Two Theories of Complex Bodies, one Lagrangean, the other Kinetic, Have a Common Ground

I have found in previous works that most special models proposed to represent bodies with some type of microstructure can be classified easily under the general umbrella of a theory where each element of the continuum is thought of as a Lagrangian system. To study phenomena in ‘kinetic’ continua I proposed an apparently different approach; the outcome is again a set of evolution equations. They mimic equations familiar in continua with affine microstructure: a Cauchy’s equation and an equation of balance of tensor moment of momentum, with the addition, however, of an equation of balance for a ‘Reynolds’ tensor’, an equation which, in a sense, shifts the boundary between kinetic and thermal properties of matter. I will show that there is no contrast between the two approaches. The latter one is based on an adequate and appropriately justified expression of the kinetic energy of the continuum, comprising the trace of the quoted Reynolds’ tensor and thus importing into the mechanical energy a term usually accounted by additional heat.     

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—Theory

This work presents an analytical framework for determining the overall constitutive response of elastomers that are reinforced by rigid or compliant fibers, and are subjected to finite deformations. The framework accounts for the evolution of the underlying microstructure, including particle rotation, which results from the finite changes in geometry that are induced by the applied loading. In turn, the evolution of the microstructure can have a significant geometric softening (or hardening) effect on the overall response, leading to the possible development of macroscopic instabilities through loss of strong ellipticity of the homogenized incremental moduli. The theory is based on a recently developed “second-order” homogenization method, which makes use of information on both the first and second moments of the fields in a suitably chosen “linear comparison composite,” and generates fairly explicit estimates—linearizing properly—for the large-deformation effective response of the reinforced elastomers. More specific applications of the results developed in this paper will be presented in Part II.     

Sur le remodelage des tissus osseux anisotropesOn the remodelling of anisotropic bone tissue

Growth (change of relaxed lengths) and remodelling (change of mechanical properties) are both involved in the morphogenesis of biological tissues. To model them is of paramount import for progressing both in scientific understanding and health technologies. We model bone tissue as a microstructured continuum, whose mechanical properties at the macroscopic scale are described by a linear, anisotropic elastic response that evolves in time. Our kinematics is rich enough to allow for the microstructural evolution, as well as for the interplay between stress, growth and remodelling. This is a unified approach to the mechanics of growth and remodelling, in which all balance laws derive from one virtual-power principle. As a first application, we study the problem of stiffness remodelling due to planar rotation of the microstructure, excluding bulk growth and all physiological response to mechanical stimuli (passive remodelling). To cite this article: A. DiCarlo et al., C. R. Mecanique 334 (2006).     

Micromechanics, macromechanics and constitutive modeling of the elasto-viscoplastic deformation of rubber-toughened glassy polymers

Under certain conditions, such as sufficiently low temperatures, high loading rates and/or highly triaxial stress states, glassy polymers display an unfavorable characteristic—brittleness. A technique used for reducing the brittleness (increasing the fracture toughness) of these materials is rubber toughening. While there is significant qualitative understanding of the mechanical behavior of rubber-toughened polymers, quantitative modeling tools for the large-strain deformation of rubber-toughened glassy polymers are largely lacking.In this paper, we develop a suite of numerical tools to investigate the mechanical behavior of rubber-toughened glassy polymers, with emphasis on rubber-toughened polycarbonate. The rubber particles are modeled as voids in view of their deformation-induced cavitation early during deformation. A three-dimensional micromechanical model of the heterogeneous microstructure is developed to study the effects of initial rubber particle (void) volume fraction on the underlying elasto-viscoplastic deformation mechanisms in the material, and how these mechanisms influence the macroscopic response of the material. A continuum-level constitutive model is developed for the large-strain elasto-viscoplastic deformation of porous glassy polymers, and it is calibrated against micromechanical modeling results for porous polycarbonate. The constitutive model can be used to study various boundary value problems involving rubber-toughened (porous) glassy polymers. As an example, the case of an axisymmetric notched bar is simulated for the case of polycarbonate with varying levels of initial porosity. The quality of the constitutive model calibration is assessed using a multi-scale modeling approach.     

Stress-driven morphological instability and catastrophic failure of microdevices

In microdevices, the competition between surface energy and elastic energy could lead at the phenomenon known as stress-driven morphological instability (MI), causing an increase of surface roughness with time. Several different mass transport mechanisms can trigger such a morphological alteration and operate simultaneously: surface and bulk diffusion, evaporation and condensation, chemical reactions. Unstable solids could eventually evolve towards crack-like surfaces thus altering mechanical, electrical and optical properties of the devices or even leading to catastrophic failures by supercritical crack propagation. In this work, a more general kinetic law is employed to estimate the onset of MI, considering the effect of the stress field on the atomic mobility. A more intuitive and straightforward approach is used to determine the stability conditions, where the rate of atomic mass motion is introduced as a stability parameter. The critical loads and wavelengths for the onset of MI, determined as a function of material parameters α and β, are compared with the limiting conditions for the supercritical crack propagation (SC) of a crack-like surface in order to asses if and under which situations catastrophic failures by SC can be observed. Two practical cases are investigated: fixed wavelength (Case I) and arbitrary rough surface with a fixed remote load (Case II). In Case I, absolute and relative threshold loads are found below which MI could never occur and a transitional wavelength over which MI would always lead to SC is introduced. In Case II, it is shown that dominant perturbation for MI would always lead to SC given enough time for the surface to evolve towards a crack-like profile. The influence of the material properties α and β on the critical parameters is also addressed.     

Interfacial energy and dissipation in martensitic phase transformations. Part II: Size effects in pseudoelasticity

This paper is a continuation of the Part I (H. Petryk, S. Stupkiewicz, Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory. J. Mech. Phys. Solids, 2010, doi:10.1016/j.jmps.2009.11.003). A fully three-dimensional model of an evolving martensitic microstructure is examined, taking into account size effects due to the interfacial energy and also dissipation related to annihilation of interfaces. The elastic micro-strain energy at microstructured interfaces is determined with the help of finite element computations and is approximated analytically. Three interface levels are examined: of grain boundaries attained by parallel martensite plates, of interfaces between austenite and twinned martensite, and of twin interfaces within the martensite phase. Minimization of the incremental energy supply, being the sum of the increments in the free energy and dissipation of the bulk and interfacial type at all levels, is used as the evolution rule, based on the theory presented in Part I. An example of the formation and evolution of a rank-three laminated microstructure of finite characteristic dimensions in a pseudoelastic CuAlNi shape memory alloy is examined quantitatively.     

A multi-scale constitutive model for the sintering of an air-plasma-sprayed thermal barrier coating, and its response under hot isostatic pressing

A micromechanical model is developed for the sintering of an air-plasma-sprayed, thermal barrier coating, and is used to make predictions of microstructure evolution under free sintering and under hot isostatic pressing. It is assumed that the splats of the coating are separated by penny-shaped cracks; the faces of these cracks progressively sinter together at contacting asperities, initially by the mechanism of plastic yield and subsequently by interfacial diffusion. Diffusion is driven by the reduction in interfacial energy at the developing contacts of the cracks and also by the local contact stress at asperities. The contact stress arises from the remote applied stress and from mechanical wedging of the rough crack surfaces. Sintering of the cracks leads to an elevation in both the macroscopic Young's modulus and thermal conductivity of the coating, and thereby leads to a degradation in thermal performance and durability. An assessment is made of the relative roles of surface energy, applied stress and crack face roughness upon the sintering response and upon the evolution of the pertinent mechanical and physical properties. The evolution in microstructure is predicted for free sintering and for hot isostatic pressing in order to provide guidance for experimental validation of the micromechanical model.     

Contact mechanics of microscopically rough surfaces with graded elasticity

The well-known Greenwood and Williamson contact theory for microscopically homogeneous rough surfaces is generalized by considering functionally graded elastic rough surfaces. In particular, two distinct cases giving rise to a non-constant Young’s modulus with depth are considered: (I) an initially plane layered (or graded) solid which is non-uniformly eroded, so that the final product is a rough surface with asperities having an elastic modulus depending on the height; (II) an initially homogeneous rough surface which receives a surface treatment or a chemical degradation which modify the elastic properties of the asperities as a function of the depth from the exposed surface. These Functionally Graded Surfaces (FGS) can be observed both in biological systems and in mechanical components. The effects of graded elasticity on the relationship between real contact area versus applied load, and on the plasticity index are quantified and illustrated with numerical examples. It will be shown that the contact response may differ up to one order of magnitude with respect to that of a homogeneous surface. Comparison between Case I and Case II also shows that, for special surface properties, the two types of grading can provide the same mechanical response.     

Force-Free States, Relative Strain, and Soft Elasticity in Nematic Elastomers

The remarkable ability of nematic elastomers to exhibit large deformations under small applied forces is known as soft elasticity. The recently proposed neo-classical free-energy density for nematic elastomers, derived by molecular-statistical arguments, has been used to model soft elasticity. In particular, the neo-classical free-energy density allows for a continuous spectrum of equilibria, which implies that deformations may occur in the complete absence of force and energy cost. Here we study the notion of force-free states in the context of a continuum theory of nematic elastomers that allows for isotropy, uniaxiality, and biaxiality of the polymer microstructure. Within that theory, the neo-classical free-energy density is an example of a free-energy density function that depends on the deformation gradient only through a nonlinear strain measure associated with the deformation of the polymer microstructure relative to the macroscopic continuum. Among the force-free states for a nematic elastomer described by the neo-classical free energy density, there is, in particular, a continuous spectrum of states parameterized by a pair of tensors that allows for soft deformations. In these force-free states the polymer microstructure is material in the sense that it stretches and rotates with the macroscopic continuum. Limitations of and possible improvements upon the neo-classical model are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

General Properties of the Nernst-Planck-Poisson-Boltzmann System Describing Electrocapillary Effects in Porous Media

The Nernst-Planck-Poisson-Boltzmann system describes the evolution of ionic concentrations and electrocapillarity effects in porous media. The aim of this paper is a theoretical study of various drift-diffusion modellings. The well-posedness of the systems is proved and some qualitative properties of the global solution are shown to be satisfied (energy law, entropy law in the weak sense of Lyapunov functions, stationary states, Maxwellian distribution, influence of an external electric field). Moreover, some explicit solutions are established in the one-dimensional case.     

Distribution of nonliner relaxation (DNLR) approach of the annealing effects in semicristalline polymers: structure–property relation for high-density polyethylene (HDPE)

This work is devoted to the numerical and experimental study of annealing effects on microstructure and mechanical properties of the high-density polyethylene (HDPE). Uniaxiale tension tests are conducted at 25 °C in order to characterize the mechanical behavior of HDPE. The influence of the annealing treatment on the material microstructure is examined by the Fourier transform infrared spectroscopy, and microstructures are characterized using differential scanning calorimetry. The distribution of nonlinear relaxation approach is adopted to describe the mechanical response of virgin and annealed HDPE. Annealing effects are incorporated into the constitutive model by introducing the microstructure (crystallinity degree) evolution on the macroscopic response of the material. The numerical predictions of the model are in good agreement with experimental results for the different states of the material.     

Homogenization-based constitutive models for magnetorheological elastomers at finite strain

This paper proposes a new homogenization framework for magnetoelastic composites accounting for the effect of magnetic dipole interactions, as well as finite strains. In addition, it provides an application for magnetorheological elastomers via a “partial decoupling” approximation splitting the magnetoelastic energy into a purely mechanical component, together with a magnetostatic component evaluated in the deformed configuration of the composite, as estimated by means of the purely mechanical solution of the problem. It is argued that the resulting constitutive model for the material, which can account for the initial volume fraction, average shape, orientation and distribution of the magnetically anisotropic, non-spherical particles, should be quite accurate at least for perfectly aligned magnetic and mechanical loadings. The theory predicts the existence of certain “extra” stresses—arising in the composite beyond the purely mechanical and magnetic (Maxwell) stresses—which can be directly linked to deformation-induced changes in the microstructure. For the special case of isotropic distributions of magnetically isotropic, spherical particles, the extra stresses are due to changes in the particle two-point distribution function with the deformation, and are of order volume fraction squared, while the corresponding extra stresses for the case of aligned, ellipsoidal particles can be of order volume fraction, when changes are induced by the deformation in the orientation of the particles. The theory is capable of handling the strongly nonlinear effects associated with finite strains and magnetic saturation of the particles at sufficiently high deformations and magnetic fields, respectively.     

水泥水化的微结构与性能演化研究进展

水泥是一类重要的工程结构材料,其力学性能依赖于水泥的水化过程．综述了水泥水化的微结构和性能演化的研究进展;评述了研究水泥水化过程的实验方法和理论方法;详细介绍和评述了水泥水化的微结构仿真模型、力学分析模型以及建立微结构．性能关系的基本方法．分析和展望了未来水泥水化的主要问题和研究方向．     

A computational study of vortex rings interacting with a constant-temperature heated wall

This study is motivated by understanding the connections between the vortical structures in impinging jets and the wall heat transfer. Of particular interest are: (1) examining how the stage of evolution of vortex pairing in the jet might influence the wall heat transfer, and (2) establishing correlations between the vortex characteristics and the Nusselt number (Nu) distribution. To this end, CFD simulations are conducted of three simplified model problems involving the interaction of isolated axisymmetric vortex rings with a flat, constant-temperature, heated wall. The cases represent three scenarios of vortex-wall interaction: before (Case I), during (Case II) and after (Case III) pairing. The results show that when two vortices concurrently interact with the wall and undergo pairing (Case II), a significant instantaneous enhancement in Nu is attained in comparison to that associated with a single vortex interacting with the wall (Cases I and III). However, Case II also leads to the largest subsequent decay in Nu enhancement due to the formation of a particularly strong secondary vortex. In all cases, a deterioration in Nu, relative to unsteady diffusion, is observed simultaneously with the enhancement. Notwithstanding this deterioration, the net effect of vortex-wall interaction on the heat transfer remains positive with Case II producing the highest heat transfer rate. An analysis is conducted to establish the connection between the instantaneous maximum and minimum Nu, the circulation and the radial and the wall-normal location of the core-centers of the vortices, the thermal boundary layer thickness, the boundary layer separation location and the wall shear stress.     

The evolution of laminates in finite crystal plasticity: a variational approach

The analysis and simulation of microstructures in solids has gained crucial importance, virtue of the influence of all microstructural characteristics on a material’s macroscopic, mechanical behavior. In particular, the arrangement of dislocations and other lattice defects to particular structures and patterns on the microscale as well as the resultant inhomogeneous distribution of localized strain results in a highly altered stress–strain response. Energetic models predicting the mechanical properties are commonly based on thermodynamic variational principles. Modeling the material response in finite strain crystal plasticity very often results in a non-convex variational problem so that the minimizing deformation fields are no longer continuous but exhibit small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. This results in fine structures that can be interpreted as the observed microstructures. In this paper, we first review the underlying variational principles for inelastic materials. We then propose an analytical partial relaxation of a Neo-Hookean energy formulation, based on the assumption of a first-order laminate microstructure, thus approximating the relaxed energy by an upper bound of the rank-one-convex hull. The semi-relaxed energy can be employed to investigate elasto-plastic models with a single as well as multiple active slip systems. Based on the minimization of a Lagrange functional (consisting of the sum of energy rate and dissipation potential), we outline an incremental strategy to model the time-continuous evolution of the laminate microstructure, then present a numerical scheme by means of which the microstructure development can be computed, and show numerical results for particular examples in single- and double-slip plasticity. We discuss the influence of hardening and of slip system orientations in the present model. In contrast to many approaches before, we do not minimize a condensed energy functional. Instead, we incrementally solve the evolution equations at each time step and account for the actual microstructural changes during each time step. Results indicate a reduction in energy when compared to those theories based on a condensed energy functional.     

Dynamic crushing of aluminum foams: Part II – Analysis

Part II of this study uses micromechanically accurate foam models to simulate and study the dynamic crushing of open-cell foams. The model starts as random soap froth generated using the Surface Evolver software to mimic the microstructure of the foams tested. The linear edges of the cellular microstructure are “dressed” with appropriate distributions of solid to match those of ligaments in the actual foams and their relative density. The ligaments are modeled as shear-deformable beams with variable cross sections discretized with beam elements in LS-DYNA, while the Al-alloy is modeled as a finitely deforming elastic–plastic material. The numerical contact algorithm of the code is used to model ligament contact and limit localized cell crushing. The quasi-static and all dynamic crushing experiments in Part I are simulated numerically. The models are shown to reproduce all aspects of the crushing behavior including the formation and evolution of nearly planar shocks, the force acting at the two ends, the shock front velocity, the strain in the crushed material behind the shock, and the energy absorbed.     

A direct approach to fiber and membrane reinforced bodies. Part II. Membrane reinforced bodies

The paper deals with membrane reinforced bodies with the membrane treated as a two-dimensional surface with concentrated material properties. The bulk response of the matrix is treated separately in two cases: (a) as a coercive nonlinear material with convex stored energy function expressed in the small strain tensor, and (b) as a no-tension material (where the coercivity assumption is not satisfied). The membrane response is assumed to be nonlinear in the surface strain tensor. For the nonlinear bulk response in Case (a), the existence of states of minimum energy is proved. Under suitable growth conditions, the equilibrium states are proved to be exactly states of minimum energy. Then, under appropriate invertibility condition of the stress function, the principle of minimum complementary energy is proved for equilibrium states. For the no-tension material in Case (b), the principle of minimum complementary energy (in the absence of the invertibility assumption) is proved. Also, a theorem is proved stating that the total energy of the system is bounded from below if and only if the loads can be equilibrated by a stress field that is statically admissible and the bulk stress is negative semidefinite. Two examples are given. In the first, we consider the elastic semi-infinite plate with attached stiffener on the boundary (Melan’s problem). In the second example, we present a stress solution for a rectangular panel with membrane occupying the main diagonal plane.     

On the definitions of failure and critical states in hypoplasticity

A granular body is said to be at failure or in a critical state if the stress state does not change while the body is continuously deformed. Within the framework of hypoplasticity, such states, generally called stationary states,are conventionally defined by the condition that an objective (the Jaumann) stress rate vanishes. However, not all stationary states attained under monotonic deformation lie within the scope of this definition. Simple shear is an example. In fact, stationary states are characterized by zero material time derivative of the stress tensor rather than zero Jaumann rate. In the present paper, we give a generalized definition of stationarity by the condition of zero material time derivative of the stress tensor. The new definition extends the set of possible stationary states and includes those which are not covered by the previous definition. Stationary states are analysed quantitatively using calibrated hypoplastic equations. It is shown numerically that, if the norm of the spin tensor is of the same order as, or smaller than, the norm of the stretching tensor, the old definition approximates all possible sationary states with sufficient accuracy.      

Homogenization of delamination growth in an ACA flip-chip joint based on micropolar theory

In the present paper the delamination mechanism of a typical internal structure of the anisotropic conductive adhesive (ACA) interconnect for electronic packaging is modeled on the basis of micropolar theory and computational homogenization. The interface is treated as a finite Representative Volume Element (RVE), across which the macroscopic deformation is expressed in terms of regularized strong displacement and rotational discontinuities. For the microstructure of the RVE, the micro-macro kinematical coupling is considered as a Taylor series expansion in the regularized macroscopic discontinuities, and, connected to that, a discontinuous fluctuation field representing the microstructural variation is included to describe delamination on the microlevel. As to the microlevel delamination modeling, on the basis of the discontinuous fluctuation field, a damage coupled to slip and dilation formulation is used to model the interface degradation. The constitutive relations are established in a thermodynamic setting, where the interfacial free energy involves internal variables of damage and plastic deformation. The parameters of the interface are calibrated so that a predefined amount of fracture energy is dissipated in mode I. In the numerical example, the response of a planar interface is considered when it is subjected to the basic modes I-II and also the non-conventional rotational discontinuity mode. Case studies on fracture and geometry parameters have also been carried out. Finally, an uncoupled thermomechanical analysis of a microsystem involving a representative ACA microstructure has been made for the understanding of the microscopic delamination during a thermal cycling procedure.