Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.     

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.     

Finite strain––isotropic hyperelasticity

This paper presents a strain energy density for isotropic hyperelastic materials. The strain energy density is decomposed into a compressible and incompressible component. The incompressible component is the same as the generalized Mooney expression while the compressible component is shown to be a function of the volume invariant J only. The strain energy density proposed is used to investigate problems involving incompressible isotropic materials such as rubber under homogeneous strain, compressible isotropic materials under high hydrostatic pressure and volume change under uniaxial tension. Comparison with experimental data is good. The formulation is also used to derive a strain energy density expression for compressible isotropic neo-Hookean materials. The constitutive relationship for the second Piola–Kirchhoff stress tensor and its physical counterpart, involves the contravariant Almansi strain tensor. The stress stretch relationship comprises of a component associated with volume constrained distortion and a hydrostatic pressure which results in volumetric dilation. An important property of this constitutive relationship is that the hydrostatic pressure component of the stress vector which is associated with volumetric dilation will have no shear component on any surface in any configuration. This same property is not true for a neo-Hookean Green’s strain–second Piola–Kirchhoff stress tensor formulation.     

非饱和颗粒材料的多孔连续体有效压力与有效广义Biot应力

多孔连续体理论框架下的非饱和多孔介质广义有效压力定义和Bishop参数的定量表达式长期以来存在争议,这也影响了对与其直接相关联的非饱和多孔介质广义Biot有效应力的正确预测.基于随时间演变的离散固体颗粒-双联液桥-液膜体系描述的Voronoi胞元模型,利用由模型获得的非饱和颗粒材料表征元中水力-力学介观结构和响应信息,文章定义了低饱和度多孔介质局部材料点的有效内状态变量:非饱和多孔连续体的广义Biot有效应力和有效压力,导出了其表达式.所导出的有效压力公式表明,非饱和多孔连续体的有效压力张量为各向异性,它不仅对非饱和多孔连续体广义Biot有效应力张量的静水应力分量的影响呈各向异性,同时也对其剪切应力分量有影响.文章表明,非饱和多孔连续体中提出的广义Biot理论和双变量理论的基本缺陷在于它们均假定反映非混和两相孔隙流体对固相骨架水力-力学效应的有效压力张量为各向同性.此外,为定义各向同性有效压力张量和作为加权系数而引入的Bishop参数并不包含对非饱和多孔连续体中局部材料点水力-力学响应具有十分重要效应的基质吸力.所导出的非饱和多孔介质广义Biot有效应力和有效压力公式(包括反映有效压力...     

Asymptotic analysis of the equilibrium equation of a fluid-saturated porous medium by the homogenization method

The homogenization of static elasticity equations describing the stress strain state of fluid-saturated porous medium is considered. In this paper, the homogenization method is used to determine the pore pressure transfer tensor, which (a coefficient in the isotropic case) is an important parameter influencing the stress-strain state of fluid-saturated rocks. It shows what a part of the pressure in the fluid is “active” in the formation of macroscopic strains.The pore pressure transfer tensor is calculated for model and real geological specimens. The dependence of this tensor on the porosity, pore shape, and Poisson ratio is investigated. The use of the computational technique for determining the effective properties of rocks shows that it is practically important in the engineering geology.     

A network theory for the thermal conductivity of an amorphous polymeric material

A model to relate the thermal conductivity tensor to the deformation of an amorphous polymeric material above the glass transition temperature is presented. The basis of the model is formed by the transient network theory for polymer melts. With this theory it is possible to calculate the average orientation of the macromolecular segments as a function of the history of the deformation. Combined with an expression which relates the thermal conductivity to the orientation of the molecules, this provides us with the information needed to calculate the heat conduction tensor. Despite the fact that the simplest possible network model is chosen, there is good agreement with the sparse, experimental results.     

Modeling of pressure-strain rate in homogeneous turbulence

Direct numerical simulations were used to study the slow pressure strain rate term in homogeneous, axisymmetric turbulence during its relaxation towards isotropy. A strong Reynolds number dependence was found on the relaxation rate. A model for the rapid pressure strain rate tensor was developed using the classical approach of expansion in the Reynolds stress tensor. The model is tensorially complete and satisfies strong realizability. Rapid distortion analysis has been used to calibrate the model coefficients. The range of applicability of the model has been tested using direct numerical simulations.     

A Formulation of Anisotropic Elastic Damage Using Compact Tensor Formalism

An anisotropic elastic-damage model for initially-isotropic materials is presented. The model is based on a pseudo-logarithmic second-order damage tensor rate. To derive the complete expression of the tangent stiffness entering the rate constitutive law, various tensor operations and derivatives of tensor functions must be developed. Such derivations have been performed in compact form. Some useful tensor derivatives and a table of tensor algebra operations are given in Appendix. This note should interest engineering researchers involved in the development of constitutive models through tensor formalism. This revised version was published online in July 2006 with corrections to the Cover Date.     

Micromechanical modeling of shock-induced chemical reactions in heterogeneous multi-material powder mixtures

Shock waves create a unique environment of high pressure, temperature, strain rate, and gradients thereof. Chemical reactions that occur in this regime can lead to the synthesis of new materials that are not possible under conventional conditions. Shock-induced chemical reactions (also known as shock synthesis) is difficult to study experimentally due to the small time and space scales over which it occurs. To aid in our understanding, numerical simulations of shock-induced chemical reactions (SICR) are performed on the mesoscopic level. The model reaction, Nb+2Si⇌NbSi₂, is chosen for this research because it has been studied extensively. Comparisons of the calculated results with the available experimental data are presented to validate the modeling process, and additional calculations are used to determine the reaction threshold.     

Study of Pressure Sensitive Plastic Flow Behaviour of Gasket Materials

The mechanical behaviour of the materials used as compressible gasket in the ultra high pressure apparatus is investigated. Materials such as pyrophyllite and talc, showing a pressure sensitive plastic flow behaviour were considered and a testing configuration was set up for studying the dependence of their plastic response on the hydrostatic component of the stress tensor, according to the Drucker-Prager criterion. A Finite Element modelling of the test was performed to evaluate the specimen response and the local stress condition, during loading. The Finite Element results were validated by comparison with those of a specific experimental characterisation. A parametric analysis was then carried out, by varying the materials constitutive behaviour, in order to build up a data base of representative curves. In this way an algorithm was developed, with the aim of determining the material constitutive behaviours by the analysis of the experimental data. The proposed procedure was then used to study the mechanical response of different gasket materials.     

Delamination in sandwich panels due to local water slamming loads

We study delamination in a sandwich panel due to transient finite plane strain elastic deformations caused by local water slamming loads and use the boundary element method to analyze motion of water and the finite element method to determine deformations of the panel. The cohesive zone model is used to study delamination initiation and propagation. The fluid is assumed to be incompressible and inviscid, and undergo irrotational motion. A layer-wise third order shear and normal deformable plate/shell theory is employed to simulate deformations of the panel by considering all geometric nonlinearities (i.e., all non-linear terms in strain–displacement relations) and taking the panel material to be St. Venant–Kirchhoff (i.e., the second Piola–Kirchhoff stress tensor is a linear function of the Green–St. Venant strain tensor). The Rayleigh damping is introduced to account for structural damping that reduces oscillations in the pressure acting on the panel/water interface. Results have been computed for water entry of (i) straight and circular sandwich panels made of Hookean materials with and without consideration of delamination failure, and (ii) flat sandwich panels made of the St. Venant–Kirchhoff materials. The face sheets and the core of sandwich panels are made, respectively, of fiber reinforced composites and soft materials. It is found that for the same entry speed (i) the peak pressure for a curved panel is less than that for a straight panel, (ii) the consideration of geometric nonlinearities significantly increases the peak hydrodynamic pressure, (iii) delamination occurs in mode-II, and (iv) the delamination reduces the hydroelastic pressure acting on the panel surface and hence alters deformations of the panel.     

多晶金属材料的三维组集式弹塑性本构模型

本文从分析多晶金属材料的细观组织在弹塑性变形中的贮能和耗能机制入手,提出一个三维组集式弹塑性本构模型。该模型将材料单元抽象成沿三维空间各方向均匀分布组件的集合体,方向组件反映材料的细观性态并在宏观上协调变形,所有方向组件的内力总效应构成宏观应力。文中导出了显式的弹塑性本构关系,并与Budiansky的复杂加载试验结果及其它塑性模型进行了对比。     

VISCOELASTIC MODELLING OF ENTRANCE FLOW USING MULTIMODE LEONOV MODEL

A simulation of planar 2D flow of a viscoelastic fluid employing the Leonov constitutive equation has been presented. Triangular finite elements with lower-order interpolations have been employed for velocity and pressure as well as the extra stress tensor arising from the constitutive equation. A generalized Lesaint–Raviart method has been used for an upwind discretization of the material derivative of the extra stress tensor in the constitutive equation. The upwind scheme has been further strengthened in our code by also introducing a non-consistent streamline upwind Petrov–Galerkin method to modify the weighting function of the material derivative term in the variational form of the constitutive equation. A variational equation for configurational incompressibility of the Leonov model has also been satisfied explicitly. The corresponding software has been used to simulate planar 2D entrance flow for a 4:1 abrupt contraction up to a Deborah number of 670 (Weissenberg number of 6·71) for a rubber compound using a three-mode Leonov model. The predicted entrance loss is found to be in good agreement with experimental results from the literature. Corresponding comparisons for a commercial-grade polystyrene, however, indicate that the predicted entrance loss is low by a factor of about four, indicating a need for further investigation. © 1997 by John Wiley & Sons, Ltd.     

A stable unstructured finite volume method for parallel large-scale viscoelastic fluid flow calculations

A new stable unstructured finite volume method is presented for parallel large-scale simulation of viscoelastic fluid flows. The numerical method is based on the side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face, while the pressure term and the extra stress tensor are defined at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance the pressure–velocity–stress coupling. The log-conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix–logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] has been implemented in order improve the limiting Weissenberg numbers in the proposed finite volume method. The time stepping algorithm used decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. The resulting algebraic linear systems are solved using the FGMRES(m) Krylov iterative method with the restricted additive Schwarz preconditioner for the extra stress tensor and the geometric non-nested multilevel preconditioner for the Stokes system. The implementation of the preconditioned iterative solvers is based on the PETSc library for improving the efficiency of the parallel code. The present numerical algorithm is validated for the Kovasznay flow, the flow of an Oldroyd-B fluid past a confined circular cylinder in a channel and the three-dimensional flow of an Oldroyd-B fluid around a rigid sphere falling in a cylindrical tube. Parallel large-scale calculations are presented up to 523,094 quadrilateral elements in two-dimension and 1,190,376 hexahedral elements in three-dimension.     

Hyperviscosity for unstructured ALE meshes

An artificial viscosity, originally designed for Eulerian schemes, is adapted for use in arbitrary Lagrangian–Eulerian simulations. Changes to the Eulerian model (dubbed ‘hyperviscosity’) are discussed, which enable it to work within a Lagrangian framework. New features include a velocity-weighted grid scale and a generalised filtering procedure, applicable to either structured or unstructured grids. The model employs an artificial shear viscosity for treating small-scale vorticity and an artificial bulk viscosity for shock capturing. The model is based on the Navier–Stokes form of the viscous stress tensor, including the diagonal rate-of-expansion tensor. A second-order version of the model is presented, in which Laplacian operators act on the velocity divergence and the grid-weighted strain-rate magnitude to ensure that the velocity field remains smooth at the grid scale. Unlike sound-speed-based artificial viscosities, the hyperviscosity model is compatible with the low Mach number limit. The new model outperforms a commonly used Lagrangian artificial viscosity on a variety of test problems.     

Mechanical characterization and modeling of polymeric materials for high-pressure sealing

The choice of a proper material is probably the most critical factor in the design of seals able to withstand extremely high pressure, and knowledge of material mechanical properties is essential for the finite-element model (FEM) simulations needed to understand and optimize seal behavior. The aim of this work is the mechanical characterization of polymeric materials for ultrahigh-pressure sealing applications (600 MPa). After a short presentation of seal design and materials commonly used, the testing of four thermoplastic materials is described: PA6, H-TPU and UHMWPE reinforced with glass or ceramic microspheres to enhance wear resistance. Uniaxial tension and compression, shear and planar tension test were performed as well as a stress relaxation test to gain information about viscoelastic effects. Experimental data are then discussed and elasto-plastic and hyperelastic constitutive models for polymeric materials reviewed, focusing on the application of these models at high pressure. The Young's modulus and yield strength are very sensitive to hydrostatic pressure for polymeric materials and a proposal for the implementation in the FEM of this effect is illustrated.     

A new second‐moment closure approach for turbulent swirling confined flows

An improved anisotropic model for the dissipation rate—ε—of the turbulent kinetic energy (k), to be used together with a non‐linear pressure‐strain correlations model, is proposed. Experimental data from the open literature for two confined turbulent swirling flows are used to assess the performance of the proposed model in comparison to the standard ε transport equation and to a linear approach to model the pressure‐strain term that appears in the exact equations for the Reynolds‐stress tensor. For the less strongly swirling flow the predictions show much more sensitivity to the εtransport equation than to the pressure‐strain model. In opposition, for the more strongly swirling flow, the results show that the predictions are much sensitive to the pressure‐strain model. Nevertheless, the improved εtransport equation together with the non‐linear pressure strain model yield predictions in good agreement with experiments in both studied cases. Copyright © 2003 John Wiley & Sons, Ltd.     

Determination of the effective elastic–plastic response of metal–ceramic composites

The mechanical response of metal–ceramic composites is analysed through a homogenization model accounting for the mechanical behaviour of the constituent materials. In order to achieve this purpose a nonlinear homogenization method based on the phase field approach has been suitably implemented into a numerical code. A prescribed homogenized strain state is applied to a unit volume element of a metal–ceramic composite with proportional loading in which all components of the strain tensor are proportional to one scalar parameter. The mechanical response of the material has been modeled by considering a von Mises plasticity model for the metal phase and a Drucker–Prager associative elastic–plastic material model for the ceramic phase. A two stages plasticity has been obtained in which inelastic strain develops in the metal phase followed by a fully plastic response. A comparison with a finite element model of the stress–strain response of an axisymmetric unit cell has been carried out with the purpose to validate the homogenization based modeling presented in the paper. Plastic parameters of a Drucker–Prager yield surface for the homogenized composite have been calculated at different materials compositions. Associative Drucker–Prager plasticity has been found to be accurate for high ceramic content.     

Theory of creep deformation with kinematic hardening for materials with different properties in tension and compression

A constitutive model for creep deformation that describes the loading-history-dependent behavior of initially isotropic materials with different properties in tension and compression under stress vector rotations limited by 50–60° is presented within a thermodynamic framework. In the proposed constitutive model a kinematic hardening rule is adopted. This model also introduces an effective equivalent stress in the creep potential that is based on the first and second invariants of the effective stress tensor, and on the joint invariant of the effective stress tensor and eigenvector associated with the maximum principal Cauchy stress. The formulation of the kinematic hardening rule is presented and discussed. All the material parameters in the model have been obtained from a series of proposed basic experiments with constant stresses. These model parameters are then used to predict the creep deformation of the aluminum alloy under multiaxial loading with constant stresses, and under non-proportional uniaxial and non-proportional multiaxial loadings for both isothermal and nonisothermal processes.