Damage and viscoplastic behavior of sintered nano-silver joints under shear loading

Engineering Fracture Mechanics 222 (2019) 106741

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Damage and viscoplastic behavior of sintered nano-silver joints under shear loading

T

⁎

Yao Yaoa,b, , He Gonga a b

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China Max-Planck-Institut für Eisenforschung GmbH, Max Planck Str. 1, D-40237 Düsseldorf, Germany

A R T IC LE I N F O

ABS TRA CT

Keywords: Sintered nano-silver Constitutive model Shear Plasticity Damage

Compared with most of the traditional solder materials, sintered nano-silver shows better hightemperature service performance and is expected to replace the traditional packaging materials in the next-generation of electronic devices. However, there exist substantial voids inside the sintered nano-silver layer after sintering, and the conventional continuum damage mechanics has certain limitation to describe the shear stress-strain relationship of sintered nano-silver materials. To analyze the void evolution and microstructure of sintered nano-silver, gold-plated sintered nano-silver specimens were tested under shear loading at different temperatures. A unified viscoplastic constitutive model is proposed to describe the properties of sintered nano-silver specimen under shear loading, in which the viscous overstress is replaced by the potential function of Gurson-Tvergaard-Needleman model and the effects of void on the yield surface of viscoplastic material is considered. To describe the influence of void evolution to the deterioration of mechanics properties at high temperatures, a damage variable is incorporated into the drag strength to indicate the damage mechanism with respect to the softening of sintered nano-silver material. The numerical predictions are compared with the experimental data, which shows that the developed model can describe the shear constitutive behavior of sintered nano-silver specimen with reasonable accuracy.

1. Introduction With the miniaturization trend of electric devices, the working environment and loading condition of electronic packaging becomes more complex. The traditional silicon-based devices are gradually replaced by high-energy wide-band gap semiconductor. The density of solder interconnections and working electrical current density are essentially increased with the miniaturization of packaging size. Extreme working temperature of the electronic packaging also causes challenging to the traditional joining materials. On the other hand, lead-free solder materials are widely applied in electronic packaging and replaced lead-rich solders due to the global promotion of environmental friendly electronic materials. However, lead-free solders also confront problems such as Joule heating melting and interlayer cracking when the interconnect size reduces to micro or nano scale, which also limits their application in the high temperature working environments. Sintered nano-silver materials are attracting more attention due to their excellent conductivity and high temperature serviceability. To understand better the effects of processing conditions on the interconnected structure, Stuckner et al. [1] investigated the effects of sintering temperature, environment and time on the microstructure. Sakamoto et al. [2] tested three different silver-filled

⁎

Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Yao).

https://doi.org/10.1016/j.engfracmech.2019.106741 Received 13 March 2019; Received in revised form 16 October 2019; Accepted 17 October 2019 Available online 21 October 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 222 (2019) 106741

Y. Yao and H. Gong

sintered joint materials to find the promising alternatives for the application in high power semiconductor devices under ultra-high temperature. Paknejad and Mannan [3] sorts out external factors (sintering pressure, metallization, sintering temperature etc.) of material properties in the processing, and proposed methods to improve the sintering process. Recent researches on nano-silver have been reviewed by Siow [4]. Factors affecting the tensile strength, shear strength and thermal fatigue properties of nano-silver have been discussed. In addition, scholars have made efforts to improve the reliability of nano-silver paste from the perspective of materials science and preparation. Li et al. [5] reduced the voids of sintered nano-silver joints by ultrasonic-assisted pressureless sintering, which improves the bonding performance of large area chips. Tan et al. [6] tested the thermal properties of nanopaste mixed with nano-copper and nano-silver. Guo et al. [7] investigated nano-silver solders with better resistance to electrochemical migration and sintering by adding nano-silver wires and nano-copper particles into nano-silver paste. Siow [8] proposed that nano-silver has potential to replace the traditional solder materials in the next generation of electronic devices. Nano-silver particles are stored in paste and mixed with organic solvents to prevent the aggregation of silver particles [9,10]. Up till now, investigation on constitutive behavior of sintered nano-silver is still insufficient. Yu et al. [11] adopted Anand model to determine the cumulative plastic strain during sintering of nano-silver. Chen et al. [12] investigated the ratcheting effect of sintered nano-silver joints through experimental and numerical analysis, and compared the simulated ratcheting behavior with the Ohno–Wang, Armstrong–Fedrick (OW–AF) and the Anand model. Current available macroscopic phenomenological constitutive models are mainly based on the continuum mechanics. Continuum damage mechanics (CDM) and extended models are effective in the failure analysis of solder materials [13,14], which considers the deterioration of mechanical properties by defining the thermodynamic damage state. Based on the phenomenological theory, McDowell et al. [15] proposed a Unified Creep and Plasticity (UCP) model to describe the constitutive relationship of rate-dependent metals and alloys. For the traditional solder materials, the UCP model can describe the stress-strain behavior under different loading conditions with reasonable accuracy. Compared with the traditional solder joint, abundant voids were formed in the nano-silver joints after sintering [16]. Because of the porous properties of sintered nano-silver materials, the cracks are prone to initiate near the voids and the adjacent voids could converge to form macro-cracks. The traditional continuum mechanics based constitutive models can hardly characterize the deformation behavior of sintered nano-silver specimen. For porous medium materials, the influence of hydrostatic pressure on the plastic properties cannot be ignored due to the evolution of voids. McClintock [17], Rice and Tracy [18] investigated the growth of voids in infinite matrix, revealing the importance of hydrostatic pressure on void growth. Gurson [19] proposed a constitutive model considering the void nucleation, growth and coalescence, which describes the influence of void evolution on the plastic deformation by defining an internal damage variable as a function of void volume fraction. Tvergaard [20,21] introduced three modified parameters into the Gurson model, which further improve the accuracy of prediction compared with the experimental results. Tvergaard and Needleman [22] divided the void volume fraction into a piecewise function on basis of the void evolution process by considering the effect of void coalescence. The work is usually referred as Gurson-Tvergaard-Needleman (GTN) model. GTN provides a meso-physical approach to analyze the elastic-plastic behavior of porous media materials, which can better describe the effects of micro-voids evolution on the ductility behavior. The damage is associated with void evolution, which defines the damage evolution with a clear physical meaning. Gurson and the extended models show reasonable accuracy in simulating medium containing voids, and have been applied to concrete, rock and other materials [23–25]. Although the GTN model can be used to analyze the yield surface evolution of porous media materials, it cannot directly describe the mechanical properties of viscoplastic materials. In the current study, a modified unified creep and plasticity model is proposed for describing the shear failure of nano-silver material. Based on the viscoplastic constitutive theory, the viscous overstress of sintered nano-silver is calculated by the yield potential function of the GTN model. It is noted that the void evolution plays a key role in deterioration of the macroscopic mechanical properties at high temperatures, Bonora [26] proposed a damage model to predict the ductile failure of materials under cycle loading, the model is adopted to characterize the damage evolution in the present study, in which the failure mechanism is affected by voids nucleation, growth and coalescence. Furthermore, the damage variable is coupled into the drag strength to represent the damage function with respect to the softening of nano-silver materials. The developed model is verified by comparing with the experimental results. 2. Experimental analysis and discussion To investigate the failure pattern of sintered nano-silver material, 16 specimens were developed for the shear test. The sintered nano-silver specimens were divided into four groups, shear experiments were carried out by using the Bose ElectroForce 3330 instrument with a strain rate of 1 × 10−2 s−1 at 0 °C, 50 °C, 100 °C, 150 °C, respectively. The schematic of thermo-mechanical coupling test device is shown in Fig. 1(a). The nano-silver paste for the experiment was provided by NBE Tech, LLC, which consists of micro and nano-scale silver particles and organic dispersants to prevent the polymerization of silver particles during storage. The shear specimens compose of two symmetric L-shaped gold-plated copper plates and nano-silver layer, which were designed and fabricated as sandwich type structure (gold-silver-gold). The specimens were heated to 280 °C under pressure-free condition and kept constant temperature for 40 min on the heating table, then cooled naturally to room temperature. The lap area of the shear specimen is 2 × 5 mm2. The thickness of lap zone is approximately 110 μm. To prevent the deviation from additional constraints produced by the fixed clamping, two holes were reserved at both ends of the specimen to apply the load. The loading holes were preset on the short limbs of shear specimen, which were riveted with the testing clevis of the tensile machine through the pin to realize the hinges, as shown in Fig. 1(b). Compared with the traditional solder materials, nano-silver paste contains additives such as diluents, adhesives and dispersants. During the sintering process, silver nanoparticles have larger specific surface area and smaller surface curvature radius, resulting in 2

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Gold-plated layer Copper substrate

2-1

3-1 4

6

5

Sintered nanosilver layer

3-2 2-2

(a)

(b)

Fig. 1. The experimental device: (a) schematic of loading system; (b) articulated shear specimens; where 1: Bose ElectroForce 3330 instrument; 2-1 and 2-2: Articulated fixture; 3-1 and 3-2: Bolt; 4: Gold-plated copper shear specimen; 5: Sintered nano-silver layer; 6: Temperature box.

lower melting point. The silver nanoparticles will contact with each other and form the sintering neck, and the sintered nano-silver layer densifies, as shown in Fig. 2(a) by Scanning Electron Microscope (SEM). However, the additives will be burned or evaporated with the increase of sintering temperature and form substantial micro-voids in the sintered layer. During the loading process, the void growth and coalescence is one of the key reasons for the deterioration of mechanical properties, as shown in Fig. 2(b). The fracture sections of sintered nano-silver specimen after shear test at 0 °C, 50 °C, 100 °C, 150 °C are shown Fig. 3. At lower temperature, the interface strength between nano-silver and gold plating is relatively low, and the interfacial failure mainly occurs in the specimens. With the increasing of temperature, the interfacial strength increases and the fraction section moves towards the inner silver layer, resulting in fracture of the silver layer. From 0 to 150 °C, the fracture section can be observed clearly at elevated temperatures, the failure mode of specimen changes from interfacial fracture to interlayer failure and shows better plasticity. The fracture morphology of the nano-silver layer at different temperatures is observed by SEM, as shown in Fig. 4. In the sintering process, silver nanoparticles contact with each other and form a sintering neck through the diffusion of silver atoms. After volatilization of the organic compounds, several silver particles form a hollow through sintering neck. Sintering neck is the bridge of force transmission in the nano-silver material and its strength is lower compared with nano-silver particles. The failure strength of sintered neck is one of the key factors determining the material strength. In the process of shearing test, the sintering neck is destroyed due to rotation and elongation of the voids, which leads to fracture of the nano-silver layer. This phenomenon can be observed on the fracture interfaces at different temperatures, as shown in Fig. 4(a)-(d). The silver nanoparticles are shown as white and the voids are black. Porosity can be determined by calculating the pixel ratio of the red line area in Fig. 4, as given in Table 1. The solid arrow represents the direction of shear stress and the dotted arrow denotes the direction of sintering neck failure. The fracture failure direction of sintered neck is consistent with the shear failure mode. For porous materials such as sintered nano-silver, the evolution of internal voids can cause damage accumulation and ultimate failure of specimen. In addition, the effect of hydrostatic pressure on the yield surface cannot be ignored, which is associated with the evolution of voids. Based on the experimentally observed failure mode of nano-silver, a constitutive model is developed to describe the stress-strain behavior of nano-silver material.

Fig. 2. Microstructure of sintered nano-silver observed by SEM: (a) polymerization of nano-silver particles; (b) internal voids of sintered nano-silver. 3

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Fig. 3. Fracture section of sintered nano-silver specimen.

Fig. 4. Fracture morphology of nano-silver layer observed by SEM: (a) 0 °C, (b) 50 °C, (c) 100 °C, (d) 150 °C.

Table 1 Internal porosity of sintered nano-silver joint. Porosity

Total pixel Void pixel Porosity (%)

Figure Fig. 4(a)

Fig. 4(b)

Fig. 4(c)

Fig. 4(d)

97,893 19,219 19.6

98,560 18,391 18.7

97,893 18,092 18.5

97,647 24,407 25.0

4

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3. A unified viscoplastic constitutive model for sintered nano-silver Compared with the non-unified creep theory, the unified creep and plasticity constitutive model attributes both plastic and creep strain to the dislocation motion, which can be determined by an integrated equation. The UCP theory has been widely adopted to analyze the mechanical properties of solder alloys [14]. The total strain rate ε ̇ is assumed to be consist of elastic strain rate and inelastic strain rate: (1)

̇ ε ̇ = ε ė + ε in

̇ incorporates plastic strain rate and creep strain rate. where the inelastic strain rate ε in The inelastic strain can be calculated by a flow rule in the unified creep and plasticity model. McDowell et al. [15] developed a phenomenological UCP model to describe the deformation of temperature and rate-dependent metal alloys, in which the viscoplastic flow rule is given as: ̇ = εklin

3 s s A ( v )nexp(B ( v )n + 1) θN 2 d d

(2)

where the material constant A , B , n can be obtained by fitting the relationship between plastic strain and stress, and s v is the viscous overstress. The isotropic hardening of materials can be represented by the drag strength in the constitutive model [27]. With the increasing of plastic strain, dislocations gradually accumulate and the internal defects continuously nucleate and growth, resulting in softening of material and larger inelastic strain at the same stress gradient. In order to describe the softening behavior, the drag strength coupled with shear damage can be defined as:

d=

d0 1 − Ds

(3)

where d 0 represents the initial drag strength; Ds is shear damage. According to the relationship between the environmental and melting temperature of materials, the value of the diffusivity parameter θ can be defined as [15]: −Q

θ=

for T ⩾

Tm 2

⎨ exp( −2Q (ln( Tm ) + 1)) for T ⩽ RG Tm 2T ⎩

Tm 2

⎧ exp( RG T )

(4)

where Q denotes the activation energy; Tm is the melting temperature; T represents temperature of working environment; RG is the gas constant. The flow direction N of inelastic strain is defined by Eq. (5) by neglecting the back stress:

N=

s ∥s∥

(5)

The stress deviation s is determined by:

s=σ−

1 tr (σ ) 3

(6)

The stress can be calculated by the Hooke's Law:

σ = (1 − Dt ) Cijkl (ε − ε in )

(7)

where Cijkl is a fourth order tensor representing the elastic matrix of material. For continuous medium, the hydrostatic pressure has negligible effect on yield stress and material hardening [28]. For viscoplastic materials, the loading equation can be expressed as:

F=

3 1 ∥s∥ − R = s v > 0 2 1−D

(8)

By neglecting the back stress, the viscous overstress can be calculated by:

sv =

3 1 ∥s∥ − R 2 1−D

(9)

where R is an isotropic hardening variable; s is the stress deviation tensor; 〈 〉 is Macauley brackets, defined as (x + ∥x∥ )/2 ; D is a damage variable. Experimental analysis suggests that the effect of hydrostatic pressure should be taken into account in the evolution of yield surface. Thus, Eq. (9) is not directly applicable to determine the viscous overstress of porous medium materials. The yield function of GTN model is defined as: 5

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Φ=(

σeq σy

3 σ )2 + 2q1 f cosh(− q2 m ) − (1 + q3 f 2 ) 2 σy

(10)

q12

for most of the metallic materials [20]; where q1, q2 , q3 are the modified parameters, Tvergarrd suggested q1 = 1.5, q2 = 1 and q3 = σm is the hydrostatic pressure; σy is the yield strength of undamaged material; σeq is the equivalent stress. The Mises equivalent stress is defined as:

3 1/2 σeq = ⎛ ss ⎞ ⎝2 ⎠

(11)

Recalling Eq. (8), the intermediate term can be regarded as the plastic yield equation, which can be translated into Eq. (12): (12)

F = Φ = sv > 0 where Φ is the yield function for plastic material. Eq. (10) can be appropriately deformed and gives:

3 σ 2 Φ = σeq − σy2 ((1 + q3 f ∗2 ) − 2q1 f ∗ cosh(− q2 m )) 2 σy

(13)

Therefore, the viscous overstress of porous materials can be expressed by:

sv =

3 σ 2 σeq − σy2 ((1 + q3 f ∗2 ) − 2q1 f ∗ cosh(− q2 m )) 2 σy

(14)

To determine the viscous overstress, the effect of voids on the inelastic properties of porous material is considered. Compared with Eq. (9), the developed Eq. (14) is more suitable for analyzing the viscoplastic properties of porous materials. 4. Damage variable and failure mechanism The correlation between adjacent voids can lead to localization of deformation when the ratio of radius and spacing of internal voids reach a certain magnitude. Consequently, the coalescence occurs between adjacent voids in two directions [29]: (1) perpendicular to the loading direction due to necking of the void spacing, as shown in Fig. 5(a); (2) shear direction due to void torsion and deformation, as revealed in Fig. 5(b). Then the cracks gradually form and result in the ultimate failure. In the GTN model, porosity is considered as a damage variable to describe the influence of void evolution on plastic properties of material. The model can accurately describe the ductile fracture associated with the propagation of voids under high stress triaxiality. The change of porosity is mainly attributed to hydrostatic pressure. If the increasing of void volume by nucleation is neglected, the material damage will not be considered in the GTN model for zero hydrostatic pressure or shear-dominated low stress triaxiality conditions. However, the void volume fraction remains unchanged for the pure shear case, damage will be accumulated due to rotation and elongation of voids [17,30]. Therefore, the traditional GTN model cannot accurately predict the damage caused by torsion and elongation of void under low stress triaxiality [31]. Thus, Eq. (13) can hardly describe the material damage caused by voids with shear or low stress triaxiality. A shear damage variable is then introduced in Eq. (14) to describe the material damage caused by void evolution in shear state [32]:

Fig. 5. Failure modes of porous materials: (a) necking failure of porous material; (b) shear failure of porous material [29]. 6

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Table 2 Parameters adopted in the modified GTN-UCP model without damage. Temperature

A

−40 °C 0 °C 25 °C 60 °C

d0

360 90 40 15

E

0.1%

0.01%

0.001%

60 64 63.5 51

56.4 59.3 56.4 41.2

53 46.6 35 22.2

n

σy

0.1%

0.01%

0.001%

21,504 16,268 13,046 8979

8 8 8 8

4 4 4 4

2 2 2 2

10.67 5.88 5.52 3.39

Table 3 Parameters adopted in the modified GTN-UCP model with damage. Temperature

A

d0

E

σy

n

α

Dcr

εth

εcr

225 °C 325 °C

30 6

38 28

3482 2986

0.51 0.11

1.8 1.8

2.0 2.0

0.95 0.94

0.023 0.023

0.17 0.17

Fig. 6. Comparison of experimental data and simulation results (0.001% strain rate).

Fig. 7. Comparison of experimental data and simulation results (0.01% strain rate).

sv =

2 ⎛ σeq ⎞ − σ 2 ((1 + q f ∗2 ) − 2q f ∗ cosh(− 3 q σm )) y 3 1 1 − D 2 2 σy s⎠ ⎝ ⎜

⎟

(15)

where Ds is the shear damage. By assuming isotropic damage [33], the distribution of microscopic defects is consistent along all the directions. To determine the damage evolution, the Helmholtz free energy function, which is concave with respect to temperature and other variables, is selected as the thermodynamic potential: 7

Engineering Fracture Mechanics 222 (2019) 106741

Y. Yao and H. Gong

Fig. 8. Comparison of experimental data and simulation results (0.1% strain rate).

Fig. 9. Stress–strain relationship associated with the damage for sintered nano-silver material.

Fig. 10. Comparison of experimental data and simulation results at (a) 225 °C and (b) 325 °C.

ψ = ψe (εe, T , D , p) + ψp (T , D , p)

(16)

where T is the temperature, p represents the accumulated inelastic strain. Based on the linear thermoelastic framework, the damage strain energy release rate Y can be obtained:

Y=ρ

∂ψe ∂D

=−

2 σeq

2E (1 − D)2

f(

σm ) σeq

(17) 8

Engineering Fracture Mechanics 222 (2019) 106741

Y. Yao and H. Gong

Constraints

Sintered nano-silver paste

Load

Copper substrates

(a) Finite element model and boundary conditions

Fig. 11. Finite element analysis to the lapped shear specimen.

Fig. 12. Comparison of numerical predictions and experimental data.

where 2

f(

σm 2 σ ) = (1 + ν ) + 3(1 − 2ν ) ⎜⎛ m ⎞⎟ σeq 3 ⎝ σeq ⎠

(18)

where ν is the Poisson’s ratio. A thermodynamic based damage evolution theory is proposed by Bonora [26], which is applicable to predict the damage evolution [14,34]. Subsequently, Yao et al. [14,34] modified the damage evolution equation by taking into account the effects of temperature and electrical current on the damage of solder materials. The acquisition of damage evolution can be calculated by a scalar convex damage dissipation potential function. According to the evolution of voids in different stages, the damage dissipation potential function is defined as follow [26]: 2

1 ⎛ Y ⎞ s0 ⎤ (Dcr − D)α − 1/ α FD = ⎡ ⎢ 2 − s0 1 − D ⎥ p2 + n / n ⎠ ⎣ ⎝ ⎦ ⎜

⎟

(19)

where s0 is a material parameter; α is a damage index, which can be modified to describe different forms of damage; p represents the cumulative equivalent plastic strain. 9

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The damage evolution rate is:

∂F Ḋ = −λ ̇ D ∂Y

(20)

where λ ̇ is the plastic multipliers defined as:

λ ̇ = p ̇ (1 − D)

(21)

The cumulative inelastic strain rate ṗ is defined as:

2 ̇ ε in ̇ ⎞ p ̇ = ⎛ ε in ⎝3 ⎠

(22)

It is assumed that the equivalent stress and cumulative inelastic strain satisfies:

σeq 1−D

= kp1/ n

(23)

where k is a material constant. Incorporating Eqs. (19), (21), (22) and (23) into Eq. (20) gives:

K2 σ ṗ α−1 Ḋ = (Dcr − D) α f ( H ) 2ES0 σeq p

(24)

Integrating Eq. (24) between [D , Dcr ] and [p , pcr ]: α

ln(p / pth ) σH ⎤ ⎫ ⎧ f( ) D = D0 + (Dcr − D0) 1 − ⎡1− ⎢ ln(εcr / εth) σeq ⎥ ⎬ ⎨ ⎣ ⎦ ⎭ ⎩

(25)

Eq. (25) is proposed to describe the material damage caused by rotation and elongation of voids during the shear process, which is incorporated into Eq. (15) in the developed constitutive model. 5. Numerical analysis Since there is no obvious yield point observed in the shear experiments of sintered nano-silver specimen, the strength corresponding to 0.2% of residual strain after unloading is assumed to be the yield shear strength. The instantaneous modulus, which is the ratio of instantaneous shear stress to shear strain, can then be calculated. The instantaneous modulus corresponding to yield point is defined as the elastic modulus. We assume that damage starts from the ultimate stress point, then the threshold strain εth is corresponding to the ultimate stress. Strain critical value εcr is taken as the ultimate strain. The critical damage Dcr is defined as:

Dcr = 1 −

τf (26)

τu

where τf is the shear stress at fracture. τu is the maximum shear stress corresponding to fracture strain, it is obtained by fitting the stress-strain relationship without damage. Material parameter A, n, α are obtained by fitting the experimental data containing damage. The void volume fraction of sintered nano-silver layer is around 10–13% [10]. The porosity of sintered nano-silver is found to be inversely proportional to the sintering temperature and time. According to the experimental conditions, the porosity is adopted as 20% in the numerical simulation. For uniaxial shear condition, the initial drag strength can be obtained based on the ultimate shear stress. The initial drag strength is set to be 1.7 times of the uniaxial shear strength from numerical analysis on the von Mises stress. The material parameters obtained from experiments are given in Tables 2 and 3. The modified GTN-UCP model is embedded into commercial software ABAQUS by a user defined subroutine (Umat), and the numerical simulations of lap-shear sintered nano-silver specimens are conducted under different conditions. The predicted shear stress-strain curves by the developed model are compared with experimental data from literature [11], as shown in Figs. 6–8. The dots lines are experimental data and the solid lines represent the simulated results. For strain rate of 0.001% and 0.01%, the experimental data are well represented by the numerical predictions. However, for strain rate of 0.1%, the predicted shear stress at 0 °C and 60 °C is less than the experimental observations. Because high strain rate and low temperature can lead to inadequate evolution of internal voids, which will increase the brittleness of silver nanoparticles and cause dispersion of experimental data [11]. At room or relatively lower temperatures, the damage of nano-silver joints is negligible due to smaller deformation. However, nano-silver material exhibits good ductility and obvious damage stage at high temperatures. For the constitutive model without damage, the relationship of stress and strain is shown in parts I and II of Fig. 9. At higher working temperature, there is no drastic fracture being observed when the stress exceeds the ultimate strength of sintered nano-silver, the stress softening process appears with damage accumulation and the material shows good ductility. To describe the influence of damage on the stress softening of sintered nano-silver, a damage variable is proposed and incorporated into the developed constitutive model. The strain is relative small before reaching the ultimate stress of the lap-shear sintered nano-silver specimen, therefore, it is assumed that the damage starts from the ultimate stress point, thus the threshold strain corresponds to the limiting stress. Considering the effects of damage, part II of Fig. 9 decreases and the stress-strain curve changes to part III due to damage accumulation at high temperature. 10

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Comparisons of numerical predictions and experimental data [35] at 225 °C and 325 °C are shown in Fig. 10. With the increasing of strain, the damage increases and the stress decreases while exceeds the threshold strain, which indicates the developed model can accurately describe the stress softening of lap-shear sintered nano-silver specimen caused by damage accumulation. Finite element analysis of sintered nano-silver layer in lapped shear specimen is conducted to verify the proposed model in structural reliability analysis, as shown in Fig. 11(a). The lap shear specimen included two copper substrates and the middle layer is sintered nano-silver. One side of the substrate is fixed and the load is applied at the other end. The applied load rate is 2 MPa/s and the nano-silver layer is tied to the copper substrate. Before the numerical simulation, the influence of the element mesh sensitivity to the numerical results was verified. The element mesh of sintered nano-silver layer is adopted as 0.1 × 0.1 × 0.025 mm, 0.025 × 0.025 × 0.025 mm, respectively. After reducing the element size, the results of numerical simulation did not change significantly, therefore, the numerical results do not show obvious mesh sensitivity and the adopted element size is reasonable. The stress-strain relationships of the lap shear specimens were simulated numerically at 25 °C, 225 °C and 325 °C, respectively, the parameters are kept as same as Tables 1 and 2. Through the finite element analysis, the stress distribution of the sintered nano-silver layer fluctuates under the shear load condition, as shown in Fig. 11(b). Along the direction of loading, the shear force at both ends of nano-silver layer is larger than that in the middle of silver layer, and the damage is prone to occur at both ends. In the experiments, we found that the nano-silver layer at both ends of the overlap area is seriously damaged, as shown in Fig. 3. For relatively lower temperature, the interfacial strength of the joint is lower and the failure begins at both ends, leading to the final interface failure. For relatively higher temperature, the failure becomes a mix mode consisting of both interfacial and interlayer failure. The numerical results are compared with the experimental data [36] in Fig. 12. Here the dots lines represent experimental data and the solid lines represent the numerical results. In general, the proposed constitutive model can predict the stress-strain relationship of lap specimens with reasonable accuracy at different temperatures. 6. Conclusion Four groups of sintered nano-silver specimens were tested subjected to shear loading at 0 °C, 50 °C, 100 °C, 150 °C, respectively. It is observed that void evolution dominates the damage due to break of the sintered neck. A unified viscoplastic constitutive model for sintered nano-silver material is developed based on the evolution of microstructure. The influence of void on failure mechanism of sintered nano-silver is considered, the viscous overstress is replaced by the GTN potential function. At high working temperature, a damage model considering ductility effect is proposed and incorporated into the drag strength to better describe the softening process. The numerical predictions are compared with the experimental data at different temperatures, which shows that the developed model can describe the shear stress-strain relationship of sintered nano-silver specimen with reasonable accuracy. Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgment The authors would like to acknowledge the financial support by the National Natural Science Foundation of China (No. 11572249, 11772257), Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No. CX201948) and the Alexander von Humboldt Foundation. 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