Report

STRESS CONCENTRATION AT NOTCHES One of the fundamental issues of designing a fatigue resistant structure (“design against fatigue”) is the consideration of stress concentration Stress concentration at geometrical notches are always present in a real structure Notches introduce inhomogeneous stress distribution with a stress concentration at the root of the notch Stress concentration factor: peak Kt nom Kt is referred as the theoretical stress concentration factor: it is based in the assumption of linear elastic material behavior Kt describes the severity of the notch and depends on the geometry of the notch configuration (shape factor of the notch) Common examples of stress concentration (a) Gear teeth (b) Shaft keyway (c) Bolt threads (d) Shaft shoulder (e) Riveted or bolted joint (f) Welded joint all these components might be subjected to cyclic loads ! DEFINITIONS: Kt peak nom K tg usual peak S alternative For the previous example: therefore: K tg nom S W Kt Kt W D Ktg Kt In general Kt is the preferred factor to indicate stress concentration following the definitions of R.E. Peterson in Stress Concentration Factors, John Wiley & Sons, New York (1974) THE “MODEL” STRESS CONCENTRATION CASE: the circular hole in an infinite sheet S r0 r 2 2 2 r0 r0 S r0 rr 1 2 1 3 2 4 2 cos 2 2 r r r 4 2 r0 r0 S r 1 3 4 2 2 sin 2 2 r r 2 4 r0 S r0 1 2 1 3 4 cos 2 2 r r S Along the edge of the circular hole: S 1 2 cos2 rr 0 max 2 0 S for = 0 ? 3S r 0 Kt 3 compression at = 0 = 30° or j = /2 – 60° from max Stress Profiles along the normal to the edge of the circular hole: We are interested in evaluating: a) Situation for compressive remote stress (–S): presence of tensile stress ? YES b) Gradient of Stress in the direction normal to the edge of the hole at the location of speak: strong gradient c) Gradient of Stress along the edge of the hole at the location of speak : slow decrease of stress along the edge of the notch d) Volume of material subjected to high Stress around the root of the notch: larger for larger notches! (significant to understand notch size effects on fatigue) a) Presence of Local Tensile Stress upon Remote Compression Stress * Fatigue Crack Growth under Compressive Stress? * Effect on Brittle Materials: Failure Criteria Based on Maximum Normal Stress for Brittle Materials C How would a cylinder of brittle material with a distribution of defects fail under compression? Spherical voids and sharp like cracks are often produced during processing of brittle materials: - Spherical voids sintering of ceramic powders (remnants of initial porosity) - Microcraks thermal expansion mismatch C b) Gradient of Stress in the direction NORMAL to the edge of the hole at the location of peak * Although the peak stress is of great importance, it is also interesting to know how fast the stress decreases away from the root of the notch Stress Gradient for the circular hole: peak d dr r0 r0 ; 2 1 2 K t The Stress Gradient at the root of a notch should give an indication of the volume of material under high stresses estimate the distance d along the normal to the root for a drop from peak to 0.9 peak (10 % decrease) for a circular hole with r0 = 2.5 mm: peak d dr r0 r0 ; 2 0.9 peak peak 1 2 3 d d = 0.1 mm= 100 mm if grain size 50 mm the depth d corresponds only to few grains The grains at the noth root surface are subjected to high loads and this is very important for fatigue c) y d) Gradient of Stress ALONG the edge of the hole at the location of peak As fatigue crack nucleation is a surface phenomenon it is of interest to know how fast the tangential stress along the edge of the notch is decreasing S * Slow decrease of the stress along the edge compared with the decrease from the edge at the location of peak * Larger notches have a larger material surface along the root of the notch very important to understand notch size effects on fatigue S Note: even when a particular case was analyzed here (circular hole in an infinite plate) the conclusions are of general validity similar peak stresses and notch root radii give comparable stress distribution around the root of an arbitrary notch Effect of notch geometry on Kt Geometrically similar specimens have the same Kt (Kt is dimensionless) … … but different stress gradients (stress gradient is not dimensionless) Larger specimens have larger volumes and larger notch surface areas of highly stressed material Reason of the existence of Notch Size Effects in Fatigue Also: Importance of surface quality (method of production / fabrication) surface defects due to manufacturing in a highly stressed region along the wall of a hole Further examples / further aspects of stress concentrators The elliptical hole in an infinite sheet S peak x0 y 0 a a S 1 2 S 1 2 b r S a a Kt 1 2 1 2 b r b2 r a S use large radii on surface parallel to applied stress to reduce stress concentration ! a/b 1/3 1 3 r/ a 9 1 1/9 Kt 1.67 3 7 Stress Concentration for an elliptical hole under biaxial loading: S A S 1 2 a b S S S B S 1 2 b a S * For the case of a thin walled pressure vessel under pressure = 0.5 and for the case of a circular hole (a = b): S A 2.5 S : biaxiality ratio compare with the square hole (dashed line) with rounded corners with r 10% of hole width: Kt = 4.04 for = 0.5 S B 0.5 S lower than 3 S for uniaxial loading * Same case but elliptical hole with b/a = 2: S A 1.5 S SB 1.5 S lower than 3 S for uniaxial loading (actually, = 1.5 S along the edge of the hole) Stress Concentration for a circular hole in a plate under pure shear: Kt 4 Fatigue cracks growing from holes in a shaft subjected to cyclic torsion ! Pin - loaded hole: Conection between a lug and a clevis: Comparison of Kt values for a lug and an open hole * Lugs are fatigue critical parts (also prone to fretting corrosion) values of d/W below 1/3 are usually avoided to keep Kt below 3.5 Superposition of notches: If a relativelly small notch is added to the root of the main notch Effect of superposition of notches: peak Kt 2 * Kt1S K t K t1 K t 2 This overestimates Kt because the small notch is not completely embedded in an homogeneous stress field of magnitude Kt1 Technique for estimating conservative limiting value for Kt for superposed notches: “Fill” the notch (cross hatched area) leaving a single deep narrow notch The theoretical stress concentration factor for the single deep narrow notch will always be greater than the Kt for the multiple notch (see Kt for Edge Notches two transparencies later) Examples of Superposition of notches: Lug with small lubrication hole to the lug hole Cross section of a fatigue crack at a sharp corner Edge notches and Corrosion Pits Corrosion pits at the material surface of an Al-alloy. Pit depth = 0.15 mm. Equivalent shape gives very high Kt values Further information of the type that can be found in the clasical handbook of R.E. Peterson, Stress Concentration Factors, 1974 Stress concentration factors for a shaft with a grove subjected to: Axial Load Bending Torsion THE FATIGUE STRENGHT OF NOTCHED SPECIMENS * STRESS – LIFE APPROACH: Notch Effects on the Fatigue Limit (Sm = 0) Similarity Principle: if Sa = Se is the fatigue limit of the smooth specimen, then Speak should give the fatigue limit SeK of the notched specimen: SeK Sa = Se S peak Se Kt SeK Se S meaning that: eK Kt … but this is not the case ! The Fatigue Strength Reduction Factor or Fatigue Notch Factor Kf is introduced: unnotchedfatigue limit Kf notchedfatigue limit In general: Kf < Kt fatigue limit of different materials are less notch sensitive to fatigue than predicted by Kt Examples: Effect of a Notch on S - N behavior (Tryon and Day, 2003) Mechanical Behaviour of Materials (Dowling, 1999) The examples illustrates a general observation for different materials: the finite life region is also less notch sensitive to fatigue than predicted by Kt In general Kf < Kt and Kt depends on geometry and mode of loading Kf also depends on material and notch size Blunting effects in soft materials: Yielding at the notch root reduces peak stress from the values predicted by Kt Fatigue strength of a notched component depends on the volume of highly stressed material near the notch also effect of stress gradient on crack growth. For engineering applications, the fatigue strength reduction fator Kf can be empirically related to the elastic stress concentration facto Kt by a Notch Sensitivity Factor defined as: q K f 1 Kt 1 K f 1 Kt 1 q 0 q 1 q = 1 material fully notch sensitive: Kf = Kt q = 0 material not notch sensitive: Kf = 1 Empirical equations for q were proposed by different authors: * Peterson (1959) * Neuber (1946) * Siebel and Stiele (1955) * Peterson assumed that fatigue damage occurs when a the stress at a point located at a critical distance ap away from the notch root is equal to the fatigue strength of a smooth specimen and obtained the following empirical equation: 1 q aP 1 r * for high strenght steels with SU > 560 MPa: Effect of notch root radius on Kf - r is the notch root radius - aP is a material constant related with material strength and ductility. Also, q can be obtained in graphical form: Peterson ´s notch sensitivity for steels * Neuber assumed that fatigue failure occurs if the average stress over a length aN from the notch root is equal to the fatigue limit of a smooth specimen and proposed the following empirical equation: q 1 aN 1 r Neuber´s Notch Sensitivity curves for Al alloys Relation with Hall-Petch? - r is the notch root radius - aN is the Neuber´s material constant related to the grain size * Siebel and Stiele (1955) introduced the Relative Stress Gradient (RSG) to characterize the effects of fatigue strength reduction (instead of using the notch radius!) 1 e max d e ( x) dx x 0 for the circular hole: d peak r0 dr r0 ; 2 1 where: 2 Kt 1 Similar dependencies are found for other geometries and for typical Kt values: 2 < Kt <5 No significant effect of Kt on Testing the Fatigue Strength of smooth and notched specimens they generated empirical curves relating Kf / Kt vs. Equation for the Curves: Kt 1 Css Kf where Css is a material constant related with Sy RSG () values calculated from Siebel and Stiele by the theory of elasticity for different notched members Examples: * STRESS – LIFE APPROACH: Notch Effect on Finite Life (S–N curve) In the Finite Life region the notch may be less sensitive to what is predicted by Kf for the Fatigue Limit Use Kf for the whole S-N curve? : to much conservative ! define a Fatigue Sensitivity Factor at a particular Life (and interpolate / extrapolate) analogously to what was made for estimating S – N curves for smooth specimens Kf for N1000 : K’f An Empirical Fatigue Notch Sensitivity Factor (q’1000) can be defined at 1000 cycles: q1000 K f 1 Kt 1 * Estimate of Fatigue Life for Notched Component: different approaches Examples: A) “Juvinall” approach LCF REGION! B) Fatigue Notch Sensitivity Factor Kf depends on Cycles to Failure Nf Example: Juvinall for the following loading cases: (1) Reversed bending loading (2) Reversed axial loading (negligible bending) (3) Reverse torsional loading