A general inelastic internal state variable model for amorphous glassy polymers

This paper presents the formulation of a constitutive model for amorphous thermoplastics using a thermodynamic approach with physically motivated internal state variables. The formulation follows current internal state variable methodologies used for metals and departs from the spring-dashpot representation generally used to characterize the mechanical behavior of polymers like those used by Ames et al. in Int J Plast, 25, 1495–1539 (2009) and Anand and Gurtin in Int J Solids Struct, 40, 1465–1487 (2003), Anand and Ames in Int J Plast, 22, 1123–1170 (2006), Anand et al. in Int J Plast, 25, 1474–1494 (2009). The selection of internal state variables was guided by a hierarchical multiscale modeling approach that bridged deformation mechanisms from the molecular dynamics scale (coarse grain model) to the continuum level. The model equations were developed within a large deformation kinematics and thermodynamics framework where the hardening behavior at large strains was captured using a kinematic-type hardening variable with two possible evolution laws: a current method based on hyperelasticity theory and an alternate method whereby kinematic hardening depends on chain stretching and material plastic flow. The three-dimensional equations were then reduced to the one-dimensional case to quantify the material parameters from monotonic compression test data at different applied strain rates. To illustrate the generalized nature of the constitutive model, material parameters were determined for four different amorphous polymers: polycarbonate, poly(methylmethacrylate), polystyrene, and poly(2,6-dimethyl-1,4-phenylene oxide). This model captures the complex character of the stress–strain behavior of these amorphous polymers for a range of strain rates.     

A new yield surface distortion model based on Baltov and Sawczuk’s model

In the present study, a new distortion yield surface model is proposed to represent compatible results with experimental observations. The proposed yield surface model is determined numerically during tension–torsion loadings by considering a kinematic hardening model and monotonic loading paths. The experimental results of yield surface determination (Khan et al. in Int J Plast 26:1432–1441, 2010; Naghdi et al. in ASME J. Appl Mech 25:201–209, 1957) represent the nosed and flattened regions in the loading and reverse loading directions, respectively. But, the Baltov and Sawczuk’s yield surface model can only predict nosed or flattened shape in both loading and reversed loading directions, depending on the sign of their model constant. Thus, the elliptic Baltov and Sawczuk’s yield surface is modified by changing the sign of this parameter continuously from loading to reverse loading direction. Relations and convexity of the new model are obtained and discussed. The new model is able to predict properly the shape of the yield surface. The experimental results are in a satisfactory agreement with the new yield surface distortion model predictions.     

Thermodynamics and thermal decomposition for shape memory effects with crystallization based on dissipation and logarithmic strain

The present effort provides a 3-D thermodynamic framework generalizing the 1-D modeling of 2-way shape memory materials described by Westbrook et al. (J. Eng. Mater. Technol. 312:041010, 2010) and Chung et al. (Macromolecules 41:184–192, 2008), while extending the strain-induced crystallization and shape memory approaches of Rao and Rajagopal (Interfaces Free Bound. 2:73–94, 2000; Int. J. Solids Struct. 38:1149–1167, 2001), Barot and Rao (Z. Angew. Math. Phys. 57:652–681, 2006), and Barot et al. (Int. J. Eng. Sci. 46:325–351, 2008) to include finite thermal expansion within a logarithmic strain basis. The free energy of newly-formed orthotropic crystallites is assumed additive, with no strains in their respective configurations of formation. A multiplicative decomposition is assumed for the assumed thermoelastic orthotropic expansional strains of the respective crystallites. The properties of the crystallites are allowed to depend both on current temperature and their respective temperatures of formation. The entropy production rate relation is written in the frame rotating with the logarithmic spin and produces stress and entropy relations incorporating the integrated configurational free energies, and a driving term for the crystallization analogous to that obtained by the previous studies of Rao et al. The salient attributes of the 1-D modeling of Westbrook et al. are recovered, and applications are discussed.     

A comparative study between two smoothing strategies for the simulation of contact with large sliding

The numerical simulation of contact problems is still a delicate matter especially when large transformations are involved. In that case, relative large slidings can occur between contact surfaces and the discretization error induced by usual finite elements may not be satisfactory. In particular, usual elements lead to a facetization of the contact surface, meaning an unavoidable discontinuity of the normal vector to this surface. Uncertainty over the precision of the results, irregularity of the displacement of the contact nodes and even numerical oscillations of contact reaction force may result of such discontinuity. Among the existing methods for tackling such issue, one may consider mortar elements (Fischer and Wriggers, Comput Methods Appl Mech Eng 195:5020–5036, 2006; McDevitt and Laursen, Int J Numer Methods Eng 48:1525–1547, 2000; Puso and Laursen, Comput Methods Appl Mech Eng 93:601–629, 2004), smoothing of the contact surfaces with additional geometrical entity (B-splines or NURBS) (Belytschko et al., Int J Numer Methods Eng 55:101–125, 2002; Kikuchi, Penalty/finite element approximations of a class of unilateral contact problems. Penalty method and finite element method, ASME, New York, 1982; Legrand, Modèles de prediction de l’interaction rotor/stator dans un moteur d’avion Thèse de doctorat. PhD thesis, École Centrale de Nantes, Nantes, 2005; Muñoz, Comput Methods Appl Mech Eng 197:979–993, 2008; Wriggers and Krstulovic-Opara, J Appl Math Mech (ZAMM) 80:77–80, 2000) and, the use of isogeometric analysis (Temizer et al., Comput Methods Appl Mech Eng 200:1100–1112, 2011; Hughes et al., Comput Methods Appl Mech Eng 194:4135–4195, 2005; de Lorenzis et al., Int J Numer Meth Eng, in press, 2011). In the present paper, we focus on these last two methods which are combined with a finite element code using the bi-potential method for contact management (Feng et al., Comput Mech 36:375–383, 2005). A comparative study focusing on the pros and cons of each method regarding geometrical precision and numerical stability for contact solution is proposed. The scope of this study is limited to 2D contact problems for which we consider several types of finite elements. Test cases are given in order to illustrate this comparative study.     

A note on oblique water entry

A minor error in Howison et al. (J. Eng. Math. 48:321–337, 2004) obscured the fact that the points at which the free surface turns over in the solution of the Wagner model for the oblique impact of a two-dimensional body are directly related to the turnover points in the equivalent normal impact problem. This note corrects some of the earlier results given in Howison et al. (J. Eng. Math. 48:321–337, 2004) and discusses the implications for the applicability of the Wagner model.     

Neumann exterior wave propagation problems: computational aspects of 3D energetic Galerkin BEM

In this work, we consider as model problem an exterior 3D wave propagation Neumann problem reformulated in terms of a space–time hypersingular boundary integral equation with retarded potentials. This latter is set in the so-called energetic weak form, recently proposed in Aimi et al. (Int J Numer Methods Eng 80:1196–1240, 2009; CMES 58:185–219, 2010), regularized as in Frangi (Int J Numer Methods Eng 45:721–740, 1999) and then approximated by the Galerkin boundary element method. Details on the discretization phase and, in particular, on the computation of integrals, double in time and double in space, constituting the elements of the final linear system matrix are given and analyzed. Various numerical results and simulations are presented and discussed.     

A numerical study of hypoelastic and hyperelastic large strain viscoplastic Perzyna type models

For the case of metals with large viscoplastic strains, it is necessary to define appropriate constitutive models in order to obtain reliable results from the simulations. In this paper, two large strain viscoplastic Perzyna type models are considered. The first constitutive model has been proposed by Ponthot, and the elastic response is based on hypoelasticity. In this case, the kinematics of the constitutive model is based on the additive decomposition of the rate deformation tensor. The second constitutive model has been proposed by García Garino et al., and the elasticresponse is based on hyperelasticity. In this case, the kinematics of the constitutive model is based on the multiplicativedecomposition of the deformation gradient tensor. In both cases, the resultant numerical models have been implemented in updated Lagrangian formulation. In this work, global and local numerical results of the mechanical response of both constitutive models are analyzed and discussed. To this end, numerical experiments are performed and different parameters of the constitutive models are tested in order to study the sensitivity of the resultantalgorithms. In particular, the evolution of the reaction forces, the effective plastic strain, the deformed shapes and the sensitivity of the numerical results to the finite element mesh discretization have been compared and analyzed. The obtained results show that both models have a very good agreement and represent very well the characteristic of the viscoplastic constitutive model.     

Slip modes and partitioning of energy during dynamic frictional sliding between identical elastic–viscoplastic solids

The effect of plasticity on dynamic frictional sliding along an interface between two identical elastic–viscoplastic solids is analyzed. The configuration considered is the same as that in Coker et al. (J Mech Phys Solids 53:884–992, 2005) except that here plane strain analyses are carried out and bulk material plasticity is accounted for. The specimens have an initial compressive stress and are subject to shear loading imposed by edge impact near the interface. The material on each side of the interface is modeled as an isotropically hardening elastic–viscoplastic solid. The interface is characterized as having an elastic response together with a rate- and state-dependent frictional law for its inelastic response. Depending on bulk material properties, interface properties and loading conditions, frictional slip along the interface can propagate in a crack-like mode, a pulse-like mode or a train-of-pulses mode. Results are presented for the effect of material plasticity on the mode and speed of frictional slip propagation as well as for the partitioning of energy components between stored elastic energy, kinetic energy, plastic dissipation in the bulk and frictional dissipation along the interface. Some parameter studies are carried out to explore the effects of varying the interface elastic stiffness and the impact velocity.     

Reduction of the number of material parameters by ANN approximation

Modern industrial standards require advanced constitutive modeling to obtain satisfactory numerical results. This approach however, is causing significant increase in number of material parameters which can not be easily obtained from standard and commonly known experimental techniques. Therefore, it is desirable to introduce procedure decreasing the number of the material parameters. This reduction however, should not lead to misunderstanding the fundamental physical phenomena. This paper proposes the reduction of the number of material parameters by using ANN approximation. Recently proposed viscoplasticity formulation for anisotropic solids (metals) developed by authors is used as an illustrative example. In this model one needs to identify 28 material parameters to handle particular metal behaviour under adiabatic conditions as reported by Glema et al (J Theor Appl Mech 48:973–1001, 2010), (Int J Damage Mech 18:205–231, 2009) and Sumelka (Poznan University of Technology, Poznan, 2009). As a result of proposed approach, authors decreased the number of material parameters to 19.     

New aspects for friction coefficients of finite granular avalanche down a flat narrow reservoir

This work examines the bulk internal friction coefficient, \(\mu \), and effective wall friction coefficient, \(\mu _w\), for finite number of nearly identical dry glass spheres in avalanche down a narrow inclined reservoir of smooth frictional bed using a validated discrete element scheme. Instantaneous deviatoric strain rate tensor \(\dot{\gamma }^d_{ij}\) and stress tensor \(\tau _{ij}\) are computed locally to evaluate a three-dimensional constitutive model developed based on the rheology of steady homogeneous surface flows. On one side, the algebraic \(\mu -I\) relation conforms to conventional relation for glass beads, \(\mu =0.34+0.31/(1+0.15/I)\) (Jop et al. in J. Fluid Mech. 541:167–192, 2005, Midi in Eur. Phys. J. E 14:341–365, 2004, Jop in Comptes Rendus Phys. 16:62–72, 2015), when the inertial number \(I>I_{c}=2\times 10^{-2}\). The assumption of collinear \(\tau _{ij}\) and \(\dot{\gamma }^d_{ij}\), however, does not hold and such misalignment agrees to the findings in non-uniform inhomogeneous flows (Cortet et al. in Europhys. Lett. 88(1):14001, 2009). Below \(I_c\), we observe a decaying \(\mu -I\) as found in slowly deforming rheology tests and a simplified model is developed in view of shear-induced dilatation upon yielding. Non-constant effective wall friction coefficient is measured to grow in time and with I towards the sphere-wall sliding friction coefficient in the contact model while preserving the depth-weakening feature as in confined steady surface flows (Richard et al. in Phys. Rev. Lett. 101:248002, 2008, Brodu et al. in Phys. Rev. E 87:022202, 2013). The fact that rotation at one sphere center can divert surface relative velocity across the contact area to render lower sliding friction is considered to develop a model describing how \(\mu _w\) drops with the ratio between rotation-induced velocity and sliding velocity, \(\varOmega \). The simulation data compares fairly well to the predicted monotonic decay of \(\mu _w\) with \(\varOmega \).     

An experimental study on condensation of refrigerant R134a in a multi-port extruded tube

In the present study, the local characteristics of pressure drop and heat transfer are investigated experimentally for the condensation of pure refrigerant R134a in two kinds of 865 mm long multi-port extruded tubes having eight channels in 1.11 mm hydraulic diameter and 19 channels in 0.80 mm hydraulic diameter. The pressure drop is measured at an interval of 191 mm through small pressure measuring ports. The local heat transfer rate is measured in every subsection of 75 mm in effective cooling length using heat flux sensors. It is found that the experimental data of frictional pressure drop agree with the correlation of Mishima and Hibiki [Trans. JMSE (B) 61 (1995) 99], while the correlations of Chisholm and Laird [Trans. ASME 80 (1958) 227], Soliman et al. [Trans. ASME, Ser. C 90 (1998) 267], and Haraguchi et al. [Trans. JSME (B) 60 (1994) 239], overpredict. As a trial, the data of local heat transfer coefficient are also compared with correlations of Moser et al. [J. Heat Transfer 120 (1998) 410] and Haraguchi et al. [Trans. JSME (B) 60 (1994) 245]. The data of high mass velocity agree with the correlation of Moser et al., while those of low mass velocity show different trends. The correlation of Haraguchi et al. shows the trend similar to the data when the shear stress in their correlation is estimated using the correlation of Mishima and Hibiki.     

A hierarchy of avalanche models on arbitrary topography

We use the non-Cartesian, topography-based equations of mass and momentum balance for gravity driven frictional flows of Luca et al. (Math. Mod. Meth. Appl. Sci. 19, 127–171 (2009)) to motivate a study on various approximations of avalanche models for single-phase granular materials. By introducing scaling approximations we develop a hierarchy of model equations which differ by degrees in shallowness, basal curvature, peculiarity of constitutive formulation (non-Newtonian viscous fluids, Savage–Hutter model) and velocity profile parametrization. An interesting result is that differences due to the constitutive behaviour are largely eliminated by scaling approximations. Emphasis is on avalanche flows; however, most equations presented here can be used in the dynamics of other thin films on arbitrary surfaces.     

Dielectric Design of Current Lead Parts for a 154 kV Superconducting Apparatus

Currently, studies on the development of high voltage superconducting machines are being conducted actively in many research institutes all over the world (Kang et al. in IEEE Trans. Appl. Supercond. 17:1493–1496, 2007). In this paper, a sub-cooled liquid nitrogen cooling method is considered to design current lead parts for a 154 kV superconducting apparatus because of its excellent dielectric and mechanical characteristics (Kang et al. in IEEE Trans. Appl. Supercond. 17:1493–1496, 2007). Insulation gases such as SF₆, CF₄, and N₂ are considered as suitable pressurization gases for enhancing the dielectric characteristics of a superconducting system in the Republic of Korea (Kang and Ko in IEEE Trans. Appl. Supercond. 21:1332–1335, 2011). Dielectric experiments are conducted on GFRP (glass fiber reinforced plastic) and insulation gases (SF₆, CF₄, and N₂) with various pressures to establish experimental criteria for designing current lead parts for superconducting applications. It is found that the criteria for calculating electrical breakdown voltage at sparkover of GFRP and insulation gases conform to the mean electric field intensity and the maximum electric field intensity, respectively. Also, the criteria for calculating the electrical breakdown voltage at sparkover according to various conditions are derived. Finally, the conceptual dielectric design of current leads for a 154 kV superconducting apparatus by applying the presented criteria is performed considering various safety factors.     

Multiaxial fatigue models for short glass fiber reinforced polyamide – Part I: Nonlinear anisotropic constitutive behavior for cyclic response

Components made of short glass fiber reinforced (SGFR) thermoplastics are increasingly used in the automotive industry, and more frequently subjected to fatigue loadings during their service life. The determination of a predictive fatigue criterion is therefore a serious issue for the designers, and requires the knowledge of the local mechanical response under a large range of environmental conditions (temperature and relative humidity). As the cyclic behavior of polymeric material is reckoned to be highly nonlinear, even at room temperature, an accurate constitutive model is a preliminary step for confident fatigue design.The injection molding process induces a complex fiber orientation distribution (FOD), which affects both the mechanical response and the fatigue life of SGFR thermoplastics. This paper presents an extension of the constitutive behavior proposed by the authors in a previous work [Launay et al., Int J Plasticity, 2011], in order to take into account the influence of the local FOD on overall anisotropic elastic and viscoplastic properties. The proposed model is written in a general 3D anisotropic framework, and is validated on tensile samples with various FOD and loading histories: monotonic tensions, creep and/or relaxation steps, cyclic loadings. In Part II of this paper [Launay et al., Int J Fatigue, 2012], this constitutive model will be applied to the simulation of different fatigue samples subjected to multiaxial cyclic loadings.     

Constitutive models for bcc engineering iron alloys exposed to thermal–mechanical fatigue

The progress in designing high temperature components relies on more accurate viscoplastic constitutive models. The capability of various models under high temperature and variable temperature conditions is investigated for two body centred cubic alloys, cast iron and ferritic stainless steel. Improvements are shown to overcome problems encountered by standard viscoplastic models. Firstly a physically based modified flow equation predicts reliably the behaviour of cast iron under thermal–mechanical loading. Secondly further improvement is proposed drawing on dislocation models to describe static recovery effects in stainless steels. Good agreement is thus obtained between experiment and model prediction under various thermal mechanical loading path.     

Comparison of fully coupled modeling and experiments for electromagnetic forming processes in finitely strained solids

In fracture and fragmentation research the technique of electromagnetic forming, which uses electromagnetic (Lorentz) body forces to shape metallic parts, is finding significant use due to the high velocity, high strain rate loading it can impart without contact on workpieces. The same process is also becoming increasingly relevant for manufacturing processes in sheet metal forming, where this technique offers several advantages: speed, repeatability, non-contact loading, reduced springback and considerable ductility increase in several metals. Current modeling techniques for these coupled electromagnetic and thermomechanical processes are not based on coupled variational principles that can simultaneously account for electromagnetic and mechanical effects. Typically, separate solutions to the electromagnetic (Maxwell) and motion (Newton) equations are combined in staggered or lock-step methods, sequentially solving the mechanical and electromagnetic problems. To address this issue, Thomas and Triantafyllidis (J Mech Phys Solids 57:1391–1416, 2009) have recently introduced a fully coupled Lagrangian (reference configuration) variational principle, involving the magnetic field potential and the displacement field as independent variables. The corresponding Euler-Lagrange equations are Maxwell’s and Newton’s equations in the reference configuration under the eddy current approximation. This novel approach is used here to simulate free expansion experiments of AA6063-T6 aluminum tubes. A viscoplastic constitutive model, developed independently by the authors (Thomas et al. Acta Mater 55:2863–2873, 2007) for necking experiments in tubes of the same aluminum alloy, is used in the simulations. The measured electric currents and tube deformation—the latter obtained by Photon Doppler Velocimetry—show reasonably good agreement with the corresponding simulations, which are obtained using a variational integration numerical scheme that results in an efficient staggered solution algorithm.     

Determination of nanoindentation size effects and variable material intrinsic length scale for body-centered cubic metals

It is well-known by now that the micro and nanoindentation hardness of metallic materials displays a strong size effect. The objective of this work is to formulate a micromechanical-based model for Temperature and Rate Indentation Size Effects (TRISE) for body centered cubic (BCC) metals encountered in nanoindentation experiments. In this regard, two physically based models are proposed here in order to capture the TRISE in single and polycrystalline materials by considering different expressions of the geometrical necessary dislocations (GNDs) density.The gradient plasticity theory formulates a constitutive framework on the continuum level that bridges the gap between the micromechanical plasticity and the classical continuum plasticity by incorporating the material length scale. A micromechanical-based model of variable material intrinsic length scale is also developed in the present work. The proposed length scale allows for variations in temperature and strain rate and its dependence on the grain size and accumulated plastic strain.The results of indentation experiments performed on niobium, tungsten, and single- and polycrystalline commercially pure iron (very similar to iron alloys) are used here to implement the aforementioned framework in order to predict simultaneously the TRISE and variable length scale at different temperatures, strain rates and various distances from the grain boundary. Numerical analysis is performed using the ABAQUS/VUMAT software with a physically based viscoplastic constitutive model.     

Analysis of nano-plates by atomistic-refined models accounting for surface free energy effect

This work presents several higher-order atomistic-refined models for the static and free vibration analysis of nano-plates. Stemming from a two-dimensional approach and thanks to a compact notation for the a priori kinematic field approximation over the plate through-the-thickness direction, a general model derivation is used where the approximation order is a free parameter of the formulation. Several higher-order plate theories can be obtained straightforwardly. Classical plate models, such as Kirchhoff’s and Reissner’s, are obtained as particular cases. The assumed constitutive equations for orthotropic materials are those derived by Dingreville et al. (J Mech Phys Solids 53:1827–1854, 2005), which account for the surface free energy effect as well as the third-order elastic constants. The resulting stiffness coefficients depend upon the thickness. The governing equations and boundary conditions are variationally obtained through the principle of virtual displacements. A Navier-type, strong form solution is adopted. Simply supported plates are, therefore, investigated. Static and free vibration analyses are carried out in order to investigate the effect of the thickness side as well as the crystallographic plane orientation on the mechanical response. Plates with different values of the side-to-thickness ratio are considered. Results are validated in terms of accuracy and computational costs toward three-dimensional FEM solutions. Numerical investigations show the advantages of refined plate models over the classical ones demonstrating that accurate results can be obtained with reduced computational costs.     

Computational homogenization of porous materials of Green type

The constitutive response of porous materials is investigated computationally. For the solid phase elasto- plastic behavior of Green type is considered, i.e. an isotropic compressible yield criterion is assumed. A wide range of material parameters and porosities from 0.1 to 30 % are investigated by means of FEM simulations of periodic ensembles of spherical pores. The dilatation of the pores and of the compressible matrix are evaluated. It is found that a large part of the total dilatation is due to plastic volume changes of the solid phase. The asymptotic stress states of the simulations are compared to analytical predictions by Shen et al. (Comput Mater Sci 62:189–194, 2012). Based on the computational data, an effective constitutive law is proposed and verified by means of additional computations. A three-scale homogenization procedure for double porous materials is proposed that depends only on the micro- and mesoscale porosity and the yield stress of the solid phase.