Nonlinear Strain Response Research Articles

Experimental validation of the spectral criterion for the diagnosis of periodic rod structures using ferroelectric strain gauges

Successive impact excitation and construction of Fourier transforms of linear and nonlinear strain responses and phase portraits of undamaged or damaged vertically suspended models of a periodic welded rod structure were used to establish the dependence of the area ratio of the phase portraits of the nonlinear and linear domains of the strain response, which characterizes the stress (damage) of the model, on the impact momentum. The dynamic strain responses were recorded using ultra-wideband (10−2–108 Hz) ferroelectric microgauges based on thin Pb(Zr0.53Ti0.47)O3 films 2 µm in thickness with a 0.8 mm2 contact area.

Time dependent behavior of ferroelectric materials undergoing changes in their material properties with electric field and temperature

This paper presents a time dependent polarization constitutive model suitable for predicting nonlinear polarization and electro-mechanical strain responses of ferroelectric materials subject to various histories of electric fields. The constitutive model is derived based on a single integral form with nonlinear (electric field and temperature dependent) integrand. The total polarization consists of the time-dependent and residual components. The residual component of the polarization is due to polarization switching in the ferroelectric materials. We use an ‘internal clock’ concept to incorporate the effect of electric field on the rate of polarization. The corresponding strain response is determined through the use of third order piezoelectric constant and/or fourth order electrostrictive constant that vary with polarization stage. It is assumed that in absence of polarization, both piezoelectric and electrostrictive constants are zero. To incorporate the effect of temperature on the overall polarization behavior all material parameters in the constitutive model are allowed to change with the ambient temperature. We present numerical studies on the effect of time, temperature, and electric field on the response of ferroelectric material followed by verification of the constitutive model. Experimental data on lead zirconate titanate (PZT) materials available in the literature are used to verify the model.

Micromechanical modeling of stress–strain behaviors with intermartensitic transformation in NiFeGa alloys

Magnetic shape memory alloys (MSMAs) NiFeGa are reported as good alternatives to Ni 2MnGa for their better mechanical behavior. Experimental data show its multistage transformation behavior. In spite of the recent progress in the development of material characteristics, no theoretical model has been developed to calculate the stress–strain behaviors of NiFeGa alloys. A model based on the micromechanical and the thermodynamic theory is presented for such a purpose. In the first plateau stage, the material is modeled as a three-phase composite, and a two-phase system for the second plateau stage. The end of the first plateau stage is considered as the origin of the second one. The kinetic equation is established in terms of the thermodynamic driving force derived from the reduction of Gibbs free energy of MSMA. The nonlinear and hysteretic strain response of NiFeGa is investigated under tensile load. The theoretical results are found to be in good agreement with experimental data.

Physically Networked Polymers: Materials that change with their environment

Polymer networks have continuity of molecular and intermolecular links throughout their structure. These links can be established by covalent, polar or dispersive forces. The choice of link depends on whether a permanent or a reversible structure is required. The continuous network and matrix phase may be different, either chemically or physically, if there is identifiably separate phases. When network formation is through a filler a percolation threshold concentration must be exceeded. Agglomeration of the filler will produce a non-linear strain response. When the matrix phase is of distinct chemistry and physically it can be a gas (aerogel), liquid (gel) or solid (dispersed or co-continuous). The emphasis of this discussion is on reversible networks. They require a reversing stimulus, typically heat, shear, pH, electrical or magnetic. The magnitude of the reversing force is important for application. Thermoplastic rubbers require reversal of physical cross-links during processing, for all other circumstances they must have a permanent network structure. Reological additives exhibit network reversal on shear and the gelling forces should be weak and potentially rapidly reversible. Dispersive forces, called hydrophobic forces when a polar medium is the matrix, meet networking criteria. Ionic or dipolar forces can be enhanced by molecular orientation formed by shear or an electrical field thus causing an increase in structure formation with a rheopectic or shear thickening effect. In solids such additives modify the viscoelastic properties. The distinction between a liquid and a solid is uncertain for amorphous materials. A feature of more recent research is the formation of self-assembling network systems, often using functionalized polymers with interacting groups having high specificity and facile reversibility. Reversibility is either self-controlled in response to changes in the environment or activated by external stimuli, such as those mentioned above. Materials with these responsive changes to their structure have been called smart materials. A reverse situation may apply where changes in the structure of the material lead to changes in the stimulus field. Changing chemical, mechanical or shear conditions may lead to a change in conductivity or light transmission, so the material can become a sensor. Nematic liquid crystalline ordering of side-chain mesogens can be controlled by electro-optical stimuli. Thermo-reversible networks are much studied, but network formation can be activated by electrical or magnetic fields. Many new materials are based upon the principles that have been discussed with the similarity that they include reversible supramolecular structures.

Modeling the Coupled Strain and Magnetization Response of Magnetic Shape Memory Alloys under Magnetomechanical Loading

This article is concerned with the modeling of the magnetic shape memory alloy (MSMA) constitutive response caused by the reorientation of martensitic variants under mechanical and magnetic fields. The presented model is able to better capture the complexity of the magnetomechanical MSMA behavior by accounting not only for the mechanism of field-induced variant reorientation, but also the magnetization rotation away from magnetic easy axes and the magnetic domain wall motion at low stress and magnetic field levels. Following the general formulation of the model, reduced versions of the constitutive equations are derived for three specific loading cases: (1) magnetic-field-induced variant reorientation at constant stress; (2) stress-induced variant reorientation at constant magnetic field; (3) variant reorientation under collinear magnetic field and stress with perpendicular bias field. For each of these cases the nonlinear and hysteretic strain and magnetization response of MSMAs are predicted and compared to experimental data where available. The relation of critical stresses and magnetic fields for the activation of the reorientation process are visualized in a variant reorientation diagram. The captured loading-history-dependent macroscopic material response is explained in detail by connecting it to the evolution of the crystallographic and magnetic microstructure as represented by a set of internal state variables.

Modeling of the variant reorientation in magnetic shape-memory alloys under complex magnetomechanical loading

This work is concerned with the modeling of the stress- and magnetic field-induced reorientation of martensitic variants in magnetic shape-memory alloys. Of special concern is the loading path dependence of the associated nonlinear reorientation strain and magnetization response. A loading case of variable magnetic field in the direction of a constant uniaxial stress with a constant perpendicular magnetic field is analyzed.

Micromechanical modeling of the stress-induced superelastic strain in magnetic shape memory alloy

A constitutive model for magnetic shape memory alloys (MSMAs) is developed through a combined consideration of micromechanical and thermodynamic theories. The kinetic equation is established in terms of the thermodynamic driving force derived from the reduction of Gibbs free energy of MSMA. An equation that balances the thermodynamic driving force with the corresponding resistive force is obtained to calculate the volume fraction of the martensite variant under different given stress, temperature and magnetic field. The nonlinear and hysteretic strain response of MSMAs is investigated for stress-induced reorientation under constant magnetic field. The theoretical results are found to be in good agreement with experimental data.

Model for field-induced reorientation strain in magnetic shape memory alloy with tensile and compressive loads

A model based on the micromechanical and the thermodynamic theory is presented for field-induced martensite reorientation in magnetic shape memory alloy (MSMA) single crystals. The influence of variants morphology and the material property to constitutive behavior is considered. The nonlinear and hysteretic strain and magnetization response of MSMA are investigated for two main loading cases, namely the magnetic field-induced reorientation of variants under constant compressive stress and tensile stress. The predicted results have shown that increasing tensile loading reduces the required field for actuation, while increasing compressive loads result in the required magnetic field growing considerably. It is helpful to design the intelligent composite with MSMA fibers.

Magnetic field-induced martensitic variant reorientation in magnetic shape memory alloys†

The magnetically induced martensitic variant reorientation process under applied mechanical load in magnetic shape memory alloys (MSMAs) is considered. Of particular interest is the associated nonlinear and hysteretic macroscopic strain response under variable applied magnetic field in the presence of stress, also known as the magnetic shape memory effect (MSME). A thermodynamically consistent phenomenological constitutive model is derived which captures the magnetic shape memory effect caused by the martensitic variant reorientation process, using internal state variables, which are chosen in consideration of the crystallographic and magnetic microstructure. The magnetic contributions to the free energy function considered in this work are the Zeeman energy and the magnetocrystalline anisotropy energy. Activation functions for the onset and termination of the reorientation process are formulated and evolution equations for the internal state variables are derived. The model is applied to a two-dimensional special case in which the application of a transverse magnetic field produces axial reorientation strain in a NiMnGa single-crystal specimen under a constant compressive axial stress. It is explicitly shown how the model parameters are obtained from experimental data. Model predictions of magnetic field-reorientation strain hysteresis loops under different applied stresses are discussed. †Dedicated to Professor Gerard Maugin on the occasion of his receiving of the 2003 SES A.C. Eringen Medal.

Experimental investigation of domain switching criterion for soft lead zirconate titanate piezoceramics under coaxial proportional electromechanical loading

In this experimental work, “soft” lead zirconate titanate specimens in an initially unpoled state were subjected to combined electromechanical loading, in which a compressive stress and a parallel, proportional electric field were applied simultaneously. By changing the ratio of stress to the electric field between tests, a series of nonlinear polarization and strain responses were obtained. An attempt has been made to explain the experimental findings by simultaneously taking into account the contributions of dielectric response, elastic deformation, piezoeffects, and irreversible domain switching. Initial domain switching surfaces in the biaxial stress and electric field space were determined based on an offset method. Several switching conditions existing in the literature were summarized and compared with the experimental data obtained in this work. Such an investigation may serve to calibrate and validate existing constitutive models concerning the large-signal nonlinear behavior of piezoceramics.

Investigations on Multi-axial Domain Switching Criteria for Piezoceramics

AbstractInitially unpoled soft PZT was subjected to a proportional, coaxial electromechanical loading. The ratio of compressive stress to electric field was changed between the experiments. From this series of nonlinear polarization and strain responses were obtained. Based on an offset method, initial domain switching states in the two-dimensional stress-electric field space were determined. In continuum mechanics, thin walled tubes are used to investigate multi-axial stress states. In this context, thin walled means a ratio of wall thickness to radius of 1:10 or thinner. However, simple linear dielectric analysis indicates an inhomogeneous electric field distribution in such geometries.Therefore, the suitability of hollow cylinders (in the range from thick to thin walled tubes) for multi-axial electromechanical experiments has to be investigated. Simulations with a finite element tool based on a phenomenological constitutive model for ferroelectric and ferroelastic hysteresis behavior were performed. The results confirm inhomogeneous distributions of electric fields and stresses after poling. A geometry variation is discussed to minimize these effects.

Phase-field simulations of ferroelectric/ferroelastic polarization switching

Polarization switching in a ferroelectric subjected to an electric field or a stress field is simulated using a phase-field model based on the time-dependent Ginzburg–Landau equation, which takes both multiple-dipole–dipole–electric and multiple-dipole–dipole–elastic interactions into account. The temporal evolution of the polarization switching shows that the switching is a process of nucleation, if needed, and growth of energy-favorite domains. Macroscopic polarization and strain are obtained by averaging polarizations and strains over the entire simulated ferroelectric. The simulation results successfully reveal the hysteresis loop of macroscopic polarization versus the applied electric field, the butterfly curve of macroscopic strain versus the applied electric field, and the macroscopically nonlinear strain response to applied compressive stresses.

Predicting the nonlinear response and failure of composite laminates: correlation with experimental results

A comprehensive comparison of laminate failure models was established to assess the state-of-the-art in laminate modeling technologies on an international level (known as the Worldwide Failure Exercise) [Compos. Sci. Technol. 58(1998) 1001,1011,1225]. In our first contribution to this Exercise (Part A), we presented a complete theoretical description of an analysis methodology and documented predictions for the laminate response and failure behavior of various laminates under a broad range of loading conditions [Compos. Sci. Technol. (in press)]. This paper represents our contribution to Part B of the Exercise where the laminate response and failure predictions for fourteen different cases are presented and compared with actual experimental test data. The cases include prediction of the effective nonlinear stress vs. strain responses of laminates, as well as, initial and final ply failure envelope predictions under multi-axial loading. Correlation between the theoretical predictions and experimental results are discussed. While reasonable correlation was achieved, the failure analysis employed by the authors was not universally accurate in predicting the laminate failure response for the broad range of test cases considered. This statement, although not surprising, is likely true for any given failure methodology as it is applied to a wide range of laminate lay-ups and loading conditions.

Development of glass fibre reinforced polyethylene pipes for pressure applications

Thermoplastic filament winding with in line melt impregnation has been investigated for the manufacture of continuous glass fibre reinforced thermoplastic pipes. With polyethylene matrixes it was found that the high melt viscosity hindered full melt impregnation and resulted in high pull forces in the case of pipe grade polyethylene. Using a lower viscosity polyethylene it was possible to operate the melt impregnation process, but the product obtained exhibited a high void content. Surprisingly, it was found that filament winding resulted in a considerable decrease in void content, to an acceptable level. It was found possible to wind good quality pipes and achieve high failure pressures that fully reflected the strength of the reinforcement.The non-linear strain response of glass–polyethylene pipes can be modelled using laminate theory modified to take account of the non-linearity of the matrix and the change in fibre angle that occurs as the pipe deforms.

Rolling contact fatigue of rails—finite element modelling of residual stresses, strains and crack initiation

Numerical models that can be used to evaluate crack initiation in rails owing to rolling contact loads are of great value. Very few models on rail fatigue assessments exist in the literature that consider the non-linear stress and strain response in the rail caused by both global dynamic track response and local wheel-rail contact loads. In the present investigation, such a tool has been developed using finite element (FE) models. The tool can be used to calculate governing fatigue conditions such as residual stresses, plastic strains and orientation of crack planes for crack initiation. The ability to perform parameter studies of, for example, track properties, different materials and load cases is inherent in the tool. The tool was used in a case study of commuter train traffic at a Swedish test site. In the FE simulations, a material model with non-linear isotropic and kinematic hardening, which was able to consider ratchetting behaviour, was used. Ratchetting material response near the railhead surface was found. A criterion for fatigue crack initiation caused by ratchetting and a criterion for fatigue crack initiation caused by low-cycle fatigue (LCF) were compared. The LCF criterion showed the lowest number of cycles to crack initiation, which, given the models used, indicates that cracks will initiate owing to LCF rather than ratchetting. Good agreement for the direction of surface cracks between the rail track at the test site and the results from the simulations was achieved.

The prediction and measurement of nonlinear strain response of composite plates

Simple analytical formulae are derived for the nonlinear strain response of composite plates to broadband acoustic excitation. The beam mode shape functions with the same boundary conditions as the plate are used. The effects of flexibility and coupling of higher modes are included. The results are compared with those from finite-element method and experimental results from acoustic tests of two types of composite plates. It is found that the coupling of higher modes contributes significantly to the strain response. The present formulae give very good predictions of nonlinear strain but there are still significant errors due to the uncertainty in acoustic spectral density and damping ratio.

Strain Measurement in the Wavy-Ply Region of an Externally Pressurized Cross-Ply Composite Ring

Ply-level strains are determined in the cross-section of an externally pressurized cross-ply (3:1 circumferential to axial fiber ratio) graphite-epoxy ring containing an isolated circumferential wavy region. A special test fixture was used which permitted measuring orthogonal displacement components in the wavy area using moire´ interferometry as the pressure was increased. Strain components were determined at selected locations in the wavy area up to approximately 90 percent of failure pressure. The study shows: (1) Large interlaminar shear strains, which are non-existent in the “perfect” ring, are present near the wave inflection points; (2) The wavy plies generate increased interlaminar normal compressive strains in both circumferential and axial plies along a radial line coinciding with maximum wave amplitude; and (3) Nonlinear strain response begins at approximately 60 percent of failure pressure.

Nonlinear strain response of two-mode fiber-optic interferometer

Experimental and theoretical investigation of a nonlinear response of a two-mode fiber interferometer to axial strain is described. It is discovered that the nonlinearity dramatically increases as the wavelength approaches the cutoff wavelength of the second-order mode (LP(11)).

Electric field forced phase switching in La-modified lead zirconate titanate stannate thin films

Electric field forced antiferroelectric to ferroelectric phase switching has been demonstrated in thin films of Pb0.97La0.02(Zr,Ti,Sn)O3 perovskites for the first time. Several compositions in the tetragonal antiferroelectric phase field of this system were prepared in thin film form by a sol-gel technique. Forward and reverse switching threshold fields of 27–103 kV/cm and 18–62 kV/cm, respectively, were determined from polarization-electric field hysteresis and incremental capacitance data. Switching times as fast as 300 ns were recorded for one of the antiferroelectric compositions. An electric field induced longitudinal strain of 0.16% was measured for a film of composition (Pb0.97La0.02)(Zr0.60Ti0.10Sn0.30)O3 using a laser ultradilatometer. These films are candidate materials for high charge storage integrated capacitors and microelectromechanical devices requiring large nonlinear strain response.

High-resolution fiber-optic low-frequency voltage sensor based on the electrostrictive effect

A minimum detectable voltage of 40 nV/ square root Hz at 1 Hz is demonstrated in a fiber-optic interferometric low-frequency voltage sensor. The device maintains good resolution below 1 Hz, providing a minimum detectable voltage of 55 nV/ square root Hz at 0.1 Hz and 129 nV/ square root Hz at 0.03 Hz. High resolution at low frequencies is attainable due to the nonlinear strain response of the electrostrictive transducer under an applied electric field, which upconverts low-frequency voltage signals as sidebands of a high-frequency carrier where 1/f noise is not dominant. The measured low-frequency data represent an improvement of >45 dB over previously reported results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>