Fracture Mechanics Elastic Plastic Fracture Mechanics Elastic Plastic Fracture Mechanics Presented by Calvin M. Stewart, PhD MECH 5390-6390 Fall 2020 Outline • Introduction to Non-Linear Materials • J-Integral • Energy Approach • As a Contour Integral • HRR-Fields • COD • J Dominance Introduction to Non-Linear Materials Introduction to Non-Linear Materials • Thus far we have restricted our fractured solids to nominally elastic behavior. • However, structural materials often cannot be characterized via LEFM. Non-Linear Behavior of Materials • Two other material responses are that the engineer may encounter are Non-Linear Elastic and Elastic-Plastic Introduction to Non-Linear Materials • Loading Behavior of the two materials is identical but the unloading path for the elastic plastic material allows for non-unique stress- strain solutions. For Elastic-Plastic materials, a generic “Constitutive Model” specifies the relationship between stress and strain as follows n tot =+ Ramberg-Osgood 0 0 0 0 Reference (or Flow/Yield) Stress (MPa) Dimensionaless Constant (unitless) 0 Reference (or Flow/Yield) Strain (unitless) n Strain Hardening Exponent (unitless) Introduction to Non-Linear Materials • Ramberg-Osgood Constitutive Model n increasing Ramberg-Osgood −n K = 00 Strain Hardening Coefficient, K = 0 0 E n tot K,,,,0 n=+ E Usually available for a variety of materials 0 0 0 Introduction to Non-Linear Materials • Within the context of EPFM two general ways of trying to solve fracture problems can be identified: 1. A search for characterizing parameters (cf. K, G, R in LEFM). 2. Attempts to describe the elastic-plastic deformation field in detail, in order to find a criterion for local failure. • Of the concepts developed for this purpose three have found general acceptance: ➢the J integral ➢The HRR Fields ➢and the Crack Opening Displacement (COD) approaches. J-Integral Energy Approach J-Integral (Energy Approach) • In 1968, J. R. Rice developed the Path Independent Integral for the approximate analysis of strain concentration by notches and cracks. • Encompasses both elastic and plastic energy of material. • Great for Large Plastic Zones (Ductile Fracture) James Robert Rice (1940 - ) • The J-Integral can be interpreted two ways 1. As a Non-Linear Energy Release, J du J=− Wdy Ti ds 2. As a Path Independent Contour Integral i dx J-Integral (Energy Approach) • Remember the Energy of Fracture Approach, G • The energy content of the plate plus the loading system, denoted as the total energy U, is written as J-Integral (Energy Approach) • Previously, we have considered only linear elastic behavior. • However, there is no reason why the equation should not be valid for elastic material behaviors that are nonlinear: The essence is that the behavior is elastic!!!! J-Integral (Energy Approach) • Furthermore, if unloading is restricted, non-linear elastic behavior is identical to elastic-plastic behavior Restrict Unload to produce identical behavior J-Integral (Energy Approach) • As before, only part of the total energy U performs work. • This part will be designated as the potential energy, Up, of the plate and its loading system and is equal to • Note: the change in surface energy does not appear. J-Integral (Energy Approach) • In deriving G, we considered a central crack and defined G as the energy available per increment of crack extension and per unit thickness. • In deriving J, we consider an edge crack and define the non-linear elastic equivalent as follows Note: For linear elastic materials J = G J-Integral As a Contour Integral J-Integral (as a Contour Integral) • We need to remember a few terms, • Strain energy density, W is the strain energy per unit volume. • The infinitesimal strain energy density dW is the work per unit volume done by the stress σij during an infinitesimal strain increment dεij. It is given by • The strain energy density for a total strain εkl is obtained by integration, i.e. J-Integral (as a Contour Integral) • The Traction vector, T is a force per unit area acting on some plane in a stressed material. • It can be expressed in terms of the stress tensor σ according to: • where n1 and n2 are the components of the unit vector n normal to the plane on which T acts. Note that the dimension of T is force per unit area J-Integral (as a Contour Integral) J-Integral (as a Contour Integral) • Consider the path around the crack tip shown below: J-Integral (as a Contour Integral) • We use the J2 deformation theory of plasticity (equivalent to non- linear elasticity). The (reversible) stress-strain response is depicted schematically below: J-Integral (as a Contour Integral) • For proportional loading J2 deformation theory and J2 flow theory (incremental theory of plasticity) give results that are comparable (i.e. for monotonic loading, stationary cracks). • Not appropriate for situations where significant unloading occurs. • The potential energy of the cracked body is • This represents the sum of the stored strain potential energy and the potential energy of the applied loading. J-Integral (as a Contour Integral) • In the previous integral: • W = strain energy density (per unit volume); recall that • dA an element of cross section A within S. • We now evaluate the derivative of the mechanical potential energy, uM, with respect to crack length. J-Integral (as a Contour Integral) • J represents the rate of change of net potential energy with respect to crack advance (per unit thickness of crack front) for a non-linear elastic solid. J also can be thought of as the energy flow into the crack tip. Thus, J is a measure of the singularity strength at the crack tip for the case of elastic-plastic material response. J-Integral (as a Contour Integral) • For the special case of a linear elastic solid, Plane Strain Plane Stress • This relationship can be used to infer an equivalent KIc value from JIc measurements in high toughness, ductile solids in which valid KIc testing will require unreasonably large test specimens J-Integral (as a Contour Integral) • Consider two different paths around the crack tip: • The J Integral is independent of the path around the crack tip. If S2 is in elastic material, HRR Field HRR Field • We now consider the Hutchinson, Rice, Rosengren (HRR) singular crack tip fields for elastoplastic material response. (Recall Williams solution assumes linear elastic material behavior). • Assume: Ramberg-Osgood Constitutive Model −n K = 00 Strain Hardening Coefficient, K = 0 0 E n tot K,,,,0 n=+ E 0 0 0 Usually available for a variety of materials σ0 is usually equal to the yield stress, HRR Field • With these assumptions, the crack tip fields (HRR field) can be derived. HRR Field • The HRR equations imply that the stress/strain field in the direct vicinity of a crack tip is completely characterized by a single parameter J. • Different geometries with identical J values can be expected to have the same stresses and strains near the crack tip, and thus show identical responses. • Therefore J can be considered as a single fracture mechanics parameter for the elastic-plastic regime (with the restriction of no unloading), analogous to K for the linear elastic regime. COD Crack Opening Displacement COD • The variation in crack tip opening displacement δt or (COD) for different material response is depicted below: • The crack opening displacement depends on distance from the crack tip. We need an operational definition for COD. COD • The definition of δt is somewhat arbitrary since the opening displacement varies as the crack tip is approached. A commonly used operational definition is based on the 45° construction depicted below (see C.F. Shih, JMPS, 1982). COD • In 1966 Burdekin and Stone provided an improved basis for the COD concept. They used the Dugdale strip yield model to find an expression for CTOD. COD • In Chapter 3, it was also shown that under LEFM conditions there are direct relations between δt and KI. Thus, for the Dugdale analysis • C is equal to 1.0 for plane stress and taken to be 2.0 for plane strain. COD • For the Irwin plastic zone analysis an analogous relation was found: • The foregoing relations between δt and KI are important because they show that in the linear elastic regime the COD approach is compatible with LEFM concepts. However, the COD approach is not basically limited to the LEFM range of applicability, since occurrence of crack tip plasticity is inherent to it. COD • The major disadvantage of the COD approach is that equation (3.19) is valid only for an infinite plate with a central crack with length 2a, and it is very difficult to derive similar formulae for practical geometries. This contrasts with the stress intensity factor and J integral concepts. • The COD approach has been developed mainly in the UK: more specifically, at the Welding Institute. The chief purpose was to find a characterizing parameter for welds and welded components of structural steels, which are difficult to simulate on a laboratory scale. Thus the COD approach is more strongly directed towards use in design of welded structures. (This, of course, does not mean that COD values cannot be used to compare and select materials.) • More details in Chapter 7 of the book. COD • Relationship between J and δt COD • Relationship between J and δt COD Plane Strain COD Importance/Applications of CTOD: • Critical CTOD as a measure of toughness. • Exp. measure of driving force. • Multiaxial fracture characterization. • Specimen size requirements for KIc and JIc testing. J Dominance J Dominance Just as for the K field, there is a domain of validity for the HRR (J-based) fields. J Dominance Under plane strain and small scale yielding conditions, it has been found that: For J dominance the uncracked ligament size b must be greater than 25 times the CTOD or ≈ 25 X J= σ0.