Topics in Ship Structures

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1 Topics in Ship Structures 8 Elastic-lastic Fracture Mechanics Reference : Fracture Mechanics by T.L. Anderson Lecture Note of Eindhoven University of Technology by Jang, Beom Seon Oen INteractive Structural Lab

2 . INTRODUCTION Contents 1. Crack-Tip Opening displacement. The J Contour Integral 3. Relationships Between J and CTOD 4. Crack-Growth Resistance Curves 5. J -Controlled Fracture 6. Crack-Tip Constraint Under Large-Scale Yielding 7. Scaling Model for Cleavage Fracture 8. Limitations of Two-arameter Fracture Mechanics

3 1. Crack-Tip-opening displacement Definition of CTOD 3 Structural steels has higher toughness than K Ic values characterized by LEFM. The crack faces moves apart prior to fracture; plastic deformation blunts an initially sharp crack. Crack-Tip-opening Displacement (CTOD) : a measure of fracture toughness. Crack-tip-opening displacement (CTOD). Estimation of CTOD from the displacement of the effective crack in the Irwin plastic zone correction.

4 1. Crack-Tip-opening displacement Relationship between CTOD and K I and G : Irwin plastic zone 4 Effective crack length = a+r y The displacement r y behind the effective crack tip for plane stress. 3 v1 v ( 1) ry 1 ry 4 ry uy KI v KI KI E E (1 v ) E (3 v) / (1 v), G (plane stress) (1 v) The Irwin plastic zone correction for plane stress. 1 K I ry YS CTOD to be related with K I or G. r r y, a a r eff y 4 K I u y YS E 4 G YS

5 1. Crack-Tip-opening displacement Relationship between CTOD and K I and G : Irwin plastic zone 5 For plane strain E (3 4 v), G (plane strain) (1 v) ( 1) r 4(1 v) r 4 r 4 r u K K K K y y y y y I I I I E E E (1 ) (1 v v ) The Irwin plastic zone correction for plane strain r y 1 K I 6 YS CTOD can be related with The Irwin plastic u y 4 KI 3 E YS 4 G 3 YS

6 1. Crack-Tip-opening displacement Relationship between CTOD and K I and G : Strip-yield model 6 CTOD in a through crack in an infinite plate subject to a remote tensile stress for plate stress. 4 r 4 K K u K K y I I y I I E E 4 YS E YS 8 YSa ln sec E YS u y 8 E K I r y Estimation of CTOD from the strip-yield model. Series expansion of the ln sec term gives 8 YSa a K, E 8 E E I YS YS YS / YS here, K I a KI G I YS E YS K G m 1 for plane stress m E m m for plane strain YS YS Oen INteractive Structural Lab

7 1. Crack-Tip-opening displacement Two most common definitions of CTOD 7 Two definitions, (a) and (b) are equivalent if the crack blunts in a semicircle. CTOD can be estimated from a similar triangles construction: Where r is the rotational factor, a dimensionless constant between and 1. The hinge model is inaccurate when displacements are primarily elastic. Alternative definitions of CTOD: (a) displacement at the original crack tip and (b) displacement at the intersection of a 9 vertex with the crack flanks. The hinge model for estimating CTOD from three-point bend specimens. Oen INteractive Structural Lab

8 1. Crack-Tip-opening displacement Two most common definitions of CTOD 8 A typical load () vs. displacement (V) curve from a CTOD The shape of the load-displacement curve is similar to a stress-strain curve. At a given point on the curve, the displacement is separated into elastic and plastic components by constructing a line parallel to the elastic loading line. The dashed line represents the path of unloading for this specimen, assuming the crack does not grow during the test. The CTOD in this specimen is estimated by K m I YS E The plastic rotational factor r p is approximately.44 for typical materials and test specimens. Determination of the plastic component of the crack-mouth-opening displacement Oen INteractive Structural Lab

9 . The J Contour Integral Introduction 9 The J contour integral has enjoyed great success as a fracture characterizing parameter for nonlinear materials. An elastic material : a unique relationship between stress and strain. An elastic-plastic material : more than one stress value for a given strain if the material is unloaded or cyclically loaded. Nonlinear elastic behavior may be valid for an elasticplastic material, provided no unloading occurs. The deformation theory of plasticity, which relates total strains to stresses in a material, is equivalent to nonlinear elasticity. The nonlinear energy release rate J could be written as a path independent line integral an energy parameter. J uniquely characterizes crack-tip stresses and strains in nonlinear materials a stress intensity parameter. Schematic comparison of the stress strain behavior of elasticplastic and nonlinear elastic materials.

10 . The J Contour Integral Nonlinear Energy Release Rate 1 Rice [4] presented J is a path-independent contour integral for the analysis of cracks. J is equal to the energy release rate in a nonlinear elastic body that contains a crack. The energy release rate for nonlinear elastic materials. J d da Π : the potential energy, A : the crack area, U : strain energy stored in the body, F : the work done by external forces. For load control, U F U U * Crack length = a+da Crack length = a U* the complimentary strain energy U * d Nonlinear energy release rate.

11 . The J Contour Integral Nonlinear Energy Release Rate If the plate is in load control, J is given by If the crack advances at a fixed displacement, F =, and J is given by d du U F U J da da J for load control is equal to J for displacement control. J in terms of load J d d a a J in terms of displacement J d du * da da du* du 1/ dd, du*, du * U d 11 J d d a a Nonlinear energy release rate.

12 . The J Contour Integral Nonlinear Energy Release Rate 1 More general version of the energy release rate. For the special case of a linear elastic material, J= G. For linear elastic Mode I, J K I E Caution when applying J to elastic-plastic material. The energy release rate : the potential energy that is released from a structure when the crack grows in an elastic material. However, much of the strain energy absorbed by an elastic-plastic material is not recovered. A growing crack in an elastic-plastic material leaves a plastic wake. Thus, the energy release rate concept has a somewhat different interpretation for elastic-plastic materials. J indicates the difference in energy absorbed by specimens with neighboring crack sizes. J du da plastic wake

13 . The J Contour Integral J as a ath-independent Line Integral 13 Consider an arbitrary counterclockwise path (Γ) around the tip of a crack The strain energy density is defined as The traction is a stress vector at a given point on the contour. Arbitrary contour around the tip of a crack. where n j are the components of the unit vector normal to Γ. J : a path-independent integral

14 . The J Contour Integral The J Contour Integral - roof Evaluate J along Γ Γ : closed contour A : Area enclosed by Γ 14 Using divergence theorem, the line integral can be converted into an areal integral. (a) To be proved in the next slide. Using the chain rule and the definition of strain energy density, the first term in square bracket in Eq. (a). Here, w = strain energy density. Applying the strain-displacement relationship and ij = ji Invoking the equilibrium condition Thus, J= for any closed contour w ui w w J* A* ij dxdy dxdy x x A* j x x x Oen INteractive Structural Lab

15 . The J Contour Integral The J Contour Integral - roof Using divergence theorem, the line integral can be converted into an areal integral. rs s cosi s sinj θ s xi yj B xi yj R F F1 dxdy F 1 dx F dy x y C 15 A C ns cos( 9) si sin( 9) sj n si n sj yi xj 1 x x, x y 1 ui ui ui ui J* wdy T * i ds wdy * ijn j ds wdy * i1 n1ds i nds x x x x ui ui w ui ui wdy * i1 dy i dx dxdy A* i1 dxdy i dxdy x x x x x y x w ui ui w ui i1 i dxdy ij dxdy x x A 1 x x x x x j x Oen INteractive Structural Lab A* *

16 . The J Contour Integral The J Contour Integral - roof Consider now two arbitrary contours. If Γ 1 and Γ are connected by segments along the crack face (Γ 3 and Γ 4 ), a closed contour is formed. The total J along the closed contour is equal to the sum of contributions from each segment: 16 On the crack face, T i =dy =. u J J wdy T ds x i 4 i Thus, J 1 = J 3. Two arbitrary contours Γ1 and Γ around the tip of a crack. Therefore, any arbitrary (counterclockwise) path around a crack will yield the same value of J; J is path-independent.

17 . The J Contour Integral J as a Nonlinear Elastic Energy Release Rate Consider a two-dimensional cracked body bounded by the curve Γ. Under quasi-static conditions and in the absence of body forces, the potential energy is 17 Consider the change in potential energy resulting from a virtual extension of the crack: (a) When the crack grows, the coordinate axis moves. Thus a derivative with respect to crack length can be written as (b) 를 (a) 에적용 (b) By applying the same assumptions

18 . The J Contour Integral J as a Nonlinear Elastic Energy Release Rate Invoking the principle of virtual work gives 18 Thus, Applying the divergence theorem and multiplying both sides by 1 leads to R w dxdy wdy wn A x ds x F F1 dxdy F 1 dx F dy x y C F w F 1, ns yi xj n y x nx ny s i s j i j Therefore, the J contour integral is equal to the energy release rate for a linear or nonlinear elastic material under quasi-static conditions.

19 . The J Contour Integral J as a Stress Intensity arameter 19 Ramberg-Osgood eq. : Inelastic stress-strain relationship for uniaxial deformation In order to remain path independent, stress strain must vary as 1/r near the crack tip. At distances very close to the crack tip, well within the plastic zone, elastic strains are small in comparison to the total strain, and the stress strain behavior reduces to a simple power law. k 1 and k are proportionality constants, which are defined more precisely below. For a linear elastic material, n = 1.

20 . The J Contour Integral J as a Stress Intensity arameter The actual stress and strain distributions are obtained by applying the appropriate boundary conditions HRR singularity named after Hutchinson, Rice, and Rosengren. The J integral defines the amplitude of the HRR singularity. I n : an integration constant that depends on n. σ ij and ε ij : dimensionless functions of n and θ. plane stress plane strain Effect of the strain-hardening exponent on the HRR integration constant. Angular variation of dimensionless stress for n = 3 and n = 13 Oen INteractive Structural Lab

21 . The J Contour Integral The Large Strain Zone 1 The HRR singularity predict infinite stresses as r, The large strains at the crack tip cause the crack to blunt, which reduces the stress triaxiality locally. The blunted crack tip is a free surface; thus xx must vanish at r =. HRR singularity does not consider the effect of the blunted crack tip on the stress fields, nor the large strains that are present near the crack tip. yy/ reaches a peak when x /J is unity twice the CTOD. The HRR singularity is invalid within this region, where the stresses are influenced by large strains and crack blunting. However, as long as there is a region surrounding the crack tip, the J integral uniquely characterizes cracktip conditions, and a critical value of J is a size independent measure of fracture toughness. Large-strain crack-tip finite element results.

22 . The J Contour Integral Laboratory Measurement of J Linear Elastic : J= G, G is uniquely related to the stress intensity factor. Nonlinear : The principle of superposition no longer applies, J is not proportional to the applied load. One option for determining J is to apply the line integral J integral in test panels by attaching an array of strain gages in a contour around the crack tip. This method can be applied to finite element analysis. A series of test specimens of the same size, geometry, and material and introduced cracks of various lengths. Multiple specimens must be tested and analyzed to determine J. Schematic of early experimental measurements of J, performed by Landes and Begley.

23 . The J Contour Integral Laboratory Measurement of J 3 J directly from the load displacement curve of a single specimen. Double-edge-notched tension panel of unit thickness. da = da = db Assuming an isotropic material that obeys a Ramberg- Osgood stress-strain law. From the dimensional analysis, Φ is a dimensionless function. For fixed material properties,

24 . The J Contour Integral Laboratory Measurement of J 4 If plastic deformation is confined to the ligament between the crack tips, we can assume that b is the only length dimension that influences Δ p. If plastic deformation is confined to the ligament between the crack tips b is the only length dimension that influences Δ p. Taking a partial derivative with respect to the ligament length d d Integrating by part fgdx fg f gdx d d d d b b b b d d d b b b d d

25 . The J Contour Integral Laboratory Measurement of J 5 Therefore 1 1 d d b b Unit thickness is assumed at the beginning of this derivation The J integral has units of energy/area.

26 . The J Contour Integral Laboratory Measurement of J An edge-cracked plate in bending Example Ω = Ω nc (angular displacement under no crack)+ Ω c (angular displacement when cracked) If the crack is deep, Ω c >> Ω nc. The energy absorbed by the plate 6 J for the cracked plate in bending can be written as U M d

27 . The J Contour Integral Laboratory Measurement of J 7 If the material properties are fixed, dimensional analysis leads to c M M F 3 b b b M Integration by parts fgdx fg f gdx M M M 1 J dm F dm F MdM M M M c 3 b M b b b b b M M M M c F M F dm M dm Md b b b b b c c c M c cm cdm MdM M c J Md b c If the crack is relatively deep Ω nc should be entirely elastic, while Ω c may contain both elastic and plastic contributions. c c ( el ) p J Md Md c( el ) b b p or J K p I Md E b p

28 . The J Contour Integral Laboratory Measurement of J 8 General Expression In general, the J integral for a variety of configurations can be written in the following form M J c Uc Md c b c Md c c η : dimensionless constant. Note the above Eq. contains the actual thickness, while the above derivations assumed a unit thickness for convenience. For a deeply cracked plate in pure bending, η =, it can be separated into elastic and plastic components.

29 3. Relationships between J and CTOD General 9 For linear elastic conditions, the relationship between CTOD and G is given by Sinc J= G for linear elastic material behavior, in the limit of small-scale yielding, m=1 for plane stress m= for plane strain where m is a dimensionless constant that depends on the stress state and material properties. It can be shown that it applies well beyond the validity limits of LEFM. Consider, for example, a strip-yield zone ahead of a crack tip, If the damage zone is long and slender (ρ >> δ), the first term in the J contour integral vanishes because dy = ρ Contour along the boundary of the strip-yield zone ahead of a crack tip Oen INteractive Structural Lab

30 3. Relationships between J and CTOD General Since the only surface tractions within ρ are in the y direction 3 T n n n T n n n T n n n Define a new coordinate system with the origin at the tip of the strip-yield zone (X = ρ x) yy Since the strip-yield model assumes yy = YS. ρ Thus the strip-yield model, which assumes plane stress conditions and a non-hardening material, predicts that m = 1 for both linear elastic and elastic plastic conditions

31 4. Crack-Growth Resistance Curves General 31 Many materials with high toughness do not fail catastrophically at a particular value of J or CTOD. Rather, these materials display a rising R curve, where J and CTOD increase with crack growth. In the initial stages, there is a small amount of apparent crack growth due to blunting. As J increases, the material at the crack tip fails locally and the crack advances further. Because the R curve is rising, the initial crack growth is usually stable, but an instability can be encountered later. One measure of fracture toughness J Ic is defined near the initiation of stable crack growth. The definition of J Ic is somewhat arbitrary. The slope of the R curve is indicative of the relative stability of the crack growth; a material with a steep R curve is less likely to experience unstable crack propagation. For J resistance curves, the slope is usually quantified by a dimensionless tearing modulus: Schematic J resistance curve for a ductile material Oen INteractive Structural Lab

32 4. Crack-Growth Resistance Curves Stable and Unstable Crack Growth 3 Instability occurs when the driving force curve is tangent to the R curve. Load control is usually less stable than displacement control. In most structures, between the extremes of load control and displacement control. The intermediate case can be represented by a spring in series with the structure, where remote displacement is fixed Driving force can be expressed in terms of an applied tearing modulus: C m Δ T : the total remote displacement, C m (the system compliance), Δ(line displacement) The slope of the driving force curve for a fixed ΔT is Stable crack growth Schematic J driving force/r curve diagram which compares load control and displacement control. J J & T T R app R Unstable crack growth J J & T T R app R Oen INteractive Structural Lab

33 4. Crack-Growth Resistance Curves Stable and Unstable Crack Growth 33 For most structure, If the structure is held at a fixed remote displacement T assuming & J depends only on load and crack length. dt da d Cmd a a 1 C d m a a a C m J J dj da d a T J J J a a a a a T J J J C a a a T m a a For load control, C m = and for displacement control, C m =, T=. 1 J J a a T Oen INteractive Structural Lab

34 4. Crack-Growth Resistance Curves Stable and Unstable Crack Growth 34 The point of instability in a material with a rising R curve depends on the size and geometry of the cracked structure; a critical value of J at instability is not a material property if J increases with crack growth. However, It is usually assumed that the R curve, including the J IC value, is a material property, independent of the configuration. This is a reasonable assumption, within certain limitations.

35 4. Crack-Growth Resistance Curves Computing J for a Growing Crack 35 The geometry dependence of a J resistance curve is influenced by the way in which J is calculated. The equations derived in Section 3..5 are based on the pseudo energy release rate definition of J and are valid only for a stationary crack. There are various ways to compute J for a growing crack. The deformation J : used to obtain experimental J resistance curves.. The cross-hatched area represents the energy that would be released if the material were elastic. In an elastic-plastic material, only the elastic portion of this energy is released; the remainder is dissipated in a plastic wake. The energy absorbed during crack growth in an nonlinear elastic-plastic material exhibits a history dependence. Schematic load-displacement curve for a specimen with a crack that grows to a 1 from an initial length a o. Oen INteractive Structural Lab

36 4. Crack-Growth Resistance Curves Computing J for a Growing Crack 36 The geometry dependence of a J resistance curve is influenced by the way in which J is calculated. The equations derived in Section 3..5 are based on the pseudo energy release rate definition of J and are valid only for a stationary crack. There are various ways to compute J for a growing crack; the deformation J and the far-field J, The deformation J used to obtain experimental J resistance curves. The cross-hatched area represents the energy that would be released if the material were elastic. In an elastic-plastic material, only the elastic portion of this energy is released; the remainder is dissipated in a plastic wake. The energy absorbed during crack growth in an nonlinear elastic-plastic material exhibits a history dependence. Schematic load-displacement curve for a specimen with a crack that grows to a 1 from an initial length a o. Oen INteractive Structural Lab

37 4. Crack-Growth Resistance Curves 37 Computing J for a Growing Crack Oen INteractive Structural Lab The deformation J - continued the J integral for a nonlinear elastic body with a growing crack is given by where b is the current ligament length. The calculation of U D(p) is usually performed incrementally, since the deformation theory load displacement curve depends on the crack size. A far-field J For a deeply cracked bend specimen, J contour integral in a rigid, perfectly plastic material By the deformation theory The J integral obtained from a contour integration is path-dependent when a crack is growing in an elastic-plastic material, however, and tends to zero as the contour shrinks to the crack tip.

38 4. Crack-Growth Resistance Curves Stable and Unstable Crack Growth 38 Ex 3.) Derive an expression for the applied tearing modulus in the double cantilever beam (DCB) specimen with a spring in series

39 5. J-Controlled Fracture Stationary Cracks 39 Small-scale yielding, K uniquely characterizes crack-tip conditions, despite the fact that the 1/ r singularity does not exist all the way to the crack tip. Similarly, J uniquely characterizes crack-tip conditions even though the deformation plasticity and small strain assumptions are invalid within the finite strain region. small-scale yielding Elastic-plastic conditions J is still approximately valid, but there is no longer a K field. As the plastic zone increases in size (relative to L), the K-dominated zone disappears, but the J-dominated zone persists in some geometries. The J integral is an appropriate fracture criterion Elastic-plastic conditions

40 5. J-Controlled Fracture Stationary Cracks Large-scale yielding the size of the finite strain zone becomes significant relative to L, and there is no longer a region uniquely characterized by J. Single-parameter fracture mechanics is invalid in large-scale yielding, and critical J values exhibit a size and geometry dependence. Large-scale yielding 4 For a given material,5 dimensional analysis leads to the following functional relationship for the stress distribution within this region:

41 5. J-Controlled Fracture J-Controlled Crack Growth 41 J-controlled conditions exist at the tip of a stationary crack (loaded monotonically and quasistatically), provided the large strain region is small compared to the in-plane dimensions of the cracked body