Linear Constitutive Relations Research Articles

Lattice structure: Scaling of strain related energy density

The strain energy density (SED) was used as the criterion to characterize stable and unstable crack growth in the fracture mechanics of linear and non-linear constitutive relations. It applies equally well for continuum mechanics problems in general. It is free from the restrictions of theories that implicate linear superposition. The energy density combines strain and stress and is not exclusive to strain alone nor stress alone. “Energy” and “density” for a point like element are the basic ingredients. The element can be microscopic and visualized as atoms or molecules. In this sense, the energy density criterion can be applied to the lattice structure and can also predict macro fracture. In this work, material is modeled with a lattice bond structure. Through the SED-based scaling law, the discrete form of SED criterion, which is specially used to lattice structure, is derived. The fracture mechanism of the lattice structure is analyzed. It is shown that the limit strain of the micro bond is related to the lattice size. The smaller the lattice size is, the larger the limit bond strain. It presents the same singularity order as the macro strain field near crack. The manifestation of scaling of strain is that micro bond strain is always much higher than the macro strain. Bond rupture is associated with the bond strain and the SED factor. The simulation results by the lattice structure with the SED criterion are almost free of lattice size dependency. It is suggested that the lattice structure can be modeled by the energy density and scaling of strain. The strain energy density approach can be applied to develop scale shifting laws in general.

Constitutive relations in optics in terms of geometric algebra

To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p-form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.

Modeling and Characterization of a Piezoelectric Energy Harvester Under Combined Aerodynamic and Base Excitations

This paper develops and validates an aero-electromechanical model which captures the nonlinear response behavior of a piezoelectric cantilever-type energy harvester under combined galloping and base excitations. The harvester consists of a thin piezoelectric cantilever beam clamped at one end and rigidly attached to a bluff body at the other end. In addition to the vibratory base excitations, the beam is also subjected to aerodynamic forces resulting from the separation of the incoming airflow on both sides of the bluff body which gives rise to limit-cycle oscillations when the airflow velocity exceeds a critical value. A nonlinear electromechanical distributed-parameter model of the harvester under the combined excitations is derived using the energy approach and by adopting the nonlinear Euler–Bernoulli beam theory, linear constitutive relations for the piezoelectric transduction, and the quasi-steady assumption for the aerodynamic loading. The resulting partial differential equations of motion are discretized and a reduced-order model is obtained. The mathematical model is validated by conducting a series of experiments at different wind speeds and base excitation amplitudes for excitation frequencies around the primary resonance of the harvester. Results from the model and experiment are presented to characterize the response behavior under the combined loading.

Exploring peculiarities of traffic flows with a viscoelastic model

In this paper, a visco-elastic traffic flow model is described and applied to explore peculiarities of traffic flows on a loop road with ramp effects numerically. Based on different expressions for traffic pressure, sound speed and relaxation time, the viscoelastic model is derived from mass and momentum conservations and a linear viscoelastic constitutive relation. Numerical simulations of loop traffic flows have been carried out, with the ramp flow rate being assumed to be randomly dependent on the local main flow at the intersection. The results revealed that the on-ramp effect can cause the occurrence of traffic shock waves, the off-ramp effect can lead to the decrease of traffic density on the loop road. The viscoelastic effect does cause the significant changes of traffic flow pattern, indicating that self-organisation of loop traffic is a crucial impacting feature.

A nonlinear formulation of piezoelectric shells with complete electro-mechanical coupling

A nonlinear piezoelectric shell model capable of accurately expressing the direct and the converse piezoelectric effect is presented. The developed theory is meant to encompass large strains, displacements and rotations that can occur in the shell as a consequence of a complete coupling between the mechanical and the electrical fields. Based on results present in the literature according to which a linear dependence of the electric potential on the shell thickness is not adequate to represent its electrical behavior, a specific structure of the field of admissible displacements is taken into account. Warping functions characterized by an ad-hoc polynomial expansion aimed at expressing their dependence on the shell through-the-thickness coordinate are here considered to describe the shear and the extensional deformability of the shell transverse fibers. Linear constitutive relations for a transversely isotropic continuum are considered. Finally, the governing equations of motion of the shell are obtained via enforcement of the virtual work theorem.

Nonlinear resonant magnetoelectric coupling effect with thermal, stress and magnetic loadings in laminated composites

Since the magnetostrictive strain and magnetization of giant magnetostrictive materials exhibit enhanced nonlinear magneto-elasto-thermal coupling characteristics under the influence of bias magnetic field, prestress and temperature, this paper adopted a 1D nonlinear magnetostrictive constitutive and the linear piezoelectric constitutive relations combined with the equivalent circuit method to establish a 1D enhanced nonlinear magneto-elasto-thermal coupling resonant magnetoelectric (ME) model for Terfenol-D/PZT/Terfenol-D laminated composites. Without considering prestress, the comparisons of curves of resonant ME coefficient peak and its corresponding resonant frequency versus temperature between the theoretical results and experimental results are in good agreement both qualitatively and quantitatively. On this basis, this paper predicted variations in resonant ME coefficient and resonant frequency under different bias magnetic fields, prestresses and temperatures. These results show that the exerted compressive stress can be reduced so as to increase the resonant ME coefficient and lower the required bias magnetic field to the maximum of ME coefficient at relatively low and stable ambient temperatures; however, in extreme temperature environment, a larger compressive stress can be exerted to suppress the dramatic attenuation of ME effect caused by extreme temperature increases. Moreover, as temperature is increased, the resonant frequency tends to increase monotonically under different prestresses.

Geometrically nonlinear static FE-simulation of multilayered magneto-electro-elastic composite structures

A fully geometrically nonlinear finite rotation shell element for static analysis of layered magneto-electro-elastic (MEE) coupled composite structures is proposed. Reissner–Mindlin first-order shear deformation (FOSD) theory with full geometrically nonlinear strain–displacement relations and finite rotations is used to obtain the variational formulation. The scalar electric and magnetic fields are assumed along with the quasi-static behavior of MEE layers. Electric and magnetic potentials are assumed to vary quadratically in the transverse direction. Three linear constitutive relations are used describe the magneto-electro-elastic coupling. The magneto-electro-elastic composite four node shell element behavior is refined by embodying an assumed natural strain (ANS) method for transverse shear strains, and an enhanced assumed strain (EAS) method for in-plane bending strains. The developed finite element model is deployed for static analysis of layered MEE structures in sensor and actuator configurations, and results are compared with the available references. Additionally, several numerical examples are simulated to show the potentiality and predictive capabilities of the proposed finite element method.

Earthquake nucleation in intact or healed rocks

AbstractEarthquakes are generated because faults lose strength with increasing slip and slip rate. Among the simplest representations of slip‐dependent strength is the linear slip‐weakening model, characterized by a linear drop to a residual friction. However, healed fault rocks often exhibit some slip strengthening before the onset of weakening. Here we investigate the effect of such a slip‐hardening phase on the initial growth of a slip patch and on the nucleation of rupture instabilities. We assume a piecewise linear strength versus slip constitutive relation. We compute stress and slip distributions for in‐plane or antiplane rupture configurations in response to an increasing, locally peaked (parabolic with curvature κ) stress profile. In contrast with the strictly linear slip‐weakening case, our calculations show that the curvature of the loading profile and the level of background stress strongly influence the nucleation size. Even for small amounts of slip hardening, we find that the critical nucleation size scales with for κ→0, i.e., crack growth remains stable up to very large crack sizes for sufficiently smooth loading profiles. Likewise, when the background stress τb is very close to the initial strength τc, the critical crack size scales with . An eigenvalue analysis shows that the nucleation length increases as the proportion of the crack undergoing slip hardening increases, irrespective of the details of the loading profile. Overall, our results indicate that earthquake nucleation sizes can significantly increase due to slip hardening (e.g., in healed fault rocks), especially when the background loading is smooth.

A two dimensions modeling of non-collocated piezoelectric patches bonded on thin structure

Abstract The system studied in this paper consists of thin structure with several piezoelectric patches bonded on its surface. The patches are used as actuators and sensors. Based on Kirchhoff-Love hypothesis, linear constitutive relations, plane stress formulation and Hamilton principle, we have developed a 2D model for this system using the finite element method. It is not a standard 2D model, since the calculation is performed on a structure that does not have symmetries that allow such easy assumptions. The originality of the work consists in the use of the concept of neutral plane to model this asymmetric system in 2D. This technique, beside good precision, saves computational time. An experimental device has been also built and tested to validate the model. The structural damping is included in the model to match the damping behavior of the real system. Optimizations of the thickness of piezoelectric patches and materials used in the thin structures are also presented in the paper.

Stochastic stability of SDOF linear viscoelastic system under wideband noise excitation

The stochastic moment stability and almost-sure stability of a single-degree-of-freedom (SDOF) viscoelastic system subject to parametric fluctuation is investigated by using the method of higher-order stochastic averaging. The stochastic parametric excitation is modeled as a wideband noise, which is taken as Gaussian white noise and real noise. The viscoelastic material is assumed to follow ordinary Maxwell linear constitutive relation. For small damping and weak stochastic fluctuation, analytical expressions are derived for the moment Lyapunov exponent and the Lyapunov exponent, which indicate moment stability and almost-sure stability respectively. The effects of various system and loading parameters on the stochastic stability are discussed. Both analytical and simulation results show that higher-order stochastic averaging improves the accuracy compared with the first-order stochastic averaging. However, results of the third-order averaging are almost overridden by those of second-order averaging and the third-order averaging involves far more calculation. It is advisable to consider a balance between accuracy achievement and calculation endeavor when using higher-order stochastic averaging.

An empirical formula for mode-II fracture energy of concrete

This paper presents an experimental investigation to understand the influence of Maximum Size of the Aggregate (MSA) and compressive strength of concrete on the Mode-II fracture parameters (Fracture energy (GIIf) and Brittleness number) of normal and high strength of concrete. An empirical formula for Mode-II fracture energy (GIIf) based on the Bazant’s size effect law is developed in terms of maximum size of aggregate and the compressive strength of concrete. A linear constitutive relationship between shear stress and slip i.e., the linear softening curve for Mode-II fracture of concrete is presented by conducting experiments on the double edge notched specimen. The experimental results indicated that increase in the grade of concrete (compressive strength) increases the brittleness of the member and Mode-II fracture energy and also increase in the maximum size of aggregate increases the Mode-II fracture energy.

ON ELASTIC STRESS WAVES IN AN IMPACTED PLATE

Elastic stress wave theory is developed and the stress waves in the impacted plate are examined in the paper. Generalized linear elasticity is adopted where the couple stress and curvature tensor are both deviatoric tensors and they meet a linear constitutive relation. It is found that there exist volumetric, rotational, and deviatoric waves in the generalized elastic solids. However, for macro-scale elastic solids only two wave modes, namely a volumetric wave and a deviatoric wave should be taken into account. Wave motion in plate impact tests is studied that a volumetric wave and a deviatoric wave are proposed. A set of exact solutions is attained for elastic stress waves in an impact plate. Excitation of stress waves at impact surface and reflection at free surface are formulated. Propagation of stress waves in the plate is analyzed in the waveforms. The predicted stress history in a ceramic plate under impact is agreed very well with the experiment measurement.

Nonlinear magnetoelectric effect in PZT/Terfenol-D nanobilayer on a substrate with surface stress

Based on a linear piezoelectric constitutive relation and a nonlinear magnetostrictive constitutive relation, a nonlinear magnetoelectric (ME) effect model for lead zirconate titanate (PZT)/Terfenol-D nanobilayer on a substrate has been developed. In this study, the nonlinear ME coefficients at bending mode for two cases (without surface stress and with surface stress) are calculated by using Gurtin-Murdoch theory. The difference between two cases and the influence of residual surface tension are discussed. At the same time, the clamping effect of the substrate on ME effect is studied by altering the thickness ratio of the substrate and selecting different substrate materials. The influences of frequency of the magnetic field, PZT volume fraction on the ME effect are investigated, respectively. Finally, the dependence of ME effect on pre-stress is presented. The results show that for the nanobilayer, both the residual surface tension and surface stress have non-ignored effects on the ME effect. Besides, the resonant frequency of the nanobilayer is very low at the bending mode, which can be enhanced by increasing the thickness ratio of the substrate. Also, the substrate can weaken the ME effect due to the clamping effect, and a more soft substrate material should be selected for large ME effect. In addition, pre-stress plays an important role in the nonlinear ME coupling effect of the model developed.

Dynamic pull-in instability of a micro-actuator made from nonlinear elasticity materials

This paper investigates the dynamic pull-in instability of a micro-actuator made from nonlinear elasticity materials. The theoretical formulations are based on Bernoulli–Euler beam theory and include the effects of both material nonlinearity and mid-plane stretching due to large deformation. By employing the Galerkin method, the nonlinear partial differential governing equation is decoupled into a set of nonlinear ordinary differential equations which are then solved using the Runge–Kutta method. Numerical results show that the linear constitutive relationship used in previous studies is valid for small deformation only, whereas for large deformation the nonlinear elasticity constitutive relationship must be used for accurate analysis. The effects of material nonlinearity, initial gap, beam length, and beam width on the pull-in instability of the micro-actuator are studied.

Insights Into Flexoelectric Solids From Strain-Gradient Elasticity

A material is said to be flexoelectric when it polarizes in response to strain gradients. The phenomenon is well known in liquid crystals and biomembranes but has received less attention in hard materials such as ceramics. Here we derive the governing equations for a flexoelectric solid under small deformation. We assume a linear constitutive relation and use it to prove a reciprocal theorem for flexoelectric materials as well as to obtain a higher-order Navier equation in the isotropic case. The Navier equation is similar to that in Mindlin's theory of strain-gradient elasticity. We also provide analytical solutions to several boundary value problems. We predict size-dependent electromechanical properties and flexoelectric modulation of material behavior. Our results can be used to interpret experiments on flexoelectric materials which are becoming increasingly sophisticated due to the advent of nanoscale probes.

General lattice model of gradient elasticity

In this paper, new lattice model for the gradient elasticity is suggested. This lattice model gives a microstructural basis for second-order strain-gradient elasticity of continuum that is described by the linear elastic constitutive relation with the negative sign in front of the gradient. Moreover, the suggested lattice model allows us to have a unified description of gradient models with positive and negative signs of the strain gradient terms. Possible generalizations of this model for the high-order gradient elasticity and three-dimensional case are also suggested.

Numerical Simulation of Hysteretic Live Load Effect in a Soil-Steel Bridge

Abstract The paper presents numerical simulation of hysteretic live load effect in a soil-steel bridge. The effect was originally identified experimentally by Machelski [1], [2]. The truck was crossing the bridge one way and the other in the full-scale test performed. At the same time, displacements and stress in the shell were measured. The major conclusion from the research was that the measured quantities formed hysteretic loops. A numerical simulation of that effect is addressed in the present work. The analysis was performed using Flac finite difference code. The methodology of solving the mechanical problems implemented in Flac enables us to solve the problem concerning a sequence of load and non-linear mechanical behaviour of the structure. The numerical model incorporates linear elastic constitutive relations for the soil backfill, for the steel shell and the sheet piles, being a flexible substructure for the shell. Contact zone between the shell and the soil backfill is assumed to reflect elastic-plastic constitutive model. Maximum shear stress in contact zone is limited by the Coulomb condition. The plastic flow rule is described by dilation angle ψ = 0. The obtained results of numerical analysis are in fair agreement with the experimental evidence. The primary finding from the performed simulation is that the slip in the interface can be considered an explanation of the hysteresis occurrence in the charts of displacement and stress in the shell.

Nonuniform Ferro-elastic Domain Switching for the Interfacial Crack

Existing theoretical researches about interfacial crack of bimaterial piezoelectric composites are confined to linear piezoelectric constitutive relation, neglecting the nonlinear part driven by domain switching. This paper attempts to extend the research of mode III interfacial crack from linear piezoelectricity to nonlinear ferroelectricity. The analysis focuses on the variation of stress intensity factor caused by domain switching, with an arbitrary three-dimensional initial poling orientation. Due to the mismatch of material properties, an asymmetric nonuniform domain switching zone is achieved around the interfacial crack under the nonuniform domain switching criterion. By employing the weight function method, analytic forms of mono-domain and multi- domain toughening effects about stationary and quasi-static steady-state crack are obtained, respectively. The conclusions come to that domain switching of mode III quasi-static steady-state growing interfacial crack has a positive effect to toughen the material. The research can provide some guidance for material selection to maximize the effect of ferro-elastic switch toughening.

Inverse analysis of a non‐linear viscous fluid based on dissipated energy measured with a simple‐shear rheometer

AbstractOscillatory rheometer measurements are used to determine the material parameters of a NEWTONian fluid model, which can be expressed by a linear constitutive relation. However, rheological materials, such as polymer melts, mixture of oils, or food paste, can only be modeled as non‐NEWTONian fluids by using non‐linear constitutive relations. Since the rheometer measures the energy loss in the induction motor due to shear loading of the viscous material, this can be used as the objective function for a regression analysis. The dissipated energy will be obtained as outlined in [1]. The goal of this work is to explain how to determine the parameters of a non‐linear material model by using the energy loss that is measured in a rheometer. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear electro-dynamic analysis of micro-actuators: Effect of material nonlinearity

This paper presents a nonlinear dynamic analysis of a micro-actuator made of nonlinear elasticity materials. The theoretical formulations are based on Bernoulli–Euler beam theory and include the effects of mid-plane stretching due to large deformation and material nonlinearity. By employing Linstedt–Poincaré perturbation method, the nonlinear governing equation is transformed into a set of linear differential equations which are then solved using Galerkin’s method. Numerical results show that the linear constitutive relationship used in previous studies is valid for small deformation only whereas for large deformation, the nonlinear elasticity constitutive relationship must be used for accurate analysis. The effects of initial gap and beam length on the nonlinear electro-dynamic behavior of the micro-actuator are also discussed.