Non-linear interface thermal conditions in three-dimensional thermoelastic contact problems

doi:10.1016/j.compstruc.2020.106354

Computers & Structures

Volume 241, December 2020, 106354

https://doi.org/10.1016/j.compstruc.2020.106354 Get rights and content

Highlights

•
A novel augmented Lagrangian formulation is proposed for 3D non-linear ITC.
•
Thermal contact conductance models and interstitial convective conditions are studied.
•
Influence of forced convection conditions and the presence of a CNT-based TIM are analyzed.
•
Influence of forced convection conditions is important for great conductive interface conditions.
•
Thermal conductance of the TIM is highly improved by the addition of CNT.

Abstract

In this paper, a new and robust formulation to solve three-dimensional thermomechanical contact problems under non-linear interface thermal contact (ITC) conditions is presented. This methodology allows us to consider non-linear thermal contact conductance, including the presence of thermal interface materials (TIM), and convective conditions at the interstitial contact region. The Boundary Element Method (BEM) is assumed in order to compute the thermoelastic influence coefficients. The interface thermomechanical contact methodology is based on an augmented Lagrangian formulation where an efficient iterative Uzawa type scheme of resolution is proposed. The methodology is validated by comparison with benchmark solutions and later on, some additional examples are presented to study the influence of several ITC conditions, i.e., the presence of TIM doped with carbon nanotubes (CNT) and/or interstitial boundary conditions, in the thermoelastic contact variables (i.e., the contact area, the contact tractions, the temperature or the thermal flux distributions) of this kind of problems.

Introduction

The thermoelastic response of two solids in contact has a greatly influence in contact problems. In thermoelastic contact problems, the temperature distribution causes thermoelastic distortion which influences the heat transfer problem and consequently, it leads not only to a non-uniform temperature and pressure distributions, but also to regions of separation. A joint conductance is created when two solids are in contact. This joint consists of numerous microcontacts formed by the highest contacting asperities and the associated microgaps (see Fig. 1). The heat flow from one surface to the other has to overcome these microgaps by means of a thermal contact resistance [1], [2], [3]. By introducing this thermal contact resistance between the contacting areas, the non-linearity inherent in the problem is increased due to the fact that the temperature field along the contact zone becomes tractions-dependent. Furthermore, the thermal boundary conditions at the interface are influenced by the mechanical contact conditions. So, the thermoelastic problems are coupled through the contact interface boundary conditions, and consequently, the steady-state solution may be non-unique and/or unstable [4]. It has been observed through many experiments [2] that if the interface microgaps are fullfilled by thermal interface materials (TIM), the thermal joint conductance increases significantly compared to the joint conductance when an interface fluid is assumed to be absent or the microgaps are filled with air. Therefore, TIMs are used to provide an effective heat conduction path between the solid surfaces due to their conformation (under pressure) to surface roughness and reasonably high thermal conductivity [5]. So, an efficient TIM should present the following properties: high conductivity and the ability to conform to the contact surfaces well [6]. For this reason, recent works [7], [8], [9], [10], [11], [12], [13], [14], [15] have studied how to increase the thermal conductivity of interface materials and lubricants through the addition of carbon nanotubes (CNTs). They have been proved to improve the thermal conductivity of TIMs.

In this context, it is necessary to develop advanced numerical formulations which make it possible to study this kind of problems. Several numerical formulations have been used to solve thermomechanical contact problems. Based on the Finite Element Method (FEM), Wriggers, Zavarise and coworkers[16], [17], [18], Johansson et al. [19], Strömberg et al. [20], [21], [22], Patunso [23], Hüeber and Wohlmuth [24] and Seitz et al. [25] have studied several kind of contact problems were friction and/or wear were involved. On the other hand, the Boundary Element Method (BEM) has been also proved to be a very suitable numerical formulation for this kind of mechanical interface interaction problems [26]. For this reason, some works [27], [28], [29], [30], [31], [32], [33] have tackled several thermoelastic contact problems using the BEM.

However, to the best author’s knowledge, those works have not considered the non-linear interface thermal conditions (i.e., interstitial conditions and TIMs) in three dimensional contact problems. For this reason, based on the works of Yovanovich and coworkers [3], [34], [35], [36], [37], [38], this work presents a formulation for non-linear interface thermal conductance conditions in 3D thermoelastic contact, valid for both conforming and non-conforming contact problems. The boundary elements technique is used to compute the thermo-elastic influence coefficients, while the contact modelling is based on an augmented Lagrangian formulation [39], [40], [41], [42]. An Uzawa iterative scheme [40], [43], [44], [45] is considered to solve the resulting thermomechanical contact problem. The proposed formulation has been applied not only to study several thermal contact conductance models in conforming and non-conforming contact, but also to study the influence of CNT-based thermal interface materials and interstitial boundary on the interface thermomechanical contact variables (i.e., thermal fluxes, temperatures or contact tractions).

Finally, the paper is organized as follows: Section 2 presents the basic governing equations and non-linear thermomechanical contact conditions. The literature on BEM formulation is quite extensive, so Section 3 describes the basic equations of the BEM to obtain the thermomechanical discrete equations. Then, the solution procedure is presented in Section 4. The methodology is validated by comparison with benchmark problems in Section 5, where additional exploration examples are presented and discussed in detail. Finally, some concluding remarks are presented in Section 6.

Section snippets

Governing equations

Let us consider two 3D isotropic bodies under quasi-static thermoelastic contact conditions. Each solid occupies the region $Ω^{l} \subset R^{3}$ ( $l = A, B$ ) with a piecewise smooth boundary $Γ^{l}$ , in a Cartesian coordinate system $(xyz)$ (See Fig. 2). Two partitions of $Γ^{l}$ are considered to define the thermal and mechanical boundary conditions. The first one divides $Γ^{l}$ into three partitions: $Γ_{θ}^{l}$ on which the temperature $\overline{θ}$ is imposed, $Γ_{q}^{l}$ with heat flux $\overline{q}$ prescribed and $Γ_{ic}^{l}$ which is defined as: $Γ_{ic}^{l} = Γ_{i}^{l} \cup Γ_{c}$ , i.e., $Γ_{c}$

Boundary element equations

The boundary integral equations for the 3D steady-state thermoelastic problem can be written for collocation points $ξ$ on $Γ^{l}$ , according to Brebbia [50] and Aliabadi [26], as: $c (ξ) θ (ξ) + \int_{Γ} Q (x, ξ) θ (x) d Γ (x) = \int_{Γ} Θ (x, ξ) q (x) d Γ (x),$ $\begin{matrix} c_{ij} (ξ) u_{j} (ξ) - \int_{Γ} U_{ij} (x, ξ) t_{j} (x) d Γ (x) + \int_{Γ} T_{ij} (x, ξ) u_{j} (x) d Γ (x) = \\ = \int_{Γ} {\overline{Q}}_{i} (x, ξ) θ (x) d Γ (x) - \int_{Γ} {\bar{Θ}}_{i} (x, ξ) q (x) d Γ (x), \end{matrix}$ where $x$ is a boundary point, $c (ξ)$ and $c_{ij} (ξ)$ are the free terms of the boundary integral equations for the thermal and the elastic problems, respectively, and $Q (x, ξ), Θ (x, ξ), T_{ij} (x, ξ)$ and $U$

Solution scheme

The resulting non-linear system (29) is solved using an iterative Uzawa’s method based on [51] for electroelastic contact problems. A similar solution schemes were presented in [40], [44], [45], [52]. To compute the variables on load step $(k)$ , $z^{(k)} = (x_{e}^{(k)}, u_{c}^{(k)}, θ_{c}^{(k)}, p_{c}^{(k)})$ , when the variables on previous instant $z^{(k - 1)}$ are known:

(I)
Initialize $z^{(0)} = z^{(k - 1)}$ and iterate using (n) index.
(II)
Solve: $[\begin{matrix} A_{x_{e}} & A_{u_{c}} & {\tilde{A}}_{θ_{c}}^{(n)} \end{matrix}] {\{\begin{matrix} x_{e} \\ u_{c} \\ θ_{c} \end{matrix}\}}^{(n + 1)} = - A_{p_{c}} p_{c}^{(n)} + F,$ where ${\tilde{A}}_{θ_{c}}^{(n)} = [\begin{matrix} A_{θ_{c}}^{A} - φ_{o} (p_{c}^{(n)}) A_{q}^{A} & φ_{o} (p_{c}^{(n)}) A_{q}^{A} \\ φ_{o} (p_{c}^{(n)}) A_{q}^{B} & A_{θ_{c}}^{B} - φ_{o} (p_{c}^{(n)}) A_{q}^{B} \end{matrix}] .$
(III)

Numerical results

In order to show the capabilities of the proposed BEM formulation for 3D thermoelastic contact problems, some numerical results are presented in this section. First, the methodology is validated solving a benchmarck problem (i.e., thermal interface conditions between two conforming surfaces in contact, according to Eq. (12), by comparison with analytical and numerical results presented in the literature [30]. Then, the presence of a TIM (Eq. (15)) between these surfaces is considered. The TIM

Summary and conclusions

This work presents a 3D boundary element formulation to solve three-dimensional thermoelastic contact problems. The non-linear interface thermomechanical contact restrictions are established by means of an Augmented Lagrangian formulation and the use of projection functions. The main novelty of this work is the use of non-linear interface thermal contact conductance conditions in the context of an augmented Lagrangian solution scheme and a boundary element methodology.

Using this numerical

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The research leading to these results has been supported by the Universidad de León and the Universidad de Sevilla.

References (55)

J.P. Gwinn et al.
Performance and testing of thermal interface materials
Microelectron J
(2003)
Shadab Shaikh et al.
The effect of a cnt interface on the thermal resistance of contacting surfaces
Carbon
(2007)
Zhidong Han et al.
Thermal conductivity of carbon nanotubes and their polymeer nanocompistes: a review
Prog Polym Sci
(2011)
G. Zavarise et al.
A numerical model for thermomechanical contact based on microscopic interface laws
Mech Res Comm
(1992)
P. Wriggers et al.
Thermomechanical contact – a rigorous but simple numerical approach
Comput Struct
(1993)
Lars Johansson et al.
Thermoelastic frictional contact problems: Modelling, finite element aproximantion and numerical realization
Comput Meth Appl Mech Eng
(1993)
Niclas Strömberg
Finite Element treatment of two-dimensioanl thermoelastic wear problems
Comput Meth Appl Mech Eng
(1999)
Peter Ireman et al.
Finite element algorithms for thermoelastic wear problems
Eur J Mech A/Solids
(2002)
Niclas Strömberg
An Eulerian approach for simulating frictional heating in disc-pad systems
Eur J Mech A/Solids
(2011)
S. Hüeber et al.
Thermo-mechanical contact problems on non-matching meshes
Comput Meth Appl Mech Eng
(2009)

G.I. Giannopoulos et al.

A BEM analysis for thermomechanical closure of interfacial cracks incorporating friction and thermal resistance

Comput Meth Appl Mech Eng

(2007)

J. Vallepuga Espinosa et al.

Boundary element method applied to three dimensional thermoelastic contact. jun

Eng Anal Bound Elem

(2012)

Gary F. Dargush et al.

Contact modeling in boundary element analysis including the simulation of thermomechanical wear

Tribol Int

(2016)

I. Ubero-Martinez et al.

The effect of conduction and convective conditions at interstitial regions on 3d thermoelastic contact problems

Eng Anal Bound Elem

(2019)

M.G. Cooper et al.

Thermal contact conductance

J Heat Mass Transf

(1969)

Majid Bahrami et al.

Effective thermal conductivity of rough spherical packed beds

Int J Heat Mass Transf

(2006)

P. Alart

A mixed formulation for frictional contact problems prone to Newton like solution methods

Comput Meth Appl Mech Eng

(1991)

J.C. Simo et al.

An augmented lagrangian treatment of contact problems involving friction

Comput Struct

(1992)

L. Rodríguez-Tembleque et al.

Anisotropic wear framework for 3d contact and rolling problems

Comput Meth Appl Mech Eng

(2012)

L. Rodríguez-Tembleque et al.

3d coupled multifield magneto-electro-elastic contact modelling

Int J Mech Sci

(2016)

L. Rodríguez-Tembleque et al.

A fem-bem fast methodology for 3d frictional contact problems

Comput Struct

(2010)

J. Dundurs

Distorsion of a body caused by free thermal expansion

Mech Res Comm

(1974)

M.G. Cooper et al.

Thermal contact conductance

Int J Heat Mass Transf

(1968)

I. Savija et al.

Thermal joint resistance of conforming rough surfaces with grease-filled interstitial gaps

J Thermophys Heat Transf

(2003)

Bahrami Majid, Culham JR, Yovanovich MM. Modelling thermal contact resistance: a scale analysis approach. J Heat Transf...

Barber JR. Thermoelasticity and contact. In: Third international congress on thermal stresses, Thermal Stresses ’99,...

J. Xu et al.

Silver nanowire array-polymer composite as thermal interface material

J Appl Phys

(2009)

Cited by (9)

Numerical study of thermal contact resistance considering spots and gap conduction effects
2024, Tribology International
Thermal contact resistance is a crucial parameter for the thermal management of electronic devices, as it directly impacts the heat dissipation efficiency across densely packed component interfaces. This paper proposes a finite element modeling approach to predict thermal contact resistance. The model is based on the optically measured surface topography of Al6061 alloy pairs using a white light interferometer. A mechanical-thermal sequential coupling approach is developed to simulate the contact area and temperature distribution of the contact model. Elastic-plastic deformable mechanics and interface heat transfer in terms of spots and gap conduction are incorporated into the approach. The research indicates that the pressure between longitudinal surfaces significantly influences heat transfer preceding the solid contact point. At an upper surface load pressure of 0.15 MPa, the thermal contact resistance is 329 mm²·K/W. With a gradual increase in load up to 0.6 MPa, the thermal resistance decreases to 244 mm²·K/W. Moreover, the transition of interstitial gas from free molecules to a continuous state results in decreased thermal conductivity. Specifically, at gas pressures of 5 Pa and 1 atm, the thermal contact resistance reduces from 557 mm²·K/W to 270 mm²·K/W. Furthermore, the numerical results, when compared to experimental data under different gas pressures, exhibit a discrepancy of less than 15%. This outcome effectively validates the accuracy of the proposed method.
A study of the radiation thermal boundary conditions influence in three-dimensional thermomechanical contact problems
2023, Engineering Analysis with Boundary Elements
Any body, by the simple fact of having a certain temperature on its surface, emits heat in the form of electromagnetic radiation. The heat flux emitted is proportional to the fourth power of the temperature of the solid, increasing the inherent non-linearity of the thermomechanical problems. This energy transfer, for systems at high temperatures, has greater relevance than the possible exchange of heat between bodies in contact by conduction or convection processes with a surrounding fluid in the interstitial areas. For this reason, this work presents a robust formulation to study three-dimensional thermomechanical contact problems where radiation boundary and interstitial conditions are simulated. The Boundary Element Method is used to compute the thermomechanical influence coefficients, and a double iterative procedure is used to ensure the achievement of the thermomechanical contact conditions. The final non-linear thermomechanical system of equations is solved using a Newton-Raphson procedure. The proposed formulation is validated with a comparison with some benchmark problems found in the literature and afterward applied to solve different three-dimensional thermomechanical contact problems where the thermomechanical variables are greatly affected by radiation boundary and interstitial conditions.
Modelling of magneto-electro-thermo-mechanical coupled behavior for the conductors under powerful impulse current and high-velocity friction
2022, Computers and Structures
Citation Excerpt :
When the boundary condition of structural deformation on the contact interface is considered, the rail and armature are treated as two regions in contact. Therefore, the given surface forces acting on the contact boundary is the contact boundary condition, and the unknown given surface forces should be addressed [38,39]. The rail-armature structure in Fig. 2 is meshed with rectangular elements.
When the conductors are placed under powerful impulse current, high-velocity friction, strong time-dependent magnetic field, such as the operating conditions for the rails of an electromagnetic launcher, the response of the structure will be strongly influenced by the extraordinary environments. Few experimental tools are directly available to these operating conditions to evaluate the reliability and durability of the conductors. Therefore, a transient multi-field coupling mathematical model is proposed to study the magnetic field, thermal field and structure deformation and the coupling characteristics between them by solving the problems of transient magnetic diffusion, transient heat transfer, quasi-static elasticity and contact mechanics in conductors with finite element method. The model validation is then presented by comparison with the experimental results. It is shown that the proposed multi-field coupling model could better describe the magneto-electro-thermo -mechanical response because of the consideration of the effects of multi-field coupling (EMC) and the interface contact pressure caused by structural deformation. The multiphysics characteristics and the velocity skin effects closely related to the velocity can be correctly described with our model. Especially, the EMC increases the magnetic flux density of the whole conductor, lowers the temperature in the high-temperature region but increases the temperature for the low-temperature region, which will make the temperature distribution more uniform in the conductor. Moreover, the initial section of the conductors is the most vulnerable to damage in terms of the thermal field and structural deformation. These results will be significant to understand the multi-field coupling mathematical modelling process and to study the conductor with extreme operating conditions.
Optimization design method of material stiffness of contact interface considering surface roughness for improving thermal contact performance
2023, International Communications in Heat and Mass Transfer
The contact interface of precision mechanical equipment, such as electronic equipment, is the crucial medium for transferring internal loads and physical properties and realizing the intended function of the equipment. The thermal contact performance of precision mechanical equipment determines its overall thermal performance. Previous studies have consistently ignored the activeness of differential distribution of contact interface material stiffness in improving thermal contact performance. Therefore, this paper creatively proposed a new idea to improve thermal contact performance by designing the material stiffness of the contact interface. Firstly, based on the precise analysis model of thermal contact performance considering surface roughness, a heuristic-based optimization design method of material stiffness of contact interface was developed, which took the contact interface temperature distribution non-uniformity (CITDN) and thermal contact resistance (TCR) as the optimization objectives. Then, based on this method, the optimization designs of electronic chip packaging and proton exchange membrane fuel cell (PEMFC) were carried out. The results showed that after optimization, the TCR and CITDN of the electronic chip package were reduced by 83.23% and 92%, respectively, and the TCR and CITDN of PEMFC were decreased by 99.78% and 88.71%, respectively. This method achieved a substantial improvement in thermal contact performance.
Numerical Study of Thermal Contact Resistance Considering Gas Pressure Dependent Conduction Effects
2023, SSRN
Boundary element analysis of three-dimensional thermomechanical orthotropic frictional contact problems
2023, AIP Conference Proceedings

View all citing articles on Scopus

View full text

Non-linear interface thermal conditions in three-dimensional thermoelastic contact problems

Highlights

Abstract

Introduction

Section snippets

Governing equations

Boundary element equations

Solution scheme

Numerical results

Summary and conclusions

Declaration of Competing Interest

Acknowledgments

Microelectron J

Carbon

Prog Polym Sci

Mech Res Comm

Comput Struct

Comput Meth Appl Mech Eng

Comput Meth Appl Mech Eng

Eur J Mech A/Solids

Eur J Mech A/Solids

Comput Meth Appl Mech Eng

Comput Meth Appl Mech Eng

Eng Anal Bound Elem

Tribol Int

Eng Anal Bound Elem

J Heat Mass Transf

Int J Heat Mass Transf

Comput Meth Appl Mech Eng

Comput Struct

Comput Meth Appl Mech Eng

Int J Mech Sci

Comput Struct

Mech Res Comm

Thermal contact conductance

Int J Heat Mass Transf

Thermal joint resistance of conforming rough surfaces with grease-filled interstitial gaps

J Thermophys Heat Transf

Silver nanowire array-polymer composite as thermal interface material

J Appl Phys