FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions Damage Mechanics-Based Models for High-Cycle Fatigue Life Prediction of Metals H.A.F. Argente dos Santos (Postdoc Fellow) Dipartimento diMeccanica Strutturale Universit`a degli Studi diPavia Pavia, December 18th, 2009 FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions FatigueLife Fatigue Life Classification Types of Fatigue (form in which fatigue occurs): mechanical; thermomechanical; creep; corrosion; rolling contact; fretting; Fatigue life (duration of the fatigue life): Low-Cycle Fatigue (LCF) (N <104−105 cycles): stresses are generally high enough to cause appreciable plastic deformation at the mesoscale (the scale of the RVE) prior to failure; High-Cycle Fatigue (HCF) (N >104−105 cycles): damage is localized at the microscale as a few micro-cracks and the material deforms primarily elastically at the mesoscale up to crack initiation. The cyclic evolution of an isolated grain can be resumed by the creation of localized plastic slip bands and the nucleation of microcracks until the creation of a mesocrack. FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions FatigueLife Fatigue Life ∆σ LCF HCF 2 σ f 104 to 105cycles Nf FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions EvolutionofFatigueCracks Stages Crack Initiation (CI): is a material surface phenomenon; micro-cracks usually start on localized shear planes at the surface; once nucleation occurs and cyclic loading continues, the micro-crack tends to grow along the plane of maximum shear stress and through the grain boundary. Stable Crack Growth (SCG): is normal to the maximum principal stress; it depends on the material as a bulk property; Unstable Crack Growth: leads to ductile or brittle fracture; very short, not important from a practical point of view. Remark: Under stress amplitudes just above the fatigue limit (HCF), the CI period may cover a large percentage of the fatigue life; for larger stress amplitudes (LCF), the SCG period can be a substantial portion of the fatigue life. FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions High-CycleFatigue Important Features of HCF HCF of metals may be regarded as a form of material degradation/damage caused by cyclic loading; Damage is controlled by mechanisms at the grain scale (microscale) and, therefore, a description at this scale is necessary; At the mesoscale most of the metallic materials can be considered isotropic and homogeneous; Microscopic plasticity should be determined by isotropic and kinematic hardening rules; Mean stress effect must be taken into account: while mean normal stresses have great effects on failure, mean shear stresses do not. FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions High-CycleFatigue Important Features of HCF (Cont.) Macroscopic plasticity is for the most part negligible, and crack initiation occurs in localized plasticity spots surrounded by a material in elastic range. Damage is localized on a microscopic scale with negligible influence on the mesoscale ⇒ Quasi-Brittle Failures; Crack initiation modeling is difficult in this fatigue regime since the scale where the mechanisms operate is not the engineering scale (mesoscale), and local plasticity and damage act simultaneously. All these features can be well characterized by means of the Theory of Damage Mechanics. FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions Two-ScaleModel Lemaitre’s Model (1994, 1999, 2005) It considers a microscopic spherical inclusion with an elasto-plastic-damage behavior embedded in a macroscopic infinite elastic matrix: Elastic(E,ν) Elastic(E,ν) Plastic(C,σf) (σ,ε) Damage(S,s,Dc) (σµ,εµ,εµe,εµp,D) RVE INCLUSION Mesoscale Localization Law Microscale FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions ThermodynamicsFramework Free Energy Free energy of the inclusion: ρϕµ = ρϕµ +ρϕµ e p Elastic part (affected by the damage variable to model the experimentally observed coupling between elasticity and damage) 1 ρϕµ(εµ−εµp,D)= (εµ−εµp)E(1−D)(εµ−εµp) e 2 Plastic part (assumes exponential isotropic hardening and linear kinematic hardening) 1 1 ρϕµp(α,r)=R∞(rµ+ be−brµ)+ 3X∞γαµ :αµ Free energy of the matrix: ρϕ(ε) = 1εEε 2 FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions ThermodynamicsFramework State Laws (Inclusion) Stress Tensor ∂ϕµ σµ = ρ = E(1−D):εµe ∂εµe Isotropic Hardening (it represents the growth in size of the yield surface) ∂ϕµ Rµ = ρ = R∞(1−e−brµ) ∂rµ Kinematic Hardening (it represents the translation of the yield surface) ∂ϕµ 2 Xµ = ρ∂αµ = 3X∞γαµ Energy Density Release ∂ϕµ 1 Yµ = −ρ = εµe :E : εµe ∂D 2 FatigueofMetals FatigueCP HCF Lem.MSModel Des.MSModel Fla.MSModel Macro.Models Conclusions ThermodynamicsFramework State Laws (Inclusion) (Cont.) The energy density release can be rewritten as σµ2 Rµ Yµ = eq ν 2E(1−D)2 with 2 σµ 2 Rµ = (1+ν)+3(1−2ν) H ν 3 (cid:18)σµ (cid:19) eq 1 σµ = 3σµD : σµD 2, σµ = 1tr(σµ), σµD = σµ−σµI eq (cid:18)2 (cid:19) H 3 H The mean stress effect can be taken into account by considering a new form of the energy density release rate which includes an additional parameter, 0 ≤ h ≤ 1, to model the micro-defects closure effect.