13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Theoretical models of fatigue damage indicators Relative temperature increment. Fatigue damage is known as energy dissipation accompanied by temperature changing. The temperature linked with energy dissipation enables us to understand the energy transformation, toughness reduction and damping vibration of materials. Therefore, the fatigue process can be qualitatively evaluated using the relative temperature increment. During fatigue tests, to avoid any possible errors induced by the environmental perturbation and the experimental system sensitivity, the relative temperature increment ΔT on the hot-spot zone of the specimen surface is used to describe fatigue damage status: ΔT =Tm - T0 (1) where Tm is the average temperature on the zone; and T0 is the initial temperature. Standard deviation of stress. Microcracks often initiate from local points due to the stress concentration. The fatigue damage distribution is not uniform when a material suffers from cyclic loading. The distribution of the local stress can be described by the standard deviation. The stress state on the hot-spot zone, due to the local high stress, enables us to qualitatively identify the critical location responsible for the final fracture. Accordingly, the economic losses caused by the sudden fatigue fracture might be greatly decreased by analyzing this damage indicator. The stress level used here is the thermoelastic stress calculated by the equation below: p T T C α σ ρ Δ = − ⋅ ⋅Δ (2) where α is the coefficient of linear expansion; Cp is the specific heat capacity; ρ is the material density; T is the absolute temperature; Δσ is the change in the sum of principal stresses; and ΔT is the change in temperature. The stress pattern can be visibly obtained using the infrared camera, and each pixel stands for a point in the selected zone Ω. Thus, the standard deviation of the stress can be written as: 2 SDS m , 1 ( ( , ) ) x y x y N σ σ σ ∈Ω = − ∑ (3) Where σm denotes the average stress in the zone Ω; σ(x, y) denotes the stress value at the point (x, y); and N denotes all the points in the zone Ω. Intrinsic dissipation. Based on the small perturbation hypotheses, fatigue test is considered as a quasi-static dissipation process. The local coupled thermomechanical equation is derived [8]: 2 2 . . . . . 2 pC T ( ): : : e k T T T T T ψ ψ ψ ψ ρ σ ρ ε ρ α ρ ε ρ α γ ε α ε α ∂ ∂ ∂ ∂ − ∇ = − − ⋅ + + + ∂ ∂ ∂ ∂ ∂ ∂ (4) where k is the heat conduction coefficient; σ denotes the stress tensor; ψ is Helmholtz free energy; ε is the strain tensor; α is internal variables; and γe is the external heat resource. The intrinsic dissipation is defined as: . . ( ): d ψ ψ σ ρ ε ρ α ε α ∂ ∂ = − − ⋅ ∂ ∂ (5) In fact, the intrinsic dissipation describes the dissipated energy due to inelastic effects, and it is an important part of the non-linear energy dissipation for materials and components subjected to fatigue loading. The infrared thermographic method can be used to quantitatively evaluate fatigue
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