An embedding method for modeling micromechanical behavior and macroscopic properties of composite materials

This paper presents a numerical method for modeling the micromechanical behavior and macroscopic properties of fiber-reinforced composites and perforated materials. The material is modeled by a finite rectangular domain containing multiple circular holes and elastic inclusions. The rectangular domain is assumed to be embedded within a larger circular domain with fictitious boundary loading represented by truncated Fourier series. The analytical solution for the complementary problem of a circular domain containing holes and inclusions is obtained by using a combination of the series expansion technique with a direct boundary integral method. The boundary conditions on the physical external boundary are satisfied by adopting an overspecification technique based on a least squares approximation. All of the integrals arising in the method can be evaluated analytically. As a result, the elastic fields and effective properties can be expressed explicitly in terms of the coefficients in the series expansions. Several numerical experiments are conducted to verify the accuracy and efficiency of the numerical method and to demonstrate its application in determination of the macroscopic properties of composite materials.     

On transformation laws for plastic deformations of materials with elastic range

Archive for Rational Mechanics and Analysis -     

Homogenization of nonlinearly elastic materials,microscopic bifurcation and macroscopic loss of rank-one convexity

Communicated by R. V. Kohn     

On stability of elastic domain during isothermal solid-solid phase transformation in a tube configuration

Under isothermal quasi-static stretching the phasetransition of a superelastic NiTi tube involves the formation(during loading) and vanishing(in unloading) of a high strain(martensite) domain.The two events are accompanied by arapid stress drop/rise due to the formation/vanishing of domain fronts.From a thermodynamic point of view,both areinstability phenomena that occur once the system reaches itscritical state.This paper investigates the stability of a shrinking cylindrical domain in a tube configuration during unloading.The energetics and thermodynamic driving force of thecylindrical domain are quantified by using an elastic inclusion model.It is demonstrated that the two domain fronts exhibit strong interaction when they come close to each other,which brings a peak in the total energy and a sign changein the thermodynamic driving force.It is proved that suchdomain front interaction plays an important role in controlling the stability of the domain and in the occurrence of stressjumps during domain vanishing.It is also shown that the process is governed by two nondimensional length scales(thenormalized tube length and normalized wall-thickness) andthat the length scale dependence of the critical domain lengthand stress jump for the domain vanishing can be quantifiedby the elastic inclusion model.     

On the problem of eversion for incompressible elastic materials

The paper contains a discussion on when eversion of cylindrical tubes and spherical shells is possible. The analysis shows that eversion of a cylindrical tube of every isotropic incompressible elastic material with no applied forces is possible assuming only the E-inequalities. This is not always true for spherical shells. Conditions are given as to when this is possible and when it is not possible.     

On copositive matrices and strong ellipticity for isotropic elastic materials

Archive for Rational Mechanics and Analysis -     

On the work and energy theorem for unbounded elastic bodies

The main scope of the paper is the statement and the proof of the work and energy theorem for elastic bodies that occupy an unbounded region of space and whose acoustic tensor A may suitably grow at large spatial distance from a fixed origin.     

On minimal representations for constitutive equations of anisotropic elastic materials

The problem of determining minimal representations for anisotropic elastic constitutive equations is proposed and investigated. For elastic constitutive equations in any given case of anisotropy, it is shown that there exist generating sets consisting of six generators and such generating sets are minimal in all possible generating sets. This fact implies that most of the established results for representations of elastic constitutive equations are not minimal and remain to be sharpened. For elastic constitutive equations in some cases of anisotropy, including orthotropy, transverse isotropy, the trigonal crystal class S ₆, and the classes C _2mh , m=1, 2, 3,..., etc., representations in terms of minimal generating sets are presented for the first time.     

Modelling of localization and propagation of phase transformation in superelastic SMA by a gradient nonlocal approach

In this work, a nonlocal phenomenological behavior model is proposed in order to describe the localization and propagation of stress-induced martensite transformation in shape memory alloy (SMA) wires and thin films. It is a nonlocal extension of an existing local model that was derived from a micromechanical-inspired Gibbs free energy expression. The proposed model uses, besides the local field of the internal variable, namely the martensite volume fraction, a nonlocal counterpart. This latter acts as an additional degree of freedom, which is determined by solving an additional partial differential equation (PDE), derived so as to be equivalent to the integral definition of a nonlocal quantity. This PDE involves an internal length parameter, dictating the global scale at which the nonlocal interactions of the underlying micromechanisms are manifested during phase transformation. Moreover, to account for the unstable softening behavior, the transformation yield force parameter is considered as a gradually decreasing function of the martensite fraction. Possible material and geometric imperfections that are responsible for localization initiation are also considered in this analysis. The obtained constitutive equations are implemented in the Abaqus® finite element code in one and two dimensions. This requires the development of specific finite elements having the nonlocal volume fraction variable as an additional degree of freedom. This implementation is achieved through the UEL user’s subroutine. The effect of martensitic localization on the superelastic global behavior of SMA wire and holed thin plate, subjected to tension loading, is analyzed. Numerical results show that the developed tool correctly captures the commonly observed unstable superelastic behavior characterized by nucleation and propagation of martensitic phase. In particular, they show the influence of the internal length parameter, appearing in the nonlocal model, on the size of the localization area and the stress nucleation peak.     

用能量图形面积法计算非线性弹性材料任意加载方式的能量释放率

从能量释放率G的定义出发,用图解法,即能量图形面积法,分别对线弹性材料和非线性弹性材料,以及在固定位移,或固定载荷,或任意加载方式的情况下,推导出能量释放率的计算公式.证明不同加载方式下的非线性弹性材料的能量释放率,在用能量图形面积法极限求导时,其数值是一样的.     

On Landau theory and symmetric energy landscapes for phase transitions

The aim of this presentation is the development of a general approach for modelling the global complex energy landscapes of phase transitions. For the sake of clarity and brevity the exposition is restricted to martensitic phase transition (i.e., diffusionless phase transitions in crystalline solids). The methods, however, are more broadly applicable. Explicit energy functions are derived for the cubic-to-tetragonal phase transition, where data are fitted for InTl. Another example is given for the cubic-to-monoclinic transition in CuZnAl. The resulting energies are defined globally, in a piecewise manner. We use splines that are twice continuously differentiable to ensure sufficient smoothness. The modular (piecewise) technique advocated here allows for modelling elastic moduli, energy barriers and other characteristics independent of each other.     

Estimates for the elastic moduli of random trigonal polycrystals and their macroscopic uncertainty

Explicit expressions of the upper and lower estimates on the macroscopic elastic moduli of random trigonal polycrystals are derived and calculated for a number of aggregates, which correct our earlier results given in Pham [Pham, D.C., 2003. Asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals. Int. J. Solids Struct. 40, 4911–4924]. The estimates are expected to predict the scatter ranges for the elastic moduli of the polycrystalline materials. The concept of effective moduli is reconsidered regarding the macroscopic uncertainty of the moduli of randomly inhomogeneous materials.     

On the asymptotic behavior of a phase-field model for elastic phase transitions

We study the asymptotic behavior of a one-dimensional, dynamical model of solid-solid elastic transitions in which the phase is determined by an order parameter. The system is composed of two coupled evolution equations, the mechanical equation of elasticity which is hyperbolic and a parabolic equation in the order parameter. Due to the strong coupling and the lack of smoothing in the hyperbolic equation, the asymptotic behavior of solutions is difficult to determine using standard methods of gradient-like systems. However, we show that under suitable assumptions all solutions approach the equilibrium set weakly, while the phase field stabilizes strongly.     

On quasi-non-destructive strength and toughness testing of elastic–plastic materials

Non-destructive evaluation of mechanical material properties, like strength and fracture toughness, is impossible for principal reasons. However, there are possibilities of quasi-non-destructive estimation methods, which can be quite useful in practice. Instrumented indentation tests are often suitable to get information about the elastic–plastic behaviour, where the indentation depth is measured as a function of indentation force. By approximate analytical methods, key parameters like ultimate tensile strength, work-hardening exponent or even yield stress can be derived from these measurements. A mobile indenter is presented here and its use in ambulant testing is described. To obtain the uniaxial stress–strain curve more directly and more exactly, the same instrument can be used for a miniature compression test, where a small pin is machined out from the surface of the material. Furthermore, to get information about the toughness of materials, a carving instrument has been developed, which allows the energy required to introduce a defined furrow to be measured and correlated with toughness parameters.     

A general approach for defining the macroscopic free energy density of saturated porous media at finite strains under non-isothermal conditions

A general approach is proposed for defining the macroscopic free energy density function (and its complement, the free enthalpy) of a saturated porous medium submitted to finite deformations under non-isothermal conditions, in the case of compressible fluid and solid constituents. Reference is made to an elementary volume treated as an ‘open system’, moving with the solid skeleton. The proposed free energy depends on the generalised strains (namely an appropriate measure of the strain of the solid skeleton and the variation in fluid mass content) and the absolute temperatures of the solid and fluid phases (which are assumed to differ from each other for the sake of generality). This macroscopic energy proves to be a potential for the generalised stresses (namely the associated measure of the total stress and the free enthalpy of the pore fluid per unit mass) and the entropies of the solid and fluid phases. In contrast with mixture theories, the resulting free energy is not the simple sum of the free energies of the single constituents. Two simplified cases are examined in detail, i.e. the semilinear theory (originally proposed for isothermal conditions and extended here to non-isothermal problems) and the linear theory. The proposed approach paves the way to the consistent non-isothermal-hyperelastic-plastic modelling of saturated porous media with a compressible fluid and solid constituents.     

On the active part of the stress for elastic materials with internal constraints

Constitutive equations for constrained elastic materials prescribe a whole set of possible stresses for each deformation. We prove the existence of a normalization procedure for determining the active part of the stress through a single-valued response function which is both frame-indifferent and invariant under the same symmetry group of the given constitutive equation.