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Abstract

In high-speed interconnect systems, high-speed backplane connectors are widely utilized. Its mechanical reliability is related to the stability of entire application systems directly. Insertion–extraction force and fatigue characteristics are important factors that affect the reliability of high-speed backplane connectors. To enhance the reliability of high-speed backplane connectors, in this study, a single CAD/CAE parametric model of the contacts is established. Not only the dynamic analysis of the insertion and extraction process but also the electro-thermal-structure coupling of the contacts are carried out using finite element analysis. By changing the insertion position of male terminal, the length of female terminal, and the thickness of male terminal, the insertion–extraction force can be optimized. According to the simulation of the insertion–extraction force and the electric-thermal-structure coupling results, both the mechanical life of this connector and the thermal-mechanical fatigue life under electrifying condition are evaluated. The results show that the mechanical life and thermal-mechanical fatigue life were 376 cycles and 286 cycles, respectively. The design solutions can provide the reference for the structural design, optimization, and fatigue life prediction of high-speed backplane connector.

Keywords

Interconnect systems high-speed backplane connectors insertion–extraction force electro-thermal-structure coupling fatigue life

Introduction

High-speed backplane connectors (HSBC) have been extensively used in communication equipment, super high performance server, giant computer, industrial computer, and high-end storage equipment.¹ Its main function is to transmit high-speed differential signals, single point signals, and high current between sub-card and backplane.² The performance of HSBC usually influences the reliability of whole system directly. Therefore, it is necessary to study the mechanical reliability of HSBC.^3,4 Insertion–extraction force is an important parameter that affects the reliability of connector. The changes of insertion–extraction force may cause contact pressure and contact resistance changes, which have shown a significant impact on the signal transmission. Furthermore, the fatigue performance is also a reflection of their mechanical reliability.^5–10 Therefore, it is important to study its mechanical fatigue and thermal-mechanical fatigue life. In this study, the mechanical reliability of the connector is evaluated from the aspects of insertion–extraction force and fatigue characteristics.

Li et al.¹¹ analyzed the insertion and extraction force of rigid and cylindrical slotted plugs. This analysis process is performed by plucking test and simulation calculation, respectively. Ke et al.¹² investigated the contact reliability and the insertion force of N electric connector. By changing structural parameters, impact facts of the length of socket, shrink range, groove width, and friction coefficient on the insertion force were studied. Not only the contact reliability is improved but also the contact resistance is reduced in the optimized structure. Hsu and Liao¹³ investigated the wear verification and analysis of metallic terminals for electronic connectors. Furthermore, the design solutions are validated by the associated experimental measurements. Huang et al.¹⁴ developed a 3D finite contact model and analyzed the mechanical behavior and fatigue life of micro-rectangular electrical connector.

In order to study the insertion and extraction characteristics of contacts in detail, it is necessary to use numerical simulation analysis approaches to get the force of contact during the insertion and extraction.^7,15–21 Then, based on the insertion–extraction force and the fatigue characteristic parameters of materials, the fatigue life can be simulated and calculated. Generally, the nonlinear dynamic simulation analysis of the insertion and extraction process can be carried out by the software ANSYS or LS-DYNA. The relative error can be controlled within the permissible range by interpolation force optimization. To evaluate the performance reliability of HSBC, two softwares are utilized here. ANSYS/nCode-Design Life was used to simulate the mechanical fatigue life, while Abaqus was used to simulate the stress and strain data of connector under current. In this study, a reliability approach and the corresponding model are provided to design high performance and high reliability electrical connectors.

Theoretical algorithm for contact problems and thermal-mechanical fatigue analysis

Augmented Lagrange method–based finite element (FE) solution

The purpose of the augmented Lagrange method is to perform penalty function to find the exact Lagrange multiplier. Compared with other methods, the augmented Lagrange method has more accurate optimal solution for solving constrained optimization problems. In nonlinear dynamic problems, the general formula of FE solution can be written as²²

$M^{t + Δ t} \overset{\cdot\cdot}{u} + (_{t}^{t} K_{L} + {}_{t}^{t}{K_{NL}}) u - {{}^{t - Δ t}Q}_{c} = {{}^{t + Δ t}Q}_{L} - {}_{t}^{t}F$ (1)

where $M$ is the element mass matrix of the shape with reference to time $t = 0$ ; ${}^{t + Δ t}{\overset{\cdot\cdot}{u}}$ is the unit node acceleration vector for time $t + Δ t$ ; $u$ is the displacement increment vector; ${}_{t}^{t}{K_{L}}$ , ${}_{t}^{t}{K_{NL}}$ , and ${}_{t}^{t}F$ represent matrices or vectors; and ${{}^{t + Δ t}Q}_{L}$ represents the equivalent node load vector. To deal with contact problems, the utilization of enhanced Lagrange method is combined with the displacement constraint equation which can be solved. For the kth point of contact, $({{}^{t + Δ t}Q}_{c})_{k}$ is as follows²²

$\begin{matrix} {({{}^{t + Δ t}Q}_{c})}_{k} = \\ - {[N_{c}^{T} (- μ \frac{{\bar{u}}_{1}}{{\bar{u}}_{T}} {{}^{t + Δ t}e}_{1} - μ \frac{{\bar{u}}_{2}}{{\bar{u}}_{T}} {{}^{t + Δ t}e}_{2} + {{}^{t + Δ t}e}_{3}) λ_{N}]}_{k} \end{matrix}$ (2)

Equation (2) is rewritten as follows

${({{}^{t + Δ t}Q}_{c})}_{k} = - {(K_{c λ})}_{k} {(λ_{N})}_{k}$ (3)

where

${(K_{c λ})}_{k} = {[N_{c}^{T} (- μ \frac{{\bar{u}}_{1}}{{\bar{u}}_{T}} {{}^{t + Δ t}e}_{1} - μ \frac{{\bar{u}}_{2}}{{\bar{u}}_{T}} {{}^{t + Δ t}e}_{2} + {{}^{t + Δ t}e}_{3})]}_{k}$ (4)

The displacement constraint equation can be expressed as

${(K_{cu})}_{k} {(u_{c})}_{k} = - {({}^{t}{\bar{g}}_{N})}_{k}$ (5)

where

${(K_{cu})}_{k} = {({{}^{t + Δ t}e}_{3}^{T} N_{c})}_{k}$ (6)

For all of n pairs of contact points, the equivalent node contact vector and the constraint equation of the system can be obtained by equations (3) and (5)

${{}^{t + Δ t}Q}_{c} = - K_{c λ} {{}^{t + Δ t}λ}_{N}$ (7)

$K_{cu} u_{c} = {}^{t}{\bar{g}}_{N}$ (8)

where

$K_{c λ} = \sum_{k = 1}^{n} {[N_{c}^{T} (- μ \frac{{\bar{u}}_{1}}{{\bar{u}}_{T}} {{}^{t + Δ t}e}_{1} - μ \frac{{\bar{u}}_{2}}{{\bar{u}}_{T}} {{}^{t + Δ t}e}_{2} + {{}^{t + Δ t}e}_{3})]}_{k}$ (9)

${{}^{t + Δ t}λ}_{N} = {[{({{}^{t + Δ t}λ}_{N})}_{1} {({{}^{t + Δ t}λ}_{N})}_{2} \dots {({{}^{t + Δ t}λ}_{N})}_{n}]}^{T}$ (10)

$K_{cu} = \sum_{k = 1}^{n} {({{}^{t + Δ t}e}_{3}^{T} N_{c})}_{k}$ (11)

${}^{t}{\bar{g}}_{N} = {[{({}^{t}{\bar{g}}_{N})}_{1} {({}^{t}{\bar{g}}_{N})}_{2} \dots {({}^{t}{\bar{g}}_{N})}_{n}]}^{T}$ (12)

Through substituting equation (7) into equations (1) and (8), the FE solution equation of the friction sliding contact state is obtained as follows

$M {}^{t + Δ t}{\overset{\cdot\cdot}{u}} + [\begin{matrix} {}_{t}^{t}{K_{L}} + {}_{t}^{t}{K_{NL}} K_{c λ} \\ K_{cu} 0 \end{matrix}] (\begin{matrix} u \\ {{}^{t + Δ t}λ}_{N} \end{matrix}) = (\begin{matrix} {{}^{t + Δ t}Q}_{L} - {}_{t}^{t}F \\ - {}^{t}{\bar{g}}_{N} \end{matrix})$ (13)

Calculation method of thermal-mechanical fatigue life

Considering the industry production experience, the actual fatigue life of HSBC is normally less than $10^{3}$ cycles. In this situation, the fatigue characteristics are analyzed using the Low Cycle Fatigue (LCF) analysis method. According to the Manson–Coffin formula, it can be expressed as

$Δ ε_{p} N^{α} = C$ (14)

where $Δ ε_{p}$ is the plastic strain range; N is the number of cycles to fatigue failure; α is the plastic index of the material, generally α = 0.3–0.8; and C is related to the section shrinkage $φ$ of the material. Their specific relationship can be denoted as

$C = \frac{1}{2} \ln \frac{100}{100 - φ} ~ \ln \frac{100}{100 - φ}$ (15)

The accuracy of fatigue life evaluation obtained by the Manson–Coffin formula is related to temperature. When the temperature is low, it has shown good consistency with the actual situation of material. When the temperature is high, the fatigue life calculated by the Manson–Coffin formula is larger than the actual value. This method can be used to verify the reliability of the data calculated by other methods.

LCF life is generally based on strain assessment.^23–31 So the thermal-mechanical fatigue life of this connector in the electric condition can be deduced using the strain life method. Based on the experiments, we can get the relationship between total strain and fatigue life. Their relationship can be described by an $ε - N$ strain life curve; the total strain-life curve is

$\frac{Δ ε}{2} = \frac{Δ ε_{e}}{2} + \frac{Δ ε_{p}}{2} = \frac{{σ'}_{f}}{E} {(2 N)}^{b} + ε'_{f} {(2 N)}^{c}$ (16)

where $σ'_{f}$ is the fatigue strength coefficient, b is the fatigue strength index, $ε'_{f}$ is the fatigue plasticity coefficient, and c is the fatigue plasticity index.

In double logarithmic coordinates, plastic strain life curve and elastic strain life curve are an approximate straight line. Based on a large number of experimental data, Manson pointed out that the determination of two straight lines, $Δ ε_{e} - N$ and $Δ ε_{p} - N$ , is based on four points.

Based on the study of the plastic strain–life curve slope and elastic strain–life slope on the double logarithmic coordinate plane, note that the fatigue strength index b varies from [−0.1, −0.15]. In this study, the fatigue strength index b is taken as −0.12. The fatigue ductility index c’s range of variation is between [−0.50, −0.70]. So the fatigue ductility index is taken as −0.60. Therefore, the common slope method parameters are calculated as follows

$\begin{matrix} b = - 0.12 \\ c = - 0.6 \\ σ_{f} = (1 + D) σ_{b} \\ D = \ln \frac{100}{100 - φ} \end{matrix}$ (17)

$Δ ε = 3.5 \frac{σ_{b}}{E} N^{- 0.12} + D^{0.6} N^{- 0.6}$ (18)

where $σ_{b}$ is the strength limit (tensile strength), $σ_{f}$ is the fatigue strength coefficient, $φ$ is the section shrinkage, E is the elastic modulus, and $Δ ε$ is the total strain.

FE modeling and simulation

The structure of HSBC

The model used in this study is a HSBC. Its contacts are mainly composed of the female terminal, the male terminal, and a rigid body base. The parametric model of the contacts is shown in Figure 1.

Figure 1.

Scheme of HSBC structure: (a) the structure of the whole model, (b) the structure of a wafer in the whole model, and (c) the structure of the simplified single simulation model.

FE modeling of contacts

Insertion force model of nonlinear contact

In this study, the kinematic analysis was carried out by ANSYS/LS-DYNA. The FE model used SOLID164 unit, base selected rigid body model. The beryllium bronze (QBe2) was used for the terminal materials. The QBe2 material properties used in bilinear material model are given in Table 1.

Table 1.

Material properties.

Material parameters	Contacts	Basement
Density (kg/mm³)	8.8 × 10⁻⁶	1.6 × 10⁻⁶
Elastic modulus (MPa)	1.13 × 10⁵	2.28 × 10³
Poisson’s ratio	0.3	0.417
Yield stress (MPa)	720 (600)	–
Shear strength (MPa)	4.9 × 10⁴	–

Because the structure of the female terminal is irregular, it is difficult to adopt the mapping meshing. Here, the free grid meshing is adopted to ensure that FE numerical calculation can be solved. According to the actual working conditions (only in one direction back and forth movement), the displacement degrees of freedom of the female and male terminals in X and Y directions were constrained, respectively. The base was fully constrained to ensure that the base during the insertion process can be always maintained in a fixed state. The X-direction displacement constraint of 2 mm/s was applied to both ends of the female terminal and the male terminal, respectively; Y and Z directions are fixed; and the base was set to rigid body. The contact pair was set as shown in Figure 2.

Figure 2.

Finite element model of contacts.

In order to analyze the change of insertion–extraction force, the contact surfaces of the female terminal, the male terminal, and the base were set to surface-to-surface contact. The rest was set to single-sided contact, automatic detection of contact surface, and contact calculation.³² According to the friction coefficient of common materials, we set the contact in the static friction coefficient as 0.2 and the dynamic friction coefficient as 0.18, respectively. Finally, a uniform axial displacement load was applied at both ends of the female and male terminals, and the speed of insertion and extraction was set as 2 mm/s.

Mechanical fatigue simulation model of contact

The results of DYNA simulation are directly introduced into the software nCode-DesignLife, which includes information such as displacement, velocity, stress, strain, and other information of each node of the contact pair during the plugging process. Since the mechanical fatigue of the HSBC is a LCF failure process, the stress–strain curve is incorporated by using the Manson–Coffin formula. The Morrow formula is chosen for the average stress correction equation. The material parameters for fatigue properties of the contact pairs are listed in Table 2.

Table 2.

Material parameters of beryllium bronze.

Material parameters	Value	Material parameters	Value
Elastic modulus (MPa), E	127,000	Fatigue strength index, b	−0.1
Yield stress (MPa), YS	1000	Fatigue ductility index, c	−0.6
Tensile limit (MPa), UTS	800	Elastic strain component, $ε_{e α}$	0.3
Fatigue strength factor (MPa), $σ'_{f}$	1300	Plastic strain component, $ε_{p α}$	0.5
Fatigue ductility coefficient, $ε'_{f}$	0.197

FE modeling of electro-thermal-structure coupling

In general, HSBC would generate heat under electrifying condition. Thus, the contact pair electro-thermal-mechanical coupling FE model was built to analyze the influence of thermal stress to the fatigue life of HSBC. The simulation process of Abaqus was divided into three stages. The first stage was to build the previous insertion–extraction force simulation to create a jointing situation between male terminal and female terminal. The second stage was to build the coupled electro and thermal analysis model to get the quantity of heat produced by electricity. The third stage was to map the temperature field interpolation gotten by electro and thermal coupling through the predefined field to targeted components in the first stage, which would be the temperature field of insertion–extraction force simulation.

The electro-thermal coupling FE model is shown in Figure 3. Besides the previous mechanical parameter, three coefficients are set: heat conduction coefficient of male and female connector of 0.16, thermal expansion coefficient of 17.7E-6, and electrical conduction coefficient of 23,200, correspondingly. The electrical conduction coefficient of setting substrate is 0.24E-3, the perveance is 0.0001, and thermal expansion coefficient is 2E-5.

Figure 3.

The electro-thermal coupling FE model. FE: finite element.

In the contact setting, the male and female terminals were surface-to-surface contact with friction coefficient of 0.2, contact heat conduction coefficient of 0.2, contact thermal generation coefficient of 1, and contact electrical conduction coefficient of 20. As shown in Figure 4, concentrated current value was evenly loaded on the 50 nodes of root of female terminal. Also, the zero potential boundary was built on the root of male terminal. Finally, we set the temperature boundary and define 35°C as the initial temperature (environmental temperature). We also take the root of male and female terminal as the heat sink point.

Figure 4.

The stress distribution of contact pair (MPa).

Results and discussion

Analysis of the insertion–extraction force

The stress distribution in totally jointing situation is shown in Figure 4. Note that the largest stress appears on the kinking place of female terminal. The maximum stress of the whole model is about 1267 MPa, which exceeds the material yield limit of 720 MPa. It will result in plastic strain and residual deformation. As shown in Figure 5, the residual stress is nearly 334 MPa. Since the maximum stress deviation significantly reduces the service life of the product, reducing the maximum stress value is necessary.

Figure 5.

The residual stress distribution (MPa).

As shown in Figure 6, different assembly dimensions of the male terminal can influence the insertion–extraction force. From the experimental data in Table 3, the maximum insertion force of the male terminal is about 0.37 N, while the maximum extraction force is about 0.17 N. In addition, the insertion–extraction force of the male terminal is most directly related to the surface roughness of the material and the positive pressure at the time of insertion. The maximum insertion and extraction force will be large if they are inserted from the base position. The greater the bending degree when the male head is inserted, the greater the positive pressure and the insertion–extraction force. Obviously, the insertion force data are the most consistent with the experimental results when the initial assembly size of the male terminal is 0.075 mm upward. The corresponding optimization design solutions are shown in Table 4.

Figure 6.

The contact force under different assembly dimensions of the male terminal.

Table 3.

Experimental data of male terminal insertion–extraction force.

	First mate	First unmate	Second mate	Second unmate	Third mate	Third unmate
Average	0.35	0.16	0.36	0.16	0.36	0.16
Minimum	0.34	0.16	0.35	0.16	0.36	0.16
Maximum	0.35	0.17	0.37	0.17	0.37	0.17

Table 4.

Comparison table of optimization results.

	Test values	Before optimization	Error (%)	After optimization	Error (%)
Mate force	0.35	0.348	0.57	0.36	2.86
Unmate force	0.17	0.10	41.8	0.168	1.18

It can be seen that in the range of 0.05–0.1 mm, the insertion–extraction force can better match the experimental data. Thus, regarding the male insertion position, it is appropriate to raise the thickness of 1/4 male to 1/2 male thickness in the actual assembly.

Figure 7 illustrates the effect on the insertion–extraction force when the thickness and the insert position of the male terminal and the thickness of the female terminal are changed. Specifically, the position of the male terminal and its thickness are taken as variables. Also, the insertion–extraction force curves are displayed in Figure 7(a), p1–p4. The p5 curve refers to the effect of female terminal’s length changes on the insertion–extraction force of the male terminal. From these curves, it can be observed that the thicker the male terminal and the greater the assembly deviation, the greater the insertion–extraction force. However, as shown in Figure 7(b), there is no direct relationship between the insertion–extraction force of the female terminal and the size and the assembly deviation of the male terminal. The p1–p4 insertion force curve basically coincides. But the length of the female terminal is longer in this situation. From the p5 curve, it can be seen that the insertion–extraction force of the female terminal is obviously smaller. When the length of the female terminal becomes longer, the female terminal rises to the same height and the deformation of the female terminal gets smaller. In general, the beam bending moment is equal to the beam by the external force relative to the centroid of the torque and algebra. Thus, the longer the length of the female terminal, the smaller the contact part’s external force.

Figure 7.

Different structure and assembly size of the contact force: (a) the contact force curve of male terminal and (b) the contact force curve of female terminal.

Depending on the size of the structure and the size of the assembly, researchers can obtain different insertion and extraction force curves, respectively. In this way, the electrical connectors can be designed under different special requirements to satisfy their mechanical reliability.

Fatigue life analysis

The result of mechanical fatigue life

The structural data of the insertion–extraction force simulation are introduced into nCode-DesignLife. Figure 8 shows the fatigue life chart obtained by FE simulations. Note that the minimum life of the entire model appears at the bend of the female terminal. The lowest value of 376 at node 402 corresponds to the maximum strain cloud obtained at the time of the previous insertion force simulation. Also, the maximum strain appears at the bend of the female terminal. Thus, the authenticity of the result can be judged.

Figure 8.

Mechanical life and maximum strain: (a) maximum strain cloud and (b) mechanical life cloud.

This simulation method can provide a reliable basis for evaluating the mechanical life of the connector. According to the simulation results, the structure can be optimized for the bending part, which can not only improve the mechanical life but also enhance the mechanical reliability of the backplane connector.

However, the result is only the mechanical insertion–extraction life of this connector, and the electric current could produce joule heat under the work of HSBC. Thus, it is essential to measure its thermal-mechanical fatigue life in the electric condition.

The calculation of thermal-mechanical fatigue life

The electro-thermal-structure coupling analysis of a single model was established in Abaqus. Figure 9 shows the temperature distribution of contact pairs under the environment temperature of 35°C and 0.7 A current loading. The temperature is concentrated and distributed on the contact part of female and male connector, and the reason for its formation is because the electric current density and electrical loss are mainly distributed on the contact part. The highest temperature reaches 71.42°C, meaning that it can be increased by 36.42°C. It is suitable for the actual working status of connector of this model.

Figure 9.

The temperature distribution of contact pair (°C).

Figure 10 shows the strain distribution in totally jointing situation. Note that the largest strain appears on the kinking place of lower surface of female connector, which is basically the same with the occurrence place of largest stress. Under the situation of no current passing, the largest total strain is 0.0345, while it is 0.0376 under the 0.7 A current.

Figure 10.

The strain distribution of contact pair (%).

After obtaining the maximum total strain, universal slope method was used to calculate the thermal-mechanical fatigue life of the connector. The tensile strength and reduction of area of materials found in brochure is 800 MPa and 50%, respectively. As shown in Figure 10, the largest total strain is 0.0345, while the largest total strain under electric condition is 0.0376. According to equation (18), the fatigue life without electric condition is 345 cycles. The fatigue life under 0.7 A current input is 286 cycles.

Based on the Manson–Coffin formula, through simplified method of double-logarithmic coordinates together with the strain data obtained by the electro-thermal-structure coupling, we can conclude that the thermal-mechanical fatigue life under electrifying condition is 286 cycles. This calculation result can be a more accurate assessment of the actual condition of HSBC fatigue life.

Conclusion

Through the FE simulation of the insertion–extraction force for contact pair, the changing rule of them is obtained. For the female terminal, the maximum insertion force is 0.28 N and the maximum extraction force is 0.09 N. For the male terminal, the maximum insertion force is 0.348 N and the maximum extraction force is 0.1 N.

Through the changing of structure assembly dimension, such as the insertion position, the thickness of male terminal, and the length of female terminal, this study obtained a series of curves for the insertion–extraction force. To conclude, in the actual work situation, properly raising the insertion position from 1/4 to 1/2 of the thickness of male terminal is feasible. Also, the main influencing factor of the insertion–extraction force of the female terminal is not the assembly dimension or the thickness of the male terminal, but the length of the female terminal. Overall, we can satisfy the working requirements under all kinds of situations by changing the specific structure assembly dimension.

The mechanical life of this connector is 376 cycles The thermal-mechanical fatigue life under electrifying condition is 286 cycles. This calculation result can be a more accurate assessment of the actual conditions of HSBC fatigue life. It can provide a reference for fatigue life prediction of HSBC.

In addition to fatigue property parameters of the material and working environment, the insertion and extraction force can also affect the fatigue life of HSBCs. The insertion and extraction forces are directly related to the structure and assembly of the connector. Thus, optimizing the model’s dimension and assembling is the basic method to improve the mechanical reliability of connector and enhance the stability of the system.

Footnotes

Handling Editor: José Correia

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This research is partially supported by the National Natural Science Foundation of China (No. 51605047),the Applied Basic Research Programs of Sichuan Province’s Science and Technology Department (No. 2019YJ0666),the Open Research Fund of Key Laboratory of Fluid and Power Machinery,Xihua University (No. szjj2013-03 and szjj2017-100),the Joint Foundation of China University of Petroleum-Beijing At Karamay (No. DK2018B00001),and the Natural Science Foundation of Guangdong Province,China (No. 611228787036).

ORCID iD

Haiqing Li

References

Archambeault

Connor

Halligan

et al . Electromagnetic radiation resulting from PCB/high-density connector interfaces. IEEE T Electromagn C 2013; 55: 614–623.

Ren

Wang

et al . Effects of temperature on fretting corrosion behaviors of gold-plated copper alloy electrical contacts. Tribol Int 2015; 83: 1–11.

Uhlig

Robertsson

. Limitations to and solutions for optical loss in optical backplanes. J Lightwave Technol 2006; 24: 1710–1724.

Johnson

. High-speed backplane connectors. In: Proceedings of the IEEE international symposium on electromagnetic compatibility, Long Beach, CA, 14–19 August 2011, pp.612–618. New York: IEEE.

Correia

Apetre

Arcari

et al . Generalized probabilistic model allowing for various fatigue damage variables. Int J Fatigue 2017; 100: 187–194.

De Jesus

AMP

Pinto

Fernandez-Canteli

et al . Fatigue assessment of a riveted shear splice based on a probabilistic model. Int J Fatigue 2010; 32: 453–462.

Ince

. Numerical validation of computational stress and strain analysis model for notched components subject to non-proportional loadings. Theor Appl Fract Mec 2016; 84: 26–37.

Zhu

Liu

Zhou

et al . Fatigue reliability assessment of turbine discs under multi-source uncertainties. Fatigue Fract Eng M 2018; 41: 1291–1305.

Meng

Zhang

Huang

. A concurrent reliability optimization procedure in the earlier design phases of complex engineering systems under epistemic uncertainties. Adv Mech Eng 2016; 8: 1–8.

10.

Kattoura

Telang

Mannava

et al . Effect of ultrasonic nanocrystal surface modification on residual stress, microstructure and fatigue behavior of ATI 718Plus alloy. Mater Sci Eng A 2018; 711: 364–377.

11.

Zhu

Chen

et al . Analysis of insertion force of electric connector based on FEM. In: Proceedings of the 21th international symposium on the physical and failure analysis of integrated circuits (IPFA), Marina Bay Sands, Singapore, 30 June–4 July 2014, pp.195–198. New York: IEEE.

12.

Zhu

et al . Contact resistance investigation of electrical connector with different shrink range. In: Proceedings of the 15th international conference on electronic packaging technology, Chengdu, China, 12–14 August 2014, pp.1146–1149. New York: IEEE.

13.

Hsu

Liao

. Wear analysis and verification of metallic terminals for electronic connectors. Eng Fail Anal 2012; 25: 71–80.

14.

Huang

Zeng

et al . Mechanical behavior and fatigue life estimation on fretting wear for micro-rectangular electrical connector. Microelectron Reliab 2016; 66: 106–122.

15.

Ince

. A mean stress correction model for tensile and compressive mean stress fatigue loadings. Fatigue Fract Eng M 2017; 40: 939–948.

16.

Meng

Liu

Yang

et al . A fluid-structure analysis approach and its application in the uncertainty-based multidisciplinary design and optimization for blades. Adv Mech Eng 2018; 10: 1–7.

17.

Zhu

Lei

Wang

. Mean stress and ratcheting corrections in fatigue life prediction of metals. Fatigue Fract Eng M 2017; 40: 1343–1354.

18.

Yuan

Wang

. An enhanced genetic algorithm-based multi-objective design optimization strategy. Adv Mech Eng 2018; 10: 1–6.

19.

. Reliability modeling of multiple performance based on degradation values distribution. Adv Mech Eng 2016; 8: 1–10.

20.

Zhu

Hao

et al . Critical plane–based multiaxial fatigue life prediction of turbine disk alloys by refining normal stress sensitivity. J Strain Anal Eng 2018; 53: 719–729.

21.

Meng

Huang

Wang

et al . Mean-value first-order saddlepoint approximation based collaborative optimization for multidisciplinary problems under aleatory uncertainty. J Mech Sci Technol 2014; 28: 3925–3935.

22.

Guan

Shu

Kang

et al . Multi-physics calculation and contact degradation mechanism evolution of GIB connector under daily cyclic loading. IEEE T Magn 2016; 52: 1–4.

23.

Meng

Zhang

Yang

. Interaction balance optimization in multidisciplinary design optimization problems. Concurrent Eng-Res A 2016; 24: 48–57.

24.

Zhu

Liu

Peng

et al . Computational-experimental approaches for fatigue reliability assessment of turbine bladed disks. Int J Mech Sci 2018; 142–143: 502–517.

25.

Meng

Yang

Zhang

et al . Structural reliability analysis and uncertainties-based collaborative design and optimization of turbine blades using surrogate model. Fatigue Fract Eng M. Epub ahead of print 15 August 2018. DOI: 10.1111/ffe.12906

26.

Zhu

Correia

et al . Evaluation and comparison of critical plane criteria for multiaxial fatigue analysis of ductile and brittle materials. Int J Fatigue 2018; 112: 279–288.

27.

Wang

Zhu

Wang

et al . High temperature fatigue and creep-fatigue behaviors in a Ni-based superalloy: damage mechanisms and life assessment. Int J Fatigue 2019; 118: 8–21.

28.

Wang

Meng

. Time-variant reliability assessment for multiple failure modes and temporal parameters. Struct Multidiscip O 2018; 58: 1705–1717.

29.

Yuan

Gong

et al . An enhanced Monte Carlo simulation-based design and optimization method and its application in the speed reducer design. Adv Mech Eng 2017; 9: 1–7.

30.

Meng

Huang

et al . Reliability-based multidisciplinary design optimization using subset simulation analysis and its application in the hydraulic transmission mechanism design. J Mech Des 2015; 137: 051402.

31.

Zhu

Foletti

Beretta

. Probabilistic framework for multiaxial LCF assessment under material variability. Int J Fatigue 2017; 103: 371–385.

32.

Zeng

Zhou

et al . Numerical modeling and optimization on micro-D electrical connector. Int J Numer Model El 2018; 31: 1–11.