MECHANISM-BASED MODELING OF DUCTILE VOID GROWTH FAILURE IN MULTILAYER STRUCTURES THONG CHEE MENG (B Eng (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 Acknowledgements Acknowledgements This section is specially dedicated to all the kindhearted individuals who granted the author precious advice, guidance and munificent resources on this Master of Engineering thesis The researches would not be made possible without their selfless devotion of time and effort Appreciation goes out to: A/Prof Cheng Li, supervisor of this research work, who has been a relentless source of motivation to the author in the exploration of Fracture Mechanics Through her enthusiasm and dedication, Prof Cheng shared much expert knowledge and endowed valuable insights to the author in his research process The author would like to express his deep felt gratitude to Prof Cheng for her teachings, encouragement and understanding throughout the project Dr Guo Tian Fu, visiting researcher from Tsinghua University, whom the author is greatly indebted to, for his invaluable supervision and facilitation in the computational aspect of the author’s research work Dr Guo has also been a humble mentor and more importantly, a sincere friend in sharing his experience and interpretation in the current field His patience and generous support is the basis of the research’s completion The author is also grateful to Chong Chee Wei, a postgraduate student, for his advice, guidance and most importantly the moral support he has given; and to Leo Chin Khim, a fellow colleague, for his assistance, encouragement and of course, friendship Sincere gratitude also extends to the technical officers and peers in the Strength of Materials Laboratory 2, and everyone else who has contributed to the completion of this thesis i List of Symbols List of Symbols SSY Small Scale Yielding Φ Gurson-Tvergaard continuum flow potential σe Mises stress σm Mean stress or hydrostatic stress σ0 Tensile Yield stress G Energy release rate K Mode I stress intensity factor Γ Crack growth resistance T T-stress, non-singular elastic stress which acts parallel to the crack plane ∆a Crack propagation length, distance between initial and current crack tip X Distance ahead of the current crack tip location D Size of Gurson cell element l1 Length of fracture process zone E Young’s modulus N Strain hardening exponent ν Poisson’s ratio f Void volume fraction in a Gurson cell fE Critical void volume fraction in a Gurson cell to trigger element extinction algorithm for the cell q1,q2 Micromechanics factors introduced by Tvergaard in the GursonTvergaard model p Void pressure in a Gurson cell x1,x2 Horizontal and vertical datum in the SSY models x1, x2, x3 Cartesian axis directions for the periodic composite models C0 Initial reinforcement volume fraction in a composite R0 Initial radius of the fiber or spherical reinforcement ii Table of Contents Table of Contents ACKNOWLDEGEMENT i LIST OF SYMBOLS ii TABLE OF CONTENTS iii SUMMARY vi LIST OF FIGURES x CHAPTER Introduction CHAPTER Literature Review 2.1 2.2 2.3 2.4 2.5 Ductile Crack Growth under Small Scale Yielding Fracture Toughness of Constrained Ductile Layer Modeling Interfacial Decohesion in a Composite Extended Gurson Model Incorporating Vapour Pressure Numerical Implementation 13 17 20 CHAPTER Ductile Crack Growth under Small Scale Yielding 24 3.1 Introduction 3.2 Problem Formulation 3.2.1 Computational Model and Boundary Conditions 3.2.2 Material Properties 3.2.3 Numerical Details 3.3 Results and Discussion 3.3.1 Effects of T-stress 3.3.2 Effects of Internal Void Vapour Pressure 3.4 Conclusion 24 25 25 26 29 31 31 39 44 iii Table of Contents CHAPTER Ductile Failure of Centerline Crack in a Constrained Ductile Layer 46 4.1 Introduction 4.2 Problem Formulation 4.2.1 Computational Model and Boundary Conditions 4.2.2 Material Properties 4.2.3 Numerical Details 4.3 Results And Discussion 4.3.1 Effects of Internal Void Vapour Pressure 4.3.2 Effects of Elastic Modulus Mismatch 4.3.3 Effects of T-stress 4.4 Conclusion 46 47 48 49 51 53 54 58 63 70 CHAPTER Two-Dimensional Modeling of Void-Induced Interfacial Decohesion in a FiberReinforced Polymeric Matrix Composite 72 5.1 Introduction 5.2 Problem Formulation 5.2.1 Plane Strain Periodic Array 5.2.2 Material Models 5.3 Numerical Results 5.3.1 Effect of Interface Damage Zone Size, D/R0 5.3.2 Effect of Interfacial Cell Element Porosity, f0 5.3.3 Effects of Void Vapour Pressure, p0/σ0 5.4 Discussion and Conclusion 72 74 74 76 78 78 91 98 104 CHAPTER Three-Dimensional Modeling of Void-Induced Interfacial Decohesion in a Spherical Particle-Reinforced Polymeric Matrix Composite 107 6.1 Introduction 6.2 Problem Formulation 6.2.1 Three-dimensional Periodic Array 6.2.2 Material Models 6.3 Numerical Results 6.3.1 Perfect Particle-Matrix Interface 6.3.2 Imperfect Particle-Matrix Interface 6.3.3 Mean stress-Mean strain Response 6.3.4 Effect of Interfacial Cell Element Porosity, f0 6.3.5 Effects of Void Vapour Pressure, p0/σ0 6.3.6 Effects of Particle Volume Fraction, C0 6.4 Discussion and Conclusion 107 108 108 111 112 113 118 119 121 127 133 137 CHAPTER Summary Of Conclusions 142 iv Table of Contents 7.1 Ductile Crack Growth under Small Scale Yielding 142 7.2 Ductile Failure of Centerline Crack in a Constrained Ductile Layer 144 7.3 2-D Modeling of Void-Induced Interfacial Decohesion in a Fiber-Reinforced Polymeric Matrix Composite 146 7.4 3-D Modeling of Void-Induced Interfacial Decohesion in a Spherical Particle-Reinforced Polymeric Matrix Composite 148 REFERENCES 146 APPENDIX A A-1 APPENDIX B B-1 APPENDIX C C-1 v Summary Summary Inspiration for the present research work comes from industrial developments in the field of electronic packaging Motivated by the moisture-induced failure phenomenon in the integrated circuit (IC) packages, commonly known as “popcorn cracking”, this literature serves to gain a deeper understanding on the micro-mechanics of void pressure-assisted ductile fracture IC packages assembly usually consists of an intricate multilayer structure A simple example is that of the thin layer of ductile adhesive (die attach), sandwiched between the much stiffer silicon die chip and die pad base In addition, many constituents in the IC packages are made from polymeric matrix composites, e.g Ag-filled epoxy being used as moulding compound or die attach These ductile and porous materials are typically susceptible to moisture absorption at the reinforcement-matrix interface and are therefore prone to fail by void vapour pressure assisted decohesion and ductile crack growth Investigation begins with a preliminary study on the mechanisms of ductile failure by void growth and coalescence Effects of internal void pressure as well as crack tip constraints, implemented via the application of T-stress, on the SSY mode I crack growth fracture toughness are studied It is found that a low constraint crack under negative T-stresses greatly elevates the fracture toughness of the material This is due to greater degree of plastic dissipation at the crack front, which effectively raises the total work of fracture for crack advancement Conversely, a highly constrained crack under positive T-stress shows no significant effect on the fracture toughness Upon vi Summary introduction of internal void pressure, high pressure levels significantly reduce the fracture resistance of the model The effect of void pressure is seen to promote void growth and pre-softening to the cell, thereby resulting in a lower work of separation in rupturing a cell during crack advancement Combined effect from high internal void pressure and restricted plastic dissipation from highly constrained crack is shown to greatly escalate cell damage and is extremely detrimental to the stability of the system Investigation next proceeds to discuss on the different fracture modes in a constraint layer system The model consists of a centerline crack in a thin ductile layer, which is sandwiched between two rigid substrates Three competing void interaction mechanisms are demonstrated in this study, namely: (i) near-tip void growth interactions, (ii) large scale cavitation spanning to a distance of several layer thickness from the crack tip, and (iii) voids cavitation at site of highest triaxialities ahead of the current crack tip Findings show that presence of void pressure significantly lowers the overall fracture toughness by diminishing the material’s work of separation, while having negligible effect on the plastic dissipation around the crack tip Therefore the size of the fracture process zone remained relatively unaffected under the different pressure levels Hence void pressure does not have much influence on the void interaction mechanism of the growing crack However, varying the elastic modulus mismatch between the substrates and ductile layer shows that a smaller modulus mismatch promotes the mechanism of near-tip void growth But when higher mismatch values are imposed, large constraint on the deformation within the layer caused the failure mechanism to shift into the second mechanism of large-scale multi-void cavitation Likewise, when a large negative Tstress value is applied, results show that void cavitation is initiated at distances in the vii Summary order of the layer’s thickness ahead of the current crack tip, thereby forming a new crack front This corresponds to the third void interaction mechanism described above The next two case studies presented in the report focus on the stress-strain behaviour of polymeric matrix composites, particularly pertaining to those used in IC packages Analysis is first conducted on a 2-D plane strain model of fiber-reinforced composite where the sole failure mechanism is reinforcement-matrix decohesion Stress contour plots under uniaxial loading indicate that stress carrying capacity of a composite relates more or less proportionately to the extent of void growth damage at the interface Peak stress carrying capacity is attained when approximately half of the interfacial surface area becomes severely softened by void growth At the same time, a 45° shear band develops fully across the cell diagonal when peak tensile strength is reached Furthermore, higher values of both interfacial porosity and internal void pressure are observed to reduce the composite’s stress carrying capacity and tensile strength They also cause macroscopic yielding to initiate earlier, especially so under the influence of internal void pressure The framework is then extended to a full 3-D study on multi-axial loading states on a spherical particle-reinforced polymeric matrix composite, with a Gurson damage constitutive model lining the reinforcement-matrix interface Under an imperfect interface susceptible to void growth and decohesion, the stress strain behaviour of the composite is seen to exhibit macroscopic yielding followed by strain hardening phase before attaining a maximum stress peak Beyond peak stress, macroscopic softening sets in, similar to the response found in the previous plane strain fiber-matrix model Effects of internal void pressure combined with an interface of high porosity again prove to greatly erode the stress carrying capacity and tensile strength of the viii Summary composite High triaxial loading like the equi-triaxiality, results in sudden “brittle”-like load release due to occurrence of massive interfacial voids cavitation upon reaching a critical strain Tensile behaviour also displays a distinctive dual-peak profile due to the subsequent strain hardening effects from the matrix after stress redistribution from the interfacial decohesion ix References [80] Tvergaard, V and Hutchinson, J W (1994) Toughness of an interface along a thin ductile layer joining elastic solids Philosophical Magazine A 70, pp 641656 [81] Tvergaard, V and Hutchinson, J W (1996) Effect of strain-dependent cohesive zone model on predictions of crack growth resistance International Journal of Solids Structures, 33, pp 3297-3308 [82] Tvergaard, V and Hutchinson, J W (1996) On the toughness of ductile adhesive joints Journal of the Mechanics and Physics of Solids, 44 (5), pp 789-800 [83] Tvergaard, V and Hutchinson, J W (2002) Two mechanisms of ductile fracture: void by void growth versus multiple void interaction International Journal of Solids and Structures, 39, pp 3581-3597 [84] Tvergaard, V., and Needleman, A (1995) Effects of nonlocal damage in porous plastic solids International Journal of Solids and Structures, 32 (8/9), pp 1063-1077 [85] Varias, A G and Shih, C F (1993) Quasi-static crack advance under a range of constraints − steady-state fields based on a characteristic length Journal of the Mechanics and Physics of Solids, 41, pp 835-861 [86] Varias, A G., Suo, Z and Shih, C F (1991) Ductile Failure of a Constrained Metal Foil Journal of the Mechanics and Physics of Solids, 39 (7), pp 963986 [87] Xia, L and Shih, C F (1995) Ductile crack growth − I A numerical study using computational cell with microstructurally-based length scales Journal of the Mechanics and Physics of Solids, 43 (2), pp 233−259 [88] Xia, L and Shih, C F (1995) Ductile crack growth − II Void nucleation and geometry effects on macroscopic fracture behavior Journal of the Mechanics and Physics of Solids, 43 (12), pp 1953−1981 [89] Xia, L and Shih C F (1996) Ductile crack growth – III Transition to cleavage fracture incorporating statistics Journal of the Mechanics and Physics of Solids, 44 (4), pp 603-639 [90] Xia, L., Shih, C F and Hutchinson, J W (1995) A computational approach to ductile crack growth under large scale yielding conditions Journal of the Mechanics and Physics of Solids, 43 (3), pp 398-413 [91] Williams, M L (1957) On the stress distribution at the base of a stationary crack Journal of Applied Mechanics, 24, pp 109-114 Page 157 Appendix A Appendix A Vapour Pressure Modeling This section is an excerpt from Guo and Cheng [33], which derives the relationship for fully vaporized moisture as a function of temperature T and void volume fraction f shown in Equation (2.2) Consider a small representative material sample containing a microvoid with void volume Vf and cell volume V The vapor pressure p at a fully vaporized state obeys the ideal gas law: pV f = mRT (A.1) Here R is the universal gas constant, m the moisture weight inside of the microvoid, and T the temperature Dividing both sides of Equation (A.1) by V leads to pf = CRT (A.2) where C = m/V is the averaged moisture concentration and f = Vf /V is the void volume fraction Equation (A.2), supplemented by moisture diffusion analysis, permits the evaluation of the initial volume fraction f0 of the void Two states (p, f, T, C) and (p0, f0, T0, C0) are related by: p T f0 C = p0 T0 f C (A.3) Page A-1 Appendix A During the process of deformation (V0 V) associated with a temperature rise ∆T ( = T − T0) it is assumed that the moisture weight m is conserved Hence p T f V0 = p0 T0 f V (A.4) For spherically symmetric void growth in an incompressible solid, the relative volume change V/V0 of the microvoid cell during mechanical deformation and thermal expansion is given by: V − f 3α∆T = e V0 − f (A.5) where α is the thermal expansion coefficient of the matrix material Substitution of this expression into Equation (A.4) yields: p T f − f −3α∆T = e p0 T0 f − f (A.6) This completes the continuum description of internal pressure for fully vaporized moisture state Page A-2 Appendix B Appendix B Total Work of Fracture Calculation from Applied K-field This section describes the calculation of the total work of fracture per unit area of crack advance from the applied analytical K-field at the remote boundary It is used for comparing with the J-integral computed from the Warp3D software, in order to ensure that small scale yielding condition is obeyed Selecting a particular node n at the remote boundary, the configurations that characterize n at any instant in the loading history include x1n, x2n, u1n, u2n The initial node position of n with respect to the horizontal and vertical axis is denoted by x1n and x2n respectively, while u1n and u2n denote the current displacement from the initial configuration, in the corresponding direction respectively The circular coordinates, R and θ, of the initial position of n is given by: R2 = x12+x22 and θ = tan(x2/x1) (B.1) From Equation (3.6), K /σ0 = K /σ = u1n − (T / σ ) +ν E /σ0 −ν R cos θ E /σ0 R ⎛θ ⎞ cos⎜ ⎟(3 − 4ν − cos θ ) 2π ⎝2⎠ ν (1 + ν ) R sin θ E /σ R ⎛θ ⎞ sin ⎜ ⎟(3 − 4ν − cosθ ) 2π ⎝2⎠ or (B.2) u n − (T / σ ) +ν E /σ (B.3) Page B-1 Appendix B where K/σ0 is the total applied K-field at current displacement of (u1n, u2n), T/σ0 is the applied T-stress at the first load step and E/σ0 denotes the elastic modulus Note that all quantities are normalized by the yield stress σ0 The current displacement (u1n, u2n) of node n from its initial coordinates (x1n, x2n) is extracted from the Warp3D software’s output modules The total K-value applied at any particular loading instant can thus be calculated from either Equation (B.2) or (B.3) From Irwin’s relationship combining with Griffith criterion, the total work of fracture for crack growth, Γ / σ (normalized by yield stress), associated with current applied K-value is: Γ /σ = −ν (K / σ )2 E /σ (B.4) Using the domain integral method (see Chapter 2, Section2.5), within a finite deformation setting, the computed J-integral values are path-independent and in close agreement with Equation (B.4) as long as the radii of the annular spatial domains are sufficiently large compared to the distance over which the crack has grown, in order for small scale yielding condition to prevail Page B-2 Appendix C Appendix C Automatic Mesh Generator Source Code This section contains the Unix Fortran 77 source code for the 2-D automatic mesh generator used to model the fiber-reinforced polymeric matrix composite under plane strain conditions The output file is in a Warp3D compatible format C==================================================================== C This program is an automatic mesh generator for a 1x1 2D void C With rigid boundary condition at void surface => rigid fiber C inclusion of a Polymeric Matrix Composite, under plane strain C conditions C C Written by Thong Chee Meng C C Version 1.1 C - user input f0, bias, no of void elements, C no of circular rings and loading axiality C - combined other input files e.g coordinates, incidences, C etc into the main warp3d input file via subroutines C C==================================================================== PROGRAM Void2D C IMPLICIT NONE C C C VARIABLES DECLARATIONS C -C User input variables DOUBLE PRECISION f0, bias, rings, v_elm INTEGER axiality, confirm CHARACTER*20 filename C C Main program variables INTEGER maxelm, maxnode, inc DOUBLE PRECISION pi, R, D, sq_D, sq_rings, ring_sum, sq_sum, & sq_sum1 C C -C C C INITIALIZATION OF PARAMETERS Page C-1 Appendix C C C C C C C 110 C 111 C C C 112 -pi=3.141592654 -USER INPUT PARAMETERS -write(*,*) ' Input the initial void f0' read(*,*) f0 write(*,*) ' Input the no of void elements (even #)' read(*,*) v_elm if (MOD(INT(v_elm),2) NE 0) goto 111 Calculate void radius and size of void elements R=DSQRT(4.0*f0/pi) D=0.5*pi*R/v_elm write(*,*) ' Input bias in element size (>1.0)' read(*,*) bias if (bias LE 1.0) goto 112 C write(*,*) ' Input no of circular rings' read(*,*) rings C C C C 113 C 114 C 115 C C C C C C Calculate no of square elements after rings ring_sum=D*((bias**rings)-1.0)/(bias-1.0) + R sq_sum=1.0-ring_sum sq_D=D*(bias**rings) if (sq_D GT sq_sum) then print*, " Too many circular rings!" goto 112 endif Calculate no of square elements sq_rings=0.0 sq_rings=sq_rings+1.0 sq_sum1=sq_D*((bias**sq_rings)-1.0)/(bias-1.0) if (sq_sum1 LT sq_sum) goto 113 sq_rings=sq_rings-1.0 write(*,*) ' Start of square rings at ', ring_sum write(*,*) ' No of square rings = ', sq_rings write(*,*) ' Is this ok? (Yes=1, No=0)' read(*,*) confirm if ((confirm.NE.0).AND.(confirm.NE.1)) goto 114 if (confirm.EQ.0) goto 112 write(*,*) ' Input loading axiality (Uniaxial=1, Biaxial=2)' read(*,*) axiality if ((axiality.NE.1).AND.(axiality.NE.2)) goto 115 write(*,*) ' Input filename' read(*,'(a)') filename -OUTPUT FILE PROCESSING -open(unit = 10, file = filename, status = 'unknown') maxnode=2*INT((rings+sq_rings+1.0)*(v_elm+1.0)) Page C-2 Appendix C maxelm=INT(v_elm*(rings+sq_rings)) C C C C C C C C C C C C C C C Void2D Information write(10,300) write(10,'(a,f5.4)') "c Void f0 = ", f0 write(10,'(a,i3)') "c No of void elements = ", INT(v_elm) write(10,'(a,f5.4)') "c Size of Gurson cell elements, D = ", D write(10,'(a,f5.4)') "c D/R_0 = ", D/R write(10,'(a,f4.2)') "c Element Bias = ", bias write(10,'(a,i3)') "c No of circular rings = ", INT(rings) write(10,'(a,i3)') "c No of square rings = ", INT(sq_rings) write(10,'(a)') "c Loading boundary conditions:" write(10,'(a)') "c - Plane strain" write(10,'(a)') "c - Rigid at void surface" if (axiality EQ 1) then write(10,'(a)') "c - Uniaxial in y-direction" else write(10,'(a)') "c - Biaxial in x- and y-direction" endif write(10,300) write(10,300) Material models -write(10,'(a,a)') "structure ", filename write(10,300) write(10,300) write(10,'(a)') "material steel" write(10,'(a,a)') " properties mises e 500.0 nu 0.3 yld_pt”, & "1.0 n_power 10.0," write(10,'(a)') " rho 0.0" write(10,300) write(10,'(a)') "material void_strip" write(10,'(a,a)') " properties gurson e 500.0 nu 0.3 yld_pt”, & " 1.0 n_power 10.0," write(10,'(12x,a)') "f_0 0.05 q1 1.25 q2 1.0 q3 1.5625”, & “ rho 0.0," write(10,'(a)') " p_0 0.0 T_0 1.0 alpha 1.0," write(10,'(a)') " killable" write(10,300) write(10,300) Elements and nodes write(10,'(a, i9)') "number of nodes", maxnode write(10,'(a, i9)') "number of elements", maxelm Coordinates -write(10,300) write(10,'(a / a)') "coordinates", "*echo off" call Coordinates(bias,v_elm,R,D,rings,sq_rings,sq_sum, & ring_sum,pi) Element type write(10,'(a)') "*echo on" write(10,300) write(10,'(a)') "elements" C Page C-3 Appendix C write(10,'(5x,a,i3,a,a)') "1 -", INT(v_elm), " type l3disop & "nonlinear material void_strip," write(10,'(24x,a)') "order 2x2x2 center_output bbar" ", C write(10,'(4x,i3,a,i5,a)') INT(v_elm+1.0), " -", maxelm, & " type l3disop nonlinear material steel," write(10,'(24x,a)') "order 2x2x2 center_output bbar" write(10,300) C C C C C C C C C C C C C C C Incidences write(10,'(a / a)') "incidences", "*echo off" inc=2*INT(v_elm+1.0) call Incidences(v_elm,rings,sq_rings,inc) write(10,300) write(10,300) Blocking call Blocking(v_elm,maxelm) Constraints -write(10,300) write(10,'(a)') "constraints" call Constraints(v_elm,rings,sq_rings,inc,maxnode,axiality) write(10,300) write(10,300) write(10,300) Loadsteps -write(10,'(a / a)') " loading disp", " nonlinear" write(10,'(a)') " step 1-800 constraints 1.0e-4" write(10,300) write(10,300) write(10,'(a / a)') "c output patran neutral", "c stop" write(10,300) write(10,'(a)') " nonlinear analysis parameters" write(10,'(a)') " solution technique direct sparse" write(10,'(a)') " maximum iterations 20" write(10,'(a,a)') " convergence test norm res tol 0.001 ", & "$ max resid tol 0.1" write(10,'(a / a)') " time step 10000", " trace solution on" write(10,'(a / a)') " extrapolate off", " adaptive on" write(10,'(a)') " consistent q-matrix off" write(10,'(a)') " bbar stabilization factor 0.0" write(10,'(a)') " batch messages off" write(10,'(a)') " cpu time limit off" write(10,'(a)') " material messages off" write(10,300) write(10,300) Crack growth parameters -write(10,'(a)') " crack growth parameters" write(10,'(a)') " type of crack growth gurson" write(10,'(a)') " print status on order 1-20" write(10,'(a)') " kill sequentially on order 1, 2, 3, 4" write(10,'(a)') " force release type traction-separation" write(10,'(a)') " crack plane normal y coord 0.0" Page C-4 Appendix C write(10,'(a,f13.12)') " cell height 0", D write(10,'(a)') " release fraction 0.1" write(10,'(a)') " critical porosity 1.0" write(10,'(a,a)') " adaptive load control on maximum ", & "porosity change 0.05" write(10,300) write(10,'(a,a)') "c compute displacements for loading disp ", & "step 1-800" write(10,300) write(10,300) write(10,'(a)') "*input from file 'step800f_inp'" write(10,300) write(10,'(a)') " stop" C C -300 format("c") C Endfile(10) Close(10) C 888 continue C print*, "Void2D mesh generation complete!" C END PROGRAM Void2D C==================================================================== C C C C C C C C C==================================================================== C Subroutines List C - blocking C - coordinates C - incidences C - boundary C==================================================================== C C DEFINING BLOCKING C -Subroutine Blocking(v_elm,maxelm) C IMPLICIT NONE C Arguements double precision v_elm integer maxelm C C Local variables integer b_1, b_2, b_3 C C C C write(10,'(a)') "blocking $ scalar" C b_1=1 b_2=INT(v_elm) Page C-5 Appendix C 116 C b_3=1 write(10,'(3i8)') b_1, b_2, b_3 b_1=b_1+1 b_3=b_3+b_2 if ((b_3+128-1) LE maxelm) then b_2=128 goto 116 else b_2=maxelm-b_3+1 if(b_2.GT.0) goto116 endif Return End C -C C C C C C -C DEFINING NODES AND COORDINATES C -Subroutine Coordinates(bias,v_elm,R,D,rings,sq_rings,sq_sum, & ring_sum,pi) C IMPLICIT NONE C Arguements double precision bias, v_elm, rings, sq_rings, sq_sum, & ring_sum, R, D, pi C C Local variables integer node, r_ct, v_ct, maxrings double precision sq_D, angle, cur_R, cur_sqR, cur_sqR1, x, y C C C C node=1 cur_sqR=0.0 cur_sqR1=0.0 maxrings=int(rings+sq_rings)+1 sq_D=sq_sum*(bias-1.0)/(bias**sq_rings-1.0) C DO 20 r_ct = 1, maxrings C if (r_ct LE INT(rings+1.0)) then cur_R = R + D*((bias**(r_ct-1))-1.0)/(bias-1.0) else cur_sqR = sq_D*((bias**(r_ct-1-INT(rings)))-1.0)/(bias-1.0) endif C C Node along x-axis -if (r_ct.EQ.maxrings) then x = 1.0 else x = cur_R + cur_sqR endif y = 0.0 C write(10,200) node, x, y, 0.0 Page C-6 Appendix C 200 C C write(10,200) node+1, x, y, D node = node+2 format(i7, 3(3x, e14.9)) Nodes between x- and y-axis -DO 21 v_ct= 1, INT(v_elm-1.0) C angle=pi/2.0/v_elm*DBLE(v_ct) C C C C C C C C For square ring portion -if (r_ct GT INT(rings+1.0)) then if (angle.LT.(pi/4.0)) then cur_sqR1 = cur_sqR/sq_sum*(1.0/DCOS(angle)-ring_sum) elseif (angle.GT.(pi/4.0)) then cur_sqR1 = cur_sqR/sq_sum*(1.0/DSIN(angle)-ring_sum) else cur_sqR1 = cur_sqR/sq_sum*(DSQRT(DBLE(2))-ring_sum) endif endif Boundary nodes IF (r_ct.EQ.maxrings) THEN if (angle.LT.(pi/4.0)) then x = 1.0 y = DTAN(angle) elseif (angle.GT.(pi/4.0)) then x = 1.0/DTAN(angle) y = 1.0 else x = 1.0 y = 1.0 endif ELSE Nodes along diagonal if (angle.EQ.(pi/4.0)) then x = (cur_R+cur_sqR1) / DSQRT(DBLE(2)) y = x else x = (cur_R+cur_sqR1) * DCOS(angle) y = (cur_R+cur_sqR1) * DSIN(angle) endif C ENDIF C write(10,200) node, x, y, 0.0 write(10,200) node+1, x, y, D node = node+2 C 21 C C Continue Node along y-axis x = 0.0 if (r_ct.EQ.maxrings) then y = 1.0 else y = cur_R + cur_sqR endif C Page C-7 Appendix C write(10,200) node, x, y, 0.0 write(10,200) node+1, x, y, D node = node+2 C 20 C Continue Return End C -C C C C C C -C DEFINING ELEMENTS AND INCIDENCES C -Subroutine Incidences(v_elm,rings,sq_rings,inc) C IMPLICIT NONE C Arguements integer inc double precision v_elm, rings, sq_rings C C Local variables integer elm, r_ct, v_ct, startnode, cur_node, maxrings C C C C elm=0 maxrings=int(rings+sq_rings) C DO 30 r_ct=1,maxrings startnode = + (r_ct-1)*inc C DO 31 v_ct=1,INT(v_elm) cur_node = startnode+2*(v_ct-1) elm=elm+1 C write(10,201) elm, & cur_node, cur_node+inc, cur_node+inc+2, cur_node+2, & cur_node+1, cur_node+inc+1, cur_node+inc+3, cur_node+3 C 201 format(9(i8)) C 31 Continue C 30 Continue C Return End C -C C C C C C -C DEFINING BOUNDARY CONDITIONS & CONSTRAINTS C Page C-8 Appendix C Subroutine Constraints(v_elm,rings,sq_rings,inc,maxnode, & axiality) C C C C C C C C C IMPLICIT NONE Arguements integer inc, maxnode, axiality double precision v_elm, rings, sq_rings Local variables integer n_ct, startbc - Plane strain condition write(10,'(a)') "c Plane Strain Condition" C 204 C write(10,204) "1 -", maxnode, "w", 0.0 format(7x, a, i8, 5x, a, 5x, e11.6) write(10,'(a)') "c" C C Symmetry about x-axis -write(10,'(a)') "c Symmetry about x-axis" C 41 C 202 C DO 41 n_ct = 1, INT(rings+sq_rings+1.0) write(10,202) (1+(n_ct-1)*inc), "v", 0.0 write(10,202) (2+(n_ct-1)*inc), "v", 0.0 Continue format(i8, 6x, a, 5x, e11.6) write(10,'(a)') "c" C C Symmetry about y-axis -write(10,'(a)') "c Symmetry about y-axis" C 42 C DO 42 n_ct = 1, INT(rings+sq_rings+1.0) write(10,202) (1+2*INT(v_elm)+(n_ct-1)*inc), "u", 0.0 write(10,202) (2+2*INT(v_elm)+(n_ct-1)*inc), "u", 0.0 Continue write(10,'(a)') "c" C C Rigid boundary condition at void surface write(10,'(a)') "c Rigid particle bc" C 43 C 203 DO 43 n_ct = 1, INT(v_elm+1.0) if (n_ct.EQ.1) then write(10,202) (1+2*(n_ct-1)), write(10,202) (2+2*(n_ct-1)), elseif (n_ct.EQ.INT(v_elm+1.0)) write(10,202) (1+2*(n_ct-1)), write(10,202) (2+2*(n_ct-1)), else write(10,203) (1+2*(n_ct-1)), write(10,203) (2+2*(n_ct-1)), endif Continue "u", "u", then "v", "v", 0.0 0.0 0.0 0.0 "u", 0.0, "v", 0.0 "u", 0.0, "v", 0.0 format(i8, 2(6x, a1, 5x, e11.6)) Page C-9 Appendix C C write(10,'(a)') "c" C C Loading axiality if (axiality EQ 1) then write(10,'(a)') "c Uniaxial loading in y dir" else write(10,'(a)') "c Biaxial loading" endif C 44 C 45 C startbc=1+inc*INT(rings+sq_rings) DO 44 n_ct = 1, INT(v_elm/2.0+1.0) if (axiality EQ 1) then write(10,202) (startbc+2*(n_ct-1)), "u", 0.0 write(10,202) (1+startbc+2*(n_ct-1)), "u", 0.0 else write(10,202) (startbc+2*(n_ct-1)), "u", 1.0 write(10,202) (1+startbc+2*(n_ct-1)), "u", 1.0 endif Continue startbc=1+inc*INT(rings+sq_rings)+2*INT(v_elm/2.0) DO 45 n_ct = 1, INT(v_elm/2.0+1.0) write(10,202) (startbc+2*(n_ct-1)), "v", 1.0 write(10,202) (1+startbc+2*(n_ct-1)), "v", 1.0 Continue write(10,'(a)') "c" C Return End C -C C==================================================================== Page C-10 [...]... multi-layer structures at the same time depending on the loading conditions Ductile failure in metallic alloys and polymeric materials constituents of the multilayer structures is commonly driven by void growth and coalescence This forms the basis of the current research work on which all investigations into the interplay of failure mechanisms in different case models are built upon Ductile fracture begins... those providing the framework upon which some case studies are based in this report Damage parameters and material variables, e.g void volume fraction and elastic modulus, has been well researched by many in the field of ductile void growth modeling However, void pressure-assisted ductile crack growth remains relatively new in this area Thus investigations begin with a preliminary study in Chapter 3... IC packaging industry, while effects of T-stress are studied for crack tip constraint imposed by geometric or mechanical considerations 2.2 Fracture Toughness of Constrained Ductile Layer Failure analysis in multilayer structures is of considerable importance for its extensive range of engineering applications, from multilayer protective coatings in the structural industry to composite laminates used... extensively on the contribution of plastic deformation to the effective work of fracture for a crack lying along one of the interfaces of a thin ductile layer joining two elastic solids However, the model is not adequate for ductile crack growth lying in the void by void crack advance regime [81] This is because the intense deformation immediate to the tip amplifies the growth of the void at the tip above what... of cavities from brittle cracking, or decohesion of inclusion, dispersoids or any second-phase sites in the matrix These cavities grow in size under the high triaxial tension which causes plastic flow in the surrounding material The intense local fields generated by the void growth in turn nucleates other neighbouring potential voids Page 1 Introduction leading to the ductile crack growth process of. .. provides a sound mechanism basis for the void growth process and contains two important microstructurally-linked parameters: the cell size and its initial void volume fraction Studies based on the porous ductile cell model is able to account for the effect of plastic straining on fracture, due to the mechanism of void growth to coalescence and due to plastic strain controlled nucleation of voids By contrast,... vaporized into pockets of microvoids when subjected to high temperatures manufacturing processing such as reflow soldering The extended Gurson model is thus useful in studying a potential failure mechanism common to most polymeric materials used in IC packaging: vapour pressure-induced ductile void growth failure The additional tractions imposed by the p in Equation (2.1) is representative of the vaporized... moisture induced in microvoids in the polymeric compounds, e.g epoxy adhesive, moulding compounds, etc Due to the increased hydrostatic stress field at the void surface, internal void vapour pressure greatly promote the growth of the voids, Page 19 Literature Review thereby resulting in severe damage to the voided matrix even before external load is applied With higher levels of internal void pressure, void. .. other void interaction mechanisms depicted by Varias et al [86] Xia and Shih’s methodology is adopted in Chapter 4 to further investigate the different failure mechanisms of a constrained ductile because the voided cells present a good prospect for the incorporation of internal pressure Presence of vapour pressure integrated with the constrained layer model provides a good framework to study moisture-induced... capture debonding damage mechanisms in a spherical particle-reinforced polymeric matrix composite Effects of interface layer porosity, void vapour pressure and reinforcement volume fraction on decohesion failure of reinforced polymeric matrix are investigated 2.4 Extended Gurson Model Incorporating Vapour Pressure In the approach of Xia and Shih [87] described in the earlier section, a porous ductile material ... many in the field of ductile void growth modeling However, void pressure-assisted ductile crack growth remains relatively new in this area Thus investigations begin with a preliminary study in. .. Summary Of Conclusions 142 iv Table of Contents 7.1 Ductile Crack Growth under Small Scale Yielding 142 7.2 Ductile Failure of Centerline Crack in a Constrained Ductile Layer 144 7.3 2-D Modeling of. .. of fracture for a crack lying along one of the interfaces of a thin ductile layer joining two elastic solids However, the model is not adequate for ductile crack growth lying in the void by void