Fatigue Endurance under Multiaxial Loadings Prof. Edgar Mamiya - PowerPoint PPT Presentation

Fatigue Endurance under Multiaxial Loadings Prof. Edgar Mamiya Prof. José Alexander Araújo Universidade de Brasília Dept of Mechanical Engineering mamiya@unb.br US-South America Workshop August 3, 2004

Goal: To propose a fatigue model capable to answer the following question: Under which conditions a structure subjected to dynamic multiaxial loads attains infinite number of cycles ( > 10 6 ) without experiencing fatigue failure? σ k l σ ij US-South America Workshop August 3, 2004

Phenomenological aspects: In the setting of high cycle fatigue, • mechanical degradation is mainly driven by localized plastic deformations at mesoscopic level, • while the corresponding macroscopic behavior is essentially elastic: US-South America Workshop August 3, 2004

Thus, in order to avoid fatigue degradation, the mechanical behavior (at mesoscopic level) has to evolve to a state of elastic shakedown. … In metals, this can be accomplished only under certain bounded values of the “shear stress amplitude” τ In our model: ( S ) = appropriate function of the history of eq the deviatoric stress tensor S describing its “amplitude” in the multidimensional sense. US-South America Workshop August 3, 2004

Tractive normal stresses also play an important role in solicitation to fatigue, by acting in mode I upon eventually pre-existing embryocracks in the material. In our model: p = maximum value of the hydrostatic max stress p along the stress path. (recalling that the hydrostatic stress is the average of the normal stress acting upon all the planes across a given material point) US-South America Workshop August 3, 2004

τ e finite life Within this setting, let us write our q fatigue endurance criterion as: κ τ + κ ≤ λ eq S ( ) p max endurance p max In what follows, we shall propose a measure of the shear stress τ amplitude within the setting of multiaxial stress paths. eq US-South America Workshop August 3, 2004

Shear stress amplitude: Not all the states belonging to the stress stress path threatens the path material point. S 2 Only those states belonging to the corresponding convex hull determine the solicitation to fatigue. convex hull S 1 Shear stress amplitude can Shear stress amplitude can be defined from quantities be defined from quantities associated with the convex hull. associated with the convex hull. US-South America Workshop August 3, 2004

p 2 The points of the stress path tangent to arbitrarily oriented prismatic hulls p belong to the convex hull: 1 = = p arg( max s ( t )), i 1 ,..., 5 q i i t 1 = = q arg( min s ( t )), i 1 ,..., 5 i i t q 2 min s 1 t ( ) t max s 1 t ( ) t US-South America Workshop August 3, 2004

p 2 The points of the stress path tangent to arbitrarily p 1 oriented prismatic hulls belong to the convex hull: = = p arg( max s ( t )), i 1 ,..., 5 i i t q = = q q arg( min s ( t )), i 1 ,..., 5 1 2 i i t US-South America Workshop August 3, 2004

p The points of the stress path p 2 1 tangent to arbitrarily oriented prismatic hulls belong to the convex hull: = = p arg( max s ( t )), i 1 ,..., 5 i i t q 2 = = q arg( min s ( t )), i 1 ,..., 5 i i q t 1 US-South America Workshop August 3, 2004

As a consequence, the set of prismatic hulls itself and its corresponding quantities: = max s ( t ), min s ( t ), i 1 ,..., 5 i i t t can be considered for the characterization of the convex hull US-South America Workshop August 3, 2004

We consider the following quantity as a measure of the shear stress amplitude: ( ) ∑ 1 / 2 τ = 2 d eq i i where: d i ( ) 1 θ = θ − θ d max max s ( ; t ) min s ( ; t ) i i i 2 θ t t Remark: θ is the orientation of the prismatic hull in the 5-dimensional space of deviatoric stresses US-South America Workshop August 3, 2004

The resulting fatigue endurance criterion is hence given by: ( ) 5 1 ∑ 2 + κ ≤ λ where: = θ − θ d p d max max s ( ; t ) min s ( ; t ) i max i i i 2 θ t t = i 1 τ e finite life q κ d i endurance θ p max US-South America Workshop August 3, 2004

Computational issues • The search for the orientation of the prismatic hull which gives the global maximum value of: 5 ∑ τ θ = 2 ( ) d i = i 1 is performed in the 5-dimensional deviatoric space. Jacobi (or Givens) rotations were considered for simplicity. On the other hand, this implies a 10-parametric rotation process. • The function τ(θ) may attain several local maxima and hence some care must be taken with respect to the maximization algorithm. US-South America Workshop August 3, 2004

Assessment Proportional and nonproportional multiaxial fatigue experiments for different materials were considered to assess the proposed criterion in predicting fatigue strength under a high number of cycles. Limiting situations of fatigue endurance reported by: set authors Material f -1 t -1 1 Nihihara & Kawamoto (1945) hard steel 313.9 196.2 2 Heindereich, Zenner & Richter (1983) 34Cr4 410 256 3 Heindereich, Zenner & Richter (1983) 34Cr4 415 256 4 Kaniut (1983) 25CrMo4 340 228 5 Mielke (1980) 25CrMo4 340 228 US-South America Workshop August 3, 2004

Error index: evaluation of limiting situations τ + κ ≤ λ fatigue endurance criterion p eq max τ + κ − λ p error index eq max = × ≤ I 100 0 λ > I 0 conservative prediction τ eq < non-conservative finite life I 0 endurance p max US-South America Workshop August 3, 2004

Nishihara & Kawamoto (1945), hard steel Proportional and nonproportional σ−τ , same frequency of excitation, no mean stress US-South America Workshop August 3, 2004

Nishihara & Kawamoto (1945), hard steel 10 error index (%) Crossland 0 Papdopoulos 1 2 3 4 5 6 7 8 9 10 -10 Current model -20 experiment -2.3% < I < 6.5% (current model) US-South America Workshop August 3, 2004

Heindereich, Zenner & Richter (1983), 34Cr4 Proportional and nonproportional σ−τ , same frequency of excitation US-South America Workshop August 3, 2004

Heindereich, Zenner & Richter (1983), 34Cr4 mean stress 10 0 error index Crossland 1 3 5 7 9 11 -10 Papadopoulos Current model -20 -30 experiment -6.4% < I < 5.2% (current model) US-South America Workshop August 3, 2004

Heindereich, Zenner & Richter (1983), 34Cr4 Nonproportional σ−τ , w τ = 4 w σ Piecewise linear σ−τ I=10 . 6% I=4 . 7% US-South America Workshop August 3, 2004

Kaniut (1983), 25CrMo4 Nonproportional σ−τ Nonproportional σ−τ, w τ = 2 w σ I = 4 . 3% I = -0 . 03% I = -0 . 31% I = -1 . 8% w σ = 4 w τ phase angle = 0 o w τ = 8 w σ phase angle = 90 o US-South America Workshop August 3, 2004

Mielke (1980), 25CrMo4 I=-0 . 04% I=-0 . 04% US-South America Workshop August 3, 2004

Mielke (1980), 25CrMo4 I=-1 . 9% I=4 . 6% US-South America Workshop August 3, 2004

Closure • A new stress based multiaxial fatigue criterion, which is very simple to implement and can be applied to a broad class of loadings, has been proposed; • Application of the proposed criterion for several different materials yielded very good predictions of fatigue endurance; • We are conducting studies in order to extend the applicability of the criterion to more ductile materials. • We are also addressing the question of fatigue endurance under conditions of severe stress gradients. US-South America Workshop August 3, 2004

Thank you !!! US-South America Workshop August 3, 2004

US-South America Workshop August 3, 2004