FRACTURE OF CRACKED MEMBERS 1. The presence of a crack in a structure may weaken it so that it fails by fracturing in two or more pieces.

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1 Aerospace Structures Fracture Mechanics: An Introduction Page 1 of 7 FRACTURE OF CRACED MEMBERS 1. The presence of a crack in a structure may weaken it so that it fails by fracturing in two or more pieces. 2. Fracture can occur at stresses below the material s yield strength, where failure would not normally be expected. FRACTURE MECHANICS 1. Study of crack propagation in bodies 2. Methodology used to aid in selecting materials and designing components to minimize the possibility of fracture. 3. It begins with the assumption that all real materials contain cracks of some size even if only submicroscopically 4. Based on three types of displacement modes: MODE I OPENING Tension MODE II SLIDING In-plane shear MODE III TEARING Out-of-plane shear FRACTURE TOUGHNESS 1. In fracture mechanics, one does not attempt to evaluate an effective stress concentration, rather a stress intensity factor 2. After obtaining, it is compared with a limiting value of that is necessary for crack propagation in that material, called c 3. The limiting value c is characteristic of the material and is called fracture toughness 4. Toughness is defined as the capacity of a material to resist crack growth

2 Aerospace Structures Fracture Mechanics: MODE I Page 2 of 7 STRESS INTENSITY FACTOR S σ y h y τ xy σ x a a x y r σ z θ x b b S 1. The stress intensity factor I characterizes the magnitude of the stresses in the vicinity of an ideal sharp crack tip in a linear-elastic and isotropic material under mode I displacement. 2. Near the crack tip the dominant terms in the stress field are: I σ x = cos θ [ 1 sin θ ] 3θ sin + (1) 2πr σ y = I cos θ [ 1 + sin θ ] 3θ sin + (2) 2πr τ xy = σ z = I cos θ 2πr 2 sin θ 3θ cos (3) { 0 (plane stress) ν(σ x + σ y ) (plane strain) (4) τ yz = τ xz = 0 (5) 3. The dimensional units of I are: [stress length], i.e., [MP a m] or [ksi in] 4. I measures the severity of the crack and it is generally expressed as: I = F S πa (6) F is a dimensionless quantity accounting for the plate/specimen geometry and relative crack size, S is the stress (σ y ) if no crack were present, a is half crack length

3 Aerospace Structures Fracture Mechanics: MODE I Page 3 of 7 It can also be observed that as a b, the plate fractures into two pieces. Dowling, in his book Mechanical Behavior of Materials (Prentice-Hall, 1999, pp ; or 1993, pp. 292), gives the following expression for a center-cracked plate with any α = a/b: F = 1 0.5α α2 1 α h b 1.5 From the above expression it can be shown that F = 1 for an infinite plate (b ) and for 0 a b 1. However, for a center-cracked plate with α 0.4, when taking F = 1, the result is accurate within 10%. CRITICAL STRESS INTENSITY FACTOR 1. The calculated I is compared to the critical stress intensity factor or fracture toughness Ic : I < Ic, material will resist crack growth without brittle fracture (safe) I = Ic, crack begins to propagate and brittle fracture occurs (fracture) 2. The critical value Ic is defined as Ic = F S C πa The following figure shows the relationship between the critical value of the remote stress and the crack length. S c σ o S C = IC F π a a t a a t is the transition crack length, and it is defined as the approximate length above which strength is limited by brittle fracture; and σ o = σ y. In other words, a t is the crack length where S C = σ y : a t = 1 ( ) 2 Ic F 2 = 1 ( ) 2 Ic (F = 1) π π σ o When a > a t strength is limited by fracture, and when 0 < a < a t yielding dominates strength. Materials with: 1. high Ic and low σ o implies long a t ; therefore small cracks are not a problem. 2. low Ic and high σ o implies short a t ; therefore small cracks can be a problem. σ o

4 Aerospace Structures Fracture Mechanics Page 4 of 7 Taken from Mechanical Behavior of Materials, by N.E. Dowling, Prentice-Hall Inc, NJ, 1993 (Will be useful for homework # 3)

5 Aerospace Structures Fracture Mechanics Page 5 of 7 Taken from Mechanical Behavior of Materials, by N.E. Dowling, Prentice-Hall Inc, NJ, 1999

6 Aerospace Structures Spring 2001 Aerospace Structures Fracture Mechanics: Mixed Modes Page 6 of 7 FRACTURE UNDER COMBINED LOADING t gives the minimum In manyweight. cases, the structure is not only subjected to tensile stress σ x but also a contour shear stress τ xs, such as web 1 in the wing box of homework problem 4. Thus, the crack is exposed to tension and shear which leads to mixed mode cracking; i.e., a mixture of mode I and mode II. loading Whenever the crack length 2a is small with respect to the web length b, the geometric factor F is ed to tensile unity stress in formulas σ x but also for the a contour stress intensity shear stress factors τ xs from. Thus, LEFM. the crack That is, is exds to mixed where mode σcracking; x and τ xs i.e., are the a mixture normal of andmode sheari stresses and mode in the II. Since web if there were no crack present. I = σ x πa and II = τ xs πa, ith respect to the web length b 1 = 195mm, the geometric factor is unity in Mixed mode fracture is complicated by the fact that the crack extension takes place at an angle rs from LEFM. with That respect is, to the I = original σ x πacrack and direction. II = τ xs If aπa crack, where propagates σ x and inτthe xs direction of the original crack, it is called self-similar crack growth. Under mixed mode fracture the crack growth is, in general, not he web if there were no crack present. Mixed mode fracture is complicated by self-similar. Various mixed mode criteria for crack growth have been proposed based on experiments. place at an angle with respect to the original crack direction. If a crack proparack, it is called self-similar crack growth. Under mixed mode fracture the A common mixed mode criterion, at the initiation of the fracture, is ( ) 2 ( ) imilar. Various mixed mode criteria for crack growth 2 I have been II proposed based + = 1 (7) ode criterion, and the one we will use in this assignment, is Ic where Ic is the fracture toughness for mode I loading only, and IIc is the fracture toughness for II mode = 1 II loading at the only. initiation The plane of fracture strain fracture toughness for mode I loading (2) is usually readily available IIc in the literature, but the mode II fracture toughness is not usually available. or mode I loading only, and Tests for mode II IIc is the fracture toughness for mode II loading are more difficult to design than for mode I. To estimate IIc knowing the ess for mode value I loading of Ic we is usually use thereadily maximum available principal the stress literature, criterion 1 but. The the maximum principal stress criterion ally available. postulates Tests for that mode crackii growth are more willdifficult occur into the design direction than perpendicular for mode I. To to the maximum principal stress in the vicinity of the crack tip. Using this criterion it is possible to estimate IIc (see Fig in we use the maximum principal stress criterion 1 Ic. The maximum principal Broek) as rowth will occur in the direction perpendicular to the maximum 3 principal stress IIc = 2 Ic = Ic (8) this criterion it is possible to estimate IIc (see Fig in Broek) as The mixed mode criterion given by Eq. (7) is plotted in the following figure: IIc = Ic (3) q. (2) is plotted in the figure at right. ses, and in turn the stress intensity tude of the total lift acting on the he 80% limit load specified for the he coordinates of the required ted by the ray 0 f.the quantity f trength. The excess strength with ref, and if it is divided by the resafety as 1 f where f f IIc Under proportional loading, the stresses, and in turn the stress intensity factors, are proportional to the magnitude of the total I II = (4) Ic + lift acting IIc 2 on the wing. The stress intensity factors at the 80% limit load specified for the damage design condition determine the coordinates of the required strength in the 1 Broek, David, Elementary engineering fracture mechanics, Martinus Nijhoff Publishers, Dordecht, The Nether- te if 0 f < lands, 1, and 1987, the pp. initiation of fracture is predicted if f = 1. The margin the margin of safety is zero if f = II IIc f 1 I II 2 + = 1 Ic I Ic IIc

7 Aerospace Structures Fracture Mechanics: Mixed Modes Page 7 of 7 plot, which is represented by the ray 0 f. The quantity f denotes the dimensionless required strength. The excess strength with respect to fracture is represented by 1 f, and if it is divided by the required strength we get the margin of safety as MS = 1 f f (9) ( I f = Ic ) 2 ( ) 2 II + (10) IIc The crack is predicted not to propagate if 0 f < 1, and the initiation of fracture is predicted if f = 1. The margin of safety is positive if 0 < f < 1, and the margin of safety is zero if f = 1.