Transient heat conduction in a medium with multiple circular cavities and inhomogeneities

A two‐dimensional transient heat conduction problem of multiple interacting circular inhomogeneities, cavities and point sources is considered. In general, a non‐perfect contact at the matrix/inhomogeneity interfaces is assumed, with the heat flux through the interface proportional to the temperature jump. The approach is based on the use of the general solutions to the problems of a single cavity and an inhomogeneity and superposition. Application of the Laplace transform and the so‐called addition theorem results in an analytical transformed solution. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. Several numerical examples are given to demonstrate the accuracy and the efficiency of the method. The approximation error decreases exponentially with the number of the degrees of freedom in the problem. A comparison of the companion two‐ and three‐dimensional problems demonstrates the effect of the dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.     

Time dependent temperature fields and stress interactions with subsurface or surface breaking defects

A fundamental solution is developed which has a jump in temperature 2T ₀ H(t–t ₀) on a semi-infinite cut (a transient thermal dislocation) together with continuous traction and displacement fields. This solution is used together with fundamental elastic dislocation solutions to study the interaction of time dependent temperature fields with subsurface or surface breaking cracks. Potential applications of this analysis to the problem of the influence of surface temperatures on delamination theories of wear and the photodisplacement and photothermal imaging methods are discussed.     

A circular inhomogeneity with interface slip and diffusion under in-plane deformation

We consider the in-plane deformation of a circular elastic inhomogeneity embedded in an infinite elastic matrix subjected to remote uniform stresses or uniform heat flow. The inhomogeneity and matrix have different material properties. The rate-dependent slip and mass transport by stress-driven diffusion concurrently occur on the inhomogeneity/matrix interface. For the remote uniform stress case, it is observed that the internal stresses within the inhomogeneity are quadratic functions of the coordinates x and y, and decay with two relaxation times. Interestingly the average mean stress within the circular inhomogeneity is in fact time-independent. As time approaches infinity, the internal stress field within the inhomogeneity becomes uniform and hydrostatic. In addition the change of strain energy due to the introduction of the circular elastic inhomogeneity is derived, containing various existing results as special cases. Furthermore, a simple condition leading to an internal uniform stress state within the inhomogeneity is found. This condition, which is independent of the elastic properties of the inhomogeneity and matrix, gives a simple relationship between the interface drag and diffusion parameters. For the remote heat flow case, the internal thermal stresses are linear functions of the coordinates x and y and decay only with a single relaxation time. Numerical results are presented to demonstrate the obtained solution and the corresponding physics.     

Coordinate-free derivation and weak formulation of a general imperfect interface model for thermal conduction in composites

In a great number of situations of practical interest, the interfaces between the constituent phases of a composite turn out to be imperfect. In the context of thermal conduction, an interface is said to be imperfect if the requirement that both the temperature and the normal heat flux be continuous across the interface is not satisfied. A powerful method based on mathematical asymptotic analysis has been proposed and developed in the literature by several authors for the derivation of linear imperfect interface models of thermal conduction. This method consists in replacing an interphase of small uniform thickness between two-phases by an imperfect interface of null thickness characterized by the temperature and normal heat flux jump relations deduced by carrying out an appropriate asymptotic analysis. The objective of the present work is threefold. Firstly, it aims to explicitly show and emphasize the key role played by Hadamard’s relation in the method. Secondly, it has the purpose of using a coordinate-free differential geometry theory and Hadamard’s relation to render the method coordinate-free. Thirdly and most importantly, the present work gives a weak formulation for the problem concerning the steady thermal conduction in a composite with the interfaces described by the general temperature and normal heat flux jump relations derived. This weak formulation is a key step toward solving the problem by the extended finite element method (XFEM) presented in a companion paper.     

A versatile interface model for thermal conduction phenomena and its numerical implementation by XFEM

A general interface model is presented for thermal conduction and characterized by two jump relations. The first one expresses that the temperature jump across an interface is proportional to the interfacial average of the normal heat flux while the second one states that the normal heat flux jump is proportional to the surface Laplacian of the interfacial average of the temperature. By varying the two scalar proportionality parameters, not only the Kapitza resistance and highly conducting interface models can be retrieved but also all the intermediate cases can be covered. The general interface model is numerically implemented by constructing its weak form and by using the level-set method and XFEM. The resulting numerical procedure, whose accuracy and robustness are thoroughly tested and discussed with the help of a benchmark problem, is shown to be efficient for solving the problem of thermal conduction in particulate composites with various imperfect interfaces.     

Interaction between a crack and a circular inhomogeneity with interface stiffness and tension

The interaction between a straight crack and a circular inhomogeneity with interface stiffness and energy is considered. The Gurtin and Murdoch model is adopted, wherein the interface between the inhomogeneity and the matrix is regarded as a material surface that possesses its own mechanical properties and surface tension. The elastostatics problem is decomposed into two complimentary problems for (1) a circular disk with unknown distributions of traction and displacements along its boundary and (2) a loaded isotropic plane containing a circular hole with unknown distributions of traction and displacements along its boundary. The unknown distributions are determined through the application of the constitutive relations at the material surface. For selected values of the dimensionless parameters that quantify the geometry, material properties and applied loading, the stress field, stress intensity factors and energy release rates are calculated using a complex boundary integral equation approach. Particular attention is paid to crack-tip shielding and anti-shielding that develops as a result of the stresses introduced by the material surface. An illustrative example involving a perforated plate loaded in tension suggests that the material surface produces a modest level of expected effective toughening.     

位错与含裂纹非理想界面圆形夹杂的干涉效应

研究了基体中任意位置的螺型位错与含裂纹非理想界面圆形弹性夹杂的干涉力学问题。运用复变函数的解析延拓技术与奇性主部分析方法,获得了该问题复势函数的一般解答。作为典型例子,求出了非理想圆形界面含一条裂纹时基体和夹杂区域复势函数的封闭形式解。计算了作用在螺型位错上的位错力。讨论了含裂纹的非理想界面以及材料失配对位错力的影响规律。结果表明:与含裂纹理想界面相比,非理想界面对位错力的影响更大,对位错的捕获能力更强。硬夹杂时,可能存在非稳定平衡点,使得非理想界面、界面裂纹和夹杂对位错的作用力为零;软夹杂时,非理想界面、界面裂纹和夹杂则始终吸引位错。     

Thermal Conductivity and Interfacial Thermal Resistance in Bilayered Nanofilms by Nonequilibrium Molecular Dynamics Simulations

A nonequilibrium molecular dynamics study of the cross-plane thermal conductivity and interfacial thermal resistance of nanoscale bilayered films is presented. The films under study are composed of argon and another material that is identical to argon except for its atomic mass. The results show that a large temperature jump occurs at the interface and that the interfacial thermal resistance plays an important role in heat conduction for the whole films. The cross-plane thermal conductivity is dependent on the average temperature. The interfacial thermal resistance is found to be dependent apparently on the atomic mass ratio of the two materials and the temperature, but to be independent of the film thickness. A linear relationship is observed between the reciprocal of the cross-plane thermal conductivity and that of the film thickness with the film thickness between 5.4 nm and 64.9 nm, which is in good agreement with results in the literature for a single film.     

Nonstationary heat conduction in three-dimensional bodies with nonsmooth inclusions

We study the problem of nonstationary heat conduction for a three-dimensional body containing a nonsmooth inclusion whose thermal contact with the matrix is perfect. By using thermal potentials determined and studied on nonsmooth surfaces, the problem is reduced to a solvable system of integral equations. The results are illustrated by analyzing, as an example, a jump of temperature in a three-dimensional body containing a circular conic inclusion.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 2, pp. 36–44, March–April, 2004.     

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.     

The interaction between a screw dislocation and a nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface

The interaction between a screw dislocation and a circular nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface is investigated. By using Riemann-Schwartz’s symmetry principle integrated with the analysis of singularity of complex functions and the conformal mapping technique, the analytical expressions of the stress field in both the circular nano-inhomogeneity and the infinite matrix, the image force acting on the screw dislocation and the stress intensity factor at the crack tip are obtained. The influence of elastic mismatch of materials, inhomogeneity size, interface stress, wedge crack opening angle and the relative location of dislocation on the image force and on the equilibrium position of the screw dislocation and the shielding effect of the screw dislocation are discussed in detail. The results show that interface stress has a significant impact on the movement of dislocations near the interface, and the effect of interface stress enhances when the inhomogeneity radius decreases. With the decrease in the wedge crack opening angle, the influence of interface stress on the movement of the screw dislocation and on the SIF enhances. With the increment of the relative shear modulus, the influence of interface stress weakens with the screw dislocation locating in the inhomogeneity and strengthens with the screw dislocation locating in the matrix. When the screw dislocation is located in the inhomogeneity, the positive (negative) interface stress increases (decreases) the shielding effect, while this phenomenon is opposite when the screw dislocation locates in the matrix.     

Analysis of a screw dislocation interacting with an elliptical nano inhomogeneity

The interaction between a screw dislocation and an elliptical nano inhomogeneity embedded in an infinite matrix is investigated. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin-Murdoch model. The stress fields inside the inhomogeneity and matrix are then solved with the complex variable and conformal mapping method. The solution is of semi-analytical nature and is verified by studying a degenerated case wherein a screw dislocation interacts with a circular nano inhomogeneity. The image force on the screw dislocation is then calculated. The influences of the elastic mismatch between the inhomogeneity and matrix, the interfacial properties, the aspect ratio of the elliptic nano inhomogeneity and the position of the screw dislocation on the image force are systematically discussed.     

Thermal Green’s functions in plane anisotropic bimaterials

Summary. Greens functions solutions are presented for the plane problems of a thermal source and a thermal dislocation near the interface between two anisotropic, perfectly bonded semi-infinite bodies with different thermo-mechanical properties. The analysis utilizes the complex potential approach for both heat conduction and thermoelastic problems, and the solutions are built on those of the homogeneous case obtained in the current work.     

层间弱粘贴复合材料层板的热弹性脱层

本文建立了一个层间弱粘贴复合材料层板热弹性脱层模型。该模型建立在两个描述层间弱粘贴的基本假设基础上。层间位移不连续由层间粘贴的物理关系来描述，表现为层间位移跳跃值与层间残余横向应力的关系；层间温度不连续由层间传热薄层来描述，并据此给出一个计算层间温度跳跃值的计算公式，表现为温度跳跃值与层间横向张开量之间的关系。在此假设基础上，根据平衡方程和静态传热方程导出了正交柱面弯曲层板热弹性脱层解。算例显示了层间弱粘贴对层板热弹性响应的影响。     

A new finite element for treating plane thermomechanical heterogeneous solids

This study is concerned with the development and implementation of a new finite element which is capable of treating the problem of interacting circular inhomogeneities in heterogeneous solids under mechanical and thermal loadings. The general form of the element, which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix, is derived explicitly using the complex potentials of Muskhelishvili and the Laurent series expansion method. The newly proposed eight‐noded plane element can be used to treat quite readily the two‐dimensional steady‐state heat conduction and thermoelastic problems of an elastic circular inclusion embedded in an elastic matrix with different thermomechanical properties. Moreover, the devised element may be applied to deal with arbitrarily and periodically located multiple inhomogeneities under general mechanical and thermal loading conditions using a very limited number of elements. The current element also enables the determination of the local and effective thermoelastic properties of composite materials with relative ease. Three numerical examples are given to demonstrate its versatility, accuracy and efficiency. Copyright © 1999 John Wiley & Sons. Ltd.     

Dislocations by partition of unity

A new finite element method for accurately modelling the displacement and stress fields produced by a dislocation is proposed. The methodology is based on a local enrichment of the finite element space by closed form solutions for dislocations in infinite media via local partitions of unity. This allows the treatment of both arbitrary boundary conditions and interfaces between materials. The method can readily be extended to arrays of dislocations, 3D problems, large strains and non‐linear constitutive models. Results are given for an edge dislocation in a hollow cylinder and in an infinite medium, for the cases of a glide plane intersecting a rigid obstacle and an interface between two materials. Copyright © 2005 John Wiley & Sons, Ltd.     

Analytical study for the estimation of thermal properties of processed meat based on hyperbolic heat conduction model

This study proposes an analytical method in conjunction with existing experimental temperature to estimate the unknown relaxation time and thermal diffusivity of processed meat based on the hyperbolic heat conduction model. This analytical method is a combination of the Laplace transform and least squares methods. The thermal contact resistance at the interface between adjacent samples at different temperatures is assumed to be negligible. The relaxation time is estimated from the temperature jump at a specific measurement location. The thermal diffusivity is determined from the definition of the dimensionless spatial coordinate and the resulting relaxation time. The results show that the relaxation time and thermal diffusivity obtained are in good agreement with the existing results. The obtained dimensionless temperature history at a specific measurement location is close to the experimental temperature data. This means that the Cattaneo–Vernottee (CV) model can be suitable for this study. The proposed analytical inverse method can be applied to determine a more accurate estimate of such problems. A comparison of the estimate obtained from CV and dual phase lag models is made.     

Solution of the plane problem for anisotropic media containing an elliptic inhomogeneity with dislocation-like interface

A general form solution of the plane problem for anisotropic media containing an elliptic inhomogeneity with a dislocation-like interface is presented. It is based on the complex series expansion of the stress functions. A general procedure for the determination of coefficients in the series using a continuity condition for the stresses at the imperfect interface and a discontinuity condition for a jump in the normal or tangential displacements at the interface is illustrated. The jump in the displacement components is described by the dislocation-like model based on the assumption that discontinuity of displacement across the interface is linearly proportional to the elastic displacement at the interface in the inhomogeneity. The model can reasonably characterize the imperfect interface from perfect bonding to complete debonding due to separate or combined effect of eigenstrains and far-field tension. Convergence of the results is obtained by truncating a finite number of terms in the series. The present solution is verified with available analytical results for the case of a perfect interface. In this paper, the pattern for the stresses in the heterogeneous anisotropic materials is shown. The effect of imperfect parameters on the distributions of stresses is also discussed. The method and procedure proposed in the paper are useful in analyzing the strength and failure of anisotropic materials containing an inhomogeneity with imperfect interface.     

Non-steady state heat conduction across an imperfect interface: A dual-reciprocity boundary element approach

The two-dimensional problem of determining the time-dependent temperature in a bimaterial with a homogeneously imperfect interface is considered. A temperature jump which is proportional to the thermal heat flux is assumed across the imperfect interface. Through the use of the corresponding steady-state Green's function for the imperfect interface, a dual-reciprocity boundary element method is derived for the numerical solution of the problem under consideration. To assess the validity and accuracy of the proposed method of solution, some specific problems are solved.