrast to other four existing ones. Moreover, the threshold damage-based method provides the higher prediction accuracy than others due to improved HCF damage determined by the damage threshold.

Hemitropic properties of thin non-centrosymmetric 3D woven textile composites

Due to their microstructural architecture, 3D woven textile composites, (3DWTC), exhibit complex macroscopic behavior, which cannot be effectively captured using classical continuum theories. One reason for this can be attributed to the ratio of the characteristic microscopic and macroscopic length scales, (d and L, respectively), which violates the separation of length scales requirement d/L<<1 in micromechanics. In these instances, higher order continuum theories, such as micropolar theory, are considered to effectively describe the macroscopic behavior of a material. In this study, the 3D macroscopic constitutive relation of 3D woven textile composite (3DWTC) is reported. The three different 3DWTC considered are referred to as thin asymmetric, thick asymmetric, and thick symmetric. The lack of centro-symmetry in the asymmetric microstructures produces an interesting anisotropic response, such as stress-curvature (or couple stress-strain) coupling effect. This is analogous to asymmetric plate structures and the coupling between the resultant forces and curvatures through the B-matrix. At the macroscopic (continuum) scale, the material is modeled as a micropolar continuum, where internal couple-stresses and curvature strains are introduced. To capture the stress-curvature coupling effect, a generalized Aero-Kuvshinski type constitutive model is considered. With this, the anisotropic micropolar properties and the additional hemitropic material constants of the 3DWTC are determined numerically and their results qualitatively summarized.

Experimental and theoretical analysis of the DMASP cantilever vibration behavior based on the MCS theory in moist environment

A micro cantilever (MC) and a probe are two main components of the atomic force microscope (AFM). The dimensions of these components are in micro scales while their oscillation amplitude is on a nanometer scale. The present study intended to not only increase the accuracy of the simulation with regard to geometric discontinuities based on the Timoshenko beam model using the modified couple stress (MCS) theory but also increase the accuracy of the prediction of a system behavior by considering the hysteresis effect into the system vibration equations based on Bouc–Wen model. Since, the classical theory disregarded the molecular structure of matter and neglected the microstructure size dependency, which raise its physical breakdowns in micro and nanoscale applications. Therefore, some micro continuum theories are developed to account for the microstructure effects by introducing additional material constants to the conventional ones. Due to the lack of the experimental results for this kind of MCs, this paper is focused on the both experimental and simulation results by utilizing the DMASP MC. The roughness of the surfaces affects the MC vibration behavior in the air medium. Therefore, surface roughness affecting van der Waals force has been considered in the air medium. Furthermore, the simulation results are compared with experimental results in the air medium with different moisture content during the surface topography. Due to the results, the best surface topography is achieved in the third vibration mode. Since the piezoelectric MC can tolerate the limited voltage, for this AFM MC type, the best surface topography is achieved in the third mode. Hence, this paper highly recommends using the third vibration mode during the surface topography. Finally, based on the EFAST method the most effective parameters on the AFM MC performance are the piezoelectric layer length, thickness and the probe length.

Critical plane–based multiaxial fatigue life prediction of turbine disk alloys by refining normal stress sensitivity

For engineering components subjected to complex multiaxial loadings, critical plane approaches like Fatemi–Socie criterion have been commonly utilized for life prediction of these components. Within the Fatemi–Socie criterion, the normal stress sensitivity parameter k is usually fitted from additional experimental data, which introduces inconvenience for practice especially under limited testing data conditions. In this regard, a simple critical plane–based damage parameter is put forward with no additional material constants, which attempts to provide a robust method for multiaxial fatigue analysis of turbine disk alloys. Using experimental datasets of TC4 and GH4169 alloys under different loadings, the proposed model provides better correlations with fatigue life of the two alloys than the models of Smith–Watson–Topper and Wang–Brown.

Evaluation and comparison of critical plane criteria for multiaxial fatigue analysis of ductile and brittle materials

This paper conducts a comparative evaluation on typical critical plane criteria, including Fatemi-Socie, Wang-Brown, modified Smith-Watson-Topper (MSWT) and proposed modified generalized strain energy (MGSE) criteria for multiaxial fatigue analysis of ductile/brittle materials. Experimental datasets of four materials under uniaxial tension, torsion and proportional/non-proportional multiaxial loadings are introduced for model comparison. This study results indicate that criteria with additional material constants yield robust life predictions for different materials. Moreover, the criteria with shear and uniaxial fatigue properties are respectively suitable for ductile and brittle materials, particularly the MGSE superior to others for ductile/brittle materials while MSWT only for brittle materials.

Multiaxial Fatigue Damage Parameter and Life Prediction without Any Additional Material Constants

Based on the critical plane approach, a simple and efficient multiaxial fatigue damage parameter with no additional material constants is proposed for life prediction under uniaxial/multiaxial proportional and/or non-proportional loadings for titanium alloy TC4 and nickel-based superalloy GH4169. Moreover, two modified Ince-Glinka fatigue damage parameters are put forward and evaluated under different load paths. Results show that the generalized strain amplitude model provides less accurate life predictions in the high cycle life regime and is better for life prediction in the low cycle life regime; however, the generalized strain energy model is relatively better for high cycle life prediction and is conservative for low cycle life prediction under multiaxial loadings. In addition, the Fatemi–Socie model is introduced for model comparison and its additional material parameter k is found to not be a constant and its usage is discussed. Finally, model comparison and prediction error analysis are used to illustrate the superiority of the proposed damage parameter in multiaxial fatigue life prediction of the two aviation alloys under various loadings.

An enhanced Brinson model with modified kinetics for martensite transformation

We propose an enhanced Brinson model with modified kinetics for martensite transformation. Two additional material constants are considered to follow the stress-temperature diagram above austenite start temperature (As) along with treatment to keep the continuity of the martensite volume fraction and the path dependency of the phase transformation. To demonstrate the performance of the proposed model, we implement this algorithm into ABAQUS user subroutine, then conduct several numerical simulations and compare their results with SMA wire experiments as well as those of three-dimensional SMA constitutive models. From the results, it turns out that the proposed model is as accurate as the three-dimensional models and shows better accuracy over original Brinson model in terms of recovery stress.

Seismic response of reduced micropolar elastic half-space

This paper explores reduced micropolar theory to simulate ground motion during an earthquake. In this theory, rotational motions are kinematically independent of translational motions. Analytical expressions for ground displacement and rotational motions due to a buried seismic source are presented in this paper. This theory requires two additional material constants which characterise the microstructure of the medium compared with linear elastic theory. Ground motions are simulated for an earthquake of magnitude (M w) 5.0. The sensitivity of ground motion to these new material constants is reported. It is observed that rotations are sensitive to microstructure of the medium. A comparison with recorded rotations of the M w 5.2 Izu peninsula, Japan event is also presented in this article.

On a second-order rotation gradient theory for linear elastic continua

A second-order rotation gradient theory for non-classical elastic continua is developed. This theory accounts for the higher-order deformation of the material structure where the material particle inside the elastic domain is idealized as a microvolume having three degrees of freedom, namely, a translation, a micro-rotation, and a higher-order micro-deformation. The associated strain energy density is a function of the infinitesimal strain tensor and the first and second gradients of the rotation tensor. It is demonstrated that for materials in nano-scale applications and because of some defects at the material structure level, a higher-order deformation measure may be needed. The second-strain gradient theory has the merit to account for the higher-order deformation of the material particle. However, this theory has limited applications because it depends on 16 additional material constants for isotropic elastic continua. By discussing some physical concepts relevant to the natures of material structures, crystallinity, and amorphousness, the second-strain gradient theory is reduced to the second-rotation gradient theory for certain types of materials. For isotropic materials, the developed second-rotation gradient theory only depends on three additional material constants instead of 16. A continuum model equipped with an atomic lattice model is then proposed to examine the applicability of the available non-classical theories and the applicability of the proposed theory for different types of materials.

A new Timoshenko beam model incorporating microstructure and surface energy effects

A new Timoshenko beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and complete boundary conditions for a Timoshenko beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure-and surface energy-dependent size effect. In addition, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, unlike existing Timoshenko beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as limiting cases and recovers the Bernoulli–Euler beam model incorporating the two effects as a special case. Also, the current model reduces to the classical Timoshenko beam model when the microstructure dependence, surface energy and Poisson’s effect are all suppressed. To demonstrate the new model, the static bending and free vibration problems of a simply supported beam are analytically solved by directly applying the general formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. In addition, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that given by the classical model, with the difference between them being significantly large for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally.

Micropolar continuum modelling of bi-dimensional tetrachiral lattices.

The in-plane behaviour of tetrachiral lattices should be characterized by bi-dimensional orthotropic material owing to the existence of two orthogonal axes of rotational symmetry. Moreover, the constitutive model must also represent the chirality inherent in the lattices. To this end, a bi-dimensional orthotropic chiral micropolar model is developed based on the theory of irreducible orthogonal tensor decomposition. The obtained constitutive tensors display a hierarchy structure depending on the symmetry of the underlying microstructure. Eight additional material constants, in addition to five for the hemitropic case, are introduced to characterize the anisotropy under Z2 invariance. The developed continuum model is then applied to a tetrachiral lattice, and the material constants of the continuum model are analytically derived by a homogenization process. By comparing with numerical simulations for the discrete lattice, it is found that the proposed continuum model can correctly characterize the static and wave properties of the tetrachiral lattice.

Micropolar modeling of planar orthotropic rectangular chiral lattices

Rectangular chiral lattices possess a two-fold symmetry; in order to characterize the overall behavior of such lattices, a two-dimensional orthotropic chiral micropolar theory is proposed. Eight additional material constants are necessary to represent the anisotropy in comparison with triangular ones, four of which are devoted to chirality. Homogenization procedures are also developed for the chiral lattice with rigid or deformable circles, all material constants in the developed micropolar theory are derived analytically for the case of the rigid circles and numerically for the case of the deformable circles. The dependences of these material constants and of wave propagation on the microstructural parameters are also examined.

AN ELASTOPLASTIC CONSTITUTIVE MODEL FOR POROUS MATERIALS

A micromechanics-based elastoplastic constitutive model for porous materials is proposed. With an assumption of modified three-dimensional Ramberg–Osgood equation for the compressible matrix material, the variational principle based on a linear comparison composite is applied to study the effective mechanical properties of the porous materials. Analytical expressions of elastoplastic constitutive relations are derived by means of micromechanics principles and homogenization procedures. It is demonstrated that the derived expressions do not involve any additional material constants to be fitted with experimental data. The model can be useful in the prediction of mechanical properties of elastoplastic porous solids.

A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects

A new Bernoulli–Euler beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equation and complete boundary conditions for a Bernoulli–Euler beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure- and surface energy-dependent size effect. In addition, Poisson’s effect is incorporated in the current model, unlike existing beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as special cases. The current model reduces to the classical Bernoulli–Euler beam model when the microstructure dependence, surface energy, and Poisson’s effect are all suppressed. To demonstrate the new model, a cantilever beam problem is solved by directly applying the general formulas derived. Numerical results reveal that the beam deflection predicted by the new model is smaller than that by the classical beam model. Also, it is found that the difference between the deflections predicted by the two models is very significant when the beam thickness is small but is diminishing with the increase of the beam thickness.

Long-term strength of metals in complex stress state (a survey)

An analytic survey of experimental data and theoretical approaches characterizing the long-term strength of metals in complex stress state is given. In Sections 2 and 3, the results of plane stress tests (with opposite and equal signs of the nonzero principal stresses, respectively) are analyzed. In Section 4, the results of inhomogeneous stress tests (thick-walled tubes under the action of internal pressures and tensile forces) are considered. All known experimental data (35 test series) are analyzed by a criterion approach. An equivalent stress σe is introduced as a characteristic of the stress state. Attention is mainly paid to the dependence of σe on the principal stresses. Statistical methods are used to obtain an expression for σe, which can be used to study various types of the complex stress state. It is shown that for the long-term strength criterion one can use the power or power-fractional dependence of the time to rupture on the equivalent stress. The methods proposed to describe the test results give a good correspondence between the experimental and theoretical values of the time to rupture. In Section 5, the possibilities of complicating the expressions for σe by using additional material constants are considered.

Calculation of the Additional Constants for fcc Materials in Second Strain Gradient Elasticity: Behavior of a Nano-Size Bernoulli-Euler Beam With Surface Effects

In addition to enhancement of the results near the point of application of a concentrated load in the vicinity of nano-size defects, capturing surface effects in small structures, in the framework of second strain gradient elasticity is of particular interest. In this framework, sixteen additional material constants are revealed, incorporating the role of atomic structures of the elastic solid. In this work, the analytical formulations of these constants corresponding to fee metals are given in terms of the parameters of Sutton-Chen interatomic potential function. The constants for ten fcc metals are computed and tabulized. Moreover, the exact closed-form solution of the bending of a nano-size Bernoulli-Euler beam in second strain gradient elasticity is provided; the appearance of the additional constants in the corresponding formulations, through the governing equation and boundary conditions, can serve to delineate the true behavior of the material in ultra small elastic structures, having very large surface-to-volume ratio. Now that the values of the material constants are available, a nanoscopic study of the Kelvin problem in second strain gradient theory is performed, and the result is compared quantitatively with those of the first strain gradient and traditional theories.

A model for natural soil with bonds

This paper presents a thermodynamically consistent constitutive model for natural soils with bonds. In the model, the free energy (the internal energy available to do work) is contributed partly by the so-called frozen or locked energy, whose evolution is assumed to be homogeneously related to the irrecoverable deformation. During loading, the bonds existing in the natural soil not only boost the dissipation rate but also liberate certain historically accumulated locked energy. Such effects, however, are diminished as loading proceeds and the bonds are destroyed. The novel aspect of the present model is that it accommodates both the Mohr–Coulomb and critical-state failure modes, and the two modes are unified through the evolution law of a thermodynamic force associated with the locked bonding energy. As compared with the classical Cam-clay models, the model contains two additional material constants, where one is proposed by Collins &amp; Kelly to improve the shape of the yield surface, and the other is dedicated to bonding evolution. The calibration procedure for the material parameters is provided. The capability of the model is demonstrated by a series of model simulations on a hypothetical bonded soil under various triaxial loading paths, and the model response is also compared with representative testing results in the literature.