Derivation of a rod theory for biphase materials with dislocations at the interface

Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of SO(3). First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of Γ-convergence.     

On Well Posedness in Continuum Interface Problems

By considering continuum interface problems like e.g. the modeling of composites, the possible loss of well posedness of the resulting Boundary Value Problem has to be considered, dependent on the choice of material laws in the bulk and in the interface. In this contribution, the problem is discussed for a bulk material connected to a rigid substrate by an interface layer. The isotropic bulk material is linear elastic while for the interface elasticity and elasticity with damage is investigated. A complex surface acoustic tensor is introduced by applying a decaying surface wave ansatz to the incremental boundary value problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the competition of elastic energy and surface energy in discrete numerical schemes

The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.      

Dislocations in nanowire heterostructures: from discrete to continuum

We discuss an atomistic model for heterogeneous nanowires, allowing for dislocations at the interface. We study the limit as the atomic distance converges to zero, considering simultaneously a dimension reduction and the passage from discrete to continuum. Employing the notion of Gamma-convergence, we establish the minimal energies associated to defect-free configurations and configurations with dislocations at the interface, respectively. It turns out that dislocations are favoured if the thickness of the wire is sufficiently large. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of strip mine excavations as dislocations in elastic rock

We propose a new model of excavation in rock based on Volterra dislocations. We develop a software package for determining the stressdeformed state in modeling excavations as dislocations, on the basis of which we study the qualitative properties of the vertical stress field of a solid problem of elasticity theory. Four figures.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 85–91.     

Large-amplitude love waves

In the context of the finite elasticity theory, we considera model for compressible solids called ‘compressible neo-Hookeanmaterial’. We show how finite-amplitude inhomogeneousplane wave solutions and finite-amplitude unattenuated solutionscan combine to form a finite-amplitude Love wave. We take alayer of finite thickness overlying a solid half-space, bothmade of different prestressed compressible neo-Hookean materials.We derive an exact solution of the equations of motion and boundaryconditions and also obtain results for the energy density andthe energy flux of the waves. Finally, we investigate the specialcase when the interface between the layer and the substrateis in a principal plane of the prestrain. A numerical exampleis given.     

Dynamics of discrete screw dislocations via discrete gradient flow

We present variational approaches (developed in [3,4,11]) to the study of statics and dynamics of screw dislocations in crystals. We model the crystal as a cubic lattice and we give the asymptotic Γ-convergence expansion of the elastic energy induced by a finite family of screw dislocations as the lattice spacing goes to zero. We show that the effective energy associated to the presence of a finite system of screw dislocations coincides with the renormalized energy, studied within the Ginzburg-Landau framework and ruling the interactions between the dislocations. As a byproduct of this analysis, we show the existence of many metastable configurations of dislocations pinned by energy barries. Using the minimizing movement approach á la De Giorgi, we introduce a discrete-in-time variational dynamics, referred to as Discrete Gradient Flow, which allows to overcome these energy barriers. More precisely, we show that lettting first the lattice spacing and then the time step of minimizing movements tend to zero, dislocations move accordingly with the gradient flow of the renormalized energy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Saint–Venant torsion of a circular bar with radial cracks incorporating surface elasticity

The Saint–Venant torsion problem of a circular cylinder containing a radial crack with surface elasticity is studied. The surface elasticity is incorporated into the crack faces by using the continuum-based surface/interface model of Gurtin and Murdoch. Both an internal crack and an edge crack are considered. By using the Green’s function method, the boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the incorporation of surface elasticity, the stresses exhibit the logarithmic singularity at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface elasticity is also solved by using a similar method. The strengths of the logarithmic singularity and the size-dependent torsional rigidity are calculated.     

Transient response of dissimilar piezoelectric layers with multiple interacting interface cracks

In this study, we examine the dynamic behavior of two bonded dissimilar piezoelectric layers containing multiple interfacial cracks subjected to electro-mechanical impact loading. The problem was formulated through Fourier transformation into singular integral equations in which the unknown variables are the jumps of displacement and electric potential across the crack surface in the Laplace transform domain. The resulting integral equations together with the corresponding single-valued conditions are solved numerically for the densities of electro-elastic dislocations on a crack surface. The dynamic field intensity factors and dynamic energy release rate (DERR) history are obtained for both permeable and impermeable crack. The stress field is also obtained for the interface crack under impact loads. The results show that the field intensity factors at the crack tips and dynamic energy release rate depend on the interfacial crack geometry, electromechanical coupling and the electric boundary conditions on the crack surface.     

Modeling,shape analysis and computation of the equilibrium pore shape near a PEM–PEM intersection

In this paper we study the equilibrium shape of an interface that represents the lateral boundary of a pore channel embedded in an elastomer. The model consists of a system of PDEs, comprising a linear elasticity equation for displacements within the elastomer and a nonlinear Poisson equation for the electric potential within the channel (filled with protons and water). To determine the equilibrium interface, a variational approach is employed. We analyze: (i) the existence and uniqueness of the electrical potential, (ii) the shape derivatives of state variables and (iii) the shape differentiability of the corresponding energy and the corresponding Euler–Lagrange equation. The latter leads to a modified Young–Laplace equation on the interface. This modified equation is compared with the classical Young–Laplace equation by computing several equilibrium shapes, using a fixed point algorithm.     

Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction

We consider a model of fluid-structure interaction in a bounded domain Ω∈ℝ² where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of Navier Stokes equation coupled on the interface to the dynamic system of elasticity. The characteristic feature of this coupled model is that the resolvent is not compact and the energy function characterizing balance of the total energy is weakly degenerated. These combined with the lack of mechanical dissipation and intrinsic nonlinearity of the dynamics render the problem of asymptotic stability rather delicate. Indeed, the only source of dissipation is the viscosity effect propagated from the fluid via interface. It will be shown that under suitable geometric conditions imposed on the geometry of the interface, finite energy function associated with weak solutions converges to zero when the time t converges to infinity. The required geometric conditions result from the presence of the pressure acting upon the solid.     

A thermo-mechanical energetic cohesive zone model accounting for Kapitza interfaces

The objective of this contribution is to study computational aspects of modeling thermo-mechanical solids containing mechanically energetic, geometrically non-coherent Kapitza interfaces under cyclic loading. The interface is termed energetic in the sense that it possesses its own energy, entropy, constitutive relations and dissipation. To date, classical thermo-mechanical cohesive zone models do not account for elastic interfaces. Therefore we propose a novel interface model that couples the classical cohesive zone formulation to the interface elasticity theory under the Kapitza assumption within a thermo-mechanical framework. In other words, such an interface model allows for discontinuities in geometry, temperature and normal stress fields, while not permitting a jump in the normal heat flux across the interface. The equations governing a fully non-linear transient problem are given. In particular, a comparison is made between the results of the classical thermo-mechanical cohesive zone model and our novel (cohesive + energetic Kapitza) interface formulation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects

A new Bernoulli–Euler beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equation and complete boundary conditions for a Bernoulli–Euler beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure- and surface energy-dependent size effect. In addition, Poisson’s effect is incorporated in the current model, unlike existing beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as special cases. The current model reduces to the classical Bernoulli–Euler beam model when the microstructure dependence, surface energy, and Poisson’s effect are all suppressed. To demonstrate the new model, a cantilever beam problem is solved by directly applying the general formulas derived. Numerical results reveal that the beam deflection predicted by the new model is smaller than that by the classical beam model. Also, it is found that the difference between the deflections predicted by the two models is very significant when the beam thickness is small but is diminishing with the increase of the beam thickness.     

Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

Computational modeling of elastic and plastic multiple cracks by the fundamental solutions

The fundamental solutions of elasticity are used to establish a numerical method for elastic and plastic multiple crack problems in two dimensions. The continuous distributions of the point forces, dislocations, and the plastic sources are used systematically to model the crack, non-crack boundary, and the plastic deformation. Use of these singularities are guided strictly by the physical interpretation of the problem. We adopt Muskhelishvili's complex variable formalism that facilitate the analytical evaluation of the integrals representing the continuous distributions of the singularities. The resulting numerical method is concise and accurate enough to be used for elastic and plastic multiple crack problems.     

Nucleation Barriers for the Cubic‐to‐Tetragonal Phase Transformation

We are interested in the phase transformation from austenite to martensite. This transformation is typically accompanied by the generation and growth of small inclusions of martensite. We consider a model from geometrically linear elasticity with sharp energy penalization for phase boundaries. Focusing on a cubic‐to‐tetragonal phase transformation, we show that the minimal energy for an inclusion of martensite scales like $\max \{ V^{2/3}, V^{9/11} \}$ in terms of the volume V. Moreover, our arguments illustrate the importance of self‐accommodation for achieving the minimal scaling of the energy. The analysis is based on Fourier representation of the elastic energy. © 2012 Wiley Periodicals, Inc.     

Effective bulk moduli of materials containing stochastically distributed nano-inhomogeneities with surface stresses

In the present contribution, a mathematical model for the investigation of the effective properties of a material with randomly distributed nano-particles is proposed. The surface effect is introduced via Gurtin-Murdoch equations describing properties of the matrix/nano-particle interface. They are added to the system of stochastic differential equations formulated within the framework of linear elasticity. The homogenization problem is reduced to finding a statistically averaged solution of the system of stochastic differential equations. These equations are based on the fundamental equations of linear elasticity, which are coupled with surface/interface elasticity accounting for the presence of surface tension. Using Green's function this system is transformed to a system of statistically non-linear integral equations. It is solved by the method of conditional moments. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix with randomly distributed spherical inhomogeneities. The radius of the nano-particles is included in the expression for the bulk moduli. As numerical examples, nano-porous aluminum and nano-porous gold are investigated assuming that only the influence of the interface effects on the effective bulk modulus is of interest. The dependence of the normalized bulk moduli of nano-porous aluminum on the pore volume fraction (for certain radii of nano-pores) are compared to and discussed in the context of other theoretical predictions. The effective Young's modulus of nano-porous gold as a function of pore radius (for fixed void volume fraction) and the normalized Young's modulus vs. the pore volume fraction for different pore radii are analyzed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interface cracks in anisotropic composites

The linear model equations of elasticity often give rise to oscillatory solutions in some vicinity of interface crack fronts. In this paper we apply the Wiener–Hopf method which yields the asymptotic behaviour of the elastic fields and, in addition, criteria to prevent oscillatory solutions. The exponents of the asymptotic expansions are found as eigenvalues of the symbol of corresponding boundary pseudodifferential equations. The method works for three‐dimensional anisotropic bodies and we demonstrate it for the example of two anisotropic bodies, one of which is bounded and the other one is its exterior complement. The common boundary is a smooth surface. On one part of this surface, called the interface, the bodies are bonded, while on the complementary part there is a crack. By applying the potential method, the problem is reduced to an equivalent system of Boundary Pseudodifferential Equations (BPE) on the interface with the stress vector as the unknown. The BPEs are defined via Poincaré–Steklov operators. We prove the unique solvability of these BPEs and obtain the full asymptotic expansion of the solution near the crack front. As a special case we consider the interface crack between two different isotropic materials and derive an explicit criterion which prevents oscillatory solutions. Copyright © 1999 John Wiley & Sons, Ltd.     

Thermomechanical coupling of geometrically non-coherent interfaces

An evacuated tube collector is composed of a vacuum glass tube forged to a metal tube creating an interface which undergoes thermomechanical cyclic loading. To numerically model such a thermomechanical and geometrically non-coherent interface, a decohesion element with mixed-mode capability, based on a normal-shear decomposition of the displacement discontinuity vector is used, exploiting exponential traction-separation laws. In addition to the classical strategies, interfacial elements are allowed to have an in-plane stretch resistance by assuming a thermohyperelastic interface Helmholtz energy, a function of the rank-deficient interface deformation gradient and temperature , as an extension of the surface/interface elasticity theory. The nonlinear governing equations are given and solved using the finite element method. The results are illustrated through a series of numerical examples for different material parameters. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a mode-III crack in the presence of surface elasticity and a prescribed non-uniform surface traction

We consider an elastic solid incorporating a mode-III crack in which the crack faces incorporate the effects of surface elasticity and are further subjected to prescribed non-uniform surface tractions. The surface elasticity is modelled using the continuum-based model of Gurtin and Murdoch. Using complex variable techniques, the corresponding problem is reduced to the solution of a first order Cauchy singular integro-differential equation which, in turn, leads to the complete solution of the aforementioned crack problem valid everywhere in the domain of interest (including at the crack tip). Finally, we note that, as a particular case of our analysis, the classical decomposition of a mode-III crack problem in linear elasticity continues to hold even in the presence of surface elasticity.