Heat Flux Intensity Factor Research Articles

Partially insulated cracks in orthotropic materials under steady state thermo-mechanical loadings

An attempt has been made to investigate the cracked composite medium with thermally conducting multiple cracks at its interface of the dissimilar orthotropic media subjected to thermal and mechanical loadings. The present article examines the governing equations, the associated differential constraints and boundary conditions those govern the temperature field and stress field. Fourier integral transforms, together with their corresponding inverses, have been employed to convert the singular integral equations into a set of linear equations by using the concept of orthogonality and completeness. The numerical technique, namely the Schmidt method, which employs a polynomial expansion of the unknown functions, is used to solve algebraic equations to determine the desired temperature and stress fields. The semi-analytical findings of the expressions for heat flux intensity factor (HFIF), stress intensity factor (SIF), and crack displacements have been mentioned in the article. Both mode-I and II types of cracks have been studied to investigate the effect of thermal conductivity. Detailed graphical representations effectively illustrate the numerical results and the interrelations between crack lengths, bonded strip width, and the partial conductivity coefficient of the crack surface. As the thermal conductivity coefficient increases, the temperature difference tends to zero. The results reveal that an increase in thermal conductivity decreases the SIFs.

Numerical modeling of heat conduction in bodies with cracks

All structural materials contain certain microdefects. Various elements of aircraft and spacecraft made from these materials are subjected to high mechanical loads and thermal effects during operation, which, in turn, can lead to the evolution of defects and total fracture of individual parts of the mechanism. Therefore, a quantitative and qualitative estimation of the stress and temperature fields in different bodies with complex configuration of external loading and temperature disturbances is needed. This paper presents a technique for numerical modeling of the temperature field in a medium weakened by a large system of partially heat penetrable cracks, including the possibility of considering it as an infinite periodic system. The method is based on a numerical algorithm, which uses the expansion of the solution into a finite series. Every term of this series is a certain analytical solution of the heat conduction theory obtained by the authors. To verify the method, the numerical results were compared with the well-known analytical solutions both for a configuration with finite number of cracks and for the periodic system. Also the heat flux intensity factors for a doubly periodic system were determined.

Transient non-Fourier thermal interactions of two parallel cracks in porous metal foam

Experimental evidence shows heat conduction in the porous media would be non-Fourier under specific circumstances. The novelty of this research lies in considerations of the non-Fourier effect and the interactions of two parallel thermally insulated cracks in the porous metal foam. The transient propagation characteristics are examined by utilizing the Dual-phase-lag model, which incorporates the certain time required to establish local thermal equilibrium between the solid framework and surrounding pores. Fourier transform and Laplace transform are employed to solve the mathematical model simulating a porous strip subjected to abrupt thermal shock. After reducing the problem to two groups of singular integral equations, the transient temperature distributions and heat flux intensity factors are determined with the help of numerical Laplace inversion. Numerical calculations are carried out to evaluate the impacts of crack positions, the relative density, phase lags, and two crack lengths on the thermal characteristics, which would contribute to a thorough understanding of the ultrafast process of porous materials operating in extreme conditions.

Displacement discontinuity method for interfacial cracks in one-dimensional hexagonal quasi-crystal coating under thermal-mechanical loading

This study derived Green’s functions of the concentrated interfacial displacement and temperature discontinuities. Green’s functions and the displacement and temperature discontinuity method were used to analyze interfacial cracks in a one-dimensional (1D) hexagonal quasi-crystal (QC) coating under thermal-mechanical loading. The boundary integral-differential equations were established for an interfacial crack, the fundamental solutions were obtained for uniformly distributed temperature, and the displacement discontinuities were applied over a line element. The near-crack-tip oscillatory singularity of stresses was eliminated by introducing the Gaussian distribution function to approximate the delta function. After this treatment, the expressions of the stress and heat flux intensity factors without oscillatory singularity, and the energy release rate (ERR) at the crack tips were obtained. The displacement and temperature discontinuity boundary element method for numerical simulation is proposed, and the influence of relevant factors is briefly discussed.

Schmidt method to study the disturbance of steady-state heat flows by an arbitrary oriented crack in bonded functionally graded strips

This article aims to investigate the impact of a partially insulated crack on vertical and horizontal components of heat flow. The crack is arbitrary oriented and its center is located at the interface of bonded finitely thick functionally graded strips. Employing the integral transformation, the thermoelastic equations are reduced to a set of singular integral equations. The unknown variables of these equations are the jump in temperature and displacements across the crack surfaces. Representing the unknowns as a series of Jacobi polynomials, the solutions of the singular integral equations are obtained by employing the Schmidt method which further helps to determine the analytical forms of crack tip heat flux intensity factors (HFIFs) and thermal stress intensity factors (TSIFs). The results of the present study are also validated for a particular case.To quantify the strength of heat flow components, the crack tip HFIFs and TSIFs as a function of crack angle, crack insulation parameter, strips’ width, and non-homogeneity parameters’ ratio are demonstrated pictorially.

Heat Concentration around a Cylindrical Interface Crack in a Composite Tube

Cracks always form at the interface of discrepant materials in composite structures, which influence thermal performances of the structures under transient thermal loadings remarkably. The heat concentration around a cylindrical interface crack in a bilayered composite tube has not been resolved in literature and thus is investigated in this paper based on the singular integral equation method. The time variable in the two-dimensional temperature governing equation, derived from the non-Fourier theory, is eliminated using the Laplace transformation technique and then solved exactly in the Laplacian domain by the employment of a superposition method. The heat concentration degree caused by the interface crack is judged quantitatively with the employment of heat flux intensity factor. After restoring the results in the time domain using a numerical Laplace inversion technique, the effects of thermal resistance of crack, liner material, and crack length on the results are analyzed with a numerical case study. It is found that heat flux intensity factor is material-dependent, and steel is the best liner material among the three potential materials used for sustaining transiently high temperature loadings.

Non-Fourier Heat Conduction of a Functionally Graded Cylinder Containing a Cylindrical Crack

The present work investigates the problem of a cylindrical crack in a functionally graded cylinder under thermal impact by using the non-Fourier heat conduction model. The theoretical derivation is performed by methods of Fourier integral transform, Laplace transform, and Cauchy singular integral equation. The concept of heat flux intensity factor is introduced to investigate the heat concentration degree around the crack tip quantitatively. The temperature field and the heat flux intensity factor in the time domain are obtained by transforming the corresponding quantities from the Laplace domain numerically. The effects of heat conduction model, functionally graded parameter, and thermal resistance of crack on the temperature distribution and heat flux intensity factor are studied. This work is beneficial for the thermal design of functionally graded cylinder containing a cylindrical crack.

On a symplectic analytical singular element for cracks under thermal shock considering heat flux singularity

In a precise numerical modelling of cracks under thermal shock, the singularity issue resulted from heat flux should also be considered in addition to the one resulted from stress. The assumptions of constant temperature distribution usually adopted in the existing studies may lead to significant error. The concerned problem involves the discretization in both space and time domains. Numerical error resulted from the singularity issues in the space domain may be accumulated in the time domain. Hence, a unified framework which integrates reliable methods for both space and time domains are desired. In the present contribution, the classic thermal stress problem is restudied under the Hamiltonian system and the eigen functions are obtained analytically. A symplectic analytical singular element (SASE) for thermal stress analysis is reformulated based on the existing ones for thermal conduction and stress analyses. The singularity issues of both stress and heat flux are considered. A unified framework is formed with the precise time domain expanding algorithm (PTDEA) for the time domain and the formulated SASE for the space domain. A self-adaptive technique is used for the PTDEA to improve the numerical efficiency. The time dependent fracture parameters i.e., heat flux intensity factors (HFITs) and the mixed mode thermal stress intensity factors (TSIFs) can be solved accurately without any post-processing. Numerical examples are given for verification and validation of the proposed method.

Study on transient heat flux intensity factor with interaction integral

Aim at determining of the heat flux intensity factor (HFIF) near a crack tip in structures under transient heat flux load, the transient JT integral is defined firstly and proved to be path-independent. Based on the path-independence of the transient JT integral, a transient interaction integral utilized to extract the transient HFIF is established by introducing a known auxiliary field. A singular isoparametric quadrilateral element with eight nodes, which is used to discretize the structure around the crack tip, by moving the midside nodes to one-fourth of the sides is presented. With this singular element, the 1/r singularity of heat flux near the crack tip can be exactly depicted. Numerical examples are carried out to verify the path-independence of the transient JT integral and the interaction integral method to extract the transient HFIF. Moreover, the transient HFIF in a structure with multiple cracks are numerically investigated by means of the presented method. The method proposed in this paper can be extended to investigate the thermo-mechanical problems.

An interaction integral method for calculating heat flux intensity factor with the XFEM

The JT integral is defined to extract the heat flux intensity factor (HFIF), which is used to describe the singularity of heat flux near the tip of a crack, and the path-independence of the JT integral is theoretically proven. Based on the JT integral, the relationship between the HFIF and an interaction integral is established through introducing auxiliary field function. The extended finite element method (XFEM), which enriches the standard finite element approximation by additional functions, is utilized to solve the temperature field of a cracked structure under specified thermal boundary condition. Numerical examples of cracked plates respectively with an edge crack and a center crack are used to verify the presented method, and the method is also used to investigate the HFIF near the tip of an inclined crack and the interaction effect of two cracks on the HFIF. The presented method provides a foundation for the investigation into the crack propagation behavior of a structure under thermo-mechanical loading.

Analysis of arbitrarily shaped planar cracks in three-dimensional isotropic hygrothermoelastic media

ABSTRACTIn the present article, a planar crack of arbitrary shape embedded in three-dimensional isotropic hygrothermoelastic media is investigated. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point and extended displacement discontinuities (EDD; including the displacement discontinuities, moisture concentration discontinuity, and the temperature discontinuity) are derived. The EDD boundary integral equations for an arbitrarily shaped, planar crack in the hygrothermoelastic medium are established in terms of the EDD. Utilizing the boundary integral equation method, the singularities of near-crack front fields are analyzed, and the stress, moisture flux, and heat flux intensity factors are all derived in terms of the EDD. As a special case, the analytical solution for a penny-shaped crack under uniform combined loadings is presented. The EDD boundary element method is proposed for numerical simulation. The numerical result for a penny-shaped crack subjected to uniform mechanical–moisture–thermal loading is compared with the analytical solution to verify the correctness of the proposed method. Two coplanar elliptical cracks subjected to combined loadings are simulated as an application, and the influences of applied loadings and the ellipticity ratio are discussed.

Extended displacement discontinuity method for analysis of penny-shaped cracks in three-dimensional thermal piezoelectric semiconductors

Abstract In this paper, the extended displacement discontinuity (EDD) boundary element method is developed to analyze a penny-shaped crack in the isotropic plane of a three-dimensional (3D) transversely isotropic thermal piezoelectric semiconductor (PSC). The generalized Almansi's theorem and the operator theory are used to obtain the general solutions under generalized loading. The fundamental solutions for extended displacement discontinuities (EDDs) applied on the penny-shaped crack surface, which include the displacement, electric potential, carrier density, and temperature discontinuities, are derived. By using the EDD boundary element method, the EDD under mechanical, electrical and heat loading near the edge of a penny-shaped crack is calculated. The stress and heat flux intensity factors of the crack are obtained. The influence of uniform and non-uniform heat loadings on the fracture of 3D transversely isotropic thermal PSCs is investigated.

Analysis of arbitrarily shaped, planar cracks in a three-dimensional transversely isotropic thermoporoelastic medium

The displacement discontinuity boundary integral equation method is extended to analyze planar cracks of arbitrary shape embedded in a three-dimensional, transversely isotropic, thermoporoelastic medium. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point, extended displacement discontinuities (including the displacement discontinuities, pore pressure discontinuity, and the temperature discontinuity) are derived. The extended displacement discontinuity boundary integral equations are established for an arbitrarily shaped, planar crack in the isotropic plane of the thermoporoelastic medium in terms of the extended displacement discontinuities. Using the boundary integral equations method, the singularities of near-crack front fields are analyzed, and the stress, fluid flux and heat flux intensity factors are derived in terms of the extended displacement discontinuities. To validate the analytical solution, the EDD boundary element method is proposed. The numerical simulation of a penny-shaped crack under combined uniform mechanical-pore pressure–thermal loadings is compared with the analytical solution to validate the correctness of the proposed method. As an application, two coplanar elliptical cracks are numerically simulated. The influences of the applied, combined mechanical-pore-pressure-thermal loadings, the crack distance, the ellipticity ratio as well as the size of cracks are all studied.

Extended displacement discontinuity method for analysis of cracks in 2D thermal piezoelectric semiconductors

The extended displacement discontinuities method has previously been used for crack analysis of elastic materials, piezoelectric media, magneto-electro-elastic media and piezoelectric semiconductors. Here, this method is extended to study cracks in two-dimensional n-type thermal piezoelectric semiconductors. The extended displacement discontinuities include the conventional displacement discontinuity, electric potential discontinuity, carrier density discontinuity, as well as temperature discontinuity across crack faces; correspondingly, the extended stresses represent conventional stress, electric displacement, electric current, and heat flux. Employing a Fourier transform, the fundamental solutions for a line crack under uniformly distributed extended displacement discontinuities on the crack faces are derived under mechanical, electrical, and heat loading. Based on the obtained fundamental solutions, an extended displacement discontinuity boundary element method is developed. The stress and heat flux intensity factors at the crack tip are calculated under different combined loadings.

Thermal stresses due to a uniform heat flow disturbed by a pair of offset parallel cracks in an infinite plane with orthotropy

Abstract This paper presents an investigation of the thermoelastic problem of two offset parallel cracks in an infinite orthotropic plane. It is assumed that the orthotropy is attributed to the unidirectional fiber-reinforcement and the insulated cracks are aligned along the fiber direction disturbing a steady-state uniform heat flow. Based on the use of Fourier integral transform method, formulation of the current crack problem is reduced to two sets of Cauchy-type singular integral equations for heat conduction and thermal stress analyses, which are solved via the expansion-collocation technique. The thermally-induced singular interactions between the two neighboring cracks are elaborated by defining and evaluating the heat-flux intensity factors and the thermal-stress intensity factors. Numerical results include the variations of such near-tip field intensity factors for various combinations of material and geometric parameters of the cracked orthotropic medium. In particular, the effect of degree of thermoelastic orthotropy on the strength of heat flux and thermal stress singularities is addressed by varying the fiber volume fraction of fibrous composites.

Analysis of an arbitrarily shaped interface crack in a three-dimensional isotropic thermal elastic bi-material. Part 2: Numerical method

The displacement discontinuity method is developed to analyze an arbitrarily shaped planar interfacial crack in a three-dimensional isotropic thermal elastic bi-material under combined thermo-mechanical loadings. The fundamental solutions for uniformly distributed displacement and temperature discontinuities applied over a triangular element are obtained via the displacement and temperature discontinuity- boundary integral-differential equation method. In order to eliminate the oscillatory singularity near the crack front, the Delta function in the fundamental solutions is approximated by the Gaussian distribution function, and accordingly, the unit Heaviside step function is replaced by the Error function. The stress and heat flux intensity factors without the oscillatory singularity as well as the Energy release rate are presented. An elliptical interface crack is analyzed as an example to validate the fundamental solutions and the proposed numerical method. The influences of thermal loading and material-mismatch on the thermo-mechanical response are studied, the difference between the homogeneous material and bi-material is also analyzed.

Analysis of an arbitrarily shaped interface cracks in a three-dimensional isotropic thermoelastic bi-material. Part 1: Theoretical solution

The displacement and temperature discontinuity boundary integral-differential equation method is developed for the analysis of interfacial cracks in a three-dimensional isotropic thermoelastic bimaterial. The fundamental solutions for a unit point temperature and elastic displacement discontinuities on the interface are derived, and the explicit expressions for the temperature and displacements and stresses are derived. The hyper-singular boundary integral-differential equations of displacement and temperature discontinuities are obtained for a planar interfacial crack of an arbitrary shape. By virtue of the obtained boundary integral-differential equations and the method proposed for interface cracks in purely elastic media, the singular indices and the singular behaviors of the near crack-tip fields are studied. Meanwhile, the stress and heat flux intensity factors as well as the energy release rate are derived in terms of displacement and temperature discontinuities.

Temperature and electric potential fields of an interface crack in a layered thermoelectric or metal/thermoelectric material

The interface crack problem in a layered thermoelectric or metal/thermoelectric material subjected to thermoelectric loadings is studied based on the nonlinear governing equations and complex variable method. The electric impermeable and heat semi-permeable crack boundary conditions are adopted. Explicit and closed-form solutions of temperature and electric potential on the interface crack surfaces, and crack tip thermoelectric fields are obtained. The influence of crack thickness to length ratio and thermal conductivity of air inside the crack on electric current density and heat flux intensity factors at crack tip is discussed. It is found that electric current density and heat flux intensity factors at the interface crack tip between metal interconnector and thermoelectric material are much bigger than those in homogenous or layered thermoelectric material. This explains why the failure (or microcrack initiation) of a thermoelectric generation/cooler element always appears at the interface between the metal electrode and thermoelectric material.

Non-Fourier heat conduction in a sandwich panel with a cracked foam core

Crack formation in cellular solids is often triggered by the presence of flaws caused by the manufacturing process that is used to build them. In thermal management applications, an imperfection embedded in the core of a sandwich panel might have a detrimental thermal impact that is often challenging to predict. This paper focuses on a theoretical study of non-Fourier heat conduction in a sandwich panel with a cracked foam core. With the aim of exploring the thermal response of a panel with porous core and skins made of a single material, we examine the role of crack position, relative density of the foam core, and other geometric parameters of the panel. Based on the assumption that the crack in the core is thermally insulated, i.e. no heat flux can pass through, we obtain the temperature distribution and heat flux intensity factor in the time domain via Fourier and Laplace transforms. The results are visualized in maps that show the influence of the size of the skin relative to that of the foam core and the crack location as well as the antagonist impact that the relative density of the foam core has on the maximum temperature and heat flux. The method presented here can be used to tailor the thermal response of sandwich panels.

Displacement and temperature discontinuity boundary integral equation and boundary element method for analysis of cracks in three-dimensional isotropic thermoelastic media

• The fundamental solutions for unit point displacement and temperature discontinuities are derived. • The displacement and temperature discontinuity boundary integral equation method is developed. • The displacement and temperature discontinuity boundary element method is proposed. • The stress and heat flux intensity factors are derived in terms of displacement and temperature discontinuities. The displacement and temperature discontinuity boundary integral equation and boundary element method are developed for the analysis of cracks in three-dimensional isotropic thermoelastic media. The fundamental solutions for unit point displacement and temperature discontinuities are derived, and the displacement and temperature discontinuity boundary integral equations are obtained for an arbitrarily shaped planar crack. The displacement and temperature discontinuity boundary integral equation method is developed to analyze singularities of near-crack border fields, and the stress and heat flux intensity factors are derived in terms of displacement and temperature discontinuities across crack faces. The fundamental solutions for a constant triangular element are obtained by integrating the fundamental solutions for unit point displacement and temperature discontinuities. On the basis of these solutions, the displacement and temperature discontinuity boundary element method is proposed. As an application, elliptical cracks are analyzed to validate the developed method.