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## Elastic-Plastic Fracture Mechanics Introduction When does one need to use LEFM a

Wyner, Linda, Food Editor has reference to this Academic Journal, PHwiki organized this Journal Elastic-Plastic Fracture Mechanics Introduction When does one need to use LEFM in addition to EPFM What is the concept of small-scale in addition to large-scale yielding Contents of this Chapter The basics of the two criteria used in EPFM: COD (CTOD), in addition to J-Integral (with H-R-R) Concept of K- in addition to J-dominated regions, plastic zones Measurement methods of COD in addition to J-integral Effect of Geometry Background Knowledge Theory of Plasticity (Yield criteria, Hardening rules) Concept of K, G in addition to K-dominated regions Plastic zone size due to Irwin in addition to Dugdal LEFM in addition to EPFM LEFM In LEFM, the crack tip stress in addition to displacement field can be uniquely characterized by K, the stress intensity factor. It is neither the magnitude of stress or strain, but a unique parameter that describes the effect of loading at the crack tip region in addition to the resistance of the material. K filed is valid as long as a small region around the crack tip. It depends on both the values of stress in addition to crack size. We noted that when a far field stress acts on an edge crack of width a then as long as mode I, plane strain case LEFM concepts are valid if the plastic zone is much smaller than the singularity zones. Irwin estimates Dugdale strip yield model: ASTM: a,B, W-a 2.5 , i.e. of specimen dimension. LEFM cont. Singularity dominated region For =0

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EPFM In EPFM, the crack tip undergoes significant plasticity as seen in the following diagram. EPFM cont. EPFM applies to elastoc-rate-independent materials, generally in the large-scale plastic de as long as mation. Two parameters are generally used: Crack opening displacement (COD) or crack tip opening displacement (CTOD). J-integral. Both these parameters give geometry independent measure of fracture toughness. Sharp crack Blunting crack y x ds

EPFM cont. Wells discovered that Kic measurements in structural steels required very large thicknesses as long as LEFM condition. — Crack face moved away prior to fracture. — Plastic de as long as mation blunted the sharp crack. Sharp crack Blunting crack Irwin showed that crack tip plasticity makes the crack behave as if it were longer, say from size a to a + rp -plane stress From Table 2.2, Set , Note: since CTOD in addition to strain-energy release rate Equation relates CTOD ( ) to G as long as small-scale yielding. Wells proved that Can valid even as long as large scale yielding, in addition to is later shown to be related to J. can also be analyzed using Dugdales strip yield model. If is the opening at the end of the strip. Consider an infinite plate with a image crack subject to a Exp in addition to ing in an infinite series, If , in addition to can be given as: In general, Alternative definition of CTOD Sharp crack Blunting crack Blunting crack Displacement at the original crack tip Displacement at 900 line intersection, suggested by Rice CTOD measurement using three-point bend specimen Vp ‘ ‘ ‘ displacement exp in addition to ing

Elastic-plastic analysis of three-point bend specimen Where is rotational factor, which equates 0.44 as long as SENT specimen. Specified by ASTM E1290-89 — can be done by both compact tension, in addition to SENT specimen Cross section can be rectangular or W=2B; square W=B KI is given by V,P CTOD analysis using ASTM st in addition to ards Figure (a). Fracture mechanism is purely cleavage, in addition to critical CTOD <0.2mm, stable crack growth, (lower transition). Figure (b). -- CTOD corresponding to initiation of stable crack growth. -- Stable crack growth prior to fracture.(upper transition of fracture steels). Figure (c) in addition to then --CTOD at the maximum load plateau (case of raising R-curve). More on CTOD The derivative is based on Dugdales strip yield model. For Strain hardening materials, based on HRR singular field. By setting =0 in addition to n the strain hardening index based on Definition of COD is arbitrary since A function as the tip is approached Based on another definition, COD is the distance between upper in addition to lower crack faces between two 45o lines from the tip. With this Definition Where ranging from 0.3 to 0.8 as n is varied from 3 to 13 (Shih, 1981) Condition of quasi-static fracture can be stated as the Reaches a critical value . The major advantage is that this provides the missing length scale in relating microscopic failure processes to macroscopic fracture toughness. In fatigue loading, continues to vary with load in addition to is a function of: Load variation Roughness of fracture surface (mechanisms related) Corrosion Failure of nearby zones altering the local stiffness response 3.2 J-contour Integral By idealizing elastic-plastic de as long as mation as non-linear elastic, Rice proposed J-integral, as long as regions beyond LEFM. In loading path elastic-plastic can be modeled as non-linear elastic but not in unloading part. Also J-integral uses de as long as mation plasticity. It states that the stress state can be determined knowing the initial in addition to final configuration. The plastic strain is loading-path independent. True in proportional load, i.e. under the above conditions, J-integral characterizes the crack tip stress in addition to crack tip strain in addition to energy release rate uniquely. J-integral is numerically equivalent to G as long as linear elastic material. It is a path-independent integral. When the above conditions are not satisfied, J becomes path dependent in addition to does not relates to any physical quantities 3.2 J-contour Integral, cont. y x ds Consider an arbitrary path ( ) around the crack tip. J-integral is defined as It can be shown that J is path independent in addition to represents energy release rate as long as a material where is a monotonically increasing with Proof: Consider a closed contour: Using divergence theorem: where w is strain energy density, Ti is component of traction vector normal to contour. Evaluate Note is only valid if such a potential function exists Again, Since Recall Evaluation of J Integral --1 (equilibrium) leads to Hence, Thus as long as any closed contour Now consider Recall On crack face, (no traction in addition to y-displacement), thus , leaving behind Thus any counter-clockwise path around the crack tip will yield J; J is path independent. Evaluation of J Integral --2 1 2 3 4 a y x 2D body bounded by In the absence of body as long as ce, potential energy Suppose the crack has a vertical extension, then (1) Note the integration is now over Evaluation of J Integral --3 Noting that (2) Using principle of virtual work, as long as equilibrium, then from eq.(1), we have Thus, Using divergence theorem in addition to multiplying by -1 Evaluation of J Integral --4 There as long as e, J is energy release rate , as long as linear or non-linear elastic material In general Potential energy; U=strain energy stored; F=work done by external as long as ce in addition to A is the crack area. a u p Evaluation of J Integral --5 -dP Displacement Complementary strain energy = 0 p For Load Control For Displacement Control The Difference in the two cases is in addition to hence J as long as both load Displacement controls are same J=G in addition to is more general description of energy release rate Evaluation of J-Integral More on J Dominance J integral provides a unique measure of the strength of the singular fields in nonlinear fracture. However there are a few important Limitations, (Hutchinson, 1993) De as long as mation theory of plasticity should be valid with small strain behavior with monotonic loading (2) If finite strain effects dominate in addition to microscopic failures occur, then this region should be much smaller compared to J dominated region Again based on the HRR singularity Based on the condition (2), we would like to evaluate the inner radius ro of J dominance. Let R be the radius where the J solutions are satisfied within 10% of complete solution. FEM shows that However we need ro should be greater than the as long as ces zone (e.g. grain size in intergranular fracture, mean spacing of voids) Numerical simulations show that HRR singular solutions hold good as long as about 20-25% of plastic zone in mode I under SSY Hence we need a large crack size (a/w >0.5) . Then finite strain region is , minimum ligament size as long as valis JIC is For J Controlled growth elastic unloading/non proportional loading should be well within the region of J dominance Note that near tip strain distribution as long as a growing crack has a logarithmic singularity which is weaker then 1/r singularity as long as a stationary crack Williams solution to fracture problem Williams in 1957 proposed Airys stress function As a solution to the biharmonic equation For the crack problem the boundary conditions are Note will have singularity at the crack tip but is single valued Note that both p in addition to q satisfy Laplace equations such that

Now, as long as the present problem. Williams Singularity 3 Applying boundary conditions, Case (i) or, Case (ii) Since the problem is linear, any linear combination of the above two will also be acceptable. Thus Though all values are mathematically fine, from the physics point of view, since Williams Singularity 4 Since U should be provided as long as any annular rising behavior in addition to R ,

Williams Singularity 4 Williams Singularity 5 Williams Singularity 6

HRR Singularity 1 HRR Singularity 2

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