FUNDAMENTALS OF FRACTURE MECHANICS ME 7772 DR. LUIS BARRALES FUNDAMENTALS OF FRACTURE MECHANICS CHAPTER 2 LINEAR ELASTIC FRACTURE MECHANICS Start point: Stress ELASTICITY: PHYSICAL FUNDAMENTS Hook’s Law Strain Reversible Deformation → No site exchange Atomic bonds as „springs“ π0 + βπ π0 ELASTICITY: PHYSICAL FUNDAMENTS Atomic bonds as, “springs” π0 π0 + βπ Sanity check: Is it possible to store enough energy in the atomic bonds? The stiffnes of a material depends mainly on the strength of the Metals and ceramics: bond covalent, ionic, metallic (→ rigid) Polymers: covalent between molecules und secondary intermolecular bonding between polymer chains (→ much lower rigidity) ATOMIC VIEW OF FRACTURE STRESS CONCENTRATION EFFECT OF FLAWS How was the idea conceived: • The theoretical cohesive strength of a material is approximately E/ π • But… experimental fracture strengths for brittle materials are typically three or four orders of magnitude below this value. • So, fracture cannot occur unless the stress at the atomic level exceeds the cohesive strength of the material. Thus, the flaws must lower the global strength by magnifying the stress locally. Elliptical hole in flat plate. MATHEMATICS OF STRESS CONCENTRATION Putting when a >> b equation becomes THE GRIFFITH ENERGY BALANCE “In 1920, Griffith applied the idea of first law of thermodynamics, when a system goes from a non equilibrium state to equilibrium, there is a net decrease in energy, to the formation of a crack.” • It may be supposed, for the present purpose, that the crack is formed by the sudden annihilation of the tractions acting on its surface. At the instant following this operation, the strains, and therefore the potential energy under consideration, have their original values; but in general, the new state is not one of equilibrium. If it is not a state of equilibrium, then, by the theorem of minimum potential energy, the potential energy is reduced by the attainment of equilibrium; if it is a state of equilibrium, the energy does not change. • A crack can form (or an existing crack can grow) only if such a process causes the total energy to decrease or remain constant. Thus the critical conditions for fracture can be defined as the point where crack growth occurs under equilibrium conditions, with no net change in total energy. THE GRIFFITH ENERGY BALANCE E = total energy. Π = potential energy supplied by the internal strain energy and external forces. Ws = work required to create new surfaces. A through-thickness crack in an infinitely wide plate subjected to a remote tensile stress. THE GRIFFITH ENERGY BALANCE where Πo is the potential energy of an uncracked plate and B is the plate thickness where γs is the surface energy of the material. THE GRIFFITH ENERGY BALANCE The fracture stress for a penny-shaped flaw where a is the crack radius and ν is Poisson’s ratio. A penny-shaped (circular) crack embedded in a solid subjected to a remote tensile stress. MODIFIED GRIFFITH EQUATION Irwin and Orowan independently modified the Griffith expression to account for materials that are capable of plastic flow. A sharp microcrack at the tip of a macroscopic crack. where γp plastic work per unit area of surface created and is typically much larger than γs. THE ENERGY RELEASE RATE • Irwin proposed an energy approach for fracture that is essentially equivalent to the Griffith model, except that Irwin’s approach is in a form that is more convenient for solving engineering problems. • Irwin defined an energy release rate G, which is a measure of the energy available for an increment of crack extension: Where Π is defined as the potential energy of an elastic body where U is the strain energy stored in the body and F is the work done by external forces. For the load-controlled : THE TWO MODES. (FIXED LOAD AND FIXED DISPLACEMENT) Cracked plate at a fixed load P. Cracked plate at a fixed displacement Δ. INSTABILITY AND THE R CURVE Schematic driving force vs. R curve diagrams (a) flat R curve and (b) rising R curve. INSTABILITY AND THE R CURVE The conditions for stable crack growth can be expressed as follows: Unstable crack growth occurs when STRESS ANALYSIS OF CRACKS • For certain cracked configurations subjected to external forces, it is possible to derive closed-form expressions for the stresses in the body, assuming isotropic linear elastic material behavior. • Westergaard , Irwin , Sneddon , and Williams were among the first to publish such solutions. σij = stress tensor k = constant fij = dimensionless function of θ in the leading term THE STRESS INTENSITY FACTOR The three modes of loading that can be applied to a crack. LOADING MODES • Mode I, loading occurs the most often and produces the most damage. Because of this, it naturally receives the most attention in research, structural design, failure analysis, etc. It is commonly called the Opening Mode. • Mode II corresponds to shearing of the crack face due to in-plane shear stresses. It probably receives the second most attention because the problem is still 2-D since all the action is in-plane. Mode II loading influences crack growth direction in a way that minimizes further Mode II loading while maximizing Mode I. • Mode III is the Tearing Mode for obvious reasons. It is driven by outof-plane shear stresses, and does not seem to occur as often as the other two. STRESS FIELDS AHEAD OF A CRACK TIP FOR MODE I AND MODE II IN A LINEAR ELASTIC, ISOTROPIC MATERIAL (SEMI-)ELLIPTICAL CRACK ππ is the surface correction factor πΈ is the crack shape factor EFFECT OF FINITE SIZE Stress concentration effects due to a through crack in finite and infinite width plates: (a) infinite plate and (b) finite plate. EFFECT OF FINITE SIZE FEM In general, K is always related to the through crack through an appropriate correction factor: π²π°,π°π°,π°π° = ππ π π Y is a dimensionless constant the depends on the geometry and mode of loading. K I SOLUTIONS FOR COMMON TEST SPECIMENS π· π π²π° = π π© πΎ πΎ K I SOLUTIONS FOR COMMON TEST SPECIMENS KI SOLUTIONS FOR COMMON TEST SPECIMENS PRINCIPLE OF SUPERPOSITION • For linear elastic materials, individual components of stress, strain and displacements are component-wise additive. This means that two normal stresses caused by different forces on the same direction can be added. • The stress intensity factors are additive as long as the mode of loading is consistent: (πππππ) π²π° (π¨) = π²π° (π©) + π²π° (πͺ) + π²π° • Note that: π²πππππ ≠ π²π° + π²π°π° + π²π°π°π° PRINCIPLE OF SUPERPOSITION For example, an edge cracked panel subject to combined axial loading π·π and three-point bending π·π experiences pure mode I conditions. This, the π²π° values can be simply added: (πππππ) π = π²π π° + π²π° π π π = [π·π ππ + π·π ππ ] πΎ πΎ π© πΎ π²π° PRINCIPLE OF SUPERPOSITION Determination of KI for a semielliptical surface crack under internal pressure p by means of the principle of superposition π²ππ° = π²ππ° − π²ππ° = π²ππ° π π − π = πΈπ π π(π) πΈ WEIGHT FUNCTIONS Evidently, the K value satisfies only one set of boundary conditions; different loading conditions result in different stress intensity values. However, if K was already determined for some specific boundary condition, this solution can be utilized to infer K for any other boundary conditions. WEIGHT FUNCTIONS We consider two arbitrary loading conditions and suppose that we know the stress (π) (π) intensity factor π²π° for loading condition (1) and we want to determine π²π° . WEIGHT FUNCTIONS (π) (π) π²π° and π²π° are related as follows: (π) π²π° = π π¬′ (π) ππ²π° π πππ πππ ΰΆ± π»π π πͺ + ΰΆ± ππ π π¨ ππ ππ πͺ π¨ πͺ: π©ππ«π’π¦ππππ« π¨: ππππ ππ : π ππππππππππππ Because the loading systems (1) and (2) are arbitrary one cannot depend upon the other. Thus, the function π ππ π¬ π πππ = (π) ππ ππ² π° Must be independent of the nature of loading system (1). This function is the weight function. If the weight function is known for a particulat loading condition, it is possible to determine π²π° for any boundary condition. WEIGHT FUNCTIONS By the principle of superposition, we can deduce that for a 2D cracked body, the stress intensity factor is: π²π° = ΰΆ± π π π π π π πͺπ p(x) is the crack face traction (equal to the normal stress acting on the crack plane when the body is uncracked) and πͺπ is the perimeter of the crack. RELATIONSHIP BETWEEN K AND G • Two parameters that describe the behavior of cracks have been introduced so far: the energy release rate G and the stress intensity factor K. • The former parameter quantifies the net change in potential energy that accompanies an increment of crack extension; the latter quantity characterizes the stresses, strains, and displacements near the crack tip. • The energy release rate describes global behavior, while K is a local parameter. For linear elastic materials, K and G are uniquely related. CRACK-TIP PLASTICITY • Linear elastic stress analysis of sharp cracks predicts infinite stresses at the crack tip. In real materials, however, stresses at the crack tip are finite because the crack-tip radius must be finite. • Inelastic material deformation, such as plasticity in metals and crazing in polymers, leads to further relaxation of crack-tip stresses. • The elastic stress analysis becomes increasingly inaccurate as the inelastic region at the crack tip grows. Simple corrections to linear elastic fracture mechanics (LEFM) are available when moderate crack-tip yielding occurs. For more extensive yielding, one must apply alternative cracktip parameters that take nonlinear material behavior into account. • The size of the crack-tip-yielding zone can be estimated by two methods: Irwin approach and the Strip-yield model. Both approaches lead to simple corrections for crack-tip yielding. IRWIN APPROACH The effective crack length is defined as the sum of the actual crack size and a plastic zone correction: First-order and second-order estimates of plastic zone size (ry and rp, respectively). The crosshatched area represents load that must be redistributed, resulting in a larger plastic zone. The effective stress intensity is obtained by inserting aeff into the K expression for the geometry of interest: Effective Mode I stress intensity factor for a through crack in an infinite plate in plane stress is given by THE STRIP-YIELD MODEL This model approximates elastic-plastic behavior by superimposing two elastic solutions: a through crack under remote tension and a through crack with closure stresses at the tip. Thus the strip-yield model is a classical application of the principle of superposition. The strip-yield model. The plastic zone is modeled by yield magnitude compressive stresses at each crack tip. THE STRIP-YIELD MODEL The stress intensities for the two crack tips are given by The closure force at a point within the strip-yield zone is equal to Thus, the total stress intensity at each crack tip resulting from the closure stresses is obtained by replacing a with a + ρ and summing the contribution from both crack tips: THE STRIP-YIELD MODEL Solving this integral gives The stress intensity from the remote tensile stress, , must balance with Kclosure. Therefore, performing a Taylor series expansion on Equation Neglecting all but the first two terms and solving for the plastic zone size gives set aeff equal to a + ρ: However, this equation tends to overestimate Keff. Actual aeff is somewhat less than a + ρ because the strip-yield zone is loaded to σYS. Burdekin and Stone obtained a more realistic estimate of Keff for the strip-yield model COMPARISON OF PLASTIC ZONE CORRECTIONS Comparison of plastic zone corrections for a through crack in plane strain. K -CONTROLLED FRACTURE Schematic test specimen and structure loaded to the same stress intensity. The crack-tip conditions should be identical in both configurations as long as the plastic zone is small compared to all relevant dimensions. Thus, both will fail at the same critical K value. K -CONTROLLED FRACTURE Crack-tip stress fields for the specimen and structure • In the singularity-dominated zone, a log-log plot of the stress distribution is linear with a slope of −1/2. • Inside of the plastic zone, the stresses are lower than predicted by the elastic solution, but are identical for the two configurations. • Outside of the singularity-dominated zone, higher order terms become significant and the stress fields are different for the structure and test specimen. K does not uniquely characterize the magnitude of the higher-order terms. PLANE STRAIN FRACTURE: FACT VS. FICTION Interrelationship between specimen dimensions, crack-tip triaxiality, and fracture toughness. Schematic variation of transverse stress and strain through the thickness at a point near the crack tip. • Three-dimensional deformation at the tip of a crack. The high normal stress at the crack tip causes material near the surface to contract, but material in the interior is constrained, resulting in a triaxial stress state. • The stress state can have a significant effect on the fracture behavior of a given material EFFECT OF THICKNESS ON APPARENT FRACTURE TOUGHNESS Variation of measured fracture toughness with specimen thickness for an unspecified alloy. Adapted from Barsom and Rolfe, Fracture and Fatigue Control in Structures. 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1987. EFFECT OF THICKNESS ON APPARENT FRACTURE TOUGHNESS Effect of specimen thickness on fracture surface morphology for materials that exhibit ductile crack growth. Effect of thickness on stress—the crack-tip stress state in the fracture process zone. MIXED-MODE FRACTURE • A propagating crack seeks the path of least resistance (or the path of maximum driving force) and need not be confined to its initial plane. • If the material is isotropic and homogeneous, the crack will propagate in such a way as to maximize the energy release rate. • What follows is an evaluation of the energy release rate as a function of propagation direction in mixed-mode problems. Only Mode I and Mode II are considered here, but the basic methodology can, in principle, be applied to a more general case where all three modes are present. PROPAGATION OF AN ANGLED CRACK Typical propagation from an initial crack that is not orthogonal to the applied normal stress. The loading for the initial angled crack is a combination of Mode I and Mode II, but the crack tends to propagate normal to the applied stress, resulting in pure Mode I loading. THE ENERGY RELEASE RATE FOR THE KINKED CRACK Plot of G(α) normalized by G(α = 0). The peak in G(α) at each β corresponds to the point where kI exhibits a maximum and kII = 0. Effect of β on the optimum propagation angle. The dashed line corresponds to propagation perpendicular to the remote principal stress. BIAXIAL LOADING Cracked plane subject to a biaxial stress state. Optimum propagation angle as a function of β and biaxialty. INTERACTION OF MULTIPLE CRACKS • The local stress field and crack driving force for a given flaw can be significantly affected by the presence of one or more neighboring cracks. • Depending on the relative orientation of the neighboring cracks, the interaction can either magnify or diminish the stress intensity factor. • An example of the former is an infinite array of coplanar cracks. When cracks are coplanar to one another, KI tends to increase due to the interaction. • An example of the later is an infinite array of parallel cracks. When cracks are parallel to one another, KI tends to decrease due to the interaction. • Consequently, multiple cracks that are parallel to one another are of less concern than multiple cracks in the same plane. COPLANAR CRACKS Coplanar cracks. Interaction between cracks results in a magnification of KI. Interaction of two identical coplanar through-wall cracks in an infinite plate. Taken from Murakami, Y., Stress Intensity Factors Handbook. Pergamon Press, New York, 1987. PARALLEL CRACKS Parallel cracks. A mutual shielding effect reduces KI in each crack. Interaction between two identical parallel through-wall cracks in an infinite plate. Taken from Murakami, Y., Stress Intensity Factors Handbook. Pergamon Press, New York, 1987.