Effect of rise, peak and fall characteristics of CZM in predicting fracture processes

Motivation

What is the relationship between the physics/mechanics of the separation process and shape of CZM? (There are as many shapes/equations as there are number of interface problems solved!)What is the physical significance of- Shape of the curve C- tmax and interface strength- Separation distance sep - Initial slopeMotivations

Terminology

Different CZM ModelsTo study the shape effect, three CZMs with different shapes were consideredExponential ModelTrapezoidal ModelBilinear Model

To study the effect of initial rising slope and the tail effect, parameters of trapezoidal model and of bilinear model are varied.

Exponential model Following the work of Xu and Needleman (1993), the interface potential is taken as

where are some characteristic distanceNormal displacement after shear separation under the conditionOf zero normal tensionNormal and shear traction are given by

The interfacial normal traction and shear traction variation are shown belowFig(a) is the variation for normal traction as a function of when is zeroFig(b) is the variation for normal traction as a function of whe n is zero

The interfacial energy variation due to normal and shear tractions are shown below

Trapezoidal modelFollowing the work of Trevegaard and Huchinson (1992), the interface tractions are given by

_1059148297.unknown

_1059148351.unknown

_1059148372.unknown

_1059148402.unknown

_1059148436.unknown

_1059148459.unknown

_1059148419.unknown

_1059148383.unknown

_1059148362.unknown

_1059148330.unknown

_1059148341.unknown

_1059148309.unknown

_1059148274.unknown

_1059148287.unknown

_1059148240.unknown

Cohesive zone parametersMaterial Al 2024-T3 alloyThe input energy in the exponential cohesive zone model are related to the interfacial stress and characteristic displacement as

The input energy is equated to material parameter For the material considered

For trapezoidal modelFor bilinear model

0.1

0.2

0.3

0.4

0.5

0.6

0.8

(m)

1.45E-6

3.08E-6

4.92E-6

7.04E-6

9.45E-6

1.23E-5

1.96E-5

(m)

1.45E-5

1.54E-5

1.64E-5

1.76E-5

1.89E-5

2.05E-5

2.45E-5

_1059166792.unknown

_1059166812.unknown

_1059166765.unknown

0.1

0.2

0.4

0.6

0.8

(m)

1.45E-6

4.9E-6

9.86E-6

1.47E-5

1.96E-5

_1059166812.unknown

_1059167680.unknown

Cohesive zone parameters Material model for the bounding materialMaterial considered are aluminum alloy Al 2024-T3

E=72 GPa, =0.33, Stress strain curve is given bywhere

a = 0.025m, b = 0.1m, h = 0.1m

24340 plane strain 4 node elements7300 cohesive elements (width of element along the crack plan is ~7E-7m28189 nodes

Effect of smax/sy on plastic dissipation(exponential model)

Effect of smax/sy on plastic dissipation

Local/spatial Energy Distribution

Variation of Cohesive EnergyThe 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 EnergyVariation of Elastic Energy in Various Patch of Elements as a Function of Crack Extension. The numbers indicate Patch numbers starting from Initial Crack TipConsiderable elastic energy is built up till the peak of 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 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 1Crack tip blunts for advancing crack tip leading to a lower levels of stress, resulting in reduced energy level in other patches.

Variation of dissipated plastic energy in various patched as a function of crack extension. The number indicate patch numbers starting from initial crack tip.

Schematic of crack initiation and propagation process in a ductile material

Variation of Elastic & Plastic Work ( )For trapezoidal model

Both bilinear and trapezoidal models predict considerably less amount of plasticity when compared to exponential modelsExponential model shows plastic energy is dissipated not only at crack initiation but also during crack growth.Both bilinear and trapezoidal models shows plastic dissipation only at crack initiation and during crack growth plastic dissipation does not take place. Plastic dissipation decreases with with reduction of initial slope (with increase in dmax1 for trapezoidal model and dmax in case of bilinear model)Exponential model shows a slightly higher stress field near the crack tip when compared to that of bilinear and trapezoidal models. This also contributes to increased plasticity in exponential models.

Tail part of trapezoidal CZM is varied keeping constant at 0.1 and varying from 0.1 to 0.8.Plastic dissipation decreases with longer tail in CZM

Chart1

00000

0.00376348720.00421182910.00480355560.00559740170.0060957265

0.02500786320.02759316240.03104068380.03571692310.0389039316

0.07790085470.0840.0921572650.07269059830.081025641

0.16466324790.15113846150.18094358970.11789401710.1350495726

0.17446837610.19583589740.23794871790.21635555560.255425641

0.21185641030.23917264960.68254358970.32192820510.3904615385

0.24901196580.30092991450.69675213680.74283760680.5472820513

0.2936341880.40202393160.78721367521.0629743590.6621264957

0.33119316240.51492649570.86584615381.21367521370.8538119658

0.36828034190.56504615380.94536752141.48793162391.1584273504

0.48869743590.66238632481.03391452991.628034188

0.56441025640.76287179491.1091965812

0.81688888891.2453333333

dmax2 = 0.1

dmax2 = 0.2

dmax2 = 0.4

dmax2 = 0.6

dmax2 = 0.8

delmax2=0.8

delmax2=0.6

delmax2=0.4

delmax2=0.2

delmax2=0.1

PW/(smax*dn*1.0E-2)

CE/(smax*dn*1.0E-2)

Variation of CE/PW with dmax=0.1 and different dmax2 (tail effect)

Sheet1

frstrev1, trevergard, dn=1.45E-5, delmax=0.1,delmax2=0.8, smax=650. Energy=0.008frstrev16, trever, delmax=0.1, delmax2=0.6,smax=650.,energy=0.008, dn=1.64E-5frstrev14,delmax=0.1,delmax2=0.4,dn=1.894E-5.energy=0.008,smax=650.frstrev12,delmax=0.1,delmax2=0.2,dn=2.24E-5.energy=0.008,smax=650.frstrev11,delmax=0.1,delmax2=0.1,dn=2.46E-5.energy=0.008,smax=650.

uaewrsepwintebewmintPWCEuaewrsepwintebewmintmpwceuaewrsepwintebewmintmpwceuaewrsepwintebewmintmpwceuaewrsepwintebewmintmpwce

00.02500000000000.02500000000000.02500000000000.02500000000000.025000000000

3.20E-052.50E-025.51E-053.67E-052.97E-125.50E-08-1.85E-050.00003676010.00000011010.00000010160.00376348723.20E-050.0255.51E-053.66E-051.55E-126.16E-08-1.85E-050.00003675320.00000012320.00000005290.00421182913.20E-050.0255.51E-053.66E-055.21E-137.03E-08-1.85E-050.00003673050.00000014050.00000001780.00480355563.20E-050.0255.51E-053.65E-050.00E+008.19E-08-1.85E-050.00003670370.000000163700.00559740173.20E-050.0255.50E-053.65E-0508.92E-08-1.85E-050.00003668830.000000178300.0060957265

8.00E-052.52E-022.89E-042.29E-041.92E-083.66E-07-6.04E-050.00022975150.00000073150.00065675210.02500786328.00E-052.51E-022.89E-042.29E-041.77E-084.04E-07-6.04E-050.00022966710.00000080710.0006058120.02759316248.00E-052.51E-022.89E-042.29E-041.58E-084.54E-07-6.05E-050.00022949790.00000090790.00054153850.03104068388.00E-052.51E-022.89E-042.28E-041.37E-085.22E-07-6.06E-050.00022933470.00000104470.00046700850.03571692318.00E-052.51E-022.89E-042.28E-041.02E-085.69E-07-6.07E-050.00022925790.00000113790.00034769230.0389039316

1.28E-042.55E-026.89E-045.85E-041.93E-071.14E-06-1.03E-040.00058777860.00000227860.00658119660.07790085471.28E-042.55E-026.88E-045.85E-041.87E-071.23E-06-1.03E-040.0005874570.0000024570.00640341880.0841.28E-042.55E-026.88E-045.84E-041.80E-071.35E-06-1.03E-040.00058719560.00000269560.0061572650.0921572651.10E-042.53E-025.15E-044.31E-047.44E-081.06E-06-8.35E-050.00043351620.00000212620.00254427350.07269059831.10E-042.53E-025.15E-044.31E-044.31E-081.19E-06-8.37E-050.000433360.000002370.0014741880.081025641

1.68E-042.60E-021.14E-031.00E-036.09E-072.41E-06-1.30E-040.00100971640.00000481640.02082735040.16466324791.59E-042.59E-021.03E-039.03E-044.65E-072.21E-06-1.26E-040.00090722080.00000442080.01589743590.15113846151.62E-042.59E-021.05E-039.29E-044.15E-072.65E-06-1.17E-040.00093489260.00000529260.01417777780.18094358971.33E-042.56E-026.25E-047.20E-041.44E-071.72E-06-9.42E-050.00053449840.00000344840.00493675210.11789401711.33E-042.56E-027.19E-046.25E-045.90E-081.98E-06-9.46E-050.00062846020.00000395020.0020170940.1350495726

1.71E-042.60E-021.17E-031.04E-036.57E-072.55E-06-1.31E-040.00104850320.00000510320.02246495730.17446837611.72E-042.61E-021.18E-031.05E-036.20E-072.86E-06-1.30E-040.00105732820.00000572820.02120683760.19583589741.72E-042.61E-021.17E-031.05E-034.94E-073.48E-06-1.21E-040.001059660.000006960.01690256410.23794871791.59E-042.60E-021.01E-038.96E-041.93E-073.16E-06-1.13E-040.00090262840.00000632840.0065914530.21635555561.59E-042.61E-021.01E-038.94E-046.88E-083.74E-06-1.14E-040.00090177120.00000747120.00235282050.255425641

1.79E-042.62E-021.27E-031.14E-038.10E-073.10E-06-1.32E-040.00114699680.00000619680.02769572650.21185641031.78E-042.62E-021.26E-031.13E-037.16E-073.50E-06-1.32E-040.00113399580.00000699580.02448205130.23917264961.78E-042.77E-021.25E-031.11E-036.01E-079.98E-06-1.35E-040.00113296440.00001996440.02053675210.68254358971.71E-042.64E-021.16E-031.04E-031.98E-074.71E-06-1.19E-040.00104741640.00000941640.00677606840.32192820511.71E-042.66E-021.16E-031.04E-036.88E-085.71E-06-1.21E-040.0010466210.0000114210.00235042740.3904615385

1.82E-042.63E-021.31E-031.17E-039.57E-073.64E-06-1.33E-040.00118188360.00000728360.03272478630.24901196581.79E-042.64E-021.28E-031.14E-038.54E-074.40E-06-1.34E-040.00115390220.00000880220.02918974360.30092991451.78E-042.78E-021.25E-031.12E-036.18E-071.02E-05-1.35E-040.001136980.000020380.02112136750.69675213681.78E-042.79E-021.24E-031.11E-032.25E-071.09E-05-1.32E-040.0011338280.0000217280.00770256410.74283760681.74E-042.72E-021.20E-031.07E-036.87E-088.00E-06-1.26E-040.0010855080.0000160080.00234803420.5472820513

1.83E-042.65E-021.33E-031.19E-031.11E-064.29E-06-1.35E-040.00120088880.00000858880.03791452990.2936341881.81E-042.68E-021.30E-031.16E-031.06E-065.88E-06-1.37E-040.00116995920.00001175920.03627350430.40202393161.81E-042.81E-021.29E-031.15E-037.27E-071.15E-05-1.38E-040.0011719260.0000230260.02485811970.78721367521.85E-042.88E-021.34E-031.20E-034.29E-071.55E-05-1.40E-040.0012267920.0000310920.01465982911.0629743591.76E-042.76E-021.09E-031.22E-036.87E-089.68E-06-1.29E-040.00097846720.00001936720.00234735040.6621264957

1.84E-042.66E-021.34E-031.20E-031.23E-064.84E-06-1.36E-040.00120888740.00000968740.04194871790.33119316241.82E-042.72E-021.31E-031.17E-031.27E-067.53E-06-1.40E-040.00118296160.00001506160.04331623930.51492649571.83E-042.84E-021.32E-031.18E-038.34E-071.27E-05-1.41E-040.0012018260.0000253260.02852649570.86584615381.89E-042.91E-021.39E-031.24E-035.93E-071.78E-05-1.44E-040.001280.00003550.02025641031.21367521371.80E-042.83E-021.27E-031.13E-031.35E-071.25E-05-1.35E-040.0011582740.0000249740.00462222220.8538119658

1.84E-042.68E-021.34E-031.20E-031.34E-065.39E-06-1.37E-040.00121587220.00001077220.04570940170.36828034191.82E-042.74E-021.32E-031.17E-031.36E-068.26E-06-1.42E-040.00118992760.00001652760.04632478630.56504615381.86E-042.86E-021.35E-031.21E-039.50E-071.38E-05-1.43E-040.0012337520.0000276520.03248888890.94536752141.98E-042.96E-021.51E-031.36E-031.01E-062.18E-05-1.49E-040.0014062220.0000435220.03459829061.48793162391.89E-042.92E-021.39E-031.24E-034.32E-071.69E-05-1.44E-040.0012761840.0000338840.0147658121.1584273504

1.85E-042.72E-021.36E-031.21E-031.67E-067.15E-06-1.40E-040.00122889440.00001429440.0570256410.48869743591.83E-042.77E-021.33E-031.18E-031.54E-069.69E-06-1.44E-040.00120397480.00001937480.05268376070.66238632481.89E-042.89E-021.39E-031.24E-031.12E-061.51E-05-1.46E-040.0012726420.0000302420.03818803421.03391452992.18E-042.99E-021.81E-031.65E-032.99E-062.76E-05-1.62E-040.0017055720.0000551720.10205128211.88622222222.09E-042.99E-021.67E-031.51E-031.49E-062.38E-05-1.58E-040.001563220.000047620.0509743591.628034188

1.86E-042.75E-021.36E-031.22E-031.88E-068.25E-06-1.43E-040.0012369090.0000165090.06423931620.56441025641.85E-042.81E-021.35E-031.20E-031.74E-061.12E-05-1.47E-040.0012221140.0000223140.05948717950.76287179491.92E-042.92E-021.43E-031.28E-031.30E-061.62E-05-1.48E-040.0013136440.0000324440.04444444441.1091965812

1.86E-042.77E-021.37E-031.23E-032.11E-069.44E-06-1.45E-040.00124597460.00001887460.07220512820.64528547011.86E-042.82E-021.36E-031.21E-031.85E-061.19E-05-1.49E-040.0012352940.0000238940.06324786320.81688888891.99E-042.95E-021.52E-031.37E-031.75E-061.82E-05-1.52E-040.0014074260.0000364260.0599658121.2453333333

1.88E-042.80E-021.39E-031.24E-032.37E-061.07E-05-1.47E-040.0012609420.0000213420.08085470090.7296410256

1.89E-042.83E-021.40E-031.25E-032.57E-061.17E-05-1.49E-040.0012760160.0000233160.08776068380.7971282051

1.90E-042.85E-021.42E-031.27E-032.76E-061.26E-05-1.51E-040.001293150.000025150.09432478630.8598290598

1.92E-042.87E-021.44E-031.29E-033.01E-061.37E-05-1.53E-040.0013163580.0000273580.10273504270.9353162393

Sheet1

dmax2 = 0.1

dmax2 = 0.2

dmax2 = 0.4

dmax2 = 0.6

dmax2 = 0.8

delmax2=0.8

delmax2=0.6

delmax2=0.4

delmax2=0.2

delmax2=0.1

PW/(smax*dn*1.0E-2)

CE/(smax*dn*1.0E-2)

Variation of CE/PW with dmax=0.1 and different dmax2 (tail effect)

Sheet2

Sheet3

Effect of Initial slope of CZMs (contd.)Exponential model shows slightly higher stress field in the bounding materialThe intensity of stress field decreases with increase in initial slope

Trapezoidal ModelTrapezoidal model shows slightly higher initial stiffness when compared to bilinear modelIn both the models stiffness reduces as the initial slope of CZMs are increased.

Chart1

00000

81.753872716240.87969399120.439286328513.625287708510.2198903037

81.763318672540.876783370320.439979394813.623750130910.2205807568

81.7592300440.874122407220.437821030313.624832899810.2193970678

79.924312599240.874691332720.437610469613.626977187310.2204253344

56.899590899640.8806179920.241324032913.625642960810.2182625299

40.350194727740.757713132618.449565658313.625673678210.2191430253

19.60921183334.22575516949.27845416913.626423361910.2185552754

7.73256741518.57473376554.753512949913.618808405910.2184968861

3.048035880212.76922497290.932311866910.22868274359.2073202752

03.191655544205.55971399227.3052126889

0001.34069931480.1620662313

000

00

dmax=0.8

dmax=0.6

dmax=0.4

dmax=0.2

dmax=0.1

dmax=0.1

dmax=0.2

dmax=0.4

dmax=0.6

dmax=0.8

a/b

(Tn/dn)/(E/a(1-n2))

Variation of stiffness of cohesive element (bilinear model)

Sheet1

frstrev11,delmax=0.1,delmax2=0.1,dn=2.46E-5.energy=0.008,smax=650.file=frstrever22, delmax1=0.2, delmax2=0.2, phi=8000j/m*m, sigmax=650dn=2.45E-5mfile=frstrever44, delmax1=0.4, delmax2=0.4, phi=8000j/m*m, sigmax=650dn=2.45E-5mfile=frstrever66, delmax1=0.6, delmax2=0.6, phi=8000j/m*m, sigmax=650dn=2.45E-5mfile=frstrever88, delmax1=0.8, delmax2=0.8, phi=8000j/m*m, sigmax=650dn=2.45E-5m

uaewrsepwintebewmintmpwceuaewrsepwintebewmintmpwceuaewrsepwintebewmintmpwceuaewrsepwintebewmintmpwceuaewrsepwintebuaewrsePwint

00.02500000000000.02500000000000.02500000000000.02500000000000.0250000000.250000

3.20E-050.0255.50E-053.65E-0508.92E-08-1.85E-050.00003668830.000000178300.00609572653.20E-050.0255.48E-053.62E-0501.67E-07-1.86E-050.00003651410.000000334100.01142358973.20E-050.0255.43E-053.56E-0503.11E-07-1.87E-050.00003620180.000000621800.02125675213.20E-050.0255.38E-053.50E-0504.45E-07-1.88E-050.00003588970.000000889700.03041777783.20E-050.0255.35E-053.45E-050.00E+005.68E-07-1.88E-057.11111111110.251.22449162391.179487179500.0388505983

8.00E-052.51E-022.89E-042.28E-041.02E-085.69E-07-6.07E-050.00022925790.00000113790.00034769230.03890393168.00E-052.50E-022.88E-042.26E-041.71E-091.04E-06-6.14E-050.00022822580.00000208580.00005832480.07130940178.00E-052.50E-022.85E-042.22E-043.97E-121.94E-06-6.26E-050.00022615240.00000388240.00000013570.13273162398.00E-050.0252.83E-042.19E-040.00E+002.78E-06-6.38E-050.00022433620.000005556200.18995555568.00E-050.0252.81E-042.16E-040.00E+003.55E-06-6.49E-0517.77777777780.257.61562393167.370940170900.2426324786

1.10E-042.53E-025.15E-044.31E-044.31E-081.19E-06-8.37E-050.000433360.000002370.0014741880.0810256411.33E-042.54E-027.16E-046.20E-045.62E-083.10E-06-9.65E-050.00062573260.00000620260.00192102560.21205470091.28E-042.51E-026.78E-045.69E-043.62E-084.98E-06-1.09E-040.00057896880.00000996880.00123897440.34081367521.28E-040.0256.73E-045.60E-046.41E-097.10E-06-1.13E-040.00057410940.00001420940.00021914530.4857914539.60E-050.0256.68E-045.52E-041.79E-109.08E-06-1.16E-0421.33333333330.2519.492458119718.86837606840.00000612310.6206632479

1.33E-042.56E-027.19E-046.25E-045.90E-081.98E-06-9.46E-050.00062846020.00000395020.0020170940.13504957261.59E-042.59E-021.00E-038.87E-046.69E-085.19E-06-1.16E-040.00089808540.00001038540.00228649570.35505641031.62E-042.55E-021.03E-039.05E-047.02E-088.49E-06-1.28E-040.00092238920.00001698920.00240136750.58082735041.51E-042.51E-029.07E-047.84E-045.02E-089.98E-06-1.24E-040.00080346040.00001996040.00171692310.68240683761.58E-042.50E-029.88E-048.41E-042.39E-081.38E-05-1.47E-0435.11111111110.2500129.693606837628.74871794870.00081572650.9448888889

1.59E-042.61E-021.01E-038.94E-046.88E-083.74E-06-1.14E-040.00090177120.00000747120.00235282050.2554256411.71E-042.63E-021.15E-031.03E-036.68E-087.24E-06-1.24E-040.0010429730.0000144730.00228444440.49480341881.72E-042.59E-021.16E-031.02E-037.20E-081.05E-05-1.34E-040.0010452620.0000210620.00246188030.72006837611.70E-042.54E-021.13E-039.91E-046.84E-081.32E-05-1.37E-040.0010178160.0000263160.00233982910.89969230771.76E-042.51E-021.20E-031.05E-035.36E-081.74E-05-1.57E-0439.17333333330.2514936.948717948735.76068376070.00183213681.188034188

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1.76E-042.76E-021.09E-031.22E-036.87E-089.68E-06-1.29E-040.00097846720.00001936720.00234735040.66212649571.80E-042.80E-021.26E-031.13E-031.27E-071.44E-05-1.38E-040.0011537140.0000287140.00433846150.98167521371.81E-042.78E-021.28E-031.12E-031.52E-071.90E-05-1.51E-040.0011621480.0000380480.00518632481.30078632481.78E-042.69E-021.22E-031.07E-039.52E-082.00E-05-1.50E-040.0011136980.0000399980.00325401711.36745299151.83E-042.60E-021.29E-031.12E-031.42E-072.19E-05-1.66E-0440.64888888890.2599939.775247863238.29059829060.00486153851.4983247863

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uaRFuaRFuauaua

00.025000.025000.02500.02500.025

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5.40E-052.50E-023.8685.40E-050.0253.858.24E-050.0255.8226.64E-050.0254.6536.64E-050.0254.62

7.00E-052.51E-025.0147.00E-052.50E-024.999.84E-050.0256.9528.24E-050.0255.7748.24E-050.0255.733

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1.74E-042.72E-0212.341.92E-042.91E-0213.311.78E-042.72E-0212.391.75E-042.62E-0212.191.80E-042.54E-0212.47

1.76E-042.76E-0212.432.08E-042.97E-0214.361.81E-042.77E-0212.551.78E-042.67E-0212.281.82E-042.58E-0212.56

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2.08E-042.95E-0214.242.03E-042.91E-0213.741.96E-042.74E-0213.42

2.14E-042.96E-0214.632.11E-042.94E-0214.372.10E-042.81E-0214.29

a/bRF/smax*ba/bRF/smax*ba/bRF/smax*ba/bRF/smax*ba/bRF/smax*b

0.2500.2500.2500.2500.250

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0.250.02424615380.250.02413846150.250.02001538460.250.01984615380.250.0197076923

0.250.03527692310.250.03290769230.250.05480.250.05433846150.250.0539538462

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0.251420.09473846150.252570.12932307690.251240.14169230770.250.12329230770.250.1224307692

0.253390.12109230770.256940.16415384620.252930.15892307690.250.14052307690.250.1395384615

0.254260.12986153850.260970.18123076920.255630.17615384620.250690.15769230770.250.1566153846

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0.271890.18984615380.291040.20476923080.271790.19061538460.26230.18753846150.25440.1918461538

0.275950.19123076920.29720.22092307690.276660.19307692310.266780.18892307690.257670.1932307692

0.282570.19461538460.299150.23723076920.280540.19569230770.271430.19092307690.259740.1943076923

0.291510.20307692310.286840.2020.276080.19353846150.262830.196

0.298790.22369230770.289390.20569230770.282050.19830769230.266540.1986153846

0.29340.21415384620.286040.20307692310.270010.2015384615

0.294910.21907692310.290790.21138461540.274360.2064615385

0.296290.22507692310.294190.22107692310.281350.2198461538

frstrev11,delmax=0.1,delmax2=0.1,dn=2.46E-5.energy=0.008,smax=650.file=frstrever22, delmax1=0.2, delmax2=0.2, phi=8000j/m*m, sigmax=650dn=2.45E-5mfile=frstrever44, delmax1=0.4, delmax2=0.4, phi=8000j/m*m, sigmax=650dn=2.45E-5mfile=frstrever66, delmax1=0.6, delmax2=0.6, phi=8000j/m*m, sigmax=650dn=2.45E-5mfile=frstrever88, delmax1=0.8, delmax2=0.8, phi=8000j/m*m, sigmax=650dn=2.45E-5m

uaTndns11e11s22e22misespeequaTndns11e11s22e22misespeequaTndns11e11s22e22misespeequaTndns11e11s22e22misespeequaTndns11e11s22e22misespeeq

00.0250000000000.02500.0250000000000.0250000000000.02500000000

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2.20E-050.025271.11.03E-06127.74.46E-04186.11.52E-031872.20E-050.025190.41.44E-0688.922.98E-04131.61.09E-03134.95.04E-050.025305.54.62E-06140.94.38E-04214.21.79E-03221.45.04E-050.025247.85.63E-06111.23.17E-04173.61.47E-031810.00E+005.04E-050.025207.96.29E-0693.62.97E-04141.31.18E-03164.5

3.80E-050.0254681.77E-06223.56.98E-04339.32.84E-03307.23.80E-050.025328.72.49E-06155.15.24E-042291.89E-03233.68.24E-052.50E-024997.55E-06235.35.45E-04387.83.37E-033309.36E-066.64E-050.025326.47.41E-061484.29E-042301.94E-032390.00E+008.24E-050.025339.71.03E-05156.15.08E-04233.61.94E-03269.9

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7.00E-052.51E-02622.63.39E-06289-7.99E-03562.11.39E-02362.31.53E-027.00E-052.50E-02604.84.58E-06283.7-7.04E-04507.65.91E-03345.22.63E-031.09E-042.50E-02649.99.93E-06288-1.00E-02567.31.60E-02367.11.88E-029.84E-050.025483.41.10E-05224.45.10E-04371.93.24E-03326.70.00E+001.14E-040.025471.31.43E-05222.55.72E-04337.22.99E-03331.25.84E-04

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1.50E-042.59E-02325.61.35E-0552.69-1.41E-02357.11.73E-02289.82.27E-021.18E-042.53E-02522.28.79E-06185.2-1.47E-02494.41.95E-02364.32.49E-021.62E-042.56E-02275.61.83E-0520.84-1.83E-02320.72.11E-02278.52.79E+021.34E-040.025649.81.48E-05274.5-1.60E-02559.12.19E-02380.42.86E-021.57E-040.025640.71.94E-05268.9-1.70E-02531.82.27E-02387.13.36E-02

1.66E-042.64E-02174.31.87E-05-42.87-1.44E-02241.51.63E-02265.22.27E-021.34E-042.54E-02447.81.10E-05131.3-1.50E-02445.51.92E-023352.49E-021.72E-042.59E-0267.072.31E-05-111.1-1.86E-02152.71.95E-02268.92.79E+021.46E-042.51E-025411.64E-05198.8-1.94E-02502.92.44E-02366.53.33E-021.62E-042.50E-02597.22.01E-05246.2-2.09E-02519.92.63E-02396.24.03E-02

1.76E-042.76E-0203.14E-05-137.1-1.44E-0296.091.48E-02268.92.27E-021.66E-042.61E-02187.11.89E-05-37.23-1.57E-02253.61.76E-02269.72.49E-021.74E-042.64E-0202.86E-05-155.3-1.86E-0291.141.88E-02281.42.79E+021.62E-042.52E-02345.61.94E-0569.91-2.00E-02373.72.35E-02292.83.33E-021.67E-042.50E-024922.09E-05181.2-2.13E-02458.22.60E-02341.14.03E-02

1.78E-042.79E-023.42E-05-130.4-1.43E-0296.21.47E-02263.92.27E-021.76E-042.73E-0203.19E-05-141.5-1.56E-0296.581.60E-02271.82.49E-021.75E-042.67E-0203.14E-05-149-1.84E-0291.411.87E-02276.62.79E+021.72E-042.55E-0298.212.31E-05-87.32-2.05E-02179.72.17E-02264.23.33E-021.79E-042.52E-0212.722.45E-05-113.7-2.22E-0298.192.27E-022494.03E-02

1.78E-042.77E-020.00E+003.49E-05-134.6-1.55E-0296.71.59E-02266.62.49E-021.74E-042.59E-0202.88E-05-160.8-2.05E-0288.462.06E-02287.93.33E-021.79E-042.54E-0202.67E-05-153.6-2.22E-0273.942.23E-022834.03E-02

1.75E-042.62E-0203.18E-05-154.1-2.03E-0288.852.05E-02282.83.33E-021.80E-042.54E-0202.78E-05-153.5-2.21E-0274.852.23E-02283.24.03E-02

frstrev11frstrev22frstrev44frstrev66frstrev88

uaTndns22e22a/bTn/dns22/e22normknormsa/bTn/dns22/e22normknormsa/bTn/dns22/e22normknormsa/bTn/dns22/e22normknormsa/bTn/dns22/e22normknorms

00.02500000.2500002.50E-0100002.50E-0100000.2500000.250000

8.00E-060.02598.633.73E-0750.483.54E-040.25264225246.463781142691.5792746781.75387271621.76600674852.50E-011.32E+081.21E+0540.8796939911.49628919672.50E-016.61E+071.19E+0520.43928632851.47280804050.254.40E+071.17E+0513.62528770851.45368770890.253.30E+071.19E+0510.21989030371.4782669635

2.20E-050.025271.11.03E-06186.11.52E-030.25264255775.416707122064.80388298681.76331867251.51072171542.50E-011.32E+081.21E+0540.87678337031.49809881242.50E-016.61E+071.20E+0520.43997939481.47920037310.254.40E+071.18E+0513.62375013091.46030139490.253.30E+071.20E+0510.22058075681.4850407775

3.80E-050.0254681.77E-06339.32.84E-030.25264242561.120208119606.59898477281.759230041.48029801092.50E-011.32E+081.21E+0540.87412240721.50060543892.50E-016.61E+071.15E+0520.43782103031.4244566840.254.40E+071.18E+0513.62483289981.46413428830.253.30E+071.20E+0510.21939706781.4908084825

5.40E-052.50E-02649.52.51E-065557.91E-030.25001258301849.2742170185.644190399179.92431259920.86864496082.50E-011.32E+081.19E+0540.87469133271.46838534882.50E-016.61E+077.54E+0420.43761046960.93371446990.254.40E+071.18E+0513.62697718731.46533045560.253.30E+071.20E+0510.22042533441.4900245234

7.00E-052.51E-02622.63.39E-06562.11.39E-020.25054183500839.9893940302.574030257456.89959089960.49880040632.50E-011.32E+088.60E+0440.880617991.06386953122.50E-016.54E+073.54E+0420.24132403290.43758968520.254.40E+071.15E+0513.62564296081.42271876160.253.30E+071.13E+0510.21826252991.3938941138

8.60E-052.51E-02588.94.54E-06535.41.72E-020.25142129673668.9126731052.082125043540.35019472770.38431270342.50E-011.32E+085.39E+0440.75771313260.66696033492.50E-015.96E+072.83E+0418.44956565830.34966872050.254.40E+078.80E+0413.62567367821.08971733070.253.30E+079.37E+0410.21914302531.1598097418

1.18E-042.54E-02490.97.88E-06472.11.81E-020.2542662314351.722562126016.753003416719.6092118330.32199350242.50E-011.10E+083.58E+0434.22575516940.44287639582.53E-012.96E+071.98E+049.2784541690.2444532150.254.40E+073.24E+0413.62642336190.400975350.253.30E+074.52E+0410.21855527540.5588836987

1.50E-042.59E-02325.61.35E-05357.11.73E-020.2592224102450.218372920630.88566641637.7325674150.25533590352.53E-015.94E+072.53E+0418.57473376550.31309868552.56E-011.50E+071.52E+044.75351294990.18809158040.254.40E+072.55E+0413.61880840590.31613852770.253.30E+072.35E+0410.21849688610.2904958567

1.66E-042.64E-02174.31.87E-05241.51.63E-020.26379339334.5121363114811.40754369833.04803588020.1833117682.54E-014.06E+072.32E+0412.76922497290.28652923662.59E-012.91E+067.82E+030.93231186690.09676283210.250693.30E+072.06E+0410.22868274350.25501237070.250142.97E+071.97E+049.20732027520.2442671618

1.76E-042.76E-0203.14E-0596.091.48E-020.2758106514.576271186400.08062694182.61E-019.88E+061.44E+043.19165554420.17786765872.64E-010.00E+004.85E+0300.06003428910.251751.78E+071.59E+045.55971399220.19718013480.250452.36E+071.76E+047.30521268890.2183284155

1.78E-042.79E-023.42E-0596.21.47E-020.2792306548.226805527200.0810434142.73E-010.00E+006.05E+0300.07493648192.67E-010.00E+004.89E+0300.06047607220.254944.25E+068.29E+031.34069931480.10258002060.252265.19E+054.33E+030.16206623130.0535417814

2.77E-010.00E+006.09E+0300.07532710160.259270.00E+004.29E+0300.05307939150.25360.00E+003.31E+0300.0409867127

0.26230.00E+004.33E+0300.05358358460.25440.00E+003.36E+0300.0416141616

Sheet1

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

000

00

00

0

dmax=0.1

dmax=0.2

dmax=0.4

dmax=0.6

dmax=0.8

PW/(smax*dn*1.E-2)

CE/(smax*dn*1.E-2)

Variation of CE/PW for bilinear model for different value of dmax

Sheet2

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

0000

0000

000

000

000

dmax=0.1

dmax=0.2

dmax=0.4

dmax=0.6

dmax=0.8

a/b

RF/(smax*b*t)

Sheet3

0.1, 0.2, 0.4, 0.6, 0.8

dmax=0.1

dmax=0.2

dmax=0.4

dmax=0.6

dmax=0.8

dn

Tn

0

50.99

141.3

233.6

282.2

337.2

425.6

504

531.8

519.9

458.2

98.19

73.94

74.85

dmax = 0.1

0.2

0.4

0.6

0.8

dmax=0.1

dmax=0.2

dmax=0.4

dmax=0.6

dmax=0.8

e22

s22

Stress-stran variation in the vicinity of crack tip (bilinear model)

dmax=0.8

dmax=0.6

dmax=0.4

dmax=0.2

dmax=0.1

dmax=0.1

dmax=0.2

dmax=0.4

dmax=0.6

dmax=0.8

a/b

(Tn/dn)/(E/a(1-n2))

Variation of stiffness of cohesive element (bilinear model)

Effect of Shape and Initial slope of CZMs on crack initiation loadsFrom LEFM (small scale yielding) the crack initiation load obtained by limiting SIF to is 80MPa

CZM

0.1

0.2

0.4

0.5

0.6

0.8

Bilinear

50.14

61.3

80.82

-

102.5

11.9

Trapezoidal

38.76

47.5

69.87

80.94

91.81

112.9

Exponential

81.8

_1055933035.unknown

Conclusion smax/sy has considerable effect on plasticity in case of exponential model and has little effect in case of trapezoidal and bilinear modelShape of CZM has considerable effect of amount of plasticity induced in the bounding material. Exponential model induces higher level of plasticity when compared to trapezoidal and bilinear models.Exponential model induces plasticity at both crack initiation and crack propagation: applicable for ductile metalsTrapezoidal and bilinear models induces plasticity only at crack initiation (even at higher smax/sy ) and not during crack propagation: applicable for brittle or metals with low ductilityShape of CZM has considerable effect on the size of plastic zone. Exponential model predict better size of plastic zone than other models.

Initial slope has several effect on fracture processBy increasing the initial slope (reducing value for dmax1 and dmax) stiffness of CZ elements increases, there by it increases plasticity.By decreasing the slope, stiffness of elements reduces and hence plasticity.Stiff initial slope induces higher stress field in the bounding material.Higher initial slope makes the wake length longer.Higher initial slope results in lesser cohesive energy consumption in the forward region, hence crack initiates at lower load. Crown shape (flat in trapezoidal, sharp point in bilinear, curved in exponential model) of CZMs has no significant effect except for trapezoidal model where more number of elements experience constant peak stress, which may favor in plastic dissipation.Flat crown also reduces critical displacement for both crack initiation and complete separationTail part of CZM also effects in plasticity growth. When trapezoidal model is used with longer tail amount of plasticity is reduced.

Conclusion(contd)

Picture to Define crack tip concept, wake and forwardDmax1is varied from 0.1 to 0.8 and dmax2 =0.8 for trapezoidal model. dmax is varied in bilinear model. Max coh strength & coh energy is same for all the models Exp mod predicts considerable plastic dissipation and other two models predicts small amount of plasticityTo establish how the energies are absorbed within the cohesive zone along its length.To provide good physics basis to postulate various types of micromechanisms active in the forward and wake regions of the crack. To establish relationship between the spatial distribution of energy flow in the fracture process zone, dissipative mechanisms that absorb this energy, and an overall manifestation of traction-separation law that embodies those details

Parameters for trapezoidal and bilinear models were the one which predicted crack initiation loads closed to K1c(see slide on Effect of Shape and Initial slope of CZMs on crack initiation loads)

Stress-strain variation in the vicinity of crack tip in the bounding material are shown for all the models