INS 339E Fracture Mechanics of Concrete Fall 2024 Lecture Notes PDF

INS 339E Fracture Mechanics of Concrete Fall 2024 Lecture Notes Introduction by Oğuz Güneş, Ph.D....

INS 339E Fracture Mechanics of Concrete Fall 2024 Lecture Notes Introduction by Oğuz Güneş, Ph.D. Assistant Professor Course originally developed by: Mehmet Ali Taşdemir, Ph.D. Professor of Civil Engineering Department of Civil Engineering Istanbul Technical University Maslak, Sariyer 34469 Istanbul / Turkey Course Outline Fracture Mechanics of Concrete Introduction Principles of Linear Elastic Fracture Mechanics (LEFM) Applications of LEFM to concrete Structure and fracture process of concrete Nonlinear fracture models for concrete Test methods for determination of fracture parameters Fracture mechanics and compressive failure Tension softening response of concrete Applications of fracture mechanics to concrete structures Applications to high performance cementitious materials INS 339E Fracture Mechanics of Concrete Fall 2024 1  Instructor Oğuz Güneş, Ph.D., Asst. Prof. Construction Materials Laboratory, Room 204 [email protected] Cell: 533-601 9441 (Please use responsibly) Office hours: MR 10:00-12:00 Teaching assistant: None INS 339E Fracture Mechanics of Concrete Fall 2024 Textbooks and References Textbooks: Shah, S.P., Swartz, S.E. and Ouyang, C. (1995), Fracture Mechanics of Concrete, John Wiley & Sons, New York. Karihaloo, B.L. (1995), Fracture Mechanics and Structural Concrete, Longman Scientific and Technical, London. References: Shah, S.P. and Taşdemir, M.A. (1994), Role of Fracture Mechanics in Concrete Technology, in Advances in Concrete Technology, 2nd Ed., V.M. Malhotra (Ed.), pp. 161-202. Bazant, Z.P. and Planas, J. (1998), Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRS Press, New York. Van Mier, J.G.M. (1997), Fracture Professes of Concrete: Assessment of Material Parameters for Fracture Models, CRS Press, New York. Kumar, S. and Barai, S.V. (2011), Concrete Fracture Models and Applications, Springer, Heidelberg. Elfgren, L. (Ed.) (1989), Fracture Mechanics of Concrete Structures: From Theory to Applications, RILEM TC 90- FMA Report, Chapman and Hall, London. Shah, S.P. and Carpinteri, A. (1991), Fracture Mechanics Test Methods for Concrete, RILEM TC 89-FMT Report, Chapman and Hall, London. Broek, D. (1987), Elementary Engineering Fracture Mechanics, Martinus Nijhoff Publishers, Dordrecht. INS 339E Fracture Mechanics of Concrete Fall 2024 2  Grading Exam 1 (Week 9) 20% Exam 2 (Week 13) 20% 2 HW - 1Lab assignment 20% Final (TBA) 40% Laboratory session – TBA Any HW assignment can be replaced by a pop-quiz any time Attendance is not required but will be rewarded, routine absences will affect grade University rules on academic honesty will be strictly enforced INS 339E Fracture Mechanics of Concrete Fall 2024 Design approach Use earlier test Use a handbook Appropriate results failure criterion Select a material Make a part Considerations: Test the material Select stronger Safety (expensive) material Economy Function Sustainability If not strong Environment enough? Energy … INS 339E Fracture Mechanics of Concrete Fall 2024 3  Why need fracture mechanics? Sudden and brittle failures at stresses below strength! Images compiled from the www INS 339E Fracture Mechanics of Concrete Fall 2024 The broad field of fracture mechanics Several disciplines are involved in the development Engineering: load-and-stress analysis Applied mechanics: crack tip stress fields and deformations Materials science: fracture processes on the scale of atoms and dislocations An understanding of the total field is necessary for successful applications (Broek, 1986, p. 8) INS 339E Fracture Mechanics of Concrete Fall 2024 4  Cracks in a structure The mere existence of fracture mechanics is a testament to that we live in an imperfect world! Existing cracks grow in time and reduce the residual strength Fracture mechanics attempts to answer the following questions: 1. What is the residual strength as a function of crack size? 2. What size of crack can be tolerated at the expected service load, i.e. what is the critical crack size? 3. How long does it take for a crack to grow from a certain initial size to the critical size? 4. What size of pre-existing flaw can be permitted at the moment the structure starts its service life? 5. How often should the structure be inspected for cracks? (Broek, 1986, p. 6) INS 339E Fracture Mechanics of Concrete Fall 2024 Stress Analysis of Solids Strain-displacement (kinematic) equations Strains 1  ui u j  1  ij =  +   ( ui , j + u j ,i )  2  x j xi  2 ε = Lu (6 equations) Equilibrium equations  ij , j =0  LTσ = 0 (3 equations) Constitutive equations Cauchy  ij = Cijkl kl stresses ε = Cσ σ = Sε (6 equations) Boundary conditions u = uˆ on u ˆ on  σnˆ = T u (Figures from wickipedia) INS 339E Fracture Mechanics of Concrete Fall 2024 5  Theoretical strength of materials Ionic bond Physical basis of modulus and strength Covalent bond INS 339E Fracture Mechanics of Concrete (Ashby and Jones, 2005, pp.46-52) Fall 2024 Theoretical strength of materials (cont’d) Orowan’s model (1949) Approximated as a sine curve (1901-1989) 2  = K sin (a − a0 ) (1) 2d da d = (2) a0 (a is close to a0, linear elastic behavior, constant E) d d d = = E  a0 =E (3) d  da / a0 da Take the derivative of (1) and substitute in (3) for a=a0 d   E d a0 = K a0 cos (a − a0 ) = E  K = (d is not known) da d d  a0 (Meyers and Chawla, 2009, pp.406-8) INS 339E Fracture Mechanics of Concrete Fall 2024 6  Theoretical strength of materials (cont’d) To determine d, the area under the curve is equated to the energy of the two surfaces created Surface energy per unit area:  Work of fracture = Surface energy a0 + d  a0  da = 2 (2 surfaces created) Substitute the expression for  , Eq. (1) a0 + d 2 a0 2d (a − a0 )da = 2 K sin 1 From calculus:  sin axdx = cos ax a Apply change of variables: y = a − a0 , dy = da d  d  d  K  sin ydy = 2 y 0 = 2  2 K = 2  d = d K cos (4) 0 d  d  K INS 339E Fracture Mechanics of Concrete Fall 2024 Theoretical strength of materials (cont’d) Orowan’s model (cont’d) Approximated as a sine curve 2  = K sin (a − a0 ) 2d E d  max = K = (5)  a0 Substitute (4) into (5) E E E K =  max =  K 2 =  max 2 =   max = a0 K a0 a0 According to Orowan’s model: range 2 More refined calculations Kd E d  Ed 1 1 = =     max = seem to indicate that:  max   −  E  a0     a0 (Erdogan, 2000)  4 13  INS 339E Fracture Mechanics of Concrete Fall 2024 7  Theoretical strength of materials (cont’d) Theoretical tensile strength calculated for various materials (Meyers and Chawla, 2009, p.408) INS 339E Fracture Mechanics of Concrete Fall 2024 Actual modulus and strength data Elastic modulus (Yield) Strength INS 339E Fracture Mechanics of Concrete (Ashby and Jones, 2005, pp. 39,109) Fall 2024 8  Theoretical vs. actual tensile strength The actual tensile strength of most materials is orders of magnitude or lower than the theoretical strength This is because real materials contain imperfections such as crystal structure dislocations, voids and cracks There is much room for improvement in the behavior and performance of materials through research on minimization of defects, optimization of microstructure, study of fracture at different scales. (van Mier, 1996, p. 6) micro-scale meso-scale macro-scale INS 339E Fracture Mechanics of Concrete Fall 2024 Stress concentrations Failure of a material is associated with the presence of high local stresses and strains in the vicinity of defects It is important to know the magnitude and distribution of stresses and strains around defects Possible causes of stress concentrations: Abrupt changes in section Contact stress at loading points Discontinuities in the material Initial stresses in the material Existing cracks in the material (Boresi and Schmidt, 2002, p. 505) INS 339E Fracture Mechanics of Concrete Fall 2024 9  Stresses around a circular hole in a plate (Meyers and Chawla, 2009, p.411) Circular hole in a uniaxially stressed plate First studied by Ernst Gustav Kirsch in 1898 Airy stress function Boundary conditions 1  1  2  rr = +  rr =  r = 0, r = a r r r 2  2  2  rr = 2 (1 + cos 2 )    = 2 r    = 2 (1 − cos 2 )  r →    1     r = −    r = 2 sin 2  r  r   Compatibility equation   1  2 1  2   2 1  1  2   4 =  2 + + 2  + + =0  r r r r  2  r 2 r r r 2  2  Sir George Biddell Airy 1801-1892 INS 339E Fracture Mechanics of Concrete Fall 2024 Stresses around a circular hole in a plate (cont’d) An acceptable stress function could be of the form: 2b  = f ( r ) cos 2 which has a general solution: 1 f (r ) = Ar 2 + Br 4 + C +D r2 Resulting stresses: where  b2  a2   6C 4 D  A = b 2 (4a 4 + a 2b 2 + b 4 ) / 4( a 2 − b 2 )3  rr = 2  1 −  −   2 A + 4 + 2  cos 2 2(b − a )  r 2  2  r r  B = − a 2b 2 / 2(a 2 − b 2 )3  b2  a2   6C  D = − a 2b 2 (a 4 + a 2b 2 + b 4 ) / 2(a 2 − b 2 )3   = 2  1 + 2  +   2 A + 12 Br 2 + 4  cos 2 2(b − a )  2 r   r   6 C 2 D   r =   2 A + 6 Br 2 − 4 − 2  sin 2  r r  INS 339E Fracture Mechanics of Concrete Fall 2024 10  Stresses around a circular hole in a plate (cont’d) For large plate: b  a 2b>>2a 4 2 1 a a A=− B=0 C=− D= 4 4 2 and the stresses are:  a 2    3a 4 4a 2   rr = 1 − 2  + 1 + 4 − 2  cos 2 2 r  2 r r  r = a   rr =  r = 0  a 2    3a 4    = 1 + 2  −  1 + 4  cos 2   =  (1 − 2 cos 2 ) 2 r  2 r    3a 4 2a 2   3  r = − 1 − 4 + 2  sin 2 r = a,  = ,    =  max = 3 2 r r  2 2 r = a,  = 0,     = −  max Stress concentration factor = Kt = =3  INS 339E Fracture Mechanics of Concrete Fall 2024 Stresses around an elliptical hole (Meyers and Chawla, 2009, p.410) Inglis generalized Kirsch’s solution in 1913  a  max =  1 + 2   b  b2 Radius of curvature: = a  a a  max =  1 + 2   2 for   a      →0   →  max Kt = = 2 a/  Kt depends more on the form than size of the cavity Sir Charles Edward Inglis 1875-1952 INS 339E Fracture Mechanics of Concrete Fall 2024 11  Stress concentration factors Stress concentration factors for circular and elliptical holes in large plates under various loading conditions. Stress distribution around a circular hole (Carpinteri, 1997, p. 652) INS 339E Fracture Mechanics of Concrete Fall 2024 Example INS 339E Fracture Mechanics of Concrete Fall 2024 12  Example INS 339E Fracture Mechanics of Concrete Fall 2024 Example (cont’d) INS 339E Fracture Mechanics of Concrete Fall 2024 13  Summary All materials contain imperfections All materials fail at stresses order of magnitudes less than their theoretical tensile strength Failure of a material is associated with the presence of high local stresses and strains in the vicinity of defects Stresses around cavities become extremely large as the aspect ratio is increased, reaching infinity for a flat cavity. Theoretically, very small stresses should cause failure for flat cavities, but it does not happen, why? (Carpinteri, 1997, p. 656) INS 339E Fracture Mechanics of Concrete Fall 2024 Griffith’s energy criterion Initially developed for glass, Griffith’s work laid the foundations of fracture mechanics Based on a thermodynamic energy balance, Griffith postulated that when a crack propagates: Elastic strain energy is released in a volume of material Two new crack surfaces are created Alan Arnold Griffith 1893-1963 Change in potential energy upon crack propagation: 2   2 a 2t U e =  ( )  2a 2t = E  2E  E E = (Plane strain)  a t 2 2 1− 2 U = U s − U e = 4a s − E' E = E (Plane stress) U 2E'  s = 0  c = OR  c  a = 2 E  s a a Detailed discussions in the next lecture... (Meyers and Chawla, 2009, p. 416) INS 339E Fracture Mechanics of Concrete Fall 2024 14  Example INS 339E Fracture Mechanics of Concrete Fall 2024 Effect of material properties on fracture (Anderson, 1995, p. 19) INS 339E Fracture Mechanics of Concrete Fall 2024 15  Fracture behavior of materials Different types of material response under uniaxial stress (Shah et al, 1995, p. 2) elastic-brittle material elastic-plastic material elastic-quasi-brittle material Typical load-deformation response of a quasi-brittle material in tension/flexure i. transition from linear to nonlinear response (point A) ii. pre-peak nonlinearity (AB) iii. post-peak tension softening response (BCD) (Karihaloo, 1995, p. 2) INS 339E Fracture Mechanics of Concrete Fall 2024 Crack tip stress distribution Stress distributions based on different material behaviors elastic-brittle material elastic-plastic material elastic-quasi-brittle material (Shah et al, 1995, pp. 2-3) INS 339E Fracture Mechanics of Concrete Fall 2024 16  Notch-sensitivity of materials “A material may be considered to be notch sensitive if the presence of a notch causes a change in the net section strength of the material calculated on the basis of the reduced cross section but neglecting the stress concentrating effect of the notch).” Notch sensitivity reduces the net section strength of the elastic-brittle material, i.e. the net section strength varies with notch length. The net section strength remains unaltered for a ductile material. (Shah et al, 1995, p. 5) INS 339E Fracture Mechanics of Concrete Fall 2024 Classification of fracture morphologies Hierarchy of fracture for concrete METALS cement paste CERAMICS mortar POLYMERS COMPOSITES concrete (Meyers and Chawla, 2009, p. 467) (Shah et al, 1995, pp. 90-91) INS 339E Fracture Mechanics of Concrete Fall 2024 17  Influence of confinement (Meyers and Chawla, 2009, p. 502) INS 339E Fracture Mechanics of Concrete Fall 2024 Size effect With increasing size: Reduction in nominal ultimate stress Ductile-brittle transition Statistical vs. energetic size effect (Carpinteri, 1997, p. 655) INS 339E Fracture Mechanics of Concrete Fall 2024 18  Why Apply Fracture Mechanics to Concrete? 1. Energy requirement for crack growth An energy-based crack propagation criterion 2. Objectivity of load and response calculations FEM A constant energy of dissipation – called the fracture energy – forms the backbone of the theory 3. Lack of yield plateau 4. Energy absorbing capability and ductility The area under the load-deformation diagram represents the energy absorbed by the structure during the failure process 5. Size effect (Karihaloo, 1995, pp. 3-9) INS 339E Fracture Mechanics of Concrete Fall 2024 19