Qiang Yu (Post-Doctoral Researcher , Department of Civil and Environmental Engineering , Northwestern University) Zdenek P. Bazant (Professor Department of Civil and Materials Science , Northwestern University ) John Bayldon ( Post-Doctoral Researcher Department of Engineering, University of Cambridge) Jia-Liang Le ( Graduate Research Assistant , Department of Civil and Environmental Engineering , Northwestern University) ASME Vol. 77 / 011011(1-8), 011012(1-7) Scaling of strength of Metal-Composite Joints Part 1-2 Rohit Singh Karakoti M.Tech(10CE65R19) Structural Engineering Division Department of Civil Engineering IIT Kharagpur Presentation Overview Background Introduction Experimental Investigation Interface Fracture Analysis General size law Numerical Evaluation of Model Parameters Interpretation of Experimental Test Conclusion Reference Background In the past four decades, extensive analytical studies have been devoted to the effect of structure size on the strength of structures made of quasibrittle materials . For a homogeneous body Williams showed the dependence of the stress singularity exponent on the angle of a corner. Size effect law is derved in the recent study of symmetrical loaded V notches corners in homogenous materials of various corner angles. In the derived law only the real part of singularity is important. Background In previous studies theoretical size effect law represent a smooth transition from quasiplastic behavior in the small-size limit to brittle linear elastic fracture mechanics behavior in the large-size limit. This asymptotic matching approach for V-notches in homogeneous body has been shown to lead to good agreement with experiments. Introduction Knowledge of the size effect on the strength of hybrid bi-material joints of steel and fiber composites is important for new designs of large lightweight ships, large fuel-efficient aircrafts, and light weight crashworthy automobiles. In the present study the size effect in the joints is energetic ( non statistical ) and caused by the presence of material fracture energy or material characteristic length in material failure criteria. fGolIntroduction The singularity exponent of stress field at the tip of the crack is complex number. For the finite corner angle the real part of the corner tip stress singularity is greater than -1/2 so energy balance is not satisfied. And to satisfy the energy release rate a finite FPZ is approximated near the corner. Introduction.. The energy release rate is found to be related with real part of the stress singularity the same is seen in the homogeneous material. The general size effect derived for the homogeneous material then is used for the hybrid joint. Experimental Investigation Three series of scaled geometrically similar specimens of symmetric double-lap joints are manufactured. In the first two test series laminate were manufactured and tested at Northwestern University. In third series with a slightly different geometry was manufactured and tested at the University of Michigan to explore the size effect for a different type of laminate The specimens of series I and II were loaded in tension through chains at both ends to ensure that the tension resultant is centric. The size ratios have been selected as 1:4:12. In series I smallest specimen was found to fail by tensile fracture . The width of the specimen in the third dimension is constant b=20mm. The specimens of series III were fixed at both ends against rotation and loaded at both ends by wedge grips and the size ratio is 1:3:4:12. The optimal selection of dimensions are determined by finite element simulation and it is ensured that the steel block would still be in the elastic range when the hybrid joint fails. Experimental Investigation. Properties of Composites Metallic part is made of 1018 cold rolled steel having elastic modulus E=200 G Pa and Poissons ratio =0.3. Composites of the hybrid joint of I and II series is fiberglass-epoxy laminates G-10/FR4 Epoxy Grade. In-plane and through-thickness material properties of G-10/FR4 Garolite are obtained by testing as E11=30.0 GPa, v13=0.17, E22=9.5 GPa, v21=0.20, and G12=3.0 GPa. vExperimental Investigation. In series III composites are made using Newport NCT301 carbon laminates properties for the uniaxial composites are: E11=125.5 GPa ,E22=9.0 Gpa and G12=5.6 Pa, = = 0.304. The adhesive used is NB1101 epoxy film adhesive. 12v 13vLoad-displacement deformation curves Interface Fracture Analysis In the lap joint both loading and structure is symmetric. There are two critical bi-material corners where singularity and stress concentration is relevant. To identify the critical corner from which the crack propagates, the stress singularity exponents is calculated. Interface Fracture Analysis For a plain strain problem governing equation for stress function is. The solution is in the form of Hence the dispacements,boundary traction , stress field in each layer of material can be repented by two function and where and is the root of characteristic equation 0 ) 2 (44112 2466 124422 =cc+c cc+ +ccyfSy xfS SxfS) ( y x f f + =) ( 1 1 z f ) ( 2 2 z fy x z j i + = j0 ) 2 ( 22266 12411 = + + + S S S S Interface Fracture Analysis. Near the crack tip displacement field can be assumed as separable hence complex potential is expressed as. Thus by imposing the boundary condition. The system of linear equations , with matrix form are forulated. The displacement singularity is solved for the condition det(K)=0.The singularity is calculated numerically for which the condition number of matrix K become very large. ) 2 , 1 .....( ) sin (cos ) ( = + = = k r z z f k k k k k ko o o0 u | |0 ) ( = v o KInterface Fracture Analysis. ggInterface Fracture Analysis. The size effect law is satisfied for large sizes in linear elastic fracture mechanics and stress field corresponding to strongest singularity is relevant. For strongest singularity stress field is represented as . The energy balance seems impossible in this condition. The approximation is done by forming a finite size FPZ whose effect on elastic field is approximated by the formation of equivalent crack whose length is proportional to length of FPZ and is taken as . Therefore at maximum load elastic field of stress is close to interface crack. q k i =kijiij r Hr )] ( Re[ u o o q=fcFPZl 2FPZlInterface Fracture Analysis. It has been shown that the interface crack must have the form. The stress intensity factor of the interface crack tip K will depend on the stress field whose magnitude is characterized by the stress intensity factor H of the corner. '2 1 q icrack + =Interface Fracture Analysis. The two stress intensity factors are related as Within the frame work of linear elastic fracture mechanics, a crack can propagate once reaches a certain critical value of fracture energy and this also represent the condition of maximum load P. Hence the nominal strength( ) can be related as kko D c EG g f f N5 . 01 =,q q ifkfif c Hc Kc 5 . 0'+=bD PmaxGeneral size effect Law The entire singular stress field is governed solely by one real stress singularity . In bi-material real part of the strongest singularity exponent matters for the energy release rate at the large size limit. General size effect Law The equation of similar type can be used to approximate the general size effect law.. Parameters and are obtained by the model on the basis of available size effect data. ) (0 0 ) / 1 ( | k|o o D DN + =| 0D) (| ooNumerical Evaluation of model parameter. is obtained by finite element analysis of model and thus can be obtained by equation. The singular stress zone is also obtained by finite element analysis at the right and left corner. By matching the stress field the stress singularity in left corner is obtained as -.219. fGfc5 .) () () (+= k k| | || offoc gEGDNumerical Evaluation of model parameter. Interpretation of experimental results Interpretation of experimental results For a fourfold size increase, the nominal strength reduction is significant 52% in series II and 40% in series III. The test data fitted by size effect equation. ) (0 0 ) / 1 ( | k|o o D DN + =Interpretation of experimental results Interpretation of experimental results.. Conclusion The present study explain well that the strength of metal-composite hybrid joints cannot be calculated with failure criteria expressed solely in terms of stress and strain as done in elastic models. The fracture mechanics models in which the failure criterion involves some type of energy or material length, must be used otherwise the strength of hybrid joint would be dangerously over estimated. Reference Bazant,Z.P..1993 STABILITY OF STRUCTURES Elastic, Inelastic, Fracture, and Damage Theories DOVER PUBLICATIONS, INC. Mineola, New York Baant, Z. P., and Yu, Q., 2006, Size Effect on Strength of Quasibrittle Structures With Reentrant Corners Symmetrically Loaded in Tension, J. Eng. Mech., 13211, pp. 11681176. Martin H Sadd 2009Elasticity Application,Theory and Numerics 2nd ed Elsevier Inc.london. Prashant Kumar 1999 Element of Fracture Mechanics First edition Wheeler Publication. Suo, Z., 1990, Singularities, Interfaces and Cracks in Dissimilar Anisotropic Media, Proc. R. Soc. London, Ser. A, 427, pp. 331358. Thank You