Micromorphic continuum. Part II: Finite deformation plasticity coupled with damage

It is demonstrated how a micromorphic plasticity theory may be formulated on the basis of multiplicative decompositions of the macro- and microdeformation gradient tensor, respectively. The theory exhibits non-linear isotropic and non-linear kinematic hardening. The yield function is expressed in terms of Mandel stress and double stress tensors, appropriately defined for micromorphic continua. Flow rules are derived from the postulate of Il’iushin and represent generalized normality conditions. Evolution equations for isotropic and kinematic hardening are introduced as sufficient conditions for the validity of the second law of thermodynamics in every admissible process. Finally, it is sketched how isotropic damage effects may be incorporated in the theory. This is done for the concept of effective stress combined with the hypothesis of strain equivalence.     

Shape memory behaviour: modelling within continuum thermomechanics

A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as the transition range between pseudoelasticity and pseudoplasticity.The material model is based on a free energy function and evolution equations for internal variables. By means of the free energy function, the energy storage during thermomechanical processes is described. Evolution equations for internal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies the Clausius–Duhem inequality.Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for uniaxial processes and for non-proportional loadings as well.     

Micromorphic continuum. Part III: Small deformation plasticity coupled with damage

Properties of the micromorphic theory proposed in Part II are discussed for the case of small deformations. Model responses for beam specimens under bending loading and plates with circular holes under tension loading are calculated by employing the finite element method. The results reported are concerned with the capabilities of the theory to predict size effects.     

Micromorphic continuum Part I: Strain and stress tensors and their associated rates

Micropolar and micromorphic solids are continuum mechanics models, which take into account, in some sense, the microstructure of the considered real material. The characteristic property of such continua is that the state functions depend, besides the classical deformation of the macroscopic material body, also upon the deformation of the microcontinuum modeling the microstructure, and its gradient with respect to the space occupied by the material body. While micropolar plasticity theories, including non-linear isotropic and non-linear kinematic hardening, have been formulated, even for non-linear geometry, few works are known yet about the formulation of (finite deformation) micromorphic plasticity. It is the aim of the three papers (Parts I, II, and III) to demonstrate how micromorphic plasticity theories may be formulated in a thermodynamically consistent way.In the present article we start by outlining the framework of the theory. Especially, we confine attention to the theory of Mindlin on continua with microstructure, which is formulated for small deformations. After precising some conceptual aspects concerning the notion of microcontinuum, we work out a finite deformation version of theory, suitable for our aims. It is examined that resulting basic field equations are the same as in the non-linear theory of Eringen, which deals with a different definition of the microcontinuum. Furthermore, geometrical interpretations of strain and curvature tensors are elaborated. This allows to find out associated rates in a natural manner. Dual stress and double stress tensors, as well as associated rates, are then defined on the basis of the stress powers. This way, it is possible to relate strain tensors (respectively, micromorphic curvature tensors) and stress tensors (respectively, double stress tensors), as well as associated rates, independently of the particular constitutive properties.     

On the modelling of anisotropic elastic and inelastic material behaviour at large deformation

The purpose of this work is the formulation and discussion of an approach to the modelling of anisotropic elastic and inelastic material behaviour at large deformation. This is done in the framework of a thermodynamic, internal-variable-based formulation for such a behaviour. In particular, the formulation pursued here is based on a model for plastic or inelastic deformation as a transformation of local reference configuration for each material element. This represents a slight generalization of its modelling as an elastic material isomorphism pursued in earlier work, allowing one in particular to incorporate the effects of isotropic continuum damage directly into the formulation. As for the remaining deformation- and stress-like internal variables of the formulation, these are modelled in a fashion formally analogous to so-called structure tensors. On this basis, it is shown in particular that, while neither the Mandel nor back stress is generally so, the stress measure thermodynamically conjugate to the plastic “velocity gradient”, containing the difference of these two stress measures, is always symmetric with respect to the Euclidean metric, i.e., even in the case of classical or induced anisotropic elastic or inelastic material behaviour. Further, in the context of the assumption that the intermediate configuration is materially uniform, it is shown that the stress measure thermodynamically conjugate to the plastic velocity gradient is directly related to the Eshelby stress. Finally, the approach is applied to the formulation of metal plasticity with isotropic kinematic hardening.     

Impact of material processing and deformation on cell morphology and mechanical behavior of polyurethane and nickel foams

The cell morphology and mechanical behavior of open-cell polyurethane and nickel foams are investigated by means of combined 3D X-ray micro-tomography and large scale finite element simulations. Our quantitative 3D image analysis and finite element simulations demonstrate that the strongly anisotropic tensile behavior of nickel foams is due to the cell anisotropy induced by the deformation of PU precursor during the electroplating and heat treatment stages of nickel foam processing. In situ tensile tests on PU foams reveal that the initial main elongation axis of the cells evolves from the foam sheet normal direction to the rolling direction of the coils. Finite element simulations of the hyperelastic behavior of PU foams based on real cell morphology confirm the observation that cell struts do not experience significant elongation after 0.15 tensile straining, thus pointing out alternative deformation mechanisms like complex strut junctions deformation. The plastic behavior and the anisotropy of nickel foams are then satisfactorily retrieved from finite element simulations on a volume element containing eight cells with a detailed mesh of all the hollow struts and junctions. The experimental and computational strategy is considered as a first step toward optimization of process parameters to tailor anisotropy of cell shape and mechanical behavior for applications in batteries or Diesel particulate filtering.     

Nonlinear deformation of elastomeric foams

The nonlinear deformation of a porous foam-type elastomeric material is studied, both theoretically and experimentally. The elastomer is modeled by the neo-Hookean material. The one-dimensional compressive behavior of the foam is analysed by using certain kinematic assumptions. The stress required to compress the foam is predicted by the model in terms of the porosity of the foam and the single constant in the neo-Hookean stress-strain form. A particular silicone foam is used as a test of the theory. The neo-Hookean constant is evaluated from a test of the homogeneous elastomer. Hence the behavior of the corresponding foam is predicted theoretically and compared with experimental results. The general results are applicable to closed-cells foams of intermediate density.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CON-TINUUM THEORIES (Ⅱ)—MICROMORPHIC CONTINUUM THEORY AND COUPLE STRESS THEORY

The purpose is to reestablish the balance laws of momentum, angular momentum and energy and to derive the corresponding local and nonlocal balance equations for micromorphic continuum mechanics and couple stress theory. The desired results for micromorphic continuum mechanics and couple stress theory are naturally obtained via direct transitions and reductions from the coupled conservation law of energy for micropolar continuum theory, respectively. The basic balance laws and equations for micromorphic continuum mechanics and couple stress theory are constituted by combining these results derived here and the traditional conservation laws and equations of mass and microinertia and the entropy inequality. The incomplete degrees of the former related continuum theories are clarified. Finally, some special cases are conveniently derived. Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931≈)     

Enhanced fracture energy and effects of temperature at spalling in ceramics: continuum damage modelling

Summary Enhanced fracture energy losses at spalling and the temperature dependence of the spalling strength of alumina ceramic bars are analysed on the basis of the experimental tests conducted both in room temperature and within the temperature range up to 1500°C at strain rates of some 500 s⁻¹. The experimental method and the measurements are first shortly outlined. The mechanical response of ceramic bars is modelled then as a heterogeneous distribution of brittle-elastic mesoelements undergoing continuum damage at the known strain history, corresponding to that registered in the experiments. The mesoelements are characterised by the values of initial damage randomly fluctuating within a given band-width superposed on a deterministic distribution, which corresponds to the fabrication conditions of the ceramic bars. The model has been tested in the evaluation of room-temperature experiments, its parameters: the average value of the initial damage, Young's modulus of the undamaged material and the energy absorption capacity in continuum damage are taken from the calibration fitting the experimental data. The registered energy losses at spalling, which exceed the static values of fracture energy by almost an order of magnitude, can be explained thus by the enhancement of the dissipation due to bulk damage, which is computed on the basis of the above parameters. The temperature change of the Young's modulus of the matrix material is taken as corresponding to the measured change of the uniaxial wave velocity in the bar, and corrected by the temperature change of the mass density. The analysis of the model shows that the drop in the spalling strength of the specimens with the increase of the temperature is phenomenologically related to the falling energy absorption capacity within the continuum damage mechanism. An explanation of this phenomenon is attempted, based on the grain-size-related mechanisms of the microfracture from pre-existing intergranular flaws distributed over the bulk of ceramics. Received 7 May 1999; accepted for publication 14 June 1999     

On the local analysis of continuum deformation

Summary The role of the cartesian representation of the deformation gradient in the non-linear theory of continuum deformation is studied. Indeed, explicit formulae for the local analysis of deformation are obtained in terms of such a representation. Further, some connections with the multiplicative strain and rotation measures are examined.

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Sommario Si studia il ruolo della rappresentazione cartesiana del gradiente di deformazione nella teoria non-lineare della deformazione di un corpo continua tridimensionale. In termini di tale rappresentazione, si ottengono formule esplicite per le variazioni locali di lunghezza, angolo, area e volume. Si esaminano inoltre alcune connessioni con le misure multiplicative di deformazione pura e di rotazione.

     

Discrete and continuum modelling of delamination processes

Summary The chaotic distributions of delaminated surface elements in laminates are described by a macrodelamination density function. Using this concept a phenomenological discrete mathematical model of the interlaminar debonding processes is proposed. For multilayered periodic composites the discrete model is approximated by a certain continuum model of delamination processes.

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Diskrete und kontinuierliche modelle der schichtentrennungsvorgänge

Übersicht Eine chaotische Verteilung von gelockerten Bereichen von den Trennflächen von Schichtwerkstoffen wird durch eine Dichtefunktion der Schichtentrennung modelliert. Diese Darstellung ermöglicht die Formulierung eines phänomenologischen diskreten Modells von Schichtentrennungsvorgängen. Für vielschichtige, periodische Verbundwerkstoffe wird das diskrete Modell durch ein kontinuierliches Modell approximiert.

     

High-velocity deformation properties of polyurethane foams

This paper presents the results of dynamic uniaxial-stress tests performed on polymer-foam material. A water-blown ester polyurethane foam designated as rigid and a castor-oil-base polyurethane foam designated as semirigid were tested in tension and compression at rates of loading from 10^?3 in./in./sec to 10³ in./in./sec at room temperature. A gas-operated medium-strain-rate machine was used for rates of loading from 10^?3 to about 10² in./in./sec. Tests at higher rates were performed on a split Hopkinson-bar device. Highspeed photographic techniques were used to study dynamic fracture.     

Discrete and continuum modelling of excavator bucket filling

Two-dimensional discrete and continuum modelling of excavator bucket filling is presented. The discrete element method (DEM) is used for the discrete modelling and the material-point method (MPM) for continuum modelling. MPM is a so-called particle method or meshless finite element method. Standard finite element methods have difficulty in modelling the entire bucket filling process due to large displacements and distortions of the mesh. The use of a meshless method overcomes this problem. DEM and MPM simulations (plane strain) of bucket filling are compared to two-dimensional experimental results. Cohesionless corn grains were used as material and the simulated force acting on the bucket and flow patterns were compared with experimental results. The corn macro (continuum) and micro (DEM) properties were obtained from shear and oedometer tests. As part of the MPM simulations, both the classic (nonpolar) and the Cosserat (polar) continuums were used. Results show that the nonpolar continuum is the most accurate in predicting the bucket force while the polar and DEM methods predict lower forces. The DEM model does not accurately predict the material flow during filling, while the polar and nonpolar methods are more accurate. Different flow zones develop during filling and it is shown that DEM, the polar and the nonpolar methods can accurately predict the position and orientation of these different flow zones.     

Size effects in the constrained deformation of metallic foams

The constrained deformation of an aluminium alloy foam sandwiched between steel substrates has been investigated. The sandwich plates are subjected to through-thickness shear and normal loading, and it is found that the face sheets constrain the foam against plastic deformation and result in a size effect: the yield strength increases with diminishing thickness of foam layer. The strain distribution across the foam core has been measured by a visual strain mapping technique, and a boundary layer of reduced straining was observed adjacent to the face sheets. The deformation response of the aluminium foam layer was modelled by the elastic-plastic finite element analysis of regular and irregular two dimensional honeycombs, bonded to rigid face sheets; in the simulations, the rotation of the boundary nodes of the cell-wall beam elements was set to zero to simulate full constraint from the rigid face sheets. It is found that the regular honeycomb under-estimates the size effect whereas the irregular honeycomb provides a faithful representation of both the observed size effect and the observed strain profile through the foam layer. Additionally, a compressible version of the Fleck-Hutchinson strain gradient theory was used to predict the size effect; by identifying the cell edge length as the relevant microstructural length scale the strain gradient model is able to reproduce the observed strain profiles across the layer and the thickness dependence of strength.     

Micromechanical modelling of the effect of plastic deformation on the mechanical behaviour in pseudoelastic shape memory alloys

Except for the recoverable strain induced by phase transformation, NiTi alloys are very ductile even in the martensite phase. The purpose of the present paper is to study the influence of permanent deformation, which results from plastic deformation of martensite, on the mechanical behaviour of pseudoelastic NiTi alloys. Based on phenomenological theory of martensitic transformation and crystal plasticity, a new three dimensional micromechanical model is proposed by coupling both the slip and twinning deformation mechanisms. The present model is implemented as User MATerial subroutine (UMAT) into ABAQUS/Standard to study the influences of plastic deformation on the stress and strain fields, and on the evolution of martensite transformation. Results show that with the increasing of plastic deformation the residual strain increases and the phase transformation stress–strain curves from the martensite to austenite become steeper and less obvious. Both characteristics, stabilisation of martensite and impedance of the reverse transformation, due to plastic deformation are captured.     

Eigenanalysis and continuum modelling of pre-twisted repetitive beam-like structures

A repetitive pin-jointed, pre-twisted structure is analysed using a state variable transfer matrix technique. Within a global coordinate system the transfer matrix is periodic, but introduction of a local coordinate system rotating with nodal cross-sections results in an autonomous transfer matrix for this Floquet system. Eigenanalysis reveals four real unity eigenvalues, indicating tension–torsion coupling, and equivalent continuum properties such as Poisson’s ratio, cross-sectional area, torsion constant and the tension–torsion coupling coefficient are determined. A variety of real and complex near diagonal Jordan decompositions are possible for the multiple (eight) complex unity eigenvalues and these are discussed in some detail. Analysis of the associated principal vectors shows that a bending moment produces curvature in the plane of the moment, together with shear deformation in the perpendicular plane, but no bending–bending coupling; the choice of a structure having an equilateral triangular cross-section is thought responsible for this unexpected behaviour, as the equivalent continuum second moments of area are equal about all cross-sectional axes. In addition, an asymmetric stiffness matrix is obtained for bending moment and shearing force coupling, and possible causes are discussed.     

Multiscale modelling of damage and failure in two-dimensional metallic foams

The fracture strength of metal foams depends sensitively on the properties of the constituent material as well as the cellular architecture. A change in microscopic properties carries over to the macroscopic scale through an alteration of the mesoscopic damage and fracture mechanisms. In this paper we study these dependencies using a modelling framework that takes all these ingredients into account. We have developed a micromechanical model based on a discrete Voronoi representation of cellular metals that incorporates power-law strain hardening and damage development of the cell wall material. The influence of the relative density and material strain hardening on the cell wall damage behavior and overall fracture response is analyzed in detail. The effect of the cellular architecture is studied by varying the cell shape anisotropy and structural randomness. We also simulate the effect of post-processing heat treatments on the solid material plastic and fracture properties and how this affects the overall fracture profile and damage development. Finally, all material and architectural effects are summarized in a strength versus ductility graph, identifying trends for improved design of metallic foams.     

Solution-precipitation creep—continuum mechanical formulation and micromechanical modelling

Solution-precipitation creep is considered to be one of the major deformation mechanisms of polycrystalline materials containing a fluid phase. Geological evidence suggests that it is particularly important for processes occurring in the earth crust at very low deviatoric stress. We develop a continuum mechanical model based on the assumption that the energy dissipated during deformation depends solely on the normal velocity of the grain boundary movement due to precipitation or solution of material and the velocity of material transport within the grain interfaces. This approach allows us to identify the driving force responsible for solution-precipitation creep in an unambiguous way. Moreover, a micromechanical approach based on an averaging assumption of a Voigt/Sachs type is given.