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Unformatted text preview: Modeling Fracture in Elasticplastic Solids Using Cohesive Zones
CHANDRAKANTH SHET
Department of Mechanical Engineering FAMUFSU College of Engineering Florida State University Tallahassee, Fl32310 Sponsored by US ARO, US Air Force
1 Outline General formulation of continuum solids LEFM EPFM Introduction to CZM Concept of CZM Literature review Motivation Atomistic simulation to evaluate CZ properties Plastic dissipation and cohesive energy dissipation studies Conclusion 2 What is CZM and why is it important In the study of solids and design of nano/micro/macro structures,
thermomechanical behavior is modeled through constitutive equations. & & Typically is a continuous function of , , f(, , ) and their history. Design is limited by a maximum value of a given parameter ( ) at any local point. What happens beyond that condition is the realm of `fracture', `damage', and `failure' mechanics. CZM offers an alternative way to view and failure in materials. Formulation of a general boundary value problem
For a generic 3D analysis the equilibrium equation is given by j ij + fi = 0 xj For a 2D problem equilibrium equation reduces to y xy x xy + + f x = 0; + + fy = 0 y x y y x where x , y and xy are the stresses within the domain . f x , f y are the body forces. Boundary conditions are given by u = u1 at 1 u = u2 = 0 at 2 l x + m xy = t2 at 3 1 3 2 x Formulation of a general boundary value problem
The strain compatibility conditions are given by
2 xy j 2 x + 2 = 2 x y x y It can be shown that the all field equation reduces to 2 b2 ( 2 + 2 x + y ) = 0 b y x If is the Airy's stress function such that 2 2 x = 2 , y = 2 , xy =  y x xy Then the governing DE is 2 2 4TM 4 +2 2 2 + 4 =0 4 x x y y y 2 y For problems with crack tip Westergaard introduced Airy's stress function as = Re[ Z] + y Im[Z]
Where Z is an analytic complex function
Z z = Re[ z ] + y Im[ z ] ; z = x + iy =  b g y yy X And Z, Z are 2 and 1 integrals of Z(z) Then the stresses are given by
nd st
2 M x = 2 = Re[Z]  y Im[Z' ] y =  a 2 y = 2 = Re[Z] + y Im[Z' ] x 2 xy = =  y Im[Z' ] xy where Z' = dZ dz Opening mode analysis or Mode I Consider an infinite plate a crack of length 2a subjected to a biaxial State of stress. Defining:
Z= (z z
2 a 2 ) y x
2a Boundary Conditions : At infinity (  z = ) x = y = , xy = 0 On crack faces ( a < x < a; y = 0 ) x = xy = 0 By replacing z by z+a , origin shifted to crack tip. Z= ( z + a) z ( z + 2a ) Opening mode analysis or Mode I And when z0 at the vicinity of the crack tip a KI Z= = 2az 2 z K I = a KI must be real and a constant at the crack tip. This is due to a Singularity given by 1 z
The parameter KI is called the stress intensity factor for opening mode I. Since origin is shifted to crack tip, it is easier to use polar Coordinates, Using z = e i Opening mode analysis or Mode I KI 3 cos 1  sin sin 2 2 2 2 r KI 3 y = cos 1 + sin sin 2 2 2 2r KI 3 xy = sin cos cos 2 2 2 2 r x =
From Hooke's law, displacement field can be obtained as 2(1  ) r 1 u= KI cos + sin 2 E 2 2 2 2 v= 2(1  ) r 1 KI sin + cos2 E 2 2 2 2 where u, v = displacements in x, y directions = (3  4 ) for plane stress problems
3 = for plane strain problems 1 + a y yy
u X Small Scale plasticity
. Irwin estimates rp = 1 KI 2 ( ) 2 ys
Singularity dominated region Dugdale strip yield model: 1 KI 2 rp = ( ) 8 ys 10 EPFM
In EPFM, the crack tip undergoes significant plasticity as seen in the following diagram. s h a rp tip
Load ratio, P/P y
1.0 Fracture Ideal elastic brittle behavior cleavage fracture P: Applied load Py: Yield load Displacement, u Blunt tip Load ratio, P/Py 1.0 Fracture Limited plasticity at crack tip, still cleavage fracture Displacement, u Load ratio, P/Py Blunt tip 1.0 Fracture Void formation & coalescence failure due to fibrous tearing
large scale blunting Displacement, u
Load ratio, P/P y 1.0 Fracture Large scale plasticity fibrous rapture/ductile failure Displacement, u EPFM
EPFM applies to elasticplasticrateindependent materials
2 Crack opening displacement (COD) or 4 KI = 2 E crack tip opening displacement (CTOD). ys Jintegral. J = ( wdy  i T ui ds ), xi w = ij 0 ij d ij y Sharp crack ds x Blunting crack More on J Dominance Limitations of J integral, (Hutchinson, 1993) (1) Deformation theory of plasticity should be valid with small strain behavior with monotonic loading (2) If finite strain effects dominate and microscopic failures occur, then this region should be much smaller compared to J dominated region Again based on the HRR singularity 1 n+ J 1 I ij = y ij ( , n ) y y I n r Based on the condition (2), inner radius ro of J dominance. ro ; 3 COD R the outer radius where the J solutions are satisfied within 10% of complete solution. ro
R HRR Singularity...1
Hutchinson, Rice and Rosenbren evaluated the character of crack tip in powerlaw hardening materials. RambergOsgood model, = + 0 0 0 n 0  Reference value of stress=yield strength, n  strainhardening exponent 0 0  , strain at yield,  dimensionless constant E
Note if elastic strains are negligible, then = y y
n ij 3 eq = y 2 ij n 1 ^ ij 3 ^ ; eq = ij y 2
15 HRR Singularity...2
stress and strain fields are given by EJ % ij = 0 2 ij ( n, ) I 0 b nr 1 n +1 0 EJ % ij = 2 ij ( n, ) E 0 b nr I
I n  Integration constant % % ,  Dimensionless functions of n and n n +1 16 HRR Integral, cont.
Note the singularity is of the strenth ( 1 ) n+1 . For the specific case of n=1 (linearly r 1 elastic), we have r singularity.
1 Note also that the HRR singularity still assumes that the strain is infinitesimal, i.e., ij = 1 ( ui , j + u j ,i ) , and not the finite strain Eij = 1 ( ui , j + u j ,i + uk ,i uk , j ) . Near the tip 2 2 E where the strain is finite, (typically ijwhen ), one needs to use the strain measure < 0.1 . Some consequences of HRR singularity In elasticplastic materials, the singular field is given by J ij = k1 r J ij = k2 r
1 n +1 1 n +1 (with n=1 it is LEFM) stress is still infinite at r = 0 . the crack tip were to be blunt then xx = 0 at r = 0 since it is now a free surface. This is not the case in HRR field. HRR is based on small strain theory and is not thus applicable in a region very close to the crack tip. HRR Integral, cont.
Large Strain Zone HRR singularity still predicts infinite stresses near the crack tip. But when the crack blunts, the singularity reduces. In fact at xx = 0 at r = 0 for a blunt crack. The following is a comparison when you consider the finite strain and crack blunting. In the figure, FEM results are used as the basis for comparison.
The peak occurs at x 0 and J decreases as x < 1. This corresponds to approximately twice the width of CTOD. Hence within this region, HRR singularity is not valid. Largestrain crack tip finite element results of McMeeking and Parks. Blunting causes the stresses to deviate from the HRR solution close to the crack tip. 18 Fracture/Damage theories to model failure Fracture Mechanics Linear solutions leads to singular fieldsdifficult to evaluate Fracture criteria based on K IC ,G IC , J IC ,CTOD,... Nonlinear domain solutions are not unique Additional criteria are required for crack initiation and propagation Basic breakdown of the principles of mechanics of continuous media Damage mechanics can effectively reduce the strength and stiffness of the material in an average sense, but cannot create new surface % E D = 1  , Effective stress = = E 1 D CZM is an Alternative method to Model Separation CZM can create new surfaces. Maintains continuity conditions mathematically, despite the physical separation. CZM represent physics of fracture process at the atomic scale. It can also be perceived at the mesoscale as the effect of energy dissipation mechanisms, energy dissipated both in the forward and the wake regions of the crack tip. Uses fracture energy(obtained from fracture tests) as a parameter and is devoid of any adhoc criteria for fracture initiation and propagation. Eliminates singularity of stress and limits it to the cohesive strength of the the material. Ideal framework to model strength, stiffness and failure in an integrated manner. Applications: geomaterials, biomaterials, concrete, metallics, composites... Conceptual Framework of Cohesive Zone Models for interfaces
t1
* u 1 * 1 t1 * * u1 Tn x = (X, t) 1
S1 max (c)
^ n
max sep s P s1 s2 N P
t
* 2 * P 2 2
P ^ n1 n2 ^ t ,T n P S2 1 2
u* 2 X , x
3 3 u2 * (d) X 1, x1 X , x
2 (a)
2 (b) S is a n in t er fa ce s u r fa ce s ep a r a t in g t wo d om a in s 1 , 2 (id en t ica l/ s ep a r a t e con s t it u t ive b eh a vior ). Aft er fr a ct u r e t h e s u r fa ce S com p r is e of u n s ep a r a t ed s u r fa ce a n d com p let ely s ep a r a t ed s u r fa ce (e.g. ); a ll m od eled wit h in t h e con cep t of CZM. S u ch a n a p p r oa ch is n ot p os s ib le in con ven t ion a l m ech a n ics of con t in u ou s m ed ia . Interface in the undeformed configuration
* * u1 1 and 2 are separated by a common boundary S, such that S1 1 and S2 2 and normals N1 1 and N 2 2 Hence in the initial configuration S = S1 = S2 N = N1 = N 2 S defines the interface between any two domains 1 is metal, 2 is ceramic, S = metal ceramic interface 1 , 2 represent grains in different orientation, S = grain boundary 1 , 2 represent same domain (1 2 =), S = internal surface yet to separate
2 t1 1 s P s1 s2 N t 2* 2 X , x
3 3 u2 * X 1, x1 X , x (a)
2 Interface in the deformed configuration
After deformation a material point X moves to a new location x, such that x = (X,t) if the interface S separates, then a pair of new surface S 1 and S 2 are created bounding a new domain such that ^ N moves to n
* * t1 * * u1 1
S1 P P 2 ^ n S2 ^ (S1 , N1 ) moves to (S 1 , n1 ) (S 1 1 * ) ^ (S2 , N 2 ) moves to (S 2 , n2 ) (S 2 2 )
* * can be considered as 3D domain made of extremely soft glue, which can be shrunk to an infinitesimally thin surface but can be expanded into a 3D domain. (d) P ^ n1 n2 ^ t ,T n P u* 2 1 2 (b) 1 2 After deformation, given by x = (X,t), if v is the velocity vector, % Then velocity gradient L is given by Constitutive Model for Bounding Domains , v % L= x % % % Decomposing L into a symmetric part D and antisymmetric part W % % % such that L = D +W % % % % % % Where, D = 1 ( L + LT ) and W= 1 ( L  LT ) 2 2 % % D is the rate of deformation tensor, and W is the spin tensor Extending hypoelastic formulation to inelastic material by additive decomposition of the rate of deformation tensor % % % D = D El + D In % % where D El and D In are elastic and inelastic part of the rate of deformation tensor The constitutive model for the domains 1 and 2 can be written as % % % = C ( D  D In ) % where C is elasticity tensor, and Jaumann rate of cauchy stress tensor.
o o Constitutive Model for Cohesive Zone u
t1
* A typical constitutive relation for * is given by T  relation such that * 1 1
S1 Tn max (c)
^ n
max sep P % ^ if < sep , n = T and % ^ if sep , n = T = 0
(d)
P ^ n1 n2 ^ * P 2 t ,T n P S2 1 2 u* 2 (b) It can be construed that when in the domain * , the stiffness Cijkl sep 0. Development of CZ ModelsHistorical Review
Figure (a) Variation of Cohesive traction (b) I  inner region, II  edge region Barenblatt (1959) was
first to propose the concept of Cohesive zone model to brittle fracture Molecular force of cohesion acting near the edge of the crack at its surface (region II ). The intensity of molecular force of cohesion `f ' is found to vary as shown in Fig.a. The interatomic force is initially zero when the atomic planes are separated by normal intermolecular distance and increases to high maximum f m = ETo / b : E /10 after that it rapidly reduces to zero with increase in separation distance. E is Young's modulus and To is surface tension
(Barenblatt, G.I, (1959), PMM (23) p. 434) Phenomenological Models The theory of CZM is based on sound principles. However implementation of model for practical problems grew exponentially for practical problems with use of FEM and advent of fast computing. Model has been recast as a phenomenological one for a number of systems and boundary value problems. The phenomenological models can model the separation process but not the effect of atomic discreteness.
Hillerborg etal. 1976 Ficticious crack model; concrete Bazant etal.1983 crack band theory; concrete Morgan etal. 1997 earthquake rupture propagation; geomaterial Planas etal,1991, concrete Eisenmenger,2001, stone fragmentation squeezing" by evanescent waves; brittlebio materials Amruthraj etal.,1995, composites Grujicic, 1999, fracture behavior of polycrystalline; bicrystals Costanzo etal;1998, dynamic fr. Ghosh 2000, Interfacial debonding; composites Rahulkumar 2000 viscoelastic fracture; polymers Liechti 2001Mixedmode, timedepend. rubber/metal debonding Ravichander, 2001, fatigue Tevergaard 1992 particlematrix interface debonding Tvergaard etal 1996 elasticplastic solid :ductile frac.; metals Brocks 2001crack growth in sheet metal Camacho &ortiz;1996,impact Dollar; 1993Interfacial debonding ceramicmatrix comp Lokhandwalla 2000, urinary stones; biomaterials Fracture process zone and CZM CZM essentially models fracture process zone
Mathematical crack tip by a line or a plane ahead of the crack tip Material crack tip subjected to cohesive traction. The constitutive behavior is given by traction y displacement relation, obtained by defining potential function of the type = ( n , t1 , t 2 ) x
where n , t1 , t 2 are normal and tangential displacement jump The interface tractions are given by Y a Tn =  , Tt1 =  , Tt 2 =  TM t 2 n t1 Following the work of Xu and Needleman (1993), the interface b potential +is taken as 1  r + ( 1  q ) ( , ) = exp   n t n n n n where q = t / r = *n / n n ( r  1) n b r  q ) n 2 t ( q exp  + 2 ( r  1) n t b b n , t are some characteristic distance *n Normal displacement after shear separation under the condition Of zero normal tension Normal and shear traction are given by b 2t n n n 2 t  q ) n (1 Tn =  exp  exp  + r 1   exp 2  n n 2 r  1) n ( n t t b M n t 2 n r  q ) n n 2 t ( Tt =  q exp  2 exp  + n t t r  1) n n ( b t b b 29 Dissipative Micromechanisims Acting in the wake and forward region of the process zone at the Interfaces of Monolithic and Heterogeneous Materia l ^ max
C W ak e o f cra ck tip
F i b r i l ( M M C b ri d g i n g M i c r o v o id c o a le s c e n c e P la s ti c zon e F o r w a rd o f c r a c k ti p
M e t a l li c
C leavag e fra c ture G r a in b r id g i n g y
N O M A T E RIA L S E P A R A T IO N B D
L O C A T IO N O F C O H E S IV E CR AC K TIP O x i d e b r i d g in g C O M P L E T E M A T E R IA L S E P A R A T IO N T h ic k n e s s o f c e ra m i c i n t e r f a c e C r a c k M e a n d e ri n g F i b ri l ( p o l y m e r s ) b r id g i n g P la s t ic w a k e E A l1
F O RW A RD max D , X sep l2
W A KE C e ram ic In tr i n s i c d i s s i p a t i o n
M A T E R IA L C R A C K T IP C O H E SIV E C R A C K TIP M A T H EM A TIC A L C R A C K T IP E x t r i n s ic d is s i p a t io n
M ic r o c r a c k i n g i n i ti a t io n M ic r o v o i d g r o w th / c o a l e s c e n c e P r e c ip it a t e s C r a c k D e f le c t io n C r ac k M e an d erin g C o n ta ct W ed g in g IN A C TIV E PL A ST IC Z O N E ( P l a s t ic w a k e ) sep
E D max
D C
FO RW A RD A C o n t a c t S u rf a c e ( f r ic ti o n ) P l a s t ic it y in d u c e d cra ck clo s u re
P l a stic W a k e D e l a m i n a tio n W A K E C or ne r a to m s F a c e c e n te re d a to m s F CC
y A C T I V E P L A S T IC Z O N E P hase tr a n s f o r m a ti o n C y c li c l o a d in d u c e d c r a c k c lo s u r e
BCC C o rn e r a t o m s B o d y c e n te r e d a tom s x E L A S T I C S IN G U L A R I T Y Z O N E I n t e r / tr a n s g r a n u la r f r a c tu r e Concept of wake and forward region in the cohesive process zone Active dissipation mechanisims participating at the cohesive process zone30 31 32 Motivation for studying CZM
CZM is an excellent tool with sound theoretical basis and computational ease. Lacks proper mechanics and physics based analysis and evaluation. Already widely used in fracture/fragmentation/failure critical issues addressed here
Scales What range of CZM parameters are valid? MPa or GPa for the traction J or KJ for cohesive energy nm or m for separation displacement What is the effect of plasticity in the bounding material on the fracture processes Energy Energy characteristics during fracture process and how energy flows in to the cohesive zone. Importance of shape of CZM 33 Atomistic simulations to extract cohesive properties
Motivation
What is the approximate scale to examine fracture in a solid Atomistic at nm scale or Grains at m scale or Continuum at mm scale Are the stress/strain and energy quantities computed at one scale be valid at other scales? (can we even define stressstrain at atomic scales?) 34 Embedded Atom Method Energy Functions
(D.J.Oh and R.A.Johnson, 1989 ,Atomic Simulation of Materials, Edts:V Vitek and D.J.Srolovitz,p 233) 233 The total internal energy of the crystal Etot = Ei
i where and
Ei Ei = F ( i ) + 1 2 jM 1 ( rij ) 5 4 3 2 Al Mg Cu i =
jM 1 f ( rij )
E n e rg y (e V ) Cutoff Distances
(4.86) (5.44) (6.10)
2 4 6 Internal energy associated with atom i Embedded Energy of atom i. 1 0 1 2 3 4 5 F ( i ) A tomic Seperation (A ) ij j f ( rij ) Contribution to electron density of ith atom and jth atom. Two body central potential between ith atom and jth atom. 35 T  Curve in Shear direction A small portion of 9(221) CSL grain bounary before And after application of tangential force Shet C, Li H, Chandra N ;Interface models for GB sliding and migration MATER SCI FORUM 3573: 577585 2001 T  Curve in Normal direction A small portion of 9(221) CSL grain boundary before And after application of normal force 37 Results and discussion on atomistic simulation
Summary complete debonding occurs when the distance of separation reaches a value of 2 o to 3 A . For 9 bicrystal tangential work of separation along the grain boundary is of 2 the order 3 J / m and normal 2work of separation is of the order 2.6 J / m . For 3 bicrystal, the work of separation ranges from 1.5 to 3.7 J / m 2 . Rose et al. (1983) have reported that the adhesive energy (work of separation) for 2 aluminum is of the order 0.5 J / m and the o separation distance 2 to 3 A Measured energy to fracture copper bicrystal with random grain boundary is of the order 54 J / m 2 and for 11 copper bicrystal the energy to fracture is more than 8000 J / m 2 Implications The numerical value of the cohesive energy is very low when compared to the observed experimental results Atomistic simulation gives only surface energy ignoring the inelastic energies due to plasticity and other micro processes. = 2 + Wp It should also be noted that the experimental value of fracture energy includes the plastic work in addition to work of separation (J.R Rice and J. S Wang, 1989) Table of surface and fracture energies of standard materials
Material Aluminium alloys Nomenclature 2024T351 2024T851 Titanium alloys T21 T68 Steel Medium Carbon High strength alloys 18 Ni (300) maraging Al 2O3 K IC MPam1/ 2
35 25.4 80 130 54 98 76 48 6.1 G IC J / m 2
14900 8000 48970 130000 12636 41617 25030 34240 J / m2
1.2 1.2 24 24 24 particle size Alumina SiC ceramics Polymers 10 m 0.11 to 1.28 m PMMA 1.21.7 220 Energy balance and effect of plasticity in the bounding material 40 Motivation It is perceived that CZM represents the physical separation process. As seen from atomistics, fracture process comprises mostly of inelastic dissipative energies. There are many inelastic dissipative process specific to each material system; some occur within FPZ, and some in the bounding material. How the energy flow takes place under the external loading within the cohesive zone and neighboring bounding material near the crack tip? What is the spatial distribution of plastic energy? Is there a link between micromechanic processes of the material and T  curve. Cohesive zone parameters of a ductile material Al 2024T3 alloy The input energy in the cohesive model are related to the interfacial stress and characteristic displacement n as e max t 2 The input energy n is equated to material parameter Based on the measured fracture value J IC t = n = max e n n = t = 8000 J / m 2 max = ult = 642MPa n = t = 4.5 X 10 6 m 42 Material model for the bounding material Elastoplastic model for Al 2024T3 Stress strain curve is given by = + E y 1/ n where y = 320MPa, = 0.01347, n = 0.217173
E=72 GPa, =0.33, and fracture parameter
K IC = 25MPa  m1/ 2 43 Numerical Formulation The numerical implementation of CZM for interface modeling with in implicit FEM is accomplished developing cohesive elements Cohesive elements are developed either as line elements (2D) or planar elements (3D)abutting bulk elements on either side, with zero thickness 1
3 5 7 Continuum elements 2 4 6 8 Cohesive element The virtual work due to cohesive zone traction in a given cohesive element can be written as
& & (T dS = n n & + Tt t ) dS The virtual displacement jump is written as = [N]{v} Where [N]=nodal shape function matrix, {v}=nodal displacement vector
& ( [N] dS = {v} d{T } + [N] d{T }) dS J = Jacobian of the transformation between the current deformed and original undeformed areas of cohesive surfaces
T T T s n t 1 J & Note: T is written as d{T} the incremental traction, ignoring time which is a pseudo quantity for rate independent material 44 Numerical formulation contd
The incremental tractions are related to incremental displacement jumps across a cohesive element face through a material Jacobian matrix as
d{T} = [C cz }d{} For two and three dimensional analysis Jacobian matrix is given by T h [C cz ] = n  Tt n n Tn t Tt t Tn n [C cz ] = Tt1 n Tt 2 n Tn t1 Tt1 t1 Tt 2 t1 Tn t 2 Tt1 t 2 Tt 2 t 2 Finally substituting the incremental tractions in terms of incremental displacements jumps, and writing the displacement jumps by means of nodal displacement vector through shape function, the tangent stiffness matrix takes the form
[K T ] = [N]T [C cz ][N] 1 dS J
s 45 Geometry and boundary/loading conditions a = 0.025m, b = 0.1m, h = 0.1m
46 Finite element mesh 28189 nodes, 24340 plane strain 4 node elements, 7300 cohesive elements (width of element along the crack plan is ~ 7x107 m 47 Global energy distribution
E w = Ee + E p + Ec
E e and E p are confined to bounding material E c is cohesive energy, a sum total of all dissipative process confined to FPZ and cannot be recovered during elastic unloading and reloading. Purely elastic analysis
The conventional fracture mechanics uses the concept of strain energy release rate M U G=J= a Using CZM, this fracture energy = G = J = 8000J / m 2 is dissipated and no plastic dissipation occurs, such that E w = Ee + Ec 48 Global energy distribution (continued)
Analysis with elastoplastic material model Two dissipative process = 8000J / m 2 Plasticity within Bounding material Issues
Fracture energy obtained from experimental results is sum total of all dissipative processes in the material for initiating and propagating fracture. Should this energy be dissipated entirely in cohesive zone? Should be split into two identifiable dissipation processes? Microseparation Process in FPZ Implications Leaves no energy for plastic work in the bounding material In what ratio it should be divided? Division is nontrivial since plastic dissipation depends on geometry, loading and other parameters as E p = E p max , n,Si = 1, 2,.. ,i y where Si represents other factors arising from the shape of the tractiondisplacement relations
49 What are the key CZM parameters that govern the energetics? max in cohesive zone dictates the stress level achievable in the bounding
material. Yield in the bounding material depends on its yield strength y and its post yield (hardening characteristics). Thus max y plays a crucial role in determining plasticity in the bounding material, shape of the fracture process zone and energy distribution. (other parameters like shape may also be important) Global energy distribution (continued)
Recoverable elastic work E e = 95 to 98% of external work
E n e rg y /( 2 . y . n . 1 .0 E 2 ) 4 3.5 3 2.5 2 Cumulative Plastic W ork Cumulative Cohesive Energy Plastic dissipation depends on max y max y 1 to 1.5 :  Elastic behavior max y 1.5 :  plasticity occurs. max y Plasticity increases with 4 3 1.5 1 0.5 0 1 2 0 20 u / n
8 40 60 80 Variation of cohesive energy and plastic energy for various max y ratios (1) max y = 1 (2) max y = 1.5 (3) max y = 2.0 (4) max y = 2.5 51 Relation between plastic work and cohesive work max y = 1.5 (very small scale plasticity), plastic energy ~ 15% of total dissipation.
Plasticity induced at the initial stages of the crack growth plasticity ceases during crack propagation. Very small error is induced by ignoring plasticity.
3 C ohesive Energy/(2. y. n . 1.0E2) 2.5 max y = 1.5 max y = 2.0 2 max y = 2.5 max y > 2.0 plastic work increases 1.5 considerably, ~100 to 200% as that of cohesive energy. For large scale plasticity problems the amount of total dissipation (plastic and 2 cohesive) is much higher than 8000J / m . Plastic dissipation very sensitive to max y
ratio beyond 2 till 3 Crack cannot propagate beyond max y > 3 and completely elastic below max y 1.5 1 0.5 0 0 1 2 3 4 Plastic Energy/( 2. y . n .1.0E2) Variation of Normal Traction along the interface
GDDWLA0P The length of cohesive zone is also max affected byy ratio. There is a direct correlation between the shape of the tractiondisplacement curve and the normal traction distribution along the cohesive zone. For lower max y ratios the tractionseparation curve flattens, this tend to increase the overall cohesive zone length. 53 Local/spatial Energy Distribution
A set of patch of elements (each having app. 50 elements) were selected in the bounding material. m The patches are approximately squares (130 ). They are spaced equally from each other. Adjoining these patches, patches of cohesive
elements are considered to record the cohesive energies. 54 Variation of Cohesive Energy
The cohesive energy in the patch increases up to point C (corresponding to max in Figure ) after which the crack tip is presumed to advance. The energy consumed by the cohesive elements at this stage is approximately 1/7 of the total cohesive energy for the present CZM. Once the point C is crossed, the patch of elements fall into the wake region. The rate of cohesive zone energy absorption depends on the slope of the T  curve and the rate at which elastic unloading and plastic dissipation takes place in the adjoining material. The curves flatten out once the entire cohesive energy is dissipated within a given zone.
Tn max GDED3V0A max sep The variation of Cohesive Energy in the Wake and Forward region as the crack propagates. The numbers indicate the Cohesive Element Patch numbers Falling Just Below the binding element patches Variation of Elastic Energy
Considerable elastic energy is built up till the peak of T  curve is reached after which the crack tip advances. After passing C, the cohesive elements near the crack tip are separated and the elements in this patch becomes a part of the wake. At this stage, the values of normal traction reduces following the downward slope of T  curve following which the stress in the patch reduces accompanied by reduction in elastic strain energy. The reduction in elastic strain energy is used up in dissipating cohesive energy to those cohesive elements adjoining this patch. The initial crack tip is inherently sharp leading to high levels of stress fields due to which higher energy for patch 1 Crack tip blunts for advancing crack tip leading to a lower levels of stress, resulting in reduced energy level in other patches. GDCD1C08 Tn max max sep Variation of Elastic Energy in Various Patch of Elements as a Function of Crack Extension. The numbers indicate Patch numbers starting from Initial Crack Tip Variation of Plastic Work (max y 2.0 )
plastic energy accumulates considerably along with elastic energy, when the local stresses bounding material exceeds the yield y After reaching peak point C on T  curve traction reduces and plastic deformation ceases. Accumulated plastic work is dissipative in nature, it remains constant after debonding. All the energy transfer in the wake region occurs from elastic strain energy to the cohesive zone The accumulated plastic work decreases up to patch 4 from that of 1 as a consequence of reduction of the initial sharpness of the crack. Mechanical work is increased to propagate the crack, during which the E cand E e does not increase resulting in increased plastic work. That increase in plastic work causes the increase in the stored work in patches 4 and beyond Tn max max sep Variation of dissipated plastic energy in various patched as a function of crack extension. The number indicate patch numbers starting from initial crack tip. Variation of Plastic Work ( max y 1.5)
max y = 1, there is no plastic dissipation.
max y = 1.5 plastic work is induced only in the first patch of element GDCD8U0A No plastic dissipation during crack growth place in the forward region Initial sharp crack tip profile induces high levels of stress and hence plasticity in bounding material. During crack propagation, tip blunts resulting reduced level of stresses leading to reduced elastic energies and no plasticity condition. Tn max max sep Variation of Plastic work and Elastic work in various patch of elements along the interface for the case of max y 1.5 . The numbers indicates the energy in various patch of elements starting from the crack tip. 58 Contour plot of yield locus around the cohesive crack tip at the various stages of crack growth. 59 Schematic of crack initiation and propagation process in a ductile material Conclusion
CZM provides an effective methodology to study and simulate fracture in solids. Cohesive Zone Theory and Model allow us to investigate in a much more fundamental manner the processes that take place as the crack propagates in a number of inelastic systems. Fracture or damage mechanics cannot be used in these cases. Form and parameters of CZM are clearly linked to the micromechanics. Our study aims to provide the modelers some guideline in choosing appropriate CZM for their specific material system. max y ratio affects length of fracture process zone length. For smaller max y ratio the length of fracture process zone is longer when compared with that of higher ratio. Amount of fracture energy dissipated in the wake region, depend on shape of the model. For example, in the present model approximately 6/7th of total dissipation takes place in the wake Plastic work depends on the shape of the crack tip in addition to max y ratio. Conclusion(contd.)
The CZM allows the energy to flow in to the fracture process zone, where a part of it is spent in the forward region and rest in the wake region. The part of cohesive energy spent as extrinsic dissipation in the forward region is used up in advancing the crack tip. The part of energy spent as intrinsic dissipation in the wake region is required J IC to complete the gradual separation process. In case of elastic material the entire fracture energy given by the of the material, and is dissipated in the fracture process zone by the cohesive elements, as cohesive energy. In case of small scale yielding material, a small amount of plastic dissipation (of the order 15%) is incurred, mostly at the crack initiation stage. During the crack growth stage, because of reduced stress field, plastic dissipation is negligible in the forward region. 62 ...
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 Fall '11
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 Mechanical Engineering, Plasticity

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