A non-affine network model for polymer melts

In this paper, a network model of polymer melts is proposed in which network junction points move non-affinely. In this non-affine motion, junction points follow particle paths as seen by an observer rotating at the fluid element's net rigid-rotation rate. The speed at which junction points move is reduced as the network segments near their maximum extensions. In order to maintain a frame invariant model, it is necessary that the vorticity be decomposed into two portions, such that, = _R + _D . The deformational vorticity, _D , arises from shear deformation and is frame invariant while the rigid vorticity, _R , is frame dependent. A constitutive equation based on this finitely extensible network strand (FENS) motion is developed. The model illustrates how rotations that cause changes in the relative orientation of a fluid element with its surroundings can be incorporated into a constitutive equation using the deformational vorticity. The FENS model predicts a shear-thinning viscosity, and the Trouton viscosity predicted by the model is finite for all elongation rates. Finally, stochastic simulation results are presented to justify a mathematical approximation used in deriving the constitutive equation.     

A new constitutive model for fibre suspensions: flow past a sphere

A new phenomenological constitutive equation for homogeneous suspensions of macrosized fibres is proposed. In the model, the local averaged orientation of the fibres is represented by a director field, which evolves in time in a manner similar to the rotation of a prolate spheroid. The stress is linear in the strain rate, but the viscosity is a fourth-order tensor that is directly related to the director field. In the limit of low-volume fractions of fibres, the model reduces properly to the leading terms of the constitutive equation for dilute suspensions of spheroids. The model has three parameters: the aspect ratio R of the fibres, the volume fraction , and A, which plays the role of the maximum-volume fraction of the fibres. Experimental shear data are used to estimate the parameter A, and the resulting model is used in a boundary-element program to study the flow past a sphere placed at the centre line of a cylinder for the whole range of volume fractions from 0.01 to near maximum volume fraction. The agreement with experimental data from Milliken et al. [1] is good.     

A continuum model for deformable,second gradient porous media partially saturated with compressible fluids

In this paper a general set of equations of motion and duality conditions to be imposed at macroscopic surfaces of discontinuity in partially saturated, solid-second gradient porous media are derived by means of the Least Action Principle. The need of using a second gradient (of solid displacement) theory is shown to be necessary to include in the model effects related to gradients of porosity. The proposed governing equations include, in addition to balance of linear momentum for a second gradient porous continuum and to balance of water and air chemical potentials, the equations describing the evolution of solid and fluid volume fractions as supplementary independent kinematical fields. The presented equations are general in the sense that they are all written in terms of a macroscopic potential Ψ$\Psi$ which depends on the introduced kinematical fields and on their space and time derivatives. These equations are suitable to describe the motion of a partially saturated, second gradient porous medium in the elastic and hyper-elastic regime. In the second part of the paper an additive decomposition for the potential Ψ$\Psi$ is proposed which allows for describing some particular constitutive behaviors of the considered medium. While the potential associated to the solid matrix deformation is chosen in the form proposed by Cowin and Nunziato (1981) and Nunziato and Cowin (1979) and the potentials associated to water and air compressibility are chosen to assume a simple quadratic form, the macroscopic potentials associated to capillarity phenomena between water and air have to be derived with some additional considerations. In particular, two simple examples of microscopic distributions of water and air are considered: that of spherical bubbles and that of coalesced tubes of bubbles. Both these cases are suitable to describe capillarity phenomena in porous media which are close to the saturation state. Finally, an example of a simple microscopic distribution of water and air giving rise to a macroscopic capillary potential depending on the second gradient of fluid displacement is presented, showing the need of a further generalization of the proposed theoretical framework accounting for fluid second gradient effects.     

In situ 3D Synchrotron Laminography Assessment of Edge Fracture in Dual-Phase Steels: Quantitative and Numerical Analysis

The mechanical performance of automotive structures made of advanced high strength steels (AHSS) is often seen reduced by the presence of cut edges. An attempt is made to assess and quantify the initial damage state and the damage evolution during mechanical testing of a punched edge and a machined edge via a recently developed 3D imaging technique called synchrotron radiation computed laminography. This technique allows us to observe damage in regions of interest in thin sheet-like objects at micrometer resolution. In terms of new experimental mechanics, steel sheets having sizes and mechanical boundary conditions of engineering relevance can be tested for the first time with in situ 3D damage observation and quantification. It is found for the investigated DP600 steel that the fracture zone of the punched edge is rough and that needle-shape voids at the surface and in the bulk follow ferrite-martensite flow lines. During mechanical in situ testing the needle voids grow from the fracture zone surface and coalesce with the sheared zone. In contrast, during in situ mechanical testing of a machined edge the damage starts away from the edge (～800μm) where substantial necking has occurred. Three-dimensional image analysis was performed to quantify the initial damage and its evolution. These data can be used as input and validation data for micromechanical damage models. To interpret the experimental findings in terms of mechanical fields, combined surface digital image correlation and 3D finite element analysis were carried out using an elasto-plastic constitutive law of the investigated DP steel. The stress triaxiality and the accumulated plastic strain were calculated in order to understand the influence of the edge profile and the hardening of the cutting-affected zone on the mechanical fields.     

A nonlocal constitutive model generated by matrix functions for polyatomic periodic linear chains

We establish a discrete lattice dynamics model and its continuum limits for nonlocal constitutive behavior of polyatomic cyclically closed linear chains being formed by periodically repeated unit cells (molecules), each consisting of \({n \geq 1}\) atoms which all are of different species, e.g., distinguished by their masses. Nonlocality is introduced by elastic potentials which are quadratic forms of finite differences of orders \({m \in \mathbf{N}}\) of the displacement field leading by application of Hamilton’s variational principle to nondiagonal and hence nonlocal Laplacian matrices. These Laplacian matrices are obtained as matrix power functions of even orders 2m of the local discrete Laplacian of the next neighbor Born-von-Karman linear chain. The present paper is a generalization of a recent model that we proposed for the monoatomic chain. We analyze the vibrational dispersion relation and continuum limits of our nonlocal approach. “Anomalous” dispersion relation characteristics due to strong nonlocality which cannot be captured by classical lattice models is found and discussed. The requirement of finiteness of the elastic energies and total masses in the continuum limits requires a certain scaling behavior of the material constants. In this way, we deduce rigorously the continuum limit kernels of the Laplacian matrices of our nonlocal lattice model. The approach guarantees that these kernels correspond to physically admissible, elastically stable chains. The present approach has the potential to be extended to 2D and 3D lattices.     

三维加锚弹塑性损务模型在溪洛渡地下厂房工程中的应用

本文根据断续裂隙岩体的损伤机制,建立了三维弹塑性损伤本构模型反映裂隙岩体的损伤变形特性。考虑断续裂隙岩体的岩锚支护效应,建立了空间损伤锚柱单元模型模拟锚杆的支护效果。最后将建立的模型应用于溪洛渡水电站地下厂房,进行了洞室群开挖弹塑性损伤及岩锚支护三维非线性有限元计算,获得了一些有益的工程结论。     

Multilevel bridged crack model for high-performance concretes

A multilevel bridged crack model is proposed. It reproduces the constitutive flexural response of reinforced concrete members with fibers. Considered are two different reinforcements: the longitudinal bars (primary reinforcement) and the fibers (secondary reinforcement) distributed in the brittle cementitious matrix. The bridging actions exerted by the reinforcements onto the crack faces are assumed to be rigid-perfectly plastic as the primary constituents. Cohesive softening applies to the fibers.From dimensional analysis, the constitutive flexural response is found to depend on three dimensionless parameters. The first , controls the extension of the process zone. The remaining two parameters, referred to as brittleness numbers N_P⁽¹⁾ and N_P⁽²⁾, are related to the reinforcement phases. Specimen size scale is basic to the global structural behaviour. It can range from ductile to brittle as characterized by the two brittleness numbers. They depend on the reinforcement phase of matrix toughness, reinforcement yielding or slippage limit, reinforcement volume fraction and global structural size.     

Experimental and Numerical Simulation of Elastomeric Outsole Bending

In this paper, we deal with the mechanical behaviour of elastomeric sole shoes subjected to bending. First, a methodology for material parameter identification is described for a visco-hyperelastic constitutive law with discontinuous damage, modelling the Mullins effect. This constitutive law was implemented in the finite element software CODE-ASTER (Abbas and Baranger 2010). Then, numerical results of the bending behaviour of a simple geometric shoe sole model are compared to experimental measurements obtained with a stereoscopic vision system. Finally, an industrial application of dynamic bending sole shoe is presented and experimental results and simulated load history are compared.     

Periodic solutions of <Emphasis Type="Italic">n</Emphasis>-dofs autonomous vibro-impact oscillators with one lasting contact phase

In the continuum structural mechanics framework, a unilateral contact condition between two flexible bodies does not generate impulsive contact forces. However, finite-dimensional systems, derived from a finite element semi-discretization in space for instance, and undergoing a unilateral contact condition, require an additional impact law: Unilateral contact occurrences then become impacts of zero duration unless (i) the impact law is purely inelastic, or (ii) the pre-impact velocity is zero. This contribution explores autonomous periodic solutions with one contact phase per period and zero pre-impact velocity [case (ii)], for any n-dof mechanical systems involving linear free-flight dynamics together with a linear unilateral contact constraint. A recent work has shown that such solutions seem to be limits of periodic trajectories with k impacts per period as k increases. Minimal analytic equations governing the existence of such solutions are proposed, and it is proven that, generically, they occur only for discrete values of the period. It is also shown that the graphs of such periodic solutions have two axes of symmetry in time. Results are illustrated on a spring–mass system and on a 4-dof two-dimensional system made of 1D finite elements. Animations of SPPs with up to 30 dofs are provided.     

Deformation of thermosetting resins at impact rates of strain. Part 2: constitutive model with rejuvenation

The constitutive responses of three glassy thermoset polymers at impact rates of strain and slower, together with measurements of adiabatic heating, were reported earlier by the authors. The results are interpreted here in the context of a constitutive model proposed previously for amorphous polymers, expanded to incorporate strain-softening and the adiabatic heating deficit. In terms of the model, both features are a natural consequence of strain-induced evolution of the glass structure, as represented by Tool's “fictive temperature”—the phenomenon of structural rejuvenation. A representation is proposed for the evolution of fictive temperature with plastic strain, motivated by an approximate treatment of the kinetics of physical ageing/rejuvenation. Formulated in this manner, the model agrees reasonably well with experimental results across the wide range of strain rates of the previous experiments, 10⁻³ to , and across most of the range of strain to failure in compression. At the highest strains, however, an additional adiabatic heating deficit appears that is not predicted by the model, either suggesting the onset of structural breakdown possibly associated with the appearance of cracks or reflecting a need for better physical understanding of large deformations in glassy polymers.     

A mathematical model and a numerical model for hyperbolic mass transport in compressible flows

A number of contributions have been made during the last decades to model pure-diffusive transport problems by using the so-called hyperbolic diffusion equations. These equations are used for both mass and heat transport. The hyperbolic diffusion equations are obtained by substituting the classic constitutive equation (Fick’s and Fourier’s law, respectively), by a more general differential equation, due to Cattaneo (C R Acad Sci Ser I Math 247:431–433, 1958). In some applications the use of a parabolic model for diffusive processes is assumed to be accurate enough in spite of predicting an infinite speed of propagation (Cattaneo, C R Acad Sci Ser I Math 247:431–433, 1958). However, the use of a wave-like equation that predicts a finite velocity of propagation is necessary in many other calculations. The studies of heat or mass transport with finite velocity of propagation have been traditionally limited to pure-diffusive situations. However, the authors have recently proposed a generalization of Cattaneo’s law that can also be used in convective-diffusive problems (Gómez, Technical Report (in Spanish), University of A Coruña, 2003; Gómez et al., in An alternative formulation for the advective-diffusive transport problem. 7th Congress on computational methods in engineering. Lisbon, Portugal, 2004a; Gómez et al., in On the intrinsic instability of the advection–diffusion equation. Proc. of the 4th European congress on computational methods in applied sciences and engineering (CDROM). Jyväskylä, Finland, 2004b) (see also Christov and Jordan, Phys Rev Lett 94:4301–4304, 2005). This constitutive equation has been applied to engineering problems in the context of mass transport within an incompressible fluid (Gómez et al., Comput Methods Appl Mech Eng, doi: 10.1016/j.cma.2006.09.016, 2006). In this paper we extend the model to compressible flow problems. A discontinuous Galerkin method is also proposed to numerically solve the equations. Finally, we present some examples to test out the performance of the numerical and the mathematical model.     

Multidimensional modeling of the stenosed carotid artery: A novel CAD approach accompanied by an extensive lumped model

This study describes a multidimensional 3D/lumped parameter(LP) model which contains appropriate inflow/outflow boundary conditions in order to model the entire human arterial trees. A new extensive LP model of the entire arterial network(48 arteries) was developed including the effect of vessel diameter tapering and the parameterization of resistance, conductor and inductor variables. A computer aided-design(CAD) algorithm was proposed to effciently handle the coupling of two or more 3D models with the LP model, and substantially lessen the coupling processing time. Realistic boundary conditions and Navier–Stokes equations in healthy and stenosed models of carotid artery bifurcation(CAB) were used to investigate the unsteady Newtonian blood flow velocity distribution in the internal carotid artery(ICA). The present simulation results agree well with previous experimental and numerical studies. The outcomes of a pure LP model and those of the coupled 3D healthy model were found to be nearly the same in both cases. Concerning the various analyzed 3D zones, the stenosis growth in the ICA was not found as a crucial factor in determining the absorbing boundary conditions.This paper demonstrates the advantages of coupling local and systemic models to comprehend physiological diseases of the cardiovascular system.     

Fatigue Life Estimation for an NBR Rubber and an Expanded Polyurethane

In this paper, we deal with the mechanical behavior of elastomeric materials subjected to high cyclic loading in cases of high strain. First, a methodology for material parameter identification is used for a constitutive visco-hyperelastic law with discontinuous damage, modeling the Mullins effect. Then a fatigue model characterized by a strain energy density-based criterion is proposed and implemented in the finite element code, Code-Aster [1]. Two kinds of elastomer are considered, an incompressible rubber and an expanded compressible polyurethane. Cyclic tensile tests were performed to identify material fatigue parameters. Finally, a numerical application using a finite element model is presented. This model is a plate perforated by a ?6 mm hole for the first sample and a ?10 mm hole for the second one. The results obtained from the finite element model are compared to experimental results.     

A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables. Part II – Validation and localization analysis

We study the mechanical failure of cemented granular materials (e.g., sandstones) using a constitutive model based on breakage mechanics for grain crushing and damage mechanics for cement fracture. The theoretical aspects of this model are presented in Part I: Tengattini et al. (2014), A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables, Part I – Theory (Journal of the Mechanics and Physics of Solids, 10.1016/j.jmps.2014.05.021). In this Part II we investigate the constitutive and structural responses of cemented granular materials through analyses of Boundary Value Problems (BVPs).The multiple failure mechanisms captured by the proposed model enable the behavior of cemented granular rocks to be well reproduced for a wide range of confining pressures. Furthermore, through comparison of the model predictions and experimental data, the micromechanical basis of the model provides improved understanding of failure mechanisms of cemented granular materials. In particular, we show that grain crushing is the predominant inelastic deformation mechanism under high pressures while cement failure is the relevant mechanism at low pressures. Over an intermediate pressure regime a mixed mode of failure mechanisms is observed. Furthermore, the micromechanical roots of the model allow the effects on localized deformation modes of various initial microstructures to be studied. The results obtained from both the constitutive responses and BVP solutions indicate that the proposed approach and model provide a promising basis for future theoretical studies on cemented granular materials.     

An anisotropic elastoplastic constitutive formulation generalised for orthotropic materials

This paper presents a finite strain constitutive model to predict a complex elastoplastic deformation behaviour that involves very high pressures and shockwaves in orthotropic materials using an anisotropic Hill’s yield criterion by means of the evolving structural tensors. The yield surface of this hyperelastic–plastic constitutive model is aligned uniquely within the principal stress space due to the combination of Mandel stress tensor and a new generalised orthotropic pressure. The formulation is developed in the isoclinic configuration and allows for a unique treatment for elastic and plastic orthotropy. An isotropic hardening is adopted to define the evolution of plastic orthotropy. The important feature of the proposed hyperelastic–plastic constitutive model is the introduction of anisotropic effect in the Mie–Gruneisen equation of state (EOS). The formulation is further combined with Grady spall failure model to predict spall failure in the materials. The proposed constitutive model is implemented as a new material model in the Lawrence Livermore National Laboratory (LLNL)-DYNA3D code of UTHM’s version, named Material Type 92 (Mat92). The combination of the proposed stress tensor decomposition and the Mie–Gruneisen EOS requires some modifications in the code to reflect the formulation of the generalised orthotropic pressure. The validation approach is also presented in this paper for guidance purpose. The \({\varvec{\psi }}\) tensor used to define the alignment of the adopted yield surface is first validated. This is continued with an internal validation related to elastic isotropic, elastic orthotropic and elastic–plastic orthotropic of the proposed formulation before a comparison against range of plate impact test data at 234, 450 and \({\mathrm {895\,ms}}^{\mathrm {-1}}\) impact velocities is performed. A good agreement is obtained in each test.