Three‐dimensional numerical modelling by XFEM of spring‐layer imperfect curved interfaces with applications to linearly elastic composite materials

The spring‐layer interface model is widely used in describing some imperfect interfaces frequently involved in materials and structures. Typically, it is appropriate for modelling a thin soft interphase layer between two relatively stiff bulk media. According to the spring‐layer interface model, the displacement vector suffers a jump across an interface whereas the traction vector is continuous across the same interface and is, in the linear case, proportional to the displacement vector jump. In the present work, an efficient three‐dimensional numerical approach based on the extended finite element method is first proposed to model linear spring‐layer curved imperfect interfaces and then applied to predict the effective elastic moduli of composites in which such imperfect interfaces intervene. In particular, a rigorous derivation of the linear spring‐layer interface model is provided to clarify its domain of validity. The accuracy and convergence rate of the elaborated numerical approach are assessed via benchmark tests for which exact analytical solutions are available. The computated effective elastic moduli of composites are compared with the relevant analytical lower and upper bounds. Copyright © 2011 John Wiley & Sons, Ltd.     

Analysis of an efficient finite element method for embedded interface problems

A stabilized finite element method based on the Nitsche technique for enforcing constraints leads to an efficient computational procedure for embedded interface problems, in which the finite element mesh need not be aligned with the interface geometry. We consider cases in which the jump of a field across the interface is given, as well as cases in which the primary field on the interface is given. Optimal rates of convergence hold. Representative numerical examples demonstrate the effectiveness of the proposed methodology.     

Correlation between fracture and damage for quasi-brittle bi-material interface cracks

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric with respect to the crack and loading is of pure Mode I, due to the differences in the elastic properties across an interface which disrupts the symmetry. The linear elastic solutions of the crack tip stress and displacement fields show an oscillatory type of singularity. This poses numerical difficulties while modeling discrete interface cracks. Alternatively, the discrete cracks may be modeled using a distributed band of micro-cracks or damage such that energy equivalence is maintained between the two systems. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in quasi-brittle bi-material interface beams. The study is aimed at large size structures made of quasi-brittle materials failing at concrete-concrete interfaces. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. It is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.     

水下压电/弹性材料复合层球壳声波反射的数值研究

采用有限单元计算软件ANSYS对无限水域中压电/弹性材料复合层球壳的声波反射进行了数值研究.研究表明,在压电层中加载适当的电压可以在一个圆锥形的流体区域内有效地减小壳体对于外来声波的反射,并且有可能完全消除任意选定空间点上的反射.     

Simulation of microstructural evolution in materials with misfitting precipitates

To predict the behavior of directional coarsening and the temporal evolution of the shape of coherent precipitates in two-phase materials, a dislocation-free model is proposed, based on a combination of statistical mechanics and linear elasticity. This model takes elastic anisotropy and isotropic interfacial energy into account. Based on an example of isolated precipitates under plane strain condition, the influence of particle size, inhomogeneity, direction and sign of external loads on the equilibrium shape will be discussed in terms of a generalized thermodynamic force acting on the interface. To simulate the morphological diffusion process of typical microstructures with several random distributed misfitting inclusions, a computational technique in form of a finite element Monte Carlo simulation is presented. Within this numerical technique, no restrictions on the particle shape or the elastic anisotropy of both phases are made.     

An efficient finite element method for embedded interface problems

A stabilized finite element method based on the Nitsche technique for enforcing constraints leads to an efficient computational procedure for embedded interface problems. We consider cases in which the jump of a field across the interface is given, as well as cases in which the primary field on the interface is given. The finite element mesh need not be aligned with the interface geometry. We present closed‐form analytical expressions for interfacial stabilization terms and simple procedures for accurate flux evaluations. Representative numerical examples demonstrate the effectiveness of the proposed methodology. Copyright © 2008 John Wiley & Sons, Ltd.     

Interface energy on effective elastic constant of piezoelectric composites with spherically anisotropy nano-particles under external uniform strain

The surfaces/interfaces effect is vital in nanocomposites. The electro-elastic surface/interface theory is introduced to predict the size-dependent effective elastic constants of piezoelectric composites with spherically anisotropy nano-particles under external uniform strain, and the analytical solution is obtained. New boundary conditions governing the surface piezoelectricity are used to analyze the surface piezoelectricity effect. The average electro-elastic coupling field of the randomly distributed nano-particles with surface/interface effect is derived by using the effective field method. In the numerical examples, the interface effect under different material constituents is analyzed. It is found that the interface effect on the effective elastic constants is significantly related to the material properties of nano-particles. The elastic constants and piezoelectric constants show different effects on the effective elastic constants.     

A general and efficient computational procedure for modelling the Kapitza thermal resistance based on XFEM

The thermal resistance of an interface between two materials, conceptualized by Kapitza, is an important physical phenomenon encountered in many situations of practical interest. The numerical treatment of this phenomenon has up to now run into difficulties due to the temperature discontinuity. In this work, a general and efficient computational procedure for modelling the Kapitza thermal resistance is proposed, which is based on the extended finite element method (XFEM) in tandem with a level-set method. The steady thermal conduction in a two-phase material with the Kapitza thermal resistance at the interface is first formulated in a variational way and then numerically treated with the proposed computational procedure. Different three-dimensional numerical examples with known analytical solutions show the high accuracy and robustness of the proposed computational procedure in capturing the temperature jump across an interface.     

An Elliptical Crack in an Anisotropic Elasic Medium Subjected to a Constant External Field

An isolated elliptical crack in an infinite orthotropic elastic medium is considered. An efficient numerical algorithm of the solution of the problem for a crack subjected to a constant external field is proposed. The calculation of the crack opening vector and the stress intensity factors on the crack edge is reduced to regular 2D-integrals. These integrals may be simply calculated numerically for an arbitrary orientation of the crack plane with respect to the principal axes of the anisotropy of the medium. Examples of the calculation of the crack opening vector and stress intensity factors are presented.     

Lagrangian formulism of elasticity with relevance to surface energy

By introducing the divergence of a vector potential into the Lagrangian, a Lagrangian framework is developed to incorporate the surface energy into elasticity. Besides the Euler-Lagrange equation and natural boundary condition, a new boundary constitutive equation is derived from the variation of the Lagrangian and configuration on which the Lagrangian is defined. On the boundary surface, explicit expression of the vector potential with respect to the field variable and surface curvature is determined. Based on this framework, an elastic model with relevance to the surface energy is established. The Young-Laplace’s formula is generalized to an elastic solid in a new fashion. Making use of this model, we investigate the surface energy effect in the radial vibration of a spherical nanoparticle. Numerical calculation shows that natural frequencies of the nanoparticle will shift down due to the surface energy. This shift is especially apparent in the vibration of soft matter nanoparticles.     

The spectral element method for elastic wave equations—application to 2‐D and 3‐D seismic problems

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two‐ and three‐dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor‐product of a Gauss–Lobatto–Legendre 1‐D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy‐momentum conserving scheme that can be put into a classical explicit‐implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two‐dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three‐dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three‐dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.     

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

Interfacial energy plays an important role in equilibrium morphologies of nanosized microstructures of solid materials due to the high interface-to-volume ratio, and can no longer be neglected as it does in conventional mechanics analysis. When designing nanodevices and to understand the behavior of materials at the nano-scale, this interfacial energy must therefore be taken into account. The present work develops an effective numerical approach by means of a hybrid smoothed extended finite element/level set method to model nanoscale inhomogeneities with interfacial energy effect, in which the finite element mesh can be completely independent of the interface geometry. The Gurtin–Murdoch surface elasticity model is used to account for the interface stress effect and the Wachspress interpolants are used for the first time to construct the shape functions in the smoothed extended finite element method. Selected numerical results are presented to study the accuracy and efficiency of the proposed method as well as the equilibrium shapes of misfit particles in elastic solids. The presented results compare very well with those obtained from theoretical solutions and experimental observations, and the computational efficiency of the method is shown to be superior to that of its most advanced competitor.     

Band gaps of elastic waves in 1-D phononic crystal with dipolar gradient elasticity

The dispersive relations of Bloch waves in the periodic laminated structure formed by periodically repeating of two different gradient elastic solids are studied in this paper. First, the various wave modes in the gradient elastic solid, which are different from those in the classical elastic solid, are formulated. Apart from the dispersive P wave and SV wave, there are two evanescent waves, which become the P type and S type surface waves at the interface of two different gradient elastic solids. Next, the continuity conditions of displacement vector, the normal derivative of the displacement vector and the monopolar and dipolar tractions across the interface between two different gradient elastic solids are used to derive the transfer matrix of the state vector in a typical single cell. At last, the Bloch theorem of Bloch waves in the periodical structure is used to give the dispersive equation. The in-plane Bloch waves and the anti-plane Bloch waves are both considered in the present work. The oblique propagation situation and the normal propagation situation are also considered, respectively. The numerical results are obtained by solving the dispersive equation. The influences of two microstructure parameters of the gradient elastic solid and the microstructure parameter ratio of two different gradient elastic solids on the dispersive relation are discussed based on the numerical results.     

Thermal contact conductance characterization via computational contact homogenization: A finite deformation theory framework

In order to predict the macroscopic thermal response of contact interfaces between rough surface topographies, a computational contact homogenization technique is developed at the finite deformation regime. The overall homogenization framework transfers macroscopic contact variables, such as surfacial stretch, pressure and heat flux, as boundary conditions on a test sample within a micromechanical interface testing procedure. An analysis of the thermal dissipation within the test sample reveals a thermodynamically consistent identification for the macroscopic thermal contact conductance parameter that enables the solution of a homogenized thermomechanical contact boundary value problem based on standard computational approaches. The homogenized contact response effectively predicts a temperature jump across the macroscale contact interface. The strong dependence of this homogenized response on macroscale solution variables of interest is demonstrated via representative three‐dimensional numerical investigations. The proposed contact homogenization framework is suitable for the analysis of similar energy transport phenomena across heterogeneous contact interfaces where the investigation of the sources for energy dissipation is of concern. Copyright © 2010 John Wiley & Sons, Ltd.     

Evaluation of energy release rate for a planar crack in heterogeneous media

A modified version of domain integral method is developed for evaluation of energy release rate with finite element solutions for problems with a 2-D crack located in a heterogeneous elastic field. The heterogeneous field considered in this work generally contains various materials, with discontinuous mechanical moduli across the interfaces. The formulation is proved to be patch-independent, in a generalized sense, and valid for problems under both small and large deformations. The results of calculation appear to be very insensitive to the crack tip finite element models when the tip is away from the material interface. However, strong dependency on the local modeling is observed in case the tip is located at the interface. Alternative studies on this particular case are thus required.     

Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis

Summary In this paper, the change of the elastic fields induced by the interface energies and the interface stresses from the reference configuration to the current configuration is considered. It is emphasized that the governing equations taking into account the interface energy effect should be established within the framework of finite deformation in the first place, and then the approximations of governing equations for a finitely deformed multi-phase elastic medium by an infinitesimal strain analysis can be formulated. Hence it can be seen that the asymmetric interface stress has to be used in the Young-Laplace equation. According to the above mentioned formalism, analytical expressions of the size-dependent effective moduli of a particle-filled composite material with interface energy effect are derived. It is shown that, different from the results obtained by previous researchers, the liquid-like surface/interface tension, as a residual stress-type term, also influences the effective property of the composite.     

The role of coherent vortices near the turbulent/non-turbulent interface in a planar jet

The role of coherent vortices near the turbulent/non-turbulent (T/NT) interface in a turbulent plane jet is analysed by a direct numerical simulation (DNS). The coherent vortices near the jet edge consist of large-scale vortical structures (LSVSs) maintained by the mean shear and intense vorticity structures (IVSs) created by the background fluctuating turbulence field. The radius of the LSVS is equal to the Taylor micro-scale R(lsvs)≈λ, while the radius of the IVS is of the order of the Kolmogorov micro-scale R(ivs)~η. The LSVSs are responsible for the observed vorticity jump at the T/NT interface, being of the order of the Taylor micro-scale. The coherent vortices in the proximity of the T/NT interface are preferentially aligned with the tangent to the T/NT interface and are responsible for the viscous dissipation of kinetic energy near the T/NT interface and to the characteristic shape of the enstrophy viscous diffusion observed at that location.     

Gradient piezoelectricity for cracks under an impact load

The flexoelectric effect on elastic waves is investigated in nano-sized cracked structures. The strain gradients are considered in the constitutive equations of a piezoelectric solid for electric displacements and the higher-order stress tensor. The governing equations with the corresponding boundary conditions are derived from the variational principle. The finite element method (FEM) is developed from the principle of virtual work. It is equivalent to the weak-form of derived governing equations in gradient elasticity. The computational method can be applied to analyze general 2D boundary value problems in size-dependent piezoelectric elastic solids with cracks under a dynamic load. The FEM formulation is implemented for strain-gradient piezoelectricity under a dynamic load.     

Parametric constitutive model of plain-weave fabric reinforced composite ply

A computational model that allows to explicitly determine orthotropic elastic constants of plain-weave fabric-reinforced composite ply as functions of microstructure parameters has been developed in this study. These relationships are not given in the form of analytical formulae (as it is in the case of approximate analytical models) but in the form of an extensive database of numerically evaluated results for different microstructure instances and a numerical scheme that interpolates the results. To build the database, a standard finite-element-based homogenization technique of a periodic representative volume element is employed. As a result, a numerical algorithm is provided that may be easily employed in FE codes as a part of a regular constitutive subroutine. Sensitivity of the composite elastic constants with respect to the microstructure parameters is also directly available from the model.