Inelastic Part Research Articles

A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys

The logarithmic or Hencky strain measure is a favored measure of strain due to its remarkable properties in large deformation problems. Compared with other strain measures, e.g., the commonly used Green–Lagrange measure, logarithmic strain is a more physical measure of strain. In this paper, we present a Hencky-based phenomenological finite strain kinematic hardening, non-associated constitutive model, developed within the framework of irreversible thermodynamics with internal variables. The derivation is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, and on the use of the isotropic property of the Helmholtz strain energy function. We also use the fact that the corotational rate of the Eulerian Hencky strain associated with the so-called logarithmic spin is equal to the strain rate tensor (symmetric part of the velocity gradient tensor). Satisfying the second law of thermodynamics in the Clausius–Duhem inequality form, we derive a thermodynamically-consistent constitutive model in a Lagrangian form. In comparison with the available finite strain models in which the unsymmetric Mandel stress appears in the equations, the proposed constitutive model includes only symmetric variables. Introducing a logarithmic mapping, we also present an appropriate form of the proposed constitutive equations in the time-discrete frame. We then apply the developed constitutive model to shape memory alloys and propose a well-defined, non-singular definition for model variables. In addition, we present a nucleation-completion condition in constructing the solution algorithm. We finally solve several boundary value problems to demonstrate the proposed model features as well as the numerical counterpart capabilities.

Identification of a macroscopic anisotropic damage model using digital image correlation and the equilibrium gap method

The aim of the work is to demonstrate how an anisotropic damage model may be identified from full field measurements retrieved during a heterogeneous test. The example of a biaxial test performed on a 3D C / C composite is used. In a first step, the displacement fields measured by classical Digital Image Correlation are used as input data of a finite difference version of the Equilibrium Gap Method. A benefit from unloadings (assumed to be elastic) is shown to retrieve a damage law. In a second step, inelastic strains can be assessed from the total measured strain and the elastic estimated strains. The constitutive parameters relative to the inelastic part of the model are then identified.

A 3D finite strain phenomenological constitutive model for shape memory alloys considering martensite reorientation

Most devices based on shape memory alloys experience both finite deformations and non-proportional loading conditions in engineering applications. This motivates the development of constitutive models considering finite strain as well as martensite variant reorientation. To this end, in the present article, based on the principles of continuum thermodynamics with internal variables, a three-dimensional finite strain phenomenological constitutive model is proposed taking its basis from the recent model in the small strain regime proposed by Panico and Brinson (J Mech Phys Solids 55:2491–2511, 2007). In the finite strain constitutive model derivation, a multiplicative decomposition of the deformation gradient into elastic and inelastic parts, together with an additive decomposition of the inelastic strain rate tensor into transformation and reorientation parts is adopted. Moreover, it is shown that, when linearized, the proposed model reduces exactly to the original small strain model.

Phenomenological model of longitudinal spin fluctuations in itinerant antiferromagnets

We present the phenomenological analysis of the spectrum of longitudinal spin fluctuations in isotropic itinerant electron antiferromagnets with account of spin anharmonicity giving rise to coupling of transverse and longitudinal normal modes. The spectrum consists of a quasielastic part forming a central peak or a dip, depending on temperature and the Landau relaxation rate. Effects of spin fluctuation coupling also give rise to an inelastic part of the spectrum which has a form of resonances or antiresonances near the magnon frequencies related to non-propagating longitudinal excitations.

Cap-and-Trade, Emissions Taxes, and Innovation

Emissions taxes and carbon caps can both lead to efficient production of energy, in the sense of controlling carbon emissions to the extent that is efficient with existing technologies. However, the regulatory policy has a second objective, which is to create incentives to develop lower-carbon technologies. With both objectives in mind, does one policy dominate the other? The answer depends partly on whether the regulated price of energy is in the elastic or inelastic part of the demand curve. It also depends on the size of the intended improvement. Under tax regulation, an innovator can always profit from diffusing the clean technology to all producers. This is not true under a carbon cap, because diffusion expands energy supply, reducing the price of energy and of allowances, and eroding the producers' willingness to pay for licenses. Under cap-and-trade regulation, the regulator has less ability to control the price of energy while ensuring productive efficiency (full diffusion). Because there is little incentive to invest in a larger improvement than will be fully diffused, cap-and-trade regulation limits innovation in a way that is avoided by a tax.

Non-destructive depth compositional profiles by XPS peak-shape analysis

The measured peak shape and intensity of the photoemitted signal in X-ray photoelectron spectroscopy (XPS) experiments (elastic and inelastic parts included) are strongly correlated, through electron-transport theory, with the depth distribution of photoelectron emitters within the analyzed surface. This is the basis of so-called XPS peak-shape analysis (also known as the Tougaard method) for non-destructive determination of compositional in-depth (up to 6-8 nm) profiles. This review describes the theoretical basis and reliability of this procedure for quantifying amounts and distributions of material within a surface. The possibilities of this kind of analysis are illustrated with several case examples related to the study of the initial steps of thin-film growth and the modifications induced in polymer surfaces after plasma treatments.

The Effect of Evolving Damage on the Finite Strain Response of Inelastic and Viscoelastic Composites

A finite strain micromechanical model is generalized in order to incorporate the effect of evolving damage in the metallic and polymeric phases of unidirectional composites. As a result, it is possible to predict the response of composites with ductile and brittle phases undergoing large coupled inelastic-damage and viscoelastic-damage deformations, respectively. For inelastic composites, both finite strain elastoplastic (time-independent) and viscoplastic (time-dependent) behaviors are considered. The ductile phase exhibits initially a hyperelastic behavior which is followed by an inelastic one, and its analysis is based on the multiplicative split of its deformation gradient into elastic and inelastic parts. The embedded damage mechanisms and their evolutions are based on Gurson’s (which is suitable for the modeling of porous materials) and Lemaitre’s finite strain models. Similarly, the polymeric phase exhibits large viscoelastic deformations in which the damage evolves according to a suitable evolution law that depends on the amount of accumulated deformation. Evolving damage in hyperelastic materials can be analyzed as a special case by neglecting the viscous effects. The micromechanical analysis is based on the homogenization technique for periodic multiphase materials, which establishes the strong form of the Lagrangian equilibrium equations. These equations are implemented together with the interfacial and periodic boundary conditions, in conjunction with the current tangent tensor of the phase. As a result, the instantaneous strain concentration tensor that relates the local deformation gradient of the phase to the externally applied deformation gradient is established. This provides also the instantaneous effective stiffness tangent tensor of the composite as well as its current response. Results are given that exhibit the effect of damage on the initial yield surfaces, response and possible failure of the composite.

A finite element model for shape memory alloys considering thermomechanical couplings at large strains

In the present work we propose a new thermomechanically coupled material model for shape memory alloys (SMA) which describes two important phenomena typical for the material behaviour of shape memory alloys: pseudoelasticity as well as the shape memory effect. The constitutive equations are derived in the framework of large strains since the martensitic phase transformation involves inelastic deformations up to 8%, or even up to 20% if the plastic deformation after the phase transformation is taken into account. Therefore, we apply a multiplicative split of the deformation gradient into elastic and inelastic parts, the latter concerning the martensitic phase transformation. An extended phase transformation function has been considered to include the tension–compression asymmetry particularly typical for textured SMA samples. In order to apply the concept in the simulation of complex structures, it is implemented into a finite element code. This implementation is based on an innovative integration scheme for the existing evolution equations and a monolithic solution algorithm for the coupled mechanical and thermal fields. The coupling effect is accurately investigated in several numerical examples including pseudoelasticity as well as the free and the suppressed shape memory effect. Finally, the model is used to simulate the shape memory effect in a medical foot staple which interacts with a bone segment.

Observation of internal-conversion electrons induced by inelastic nuclear resonant scattering

Measurements of the internal-conversion electron emission due to the inelastic nuclear resonant excitation are reported. Fe 57 thin films of 20 and 1.3 nm thickness were deposited on Si(1 1 1), and the internal-conversion electrons were measured as a function of the photon energy. From the inelastic part of the spectra, the phonon density of states was obtained. Whereas the phonon density of states of 20-nm thick film resembles that of bulk α -Fe, the 1.3-nm thick film revealed an obvious softening of the acoustic mode.

Experimental investigation of the P-Δ effect of low rise unbraced steel frames

In this study, the influences of the P-Δ effect on the behavior of low rise unbraced steel frames and the validity of factor B2, which is suggested by KBC 2008 (draft) to consider the P-Δ effect, were evaluated using the experimental approach. Seven specimens, one story-one bay unbraced steel frames, were tested under static loading conditions. The main parameters of the tests were the stiffness ratio of the column to the beam, and the axial load ratio. As revealed by the experimental research that was conducted, the axial load and the decrement of frame stiffness are the main factors that govern the P-Δ effect and that influence the P-Δ effect on the frame behavior. The story stiffness of the frame, which is related to factor B2, was dramatically decreased when the nonlinear behavior started. Factor B2 was compared with the moment amplification factor obtained in the test that had been conducted, and it was found that factor B2, which is based on the linear elastic theory, does not properly estimate the P-Δ effect of an unbraced frame when its assumptions are not satisfied. Thus, careful consideration is needed to apply factor B2 if the design is made using the inelastic part of the material.

Methyl Group Dynamics in Polycrystalline and Liquid Ubiquinone Q0 Studied by Neutron Scattering

We present a quasi-elastic neutron scattering (QENS) study on the methyl group dynamics of ubiquinone Q(0) in the solid and liquid state. For solid ubiquinone Q(0), the dynamics can be described with three Lorentzian functions in the framework of a jump model among three equidistant sites on a circle. According to the known molecular structure of Q(0) in the solid state, this is consistent with three nonequivalent methyl groups in the molecule. From the temperature-dependent analysis of the QENS spectra, the activation energies were determined. The barrier heights could be evaluated from librational bands in the inelastic part of the spectra. The results from neutron spectroscopy are compared to Gaussian 03 calculations leading to an assignment of the activation energies to the different methyl groups in Q(0). The dynamics of Q(0) in the liquid state is evaluated with a scattering function taking into account three different molecular motions. It is demonstrated that the temperature dependence of the long-range diffusion and isotropic rotational diffusion exhibit an Arrhenius-like behavior, whereas the process of methyl group rotation in the liquid phase is virtually free of a barrier.

Thermomechanically coupled modelling of shape memory alloys in the framework of large strains

AbstractIn the present work we propose a new thermomechanically coupled material model for shape memory alloys (SMA) which includes the two main effects observed for these materials, pseudoelasticity and the shape memory effect. The constitutive equations are derived in the framework of large strains since the martensitic phase transformation involves inelastic deformations up to 10%, or even up to 20% if the plastic deformation after the phase transformation is taken into account. Therefore we apply a multiplicative split of the deformation gradient into elastic and inelastic parts, the latter concerning the martensitic phase transformation. An extended phase transformation function has been considered to include the tension‐compression asymmetry of textured SMA samples. In order to achieve a validation of the model by means of multiaxial experiments as well as to finally apply the concept in the simulation of complex structures, it is implemented into a finite element code. This implementation is based on an innovative integration scheme for the existing evolution equations and a monolithic solution algorithm for the coupled mechanical and thermal fields. The coupling effect is accurately investigated in several numerical examples including pseudoelasticity as well as the free and the suppressed shape memory effect. Finally, the model is used to simulate the pseudoelastic behaviour of an intracranial stent and the shape memory effect in a medical foot staple. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On plastic incompressibility within time-adaptive finite elements combined with projection techniques

This article treats the interpretation of quasi-static finite elements applied to constitutive equations of evolutionary-type as a solution scheme to solve globally differential-algebraic equations. This concept is applied to finite strain viscoplasticity based on a model with non-linear kinematic hardening under the assumption of plastic incompressibility. The model is based on multiple multiplicative decomposition both for the deformation gradient into an elastic and an inelastic part as well as for the inelastic part into a kinematic hardening (energy storage) and a dissipative part. Both intermediate configurations are described by inelastic right Cauchy–Green tensors satisfying inelastic incompressibility in the theoretical context. The attention in view of the numerical treatment within finite elements is focused on diagonally implicit Runge–Kutta methods which destroy the assumption of plastic incompressibility during the time-integration due to an additive structure of the integration step. In combination with a Multilevel-Newton algorithm these algorithms embed the classical strain-driven radial-return method. To this end, a concept of geometric numerical integration is applied, where the plastic incompressibility condition is taken into account as an additional side-condition. Since the literature states large integration errors if the side-condition is not taken into account, a particular focus lies on the application of a time-adaptive procedure. Accordingly, the article investigates (i) the algorithmic treatment of kinematic hardening within time-adaptive finite elements, (ii) the influence of the Perzyna-type viscoplasticity approach in view of an order reduction phenomenon, and (iii) the influence of taking into account the exact fulfillment of plastic incompressibility using a projection method having the advantage of simple implementation.

Quasielastic and inelastic neutron scattering study of methyl group rotation in solid and liquid pentafluoroanisole and pentafluorotoluene

The rotational motion of the methyl group in pentafluoroanisole (PFA) and in pentafluorotoluene (PFT), respectively, was investigated by quasielastic neutron scattering (QENS). For solid PFA, the rotation can be described by a model for uniaxial rotational jumps between three equidistant sites on a circle. Similar to the molecular structure of alpha-toluene, two nonequivalent methyl groups in the unit cell with two different rotational barriers were found for solid PFT. From the analysis of the quasielastic scattering, the activation energies were determined. The barrier heights could be evaluated from bands in the inelastic part of the spectra. The methyl group dynamics in the liquid state is evaluated for both substances using different scattering functions, which are discussed. An empirical model for the description of the contribution of methyl groups in liquids of small organic molecules to the QENS spectra is presented. It is demonstrated that the process of methyl group rotation in the liquid phase is nearly free of a barrier.

Modeling of nonlinear viscoelasticity at large deformations

A constitutive model of finite strain viscoelasticity, based on the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts, is presented. The nonlinear response of rubbers, manifested by the rate effect, cycling loading and stress relaxation tests was captured through the introduction of two internal variables, namely the constitutive spin and the back stress tensor. These parameters, widely used in plasticity, are applied in this work to model the nonlinear viscoelastic behaviour of rubbers. The experimental results, obtained elsewhere, related with shear deformation in monotonic and cyclic loading, as well as stress-relaxation, were simulated with a good accuracy.

Finite strain viscoplasticity with nonlinear kinematic hardening: Phenomenological modeling and time integration

This article deals with a viscoplastic material model of overstress type. The model is based on a multiplicative decomposition of the deformation gradient into elastic and inelastic part. An additional multiplicative decomposition of inelastic part is used to describe a nonlinear kinematic hardening of Armstrong–Frederick type. Two implicit time-stepping methods are adopted for numerical integration of evolution equations, such that the plastic incompressibility constraint is exactly satisfied. The first method is based on the tensor exponential. The second method is a modified Euler-backward method. Special numerical tests show that both approaches yield similar results even for finite inelastic increments. The basic features of the material response, predicted by the material model, are illustrated with a series of numerical simulations.

Characterization of mechanical behavior of a porcine pulmonary artery strip using a randomized uniaxial stretch and stretch-rate protocol

BackgroundMuch of the experimental work in soft tissue mechanics has been focused on fitting approximate relations for specific tissue types from aggregate data on multiple samples of the tissue. Such relations are needed for modeling applications and have reasonable predictability – especially given the natural variance in specimens. There is, however, much theoretical and experimental work to be done in determining constitutive behaviors for particular specimens and tissues. In so doing, it may be possible to exploit the natural variation in tissue ultrastructure – so to relate ultrastructure composition to tissue behavior. Thus, this study focuses on an experimental method for determining constitutive behaviors and illustrates the method with analysis of a porcine pulmonary artery strip. The method characterizes the elastic part of the response (implicitly in terms of stretch) and the inelastic part in terms of short term stretch history (i.e., stretch-rate) Ht2, longer term stretch history Ht1, and time since the start of testing T.MethodsA uniaxial testing protocol with a random stretch and random stretch-rate was developed. The average stress at a particular stretch was chosen as the hyperelastic stress response, and deviation from the mean at this particular stretch is chosen as the inelastic deviation. Multivariable Linear Regression Analysis (MLRA) was utilized to verify if Ht2, Ht1, and T are important factors for characterizing the inelastic deviation. For acquiring Ht2 and Ht1, an integral function type of stretch history was employed with time constants chosen from the relaxation spectrum of an identical size strip from the same tissue with the same orientation. Finally, statistical models that characterize the inelasticity were developed at various, nominal values of stretch, and their predictive capability was examined.ResultsInelastic deviation from hyperelasticity was high (31%) for low stretch and declined significantly with increasing stretch to a nadir of 3.6% for a stretch of 1.7. The inelastic deviation then increased with increasing stretch at the same point in the stress-strain curve where stiffness began to increase strikingly. MLRA showed that T is a major inelastic parameter at low deformation. For moderate and high deformations, Ht2 and Ht1 were dominant.DiscussionA randomized uniaxial testing protocol was applied to a strip of porcine pulmonary artery to characterize the elasticity and inelasticity of a soft tissue. We were successful in determining the elastic response and the factors that gave rise to the inelastic deviation. This investigation seeks methods to better define, phenomenologically, the elastic and inelastic behavior of soft tissues.

Representation of elastomers in industrial applications

AbstractA material model is presented to describe inelastic effects in elastomeric components. In static cases, stress softening and permanent deformations are identified as cause of inelasticity. Both effects are covered separately in the constitutive law. This permits the largely modular design of a material description on the basis of user interfaces in commercial FE programs and allows for easier parameter adjustment as the effects can be clearly allocated and identified. The hyperelastic part of the material answer is defined by any freely chosen function of the strain energy, while the inelastic part is of a non‐linear, isotropic hardening type. The presented model is realized in the ABAQUS Finite Element system. The efficiency of the implementation is shown on an example from the hydraulic sealing technology. Further on, we discuss a modification applied to finite viscoelasticity to represent amplitude dependence as observed our in experiments. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Anisotropic magnetic field dependence of the magnetization dynamics inUPd2Al3

The magnetization dynamics of the magnetic superconductor $\mathrm{U}{\mathrm{Pd}}_{2}{\mathrm{Al}}_{3}$ has been investigated by inelastic neutron scattering in the normal and superconducting, antiferromagnetically ordered state under a magnetic field applied parallel to the hexagonal $c$ axis. Within the available resolution, the dynamic response below $2.5\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ is insensitive to the applied field on the scale of the field dependence reported by E. Blackburn et al. [Phys. Rev. B 74, 024406 (2006)] in which the field was applied in the basal plane. Our results support that the previously reported field dependent quasielastic contribution is related to the dynamics of the rotation of the magnetic moment. The changes observed in the inelastic part of the excitation spectrum are consistent with those expected from spin-wave theory. Interestingly, the anisotropic field dependence of the normal-state response may be correlated with the superconducting properties of this material.

Seismic analysis of inelastic moment‐resisting frames Part I: Modified force analogy method for end offsets

AbstractAn analytical method of performing inelastic structural dynamic analysis based on the force analogy method to study real moment‐resisting frames with plastic hinge offsets from member ends and rigid panel zones is proposed. New stiffness matrices used in the force analogy method for members with two ends being rigid are theoretically derived using the unit displacement method. Through static condensation, each term in the stiffness matrices is found to be modified simply through multiplying by a scale factor. This is an enhancement to the original force analogy method or other conventional methods, where the condensed form reduces the size of the structural problem by eliminating the unnecessary additional degrees of freedom that must be placed at the plastic hinges. But more importantly, this is a significant improvement on other conventional methods where the present inelastic analysis procedure is more of a theoretical nature rather than numerical. Numerical simulation for seismic analysis is performed on a single degree of freedom system, and the results show that plastic hinge end offsets have significant effects on the seismic response and therefore they should be carefully considered in dynamic analysis. The companion paper further discusses the influence of panel zone deformation on the dynamic response based on the method proposed in this part of the research. Copyright © 2007 John Wiley & Sons, Ltd.