thermal stress analysis of chip scale packaging by using novel multiscale … · 2019. 7. 30. ·...

Research ArticleThermal Stress Analysis of Chip Scale Packaging by Using NovelMultiscale Interface Element Method

D S Liu Y W Chen and C Y Tsai

Department of Mechanical Engineering and Advanced Institute of Manufacturing with High-Tech InnovationsNational Chung Cheng University Chiayi Taiwan

Correspondence should be addressed to Y W Chen ywc0314gmailcom

Received 3 August 2018 Accepted 3 October 2018 Published 21 October 2018

Academic Editor Fazal M Mahomed

Copyright copy 2018 D S Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this study a multiscale interface element method (MIEM) is developed to evaluate the interfacial peel and shear stressdistributions in multilayer packaging structures A newly developed polygon interface element contained many sides that couldeasily connect with many other polygon elements of much smaller scale Simple model and less computation can obtain accuratestress distributions After verification of the validity of the developedmodelMIEM is applied to analyze thermal stress distributionof the chip scale package (CSP) In general a good qualitative agreement has been observed between the various results MoreoverMIEM can directly obtain the stress value at the interface which is also one of the advantages of this method in dealing withmultilayer structures

1 Introduction

Higher integration has become a trend in modern electronicpackaging One of the major challenges is to improve thethermomechanical reliability when the package is subjectedto the thermal loads However the conjunct interfaces nearthe free edge always suffer high stress gradients and evencan generate cracks because the packaging components arefabricated with different thermal and mechanical properties[1 2] Suhir [3] predicted the thermal stresses in the layeredstructures The method provided an easy-to-use estimationmethod for interfacial stresses between layers Pao et al[4 5] developed Suhirrsquos model for extension to multilayerstructures The proposed model can estimate the total forceproduced by the solder joint from the equation of forcebalance Oda and Sakamoto [6] proposed a special finite-element method for multiple laminated structures From theresults of the basic IC model the influences of varying chipsize and package thickness for the stress distributions arepresented Ye et al [7] analyzed a simple insulated gate bipolartransistor (IGBT) model and obtained the stress and strainin the solder layer From the results the thermal cycling hadsignificant effect on the solder layer reliability Ghorbani andSpelt [8 9] developed a two-dimensional analytical model

to investigate the interfacial thermal stresses in the solderjoints of a leadless chip resistor (LCR) And the results werecompared with finite element analysis

Finite element method (FEM) has been widely usedto solve engineering problems in recent decades But fordisplacement discontinuity problems the stability of numer-ical results will not be ensured Zhuo et al [10] presentedthe interface stress element method (ISEM) for solvingthe problem of discontinuous media mechanics The ideacame from rigid body spring model (RBSM) presented byKawai [11] Zhuo and Zhang [12] applied ISEM to ana-lyze engineering problems the ISEM is more accurate andapplicable compared with the traditional FEM Zhang etal [13] proposed an ISEM-FEM-IEM hybrid model whichsolved the conflict between computational accuracy andefficiency These examples demonstrate the applicability andadaptability of the model to the engineering In additionZhang et al [14] performed algorithm of ISEM for blockelements of arbitrary shape using natural coordinate Thenumerical results show that the method was feasible andaccurate

As mentioned above many studies have shown the use ofISEM in analyzing the engineering problems In this studyISEM is applied to analyze the thermal stress distribution of

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4962498 10 pageshttpsdoiorg10115520184962498

2 Mathematical Problems in Engineering

y

1C

2C

1e

2e

A

1e

2e

1C

2C

Block element 1

Block element 2

Interfaceelement

x

s(2)

s(2)

n

(2)

n

(2)

s(1)

s(1)

n

(1)

n

(1)

Figure 1 Schematic diagram of IEM model

multilayer structures Not only that a new polygon interfaceelement is developed called multiscale interface elementmethod (MIEM) which could easily connect with manyother polygon elements of much smaller scale Finally thedeveloped model was applied to analyze thermal stress dis-tribution of the chip scale package (CSP) In this case MIEMdisplay the accuracy of analyzing structural singularities

2 Derivations of Interface-Element Method

21 e Governing Equation The interface element method(IEM) is based upon the discipline of stress vector (tractions)continuity on the interfaces by assuming rigid element andsatisfying interfacial constraint equations In IEM modela complicated domain can be subdivided into a series ofsmaller regions named block element The block elementis undeformable and it can be separated and producesrigid body motion when an external load is applied Thedeformations are accumulated on the conjunct interface ofblock elements The internal energy caused by deformationof the interface area can be related to the work done by thesurface traction around the block element For equilibriumsolution it satisfies the fact that

∭Ω120590119894119895120575119890119894119895119889Ω = minus∬

1198780

119879119894120575119906119894119889119878= ∭

Ω119891119894120575119906119894119889Ω +∬

119878120590

119901119894120575119906119894119889119878(1)

The discretization form can bewritten as (2) whereΩ119890 119878119890120590 1198781198900and 119879119894 denote the element domain stress boundary interfaceregion and interfacial stress

minus 119899sum119890=1

[∬1198781198900

119879119894120575119906119894119889119878]

= 119899sum119890=1

[∭Ω119890119891119894120575119906119894119889Ω +∬

119878119890120590

119901119894120575119906119894119889119878](2)

22 Displacement Distribution in Interfacial Element A four-node 2D interface element for illustrated purpose is shownin Figure 1 in which 997888119899 and 997888119904 represent the local coordinatesystem that are normal direction and shear direction to the

interface surfaceThe displacement field of the block elementcould be expressed by the rigid body motion translationaland rotational such as

119906 = 119873 times 119906119892119906 = [119906 V]119879119906119892 = [119906119892 V119892 120579]119879

119873 = [1 0 119910119892 minus 1199100 1 119909 minus 119909119892]

(3)

where 119906 denotes the displacement vector at any point in theblock element 119906119892 is the degree of freedom at the mass ofcenter and119873 is the shape function With respect to the localsystem the relative displacements at interface element arethen derived as

1205751119897119900119888119886119897 + 1205752119897119900119888119886119897 = [1205751119899 + 1205752119899 1205751119904 + 1205752119904 ]119879

= minus [11987111199061 + 11987121199062]= minus1198711 [1198731

1199061119892 minus11987321199062119892]

(4)

In the above equation 120575119899 is normal displacement 120575119904 is theshear displacement and L is the coordinate transformationmatrix which reads

119871 = [[cos (997888119899 997888119909) cos (997888119899 997888119910)cos (997888119904 997888119909) cos (997888119904 997888119910)]]

1198711 = minus1198712 = 119871

(5)

23 Stress Distribution in Interfacial Element A measure ofthe relative displacement of the block element is referred to asstrain on interface As shown in Figure 2 for any point M oninterface surface the normal strain contributed from blockelement 1 is expressed as the change in length 1205751119899 per unit ofthe original length 1198751119872 = ℎ1 in the local system where 1198751119872is the normal distance from the centroid of block element 1to point M The shear strain is evaluated from the ratio ofslide distance 1205751119904 from block element 1 to the original lengthℎ1 The strain in the region of block element 2 is calculatedin a similar way In summary the components of strain aredefined as follows

120576119894119899 = 120575119894119899ℎ119894120574119894119904 = 120575119894119904ℎ119894

(6)

In plane stress situation the components of stress can beevaluated by the following equation

120590119894119899 = 119864119894120576119894119899 = 119864119894 120575119894119899ℎ119894120591119894119904 = 119866119894120574119894119904 = 1198642 (1 + V119894)

120575119894119904ℎ119894(7)

Mathematical Problems in Engineering 3

Arsquo

x

y

P1

P2

h1

h2

M

C1

C2

A B

Brsquo

BrdquoArdquoP1 P2

M1 M2

M2

M1P1

P2

h1 h2

1n

2n

2s

1s

Figure 2 Schematic diagram of deformation

where 120590119894119899 and 120591119894119904 are the normal stress and shear stresscontributed by block element iThe interface stress should bebalanced due to the system being in an equilibrium state Itsatisfied

1205901119899 = 1205902119899 = 1205901198991205911119904 = 1205912119904 = 120591119904 (8)

Combining (4) (6) and (8) gives constitutive equation of theinterface element

1205751119899 + 1205752119899 = 1205901119899ℎ11198641 + 1205902119899ℎ21198642 = 1205901198991198642ℎ1 + 1198641ℎ211986411198642120590119899 = 119864111986421198642ℎ1 + 1198641ℎ2 (120575

1119899 + 1205752119899)

1205751119904 + 1205752119904 = 2 (1 + V1) ℎ112059111199041198641 + 2 (1 + V2) ℎ212059121199041198642= 120591119904 2 (1 + V1) ℎ11198642 + 2 (1 + V2) ℎ2119864111986411198642

120591119904 = 119864111986422 (1 + V1) ℎ11198642 + 2 (1 + V2) ℎ21198641 (1205751119904 + 1205752119904 )

(9)

Representing the above equations in matrix form the inter-face stress

119879 = 119863120575 = minus119863 (11987111199061 + 11987121199062)= minus119863(119871111987311199061119892 + 119871211987321199062119892)= minus1198631198711 (11987311199061119892 minus 11987321199062119892) = minus119863119871119873lowast119906lowast119892

(10)

where 119879 indicates the interface stress 119863 is the elasticitymatrix and 120575 is displacement matrix in local coordinatesystem

119879 = [120590119899 120591119904]119879 120575 = [120575119894119899 120575119894119904]119879 119863 = [119889119899 0

0 119889119904] 119873lowast = [1198731 minus1198732] 119906lowast119892 = [1199061119892 1199062119892]119879 119889119899 = 119864111986421198641ℎ2 + 1198642ℎ1119889119904 = 1198641119864221198641ℎ2 (1 + ]2) + 21198642ℎ1 (1 + ]1)

(11)

Applying (11) into (2) gives the interface element domainintegral the element stiffness matrix for elasticity and thetraction term which can be evaluated Finally the governingequation of IEM is expressed as

119870119880 = 119877119870 = 119899sum119890=1

119870119890 119870119890 = int119904119895

119873lowast119879119871119879119863119871119873lowast119889119878119880 = [1199061119892 1199062119892 1199063119892 sdot sdot sdot]119879

119877 = 119899sum119890=1

119877119890 119877119890 = ∬Ω119890

119873lowast119879119891119889Ω + int119878119890120590

119873lowast119879119875119889119878

(12)

24 ermal Effects on Interfacial Stress For an isotropicmaterial temperature change results in a body expansion


or shrinkage but no distortion In other words temperaturechange affects the normal strains but not shear strain Thusby adding the thermal strain to (6) the total strain isexpressed as

120576119894119899 = 120575119894119899ℎ119894 minus 120572119894Δ119879120574119894119904 = 120575119894119904ℎ119894

(13)

in which 120572 is the coefficient of thermal expansion (CTE) andΔ119879 indicates the temperature change Further substitution of(13) into (7) yields

120590119894119899 = 119864119894120576119894119899 = 119864119894 120575119894119899ℎ119894 minus 119864119894120572119894Δ119879120591119894119904 = 119866119894120574119894119904 = 1198642 (1 + V119894)

120575119894119904ℎ119894(14)

Following the derived procedure described in Section 23 willresult in interface stress with thermal effects as

119879 = minus119863119871119873lowast119906lowast119892 minus 119863119879Δ119879119863119879 = [

[11986411198642 (1205721ℎ1 + 1205722ℎ2)1198641ℎ2 + 1198642ℎ10

]]

(15)

where119863119879 is the thermal elasticity matrixThe stiffnessmatrixand force matrix become119870119880 = 119877119870 = 119899sum119890=1

119870119890 119870119890 = int119904119895


119877 = 119899sum119890=1

119877119890119877119890 = ∬

Ω119890119873lowast119879119891119889Ω + int

119878119890120590

119873lowast119879119875119889119878 minus int1198781198900

119873lowast119879119871119879119863119879Δ119879119889119878

(16)

25 Boundary Condition The fixed boundary conditionrestricted the interface deformation in the region of blockelement 2 those terms are neglected in the elasticity matrixie

119863 = [[[[

1198641ℎ1 00 11986412ℎ1 (1 + V1)

]]]]

(17)

The often used symmetry boundary condition corresponds toa zero shear stress (119889119904 = 0) which gives

119863 = [[1198641ℎ1 00 0]]

(18)

1

2

34

5

6

n

g

x

y

Figure 3 The polygon element

All of the derivations of IEM fundamental equations aboveare based on plane stress condition For the plane straincondition Youngrsquos modulus Poisson ratio and CTE inmatrices119863 and119863119879 are replaced by

119864 = 1198641 minus V2

V = V1 minus V

120572 = 119886 (1 + V)(19)

26 Polygon Interface Element Formulation In the interfaceelement method the block element need not necessarilybe limited to the shape of the triangle or quadrangle Anarbitrary polygon block element is available The position ofcentroid of block element can be calculated by partition ofelement to be n-2 triangle as shown in Figure 3 The positionof centroid for each triangle is easy to compute by takingaverage of the coordinate of the 3 points ie

119909119894119892 = (1199091198941 + 1199091198942 + 1199091198943)3119910119894119892 = (1199101198941 + 1199101198942 + 1199101198943)3

(20)

and the areas for each triangle and the polygon are given as

119886119894 = 12 det

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091198941 11991011989411 1199091198942 11991011989421 1199091198943 1199101198943

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119860 = 119899minus2sum119894=1

119886119894(21)

The coordinates of the centroid of polygon element are

119883119892 = (sum119899minus2119894=1 119909119892119894119886119894)119860119884119892 = (sum119899minus2119894=1 119910119892119894119886119894)119860

(22)

A novel method called multiscale interface elementmethod (MIEM) is proposed in this paper based on the


Table 1 Geometrical parameters and material properties

Layer Material Thickness(mm)

CTE(ppm∘F)

YoungrsquosModulus(GPa)

Poissonratio

1 Al 1626 231 70 03452 Cu 0635 165 126 03433 Solder 0152 284 2217 04004 BeO 0635 63 345 03005 Si 0254 26 120 0420

(1)

(2)

(4)

1

2 3

4

56

(1)

(2)

(3)(3)

(4)

1

2 3

4

5

6

Figure 4 Multiscale interface element

0 05 1 15 2

0

02

04

06

08

1

12

14

16

18

2

Figure 5 The multiscale interface element model

concept of polygon element As shown in Figure 4 the nodes3 4 5 and 6 in element 1 are rearranged in a straight line andthe shape of the polygon element 1 looks to be a quadranglewhich connects three elements with one edge The advancedmesh technique allows the triangle or quadrangle element toconnect more than one element on each edge with multipleinterfaces The method can rapidly transform the elementdensity from fine to coarse to perform amultiscale modelingas shown in Figure 5

3 Numerical Examples

31 Five-Layered Stack Structure In this section two exam-ples are used to verify the validity of the developed model

Si

BeO

Cu

Al

Solder

5

4

2

1

3

2L

Figure 6 The five-layered stack structure [4]

0 654321X (mm)

0

05

1

15

2

25

3

35

Y (m

m)


The first example is a five-layered stack structure that wasoriginally considered by Pao [4] as shown in Figure 6 Thevalues of the geometrical parameters and material propertiesare listed in Table 1 The uniform temperature decrementis -65∘C in this example Half of the model is constructeddue to geometric symmetry about the vertical axis in thispaper Figure 7 depicts the half of model created using theMIEM refining the local mesh which is very flexible at theinterface and near the free end The interfacial peel andshear stress at layer 1 and 2 conjunct interfaces (AlCu) are


IEMAnalytical Pao[4]FEM

minus10

0

10

20

30

40

50

60

70

80

90

100

Inte

rfaci

al p

eel s

tress

(MPa

)

1 2 3 4 5 60Distance from center (mm)

Figure 8 The interfacial peel stresses at layer 1 and 2 conjunctinterfaces (AlCu)


0

2

4

6

8

10

12

Inte

rfaci

al sh

ear s

tress

(MPa

)


Figure 9 The interfacial shear stresses at layer 1 and 2 conjunctinterfaces (AlCu)

presented in Figures 8 and 9 respectively Figures 10 and 11present the interfacial peel and shear stress at layer 3 and4 conjunct interfaces (SolderBeO) respectively The resultsindicate that the stress values calculated at free end by Pao aregenerally higher than those calculated by MIEM and FEM


minus20

0

20

40

60

80

100

120

140

Inte

rfaci

al p

eel s

tress

(MPa

)


Figure 10 The interfacial peel stresses at layer 3 and 4 conjunctinterfaces (SolderBeO)



0

10

20

30

40

50

60

70

Inte

rfaci

al sh

ear s

tress

(MPa

)

Figure 11 The interfacial shear stresses at layer 3 and 4 conjunctinterfaces (SolderBeO)

Because Paorsquos model assumes each layer as a Bernoulli beamTimoshenko effect is ignored

32 LCR Model The second example is a leadless chipresistor (LCR) as shown in Figure 12 Ghorbani [9] providedan analytical solution to this problem and his data is used for



Layer Width(mm)

Thickness(mm)

CTE(ppm∘C)


Poisson ratio

Resistor 325 065 28 13100 030Solder Joints 076 012 220 6454 040Substrate 325 123 170 2200 028

Resistor

Substrate

Solder Joints

L L

Figure 12 Schematic of an LCR assembly [9]

0 35325215105X (mm)

0

02

04

06

08

1

12

14

16

18

2

Y (m

m)

Figure 13 The interface element model

this work to directly compare the results The values of thegeometrical parameters and material properties are listed inTable 2The shear modulus is determined from119866119894 = 1198641198942(1+]119894) resulting from all materials being assumed to be isotropicand homogeneous Figure 13 depicts the model created usingthe MIEM the grid size is divided into 13 of the originalat the interface of solder joints to improve the accuracyof the solution Under a uniform temperature increase of75∘C the interfacial peel and shear stress distributions atthe lower conjunct interfaces in the sample are presented inFigures 14 and 15 respectively It can be seen that Ghorbanirsquosanalytical model predicts shear stresses that reach zero at thefree end this condition is not satisfied by the MIE and FEmodels The results by MIEM are consistent with the trendof FEM but these values are closer to the analytical solution

1minus100minus90minus80minus70minus60minus50minus40minus30minus20minus10

01020304050

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM Ghorbani [9]Analytical Ghorbani [9]

Lower interface

Upper interface

Upper interface

04020 06 08minus04minus06minus08 minus02minus1Normalized distance from center (xl)

Figure 14 Interfacial peel stresses at upper and lower conjunctinterfaces

Overall those are reasonably in agreement with the solutionsof Ghorbani [9]

4 Applying Multiscale InterfaceElement Method to CSP

Figure 16 shows the chip scale package (CSP) model schemaThe half-length of the structure is considered and the valuesof the geometrical parameters and material properties arelisted in Table 3 The multiscale interface element methodis employed to analyze thermomechanical behaviors of CSPThe MIEM model for the CSP is depicted in Figure 17 Thethermal stress distribution of conjunct interfaces of the lasttwo solder joints and another layer is investigated in thisstudy because these conjunct interfaces are most likely tocrack Figure 18 depicts the FEMmodel with 26600 elementsfor the CSP A 4-node biquadratic plane stress quadrilateralelement with reduced integration (CPS4R in ABAQUS) isused The numerical results of MIEM model are verified bycomparing with FEM model as shown in Figures 19ndash22The solid and dashed lines denote the numerical solutions


Table 3 The geometrical parameters and material properties for CSP model

Layer Thickness(mm)

Elastic Modulus(GPa)

Shear Modulus(GPa)

CTE(ppm∘C) Poissonrsquos ratio

1 FR4 PCB 12 22 86 18 0282 Solder joint 028 222 79 284 043 FR4 Substrate 025 22 86 18 028

4 Underfill 005 5 19 60 035Solder bump 222 79 284 04

5 Si die 05 160 615 28 03

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1Normalized distance from center (xl)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

Inte

rfaci

al sh

ear s

tress

(MPa

)


Upper interface

Upper interfaceLower interface

Lower interface

Figure 15 Interfacial shear stresses at upper and lower conjunctinterfaces

Underfill with Solder Bumps

Die

FR4 Printed Circuit Board

FR4 Substrate

Solder Joint

L

Figure 16 The schematic of chip scale package (CSP) model

by using MIEM and FEM respectively Figures 19 and 20present the interfacial peel stress and shear stress on FR4PCB and solder joint conjunct interfaces Figures 21 and 22present the interfacial peel stress and shear stress on FR4substrate and solder joint conjunct interfaces These analysesare conducted with a temperature range of 165∘C (-40∘C to

0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5

Figure 17 The MIEM model

0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5

Figure 18 The FEMmodel (total number of elements 26600)

125∘C) As shown in these results the solutions of these twomodels are consistent

From the shear stress distribution the final point ofcalculated stress curve by MIEM is closer to the singularitythan the FEM and the value is higher because MIEMhas the characteristics of the smaller interface mesh Fromthe peel stress distribution a good agreement has beenobtained between the two sets of results MIEM can directlyobtain the stress value at the interface which is also oneof the advantages of this method in dealing with multilayerstructures



minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM

Figure 19 Interfacial peel stresses on FR4 PCB and solder jointconjunct interfaces


minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM

Figure 20 Interfacial shear stresses on FR4 PCB and solder jointconjunct interfaces

5 Conclusions

This research developed amultiscale interface element modelto evaluate the interfacial stresses in multilayer packagingstructures The peel and shear stress functions are calculatedin terms of interface element fractional distribution Two

IEMFEM

36 38 4 42 4434Distance from center (mm)

minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa

)Figure 21 Interfacial peel stresses on FR4 substrate and solder jointconjunct interfaces

IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)


Figure 22 Interfacial shear stresses on FR4 substrate and solderjoint conjunct interfaces

typical examples are used to verify the validity of the devel-opedmodelThepresent analyses provide better accuracy andrationality in interfacial peel and shear stresses

The developedmodel was then applied to analyze thermalstress distributions of the solder joint conjunct interfaces in aCSP model Using this efficient numerical technique a veryfine mesh pattern can be established around each conjunct


interface without increasing the degree of freedom of theglobal FEM solution In general the results presented inthis study have shown that the proposed MIEM algorithmprovides a fast direct and accurate tool for simulating thestress analysis of multilayer structures

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from The Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project by theMinistry of Education (MOE) in Taiwan This research wasalso supported by ROC MOST Foundation Contract NoMOST107-3017-F-194 -001

References

[1] D S Liu C Y Ni T C Tsay and C Y Kao ldquoThermal stressanalysis and design optimization of direct chip attach (DCA)and chip scale package (CSP) in flip chip technologyrdquo ASMEJournal of Electronic Packaging vol 113 no 3 pp 240ndash243 2002

[2] H Dou M Yang Y Chen and Y Qiao ldquoAnalysis of thestructure evolution and crack propagation of Cu-Filled TSVafter thermal shock testrdquo in Proceedings of the 18th InternationalConference on Electronic Packaging Technology ICEPT 2017 pp611ndash614 China August 2017

[3] E Suhir ldquoInterfacial stresses in bimetal thermostatsrdquo Journal ofApplied Mechanics vol 56 no 3 pp 595ndash600 1989

[4] Y-H Pao and E Eisele ldquoInterfacial shear and peel stresses inmultilayered thin stacks subjected to uniform thermal loadingrdquoJournal of Electronic Packaging Transactions of the ASME vol113 no 2 pp 164ndash172 1991

[5] Y-H Pao S Badgley R Govila L Baumgartner R Allor andR Cooper ldquoMeasurement of mechanical behavior of high leadlead-tin solder joints subjected to thermal cyclingrdquo Journal ofElectronic Packaging Transactions of the ASME vol 114 no 2pp 135ndash144 1992

[6] J Oda and J Sakamoto ldquoApplications of FEM for multiplelaminated structure in electronic packagingrdquo Finite Elements inAnalysis and Design vol 30 no 1-2 pp 147ndash162 1998

[7] H YeM Lin andC Basaran ldquoFailuremodes and FEM analysisof power electronic packagingrdquo Finite Elements in Analysis andDesign vol 38 no 7 pp 601ndash612 2002

[8] H R Ghorbani and J K Spelt ldquoInterfacial thermal stresses intrilayer assembliesrdquo Journal of Electronic Packaging Transac-tions of the ASME vol 127 no 3 pp 314ndash323 2005

[9] H R Ghorbani and J K Spelt ldquoInterfacial thermal stressesin solder joints of leadless chip resistorsrdquo MicroelectronicsReliability vol 46 no 5-6 pp 873ndash884 2006

[10] J S Zhuo Q Zhang and N Zhao ldquoInterface stress elementmethods for deformable body with discontinuous media such

as rock massrdquo in Proceedings of the 8th International Society forRock Mechanics pp 939ndash941 1995

[11] T Kawai ldquoA new discrete model for analysis of solid mechanicsproblemrdquo Seisan Kenkyu vol 29 pp 204ndash207 1977

[12] J S Zhuo and Q Zhange Interface Element Method for Dis-continuous Mechanics Beijing Science Publishing Company2000

[13] Q Zhang Z-B Zhou and J-S Zhuo ldquoMixed model for parti-tioned interface stress element method-finite element method-infinite element methodrdquo Jisuan Lixue XuebaoChinese Journalof Computational Mechanics vol 22 no 1 pp 8ndash12 2005

[14] Q Zhang J Zhuo and X Xia ldquoThe partitioned mixed modelof finite element method and interface stress element methodwith arbitrary shape of discrete block elementrdquo MathematicalProblems in Engineering Article ID 950696 6 pages 2013

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y

1C

2C

1e

2e

A

1e

2e

1C

2C

Block element 1

Block element 2

Interfaceelement

x

s(2)

s(2)

n

(2)

n

(2)

s(1)

s(1)

n

(1)

n

(1)

Figure 1 Schematic diagram of IEM model

multilayer structures Not only that a new polygon interfaceelement is developed called multiscale interface elementmethod (MIEM) which could easily connect with manyother polygon elements of much smaller scale Finally thedeveloped model was applied to analyze thermal stress dis-tribution of the chip scale package (CSP) In this case MIEMdisplay the accuracy of analyzing structural singularities

2 Derivations of Interface-Element Method

21 e Governing Equation The interface element method(IEM) is based upon the discipline of stress vector (tractions)continuity on the interfaces by assuming rigid element andsatisfying interfacial constraint equations In IEM modela complicated domain can be subdivided into a series ofsmaller regions named block element The block elementis undeformable and it can be separated and producesrigid body motion when an external load is applied Thedeformations are accumulated on the conjunct interface ofblock elements The internal energy caused by deformationof the interface area can be related to the work done by thesurface traction around the block element For equilibriumsolution it satisfies the fact that

∭Ω120590119894119895120575119890119894119895119889Ω = minus∬

1198780

119879119894120575119906119894119889119878= ∭

Ω119891119894120575119906119894119889Ω +∬

119878120590

119901119894120575119906119894119889119878(1)

The discretization form can bewritten as (2) whereΩ119890 119878119890120590 1198781198900and 119879119894 denote the element domain stress boundary interfaceregion and interfacial stress

minus 119899sum119890=1

[∬1198781198900

119879119894120575119906119894119889119878]

= 119899sum119890=1

[∭Ω119890119891119894120575119906119894119889Ω +∬

119878119890120590

119901119894120575119906119894119889119878](2)

22 Displacement Distribution in Interfacial Element A four-node 2D interface element for illustrated purpose is shownin Figure 1 in which 997888119899 and 997888119904 represent the local coordinatesystem that are normal direction and shear direction to the

interface surfaceThe displacement field of the block elementcould be expressed by the rigid body motion translationaland rotational such as

119906 = 119873 times 119906119892119906 = [119906 V]119879119906119892 = [119906119892 V119892 120579]119879

119873 = [1 0 119910119892 minus 1199100 1 119909 minus 119909119892]

(3)

where 119906 denotes the displacement vector at any point in theblock element 119906119892 is the degree of freedom at the mass ofcenter and119873 is the shape function With respect to the localsystem the relative displacements at interface element arethen derived as

1205751119897119900119888119886119897 + 1205752119897119900119888119886119897 = [1205751119899 + 1205752119899 1205751119904 + 1205752119904 ]119879

= minus [11987111199061 + 11987121199062]= minus1198711 [1198731

1199061119892 minus11987321199062119892]

(4)

In the above equation 120575119899 is normal displacement 120575119904 is theshear displacement and L is the coordinate transformationmatrix which reads

119871 = [[cos (997888119899 997888119909) cos (997888119899 997888119910)cos (997888119904 997888119909) cos (997888119904 997888119910)]]

1198711 = minus1198712 = 119871

(5)

23 Stress Distribution in Interfacial Element A measure ofthe relative displacement of the block element is referred to asstrain on interface As shown in Figure 2 for any point M oninterface surface the normal strain contributed from blockelement 1 is expressed as the change in length 1205751119899 per unit ofthe original length 1198751119872 = ℎ1 in the local system where 1198751119872is the normal distance from the centroid of block element 1to point M The shear strain is evaluated from the ratio ofslide distance 1205751119904 from block element 1 to the original lengthℎ1 The strain in the region of block element 2 is calculatedin a similar way In summary the components of strain aredefined as follows

120576119894119899 = 120575119894119899ℎ119894120574119894119904 = 120575119894119904ℎ119894

(6)

In plane stress situation the components of stress can beevaluated by the following equation

120590119894119899 = 119864119894120576119894119899 = 119864119894 120575119894119899ℎ119894120591119894119904 = 119866119894120574119894119904 = 1198642 (1 + V119894)

120575119894119904ℎ119894(7)


Arsquo

x

y

P1

P2

h1

h2

M

C1

C2

A B

Brsquo

BrdquoArdquoP1 P2

M1 M2

M2

M1P1

P2

h1 h2

1n

2n

2s

1s



1205901119899 = 1205902119899 = 1205901198991205911119904 = 1205912119904 = 120591119904 (8)


1205751119899 + 1205752119899 = 1205901119899ℎ11198641 + 1205902119899ℎ21198642 = 1205901198991198642ℎ1 + 1198641ℎ211986411198642120590119899 = 119864111986421198642ℎ1 + 1198641ℎ2 (120575

1119899 + 1205752119899)

1205751119904 + 1205752119904 = 2 (1 + V1) ℎ112059111199041198641 + 2 (1 + V2) ℎ212059121199041198642= 120591119904 2 (1 + V1) ℎ11198642 + 2 (1 + V2) ℎ2119864111986411198642

120591119904 = 119864111986422 (1 + V1) ℎ11198642 + 2 (1 + V2) ℎ21198641 (1205751119904 + 1205752119904 )

(9)



(10)


119879 = [120590119899 120591119904]119879 120575 = [120575119894119899 120575119894119904]119879 119863 = [119889119899 0


(11)


119870119880 = 119877119870 = 119899sum119890=1

119870119890 119870119890 = int119904119895


119877 = 119899sum119890=1

119877119890 119877119890 = ∬Ω119890

119873lowast119879119891119889Ω + int119878119890120590

119873lowast119879119875119889119878

(12)




120576119894119899 = 120575119894119899ℎ119894 minus 120572119894Δ119879120574119894119904 = 120575119894119904ℎ119894

(13)


120590119894119899 = 119864119894120576119894119899 = 119864119894 120575119894119899ℎ119894 minus 119864119894120572119894Δ119879120591119894119904 = 119866119894120574119894119904 = 1198642 (1 + V119894)

120575119894119904ℎ119894(14)



[11986411198642 (1205721ℎ1 + 1205722ℎ2)1198641ℎ2 + 1198642ℎ10

]]

(15)


119870119890 119870119890 = int119904119895


119877 = 119899sum119890=1

119877119890119877119890 = ∬

Ω119890119873lowast119879119891119889Ω + int

119878119890120590

119873lowast119879119875119889119878 minus int1198781198900

119873lowast119879119871119879119863119879Δ119879119889119878

(16)


119863 = [[[[

1198641ℎ1 00 11986412ℎ1 (1 + V1)

]]]]

(17)


119863 = [[1198641ℎ1 00 0]]

(18)

1

2

34

5

6

n

g

x

y



119864 = 1198641 minus V2

V = V1 minus V

120572 = 119886 (1 + V)(19)


119909119894119892 = (1199091198941 + 1199091198942 + 1199091198943)3119910119894119892 = (1199101198941 + 1199101198942 + 1199101198943)3

(20)


119886119894 = 12 det

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091198941 11991011989411 1199091198942 11991011989421 1199091198943 1199101198943

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119860 = 119899minus2sum119894=1

119886119894(21)



(22)





CTE(ppm∘F)


Poissonratio


(1)

(2)

(4)

1

2 3

4

56

(1)

(2)

(3)(3)

(4)

1

2 3

4

5

6


0 05 1 15 2

0

02

04

06

08

1

12

14

16

18

2





Si

BeO

Cu

Al

Solder

5

4

2

1

3

2L


0 654321X (mm)

0

05

1

15

2

25

3

35

Y (m

m)





minus10

0

10

20

30

40

50

60

70

80

90

100

Inte

rfaci

al p

eel s

tress

(MPa

)




0

2

4

6

8

10

12

Inte

rfaci

al sh

ear s

tress

(MPa

)





minus20

0

20

40

60

80

100

120

140

Inte

rfaci

al p

eel s

tress

(MPa

)





0

10

20

30

40

50

60

70

Inte

rfaci

al sh

ear s

tress

(MPa

)






Layer Width(mm)

Thickness(mm)

CTE(ppm∘C)


Poisson ratio


Resistor

Substrate

Solder Joints

L L


0 35325215105X (mm)

0

02

04

06

08

1

12

14

16

18

2

Y (m

m)




01020304050

Inte

rfaci

al p

eel s

tress

(MPa

)


Lower interface

Upper interface

Upper interface








Layer Thickness(mm)


Shear Modulus(GPa)




5 Si die 05 160 615 28 03


minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

Inte

rfaci

al sh

ear s

tress

(MPa

)


Upper interface


Lower interface



Die


FR4 Substrate

Solder Joint

L



0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5


0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5






minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM



minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM


5 Conclusions


IEMFEM


minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa


IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)







Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018









Arsquo

x

y

P1

P2

h1

h2

M

C1

C2

A B

Brsquo

BrdquoArdquoP1 P2

M1 M2

M2

M1P1

P2

h1 h2

1n

2n

2s

1s



1205901119899 = 1205902119899 = 1205901198991205911119904 = 1205912119904 = 120591119904 (8)


1205751119899 + 1205752119899 = 1205901119899ℎ11198641 + 1205902119899ℎ21198642 = 1205901198991198642ℎ1 + 1198641ℎ211986411198642120590119899 = 119864111986421198642ℎ1 + 1198641ℎ2 (120575

1119899 + 1205752119899)

1205751119904 + 1205752119904 = 2 (1 + V1) ℎ112059111199041198641 + 2 (1 + V2) ℎ212059121199041198642= 120591119904 2 (1 + V1) ℎ11198642 + 2 (1 + V2) ℎ2119864111986411198642

120591119904 = 119864111986422 (1 + V1) ℎ11198642 + 2 (1 + V2) ℎ21198641 (1205751119904 + 1205752119904 )

(9)



(10)


119879 = [120590119899 120591119904]119879 120575 = [120575119894119899 120575119894119904]119879 119863 = [119889119899 0


(11)


119870119880 = 119877119870 = 119899sum119890=1

119870119890 119870119890 = int119904119895


119877 = 119899sum119890=1

119877119890 119877119890 = ∬Ω119890

119873lowast119879119891119889Ω + int119878119890120590

119873lowast119879119875119889119878

(12)




120576119894119899 = 120575119894119899ℎ119894 minus 120572119894Δ119879120574119894119904 = 120575119894119904ℎ119894

(13)


120590119894119899 = 119864119894120576119894119899 = 119864119894 120575119894119899ℎ119894 minus 119864119894120572119894Δ119879120591119894119904 = 119866119894120574119894119904 = 1198642 (1 + V119894)

120575119894119904ℎ119894(14)



[11986411198642 (1205721ℎ1 + 1205722ℎ2)1198641ℎ2 + 1198642ℎ10

]]

(15)


119870119890 119870119890 = int119904119895


119877 = 119899sum119890=1

119877119890119877119890 = ∬

Ω119890119873lowast119879119891119889Ω + int

119878119890120590

119873lowast119879119875119889119878 minus int1198781198900

119873lowast119879119871119879119863119879Δ119879119889119878

(16)


119863 = [[[[

1198641ℎ1 00 11986412ℎ1 (1 + V1)

]]]]

(17)


119863 = [[1198641ℎ1 00 0]]

(18)

1

2

34

5

6

n

g

x

y



119864 = 1198641 minus V2

V = V1 minus V

120572 = 119886 (1 + V)(19)


119909119894119892 = (1199091198941 + 1199091198942 + 1199091198943)3119910119894119892 = (1199101198941 + 1199101198942 + 1199101198943)3

(20)


119886119894 = 12 det

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091198941 11991011989411 1199091198942 11991011989421 1199091198943 1199101198943

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119860 = 119899minus2sum119894=1

119886119894(21)



(22)





CTE(ppm∘F)


Poissonratio


(1)

(2)

(4)

1

2 3

4

56

(1)

(2)

(3)(3)

(4)

1

2 3

4

5

6


0 05 1 15 2

0

02

04

06

08

1

12

14

16

18

2





Si

BeO

Cu

Al

Solder

5

4

2

1

3

2L


0 654321X (mm)

0

05

1

15

2

25

3

35

Y (m

m)





minus10

0

10

20

30

40

50

60

70

80

90

100

Inte

rfaci

al p

eel s

tress

(MPa

)




0

2

4

6

8

10

12

Inte

rfaci

al sh

ear s

tress

(MPa

)





minus20

0

20

40

60

80

100

120

140

Inte

rfaci

al p

eel s

tress

(MPa

)





0

10

20

30

40

50

60

70

Inte

rfaci

al sh

ear s

tress

(MPa

)






Layer Width(mm)

Thickness(mm)

CTE(ppm∘C)


Poisson ratio


Resistor

Substrate

Solder Joints

L L


0 35325215105X (mm)

0

02

04

06

08

1

12

14

16

18

2

Y (m

m)




01020304050

Inte

rfaci

al p

eel s

tress

(MPa

)


Lower interface

Upper interface

Upper interface








Layer Thickness(mm)


Shear Modulus(GPa)




5 Si die 05 160 615 28 03


minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

Inte

rfaci

al sh

ear s

tress

(MPa

)


Upper interface


Lower interface



Die


FR4 Substrate

Solder Joint

L



0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5


0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5






minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM



minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM


5 Conclusions


IEMFEM


minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa


IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)







Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018










120576119894119899 = 120575119894119899ℎ119894 minus 120572119894Δ119879120574119894119904 = 120575119894119904ℎ119894

(13)


120590119894119899 = 119864119894120576119894119899 = 119864119894 120575119894119899ℎ119894 minus 119864119894120572119894Δ119879120591119894119904 = 119866119894120574119894119904 = 1198642 (1 + V119894)

120575119894119904ℎ119894(14)



[11986411198642 (1205721ℎ1 + 1205722ℎ2)1198641ℎ2 + 1198642ℎ10

]]

(15)


119870119890 119870119890 = int119904119895


119877 = 119899sum119890=1

119877119890119877119890 = ∬

Ω119890119873lowast119879119891119889Ω + int

119878119890120590

119873lowast119879119875119889119878 minus int1198781198900

119873lowast119879119871119879119863119879Δ119879119889119878

(16)


119863 = [[[[

1198641ℎ1 00 11986412ℎ1 (1 + V1)

]]]]

(17)


119863 = [[1198641ℎ1 00 0]]

(18)

1

2

34

5

6

n

g

x

y



119864 = 1198641 minus V2

V = V1 minus V

120572 = 119886 (1 + V)(19)


119909119894119892 = (1199091198941 + 1199091198942 + 1199091198943)3119910119894119892 = (1199101198941 + 1199101198942 + 1199101198943)3

(20)


119886119894 = 12 det

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1199091198941 11991011989411 1199091198942 11991011989421 1199091198943 1199101198943

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119860 = 119899minus2sum119894=1

119886119894(21)



(22)





CTE(ppm∘F)


Poissonratio


(1)

(2)

(4)

1

2 3

4

56

(1)

(2)

(3)(3)

(4)

1

2 3

4

5

6


0 05 1 15 2

0

02

04

06

08

1

12

14

16

18

2





Si

BeO

Cu

Al

Solder

5

4

2

1

3

2L


0 654321X (mm)

0

05

1

15

2

25

3

35

Y (m

m)





minus10

0

10

20

30

40

50

60

70

80

90

100

Inte

rfaci

al p

eel s

tress

(MPa

)




0

2

4

6

8

10

12

Inte

rfaci

al sh

ear s

tress

(MPa

)





minus20

0

20

40

60

80

100

120

140

Inte

rfaci

al p

eel s

tress

(MPa

)





0

10

20

30

40

50

60

70

Inte

rfaci

al sh

ear s

tress

(MPa

)






Layer Width(mm)

Thickness(mm)

CTE(ppm∘C)


Poisson ratio


Resistor

Substrate

Solder Joints

L L


0 35325215105X (mm)

0

02

04

06

08

1

12

14

16

18

2

Y (m

m)




01020304050

Inte

rfaci

al p

eel s

tress

(MPa

)


Lower interface

Upper interface

Upper interface








Layer Thickness(mm)


Shear Modulus(GPa)




5 Si die 05 160 615 28 03


minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

Inte

rfaci

al sh

ear s

tress

(MPa

)


Upper interface


Lower interface



Die


FR4 Substrate

Solder Joint

L



0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5


0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5






minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM



minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM


5 Conclusions


IEMFEM


minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa


IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)







Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018











CTE(ppm∘F)


Poissonratio


(1)

(2)

(4)

1

2 3

4

56

(1)

(2)

(3)(3)

(4)

1

2 3

4

5

6


0 05 1 15 2

0

02

04

06

08

1

12

14

16

18

2





Si

BeO

Cu

Al

Solder

5

4

2

1

3

2L


0 654321X (mm)

0

05

1

15

2

25

3

35

Y (m

m)





minus10

0

10

20

30

40

50

60

70

80

90

100

Inte

rfaci

al p

eel s

tress

(MPa

)




0

2

4

6

8

10

12

Inte

rfaci

al sh

ear s

tress

(MPa

)





minus20

0

20

40

60

80

100

120

140

Inte

rfaci

al p

eel s

tress

(MPa

)





0

10

20

30

40

50

60

70

Inte

rfaci

al sh

ear s

tress

(MPa

)






Layer Width(mm)

Thickness(mm)

CTE(ppm∘C)


Poisson ratio


Resistor

Substrate

Solder Joints

L L


0 35325215105X (mm)

0

02

04

06

08

1

12

14

16

18

2

Y (m

m)




01020304050

Inte

rfaci

al p

eel s

tress

(MPa

)


Lower interface

Upper interface

Upper interface








Layer Thickness(mm)


Shear Modulus(GPa)




5 Si die 05 160 615 28 03


minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

Inte

rfaci

al sh

ear s

tress

(MPa

)


Upper interface


Lower interface



Die


FR4 Substrate

Solder Joint

L



0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5


0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5






minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM



minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM


5 Conclusions


IEMFEM


minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa


IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)







Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018










minus10

0

10

20

30

40

50

60

70

80

90

100

Inte

rfaci

al p

eel s

tress

(MPa

)




0

2

4

6

8

10

12

Inte

rfaci

al sh

ear s

tress

(MPa

)





minus20

0

20

40

60

80

100

120

140

Inte

rfaci

al p

eel s

tress

(MPa

)





0

10

20

30

40

50

60

70

Inte

rfaci

al sh

ear s

tress

(MPa

)






Layer Width(mm)

Thickness(mm)

CTE(ppm∘C)


Poisson ratio


Resistor

Substrate

Solder Joints

L L


0 35325215105X (mm)

0

02

04

06

08

1

12

14

16

18

2

Y (m

m)




01020304050

Inte

rfaci

al p

eel s

tress

(MPa

)


Lower interface

Upper interface

Upper interface








Layer Thickness(mm)


Shear Modulus(GPa)




5 Si die 05 160 615 28 03


minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

Inte

rfaci

al sh

ear s

tress

(MPa

)


Upper interface


Lower interface



Die


FR4 Substrate

Solder Joint

L



0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5


0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5






minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM



minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM


5 Conclusions


IEMFEM


minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa


IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)







Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018










Layer Width(mm)

Thickness(mm)

CTE(ppm∘C)


Poisson ratio


Resistor

Substrate

Solder Joints

L L


0 35325215105X (mm)

0

02

04

06

08

1

12

14

16

18

2

Y (m

m)




01020304050

Inte

rfaci

al p

eel s

tress

(MPa

)


Lower interface

Upper interface

Upper interface








Layer Thickness(mm)


Shear Modulus(GPa)




5 Si die 05 160 615 28 03


minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

Inte

rfaci

al sh

ear s

tress

(MPa

)


Upper interface


Lower interface



Die


FR4 Substrate

Solder Joint

L



0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5


0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5






minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM



minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM


5 Conclusions


IEMFEM


minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa


IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)







Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018










Layer Thickness(mm)


Shear Modulus(GPa)




5 Si die 05 160 615 28 03


minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

Inte

rfaci

al sh

ear s

tress

(MPa

)


Upper interface


Lower interface



Die


FR4 Substrate

Solder Joint

L



0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5


0

05

1

15

2

25

0 05 1 15 2 25 3 35 4 45 5






minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM



minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM


5 Conclusions


IEMFEM


minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa


IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)







Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018










minus40

minus30

minus20

minus10

0

10

20

30

40

Inte

rfaci

al p

eel s

tress

(MPa

)

IEMFEM



minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)

IEMFEM


5 Conclusions


IEMFEM


minus30

minus20

minus10

0

10

20

Inte

rfaci

al p

eel s

tress

(MPa


IEMFEM

minus60

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

60

Inte

rfaci

al sh

ear s

tress

(MPa

)







Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018










Data Availability




Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








thermal stress analysis of chip scale packaging by using novel multiscale … · 2019. 7. 30. ·...

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