Lifetime-Oriented Structural Design Concepts- P11 pps

258 3 Deterioration of Materials and Structures [386] within the mean part of the yield condition is a scaling factor, that leads to total material softening for f ∗ q A =1. Nonlinear isotropic hardening is considered by the following relation ˆ ¯q = J p J e [σ Y 0 + βH iso ˆα + β (σ Y ∞ − σ Y 0 )(1 − exp(−δ iso ˆα))] , (3.93) where ˙ ˆα=˙γ[( ˆ Σ  − ˆ ¯κ): ˆ N  ]/[(1 −f ∗ q A ) ˆ ¯q] describes the evolution of the scalar isotropic hardening parameter ˆα and the determinants J e and J p transform from Cauchy- to Kirchhoff-stresses. Cyclic loading is accounted for by an advanced kinematic hardening model [87] using a superposition of at most four kinematic hardening tensors ˆ ¯κ=J p J e (1 −f ∗ q A )  4 i=1 ˆ κ i with the following assumption for the material time derivative of the back stress tensor ˙ ˆκ i =(1− β)  c i ˆ D p  − b i ζ  δ kin ˙γ ˆ κ i +(1− δ kin )( ˆ κ i : ˆ ˜ N) ˆ D p   , (3.94) where ζ=1 for i=1,2,3 or < 1 − ¯κ || ˆ κ 4 || > for i=4, ˆ D p =˙γF, ( ˆ Σ− ˆ κ) =˙γ ˆ ˜ N is the symmetric plastic strain rate, ˆ D p  = ˆ G −1 ˆ D p ˆ G −1 , ˆ ˜ N describes the symmetric gradient of the yield surface, || ˆ κ||=  3/2tr( ˆ κ ˆ κ  ) is the norm of the back stress tensor ˆ κ and ˆ κ  = ˆ G ˆ κ ˆ G,¯κ, δ kin are model parameters and β controls the decomposition of isotropic and kinematic hardening. The evolution of the void volume fraction f is described by ˙ f = ˙ f growth + ˙ f nucl = q C (1 − f)tr( ˆ D p ˆ G −1 )+ ˙ f nucl , (3.95) which is related to f ∗ in (3.92) according to (3.98). Note that an additional material coefficient q C is introduced in (3.95), which is necessary to calibrate the Gurson model according to the results from unit cell analyses [387]. ˙ f nucl in (3.95) represents a nucleation law according to [197] with ˙ f nucl = f n s n √ 2 π exp  − 1 2   p −  n s n  2  ˙ p , (3.96) where f n , s n and  n are model parameters and ˙ p is given by ˙ p =  3/2( ˆ D p ˆ G −1 ) 2 : ˆ I . (3.97) To describe the physical process of void nucleation adequately, the evolution of ˙ p is only defined for loading. In case of unloading no nucleation of micropores is considered. For the consideration of the coalescence of the micropores the phenomenological law according to [797] is used, f ∗ = ⎧ ⎪ ⎨ ⎪ ⎩ f for f ≤ f c f c + K(f −f c )forf>f c with K = f ∗ u − f c f f − f c and f ∗ u = 1 q A , (3.98) 3.3 Modelling 259 Fig. 3.126. Numerical and experimental data for (a) material softening and (b) ratcheting effect wherein f is transformed into f ∗ and an acceleration of the evolution of f is driven by a scalar factor K defined by f c and f f according to (3.98). In addition to the void volume fraction f ∗ , an additional variable S is used to characterize the void shape. The evolution of S is described by an equation proposed by [258]. It is implemented into the Gurson-model by a modification of the material parameter q B [386]. The implementation of the model is based upon the return-map algorithm and a consistent linearization procedure [743]. Because of the anisotropy in- duced by the kinematic hardening, the iterative solution involves 8 unknowns (the components of the symmetric gradient ˆ ˜ N, the plastic multiplier ˙γ and the void volume fraction f). The proposed macroscopic elasto-plastic damage model has the ability to replicate all typical phenomena of cyclic plasticity such as the Bauschinger- effect, ratcheting or mean stress relaxation, cyclic hardening or softening [386]. In the following, a comparison of numerical and experimental data shows the efficiency in case of simulating the effects of material softening and ratcheting. To this end, a cyclically loaded hollow cylindrical specimen of CS 1026 [86] is re-analysed numerically. Using the isotropic hardening law according to (3.93) the stress amplitude of the first 25 load cycles can be simulated in good agreement to the experimental results, as Figure 3.126(a) shows. The use of the Bari-Hassan-type of kinematic hardening rule allows for the simulation of the ratcheting effect, which is demonstrated by the evolution of the radial strain in case of biaxial loading in Figure 3.126(b). 3.3.1.2.1.2 Model Validation The following analysis are performed for 20MnMoNi55, a low alloy steel typically used for structures such as pressure vessels. For this special type of material a calibration leads to the model parameters presented in Table 3.20 [386]. After the calibration procedure, the micropore damage model is validated according to Figure 3.125 by means of fatigue tests. Therefore, results from 260 3 Deterioration of Materials and Structures Table 3.20. Parameter of the elasto-plastic micropore damage model for 20Mn- MoNi55 E=204 [GPa] β=0.5 σ Y 0 =220 [MPa] δ iso =25 [-] ν=0.3 [-] σ Y ∞ =410 [MPa] H iso =0 [MPa] b 1 =25000 [-] b 2 =500 [-] b 3 =5 [-] b 4 =5000 [-] δ kin =0.18 [-] c 1 =500000 [MPa] c 2 =60000 [MPa] c 3 =3000 [MPa] c 4 =100000 [MPa] ˘κ=0 [MPa] f 0 =0.01 [-] S 0 =0.0 [-] q A =1.85 [-] q B =0.48 [-] q C =1.4 [-] f n =0.08 [-]  n =3.0 [-] s n =1.0 [-] f krit =0.09 [-] f Bruch =0.14 [-] Fig. 3.127. Low Cycle Fatigue in metals: Numerical and experimental results for cyclically loaded round notched bar with (a) 2mm notch radius and (b) 10mm notch radius cyclically loaded round notched bars with two different notch radii (2mm and 10mm) performed by [638] are re-analyzed. To study the damage evolution due to low cycle fatigue, the results are compared in terms of the degradation of the peak reaction force in Figure 3.127. The chosen set of material specific model parameters result in a good agree- ment of experimental and numerical results. In particular the strong change of the slope of the reaction force curve, when coalescence becomes the dominant damage mechanism, is simulated by the micropore damage model in a good manner for both cases. This final change state can be correlated to the life time of the structure, which is reasonably well predicted (see Table 3.21). It should be noted that he numerical simulations also allow a localization of the position of damage accumulation in accordance with the experimental observations. For the smaller notch radius the micropore damage initiates and starts to accumulate from the notch root (Fig.3.128(a,b)). In contrast, for the specimen with the larger notch radius a nearly homogeneous damage accumu- lation initiating from the interior of the specimen is observed(Fig.3.128(c,d)). 3.3 Modelling 261 Table 3.21. Low Cycle Fatigue in metals: Number of load cycles until failure ob- tained from numerical simulations and experiments 2 mm notch radius 10 mm notch radius Experiment Num. Model Experiment Num. Model Failure initiation 17 cycles 24 cycles 31 cycles 35 cycles Life-time 23 cycles 28 cycles 43 cycles 40 cycles ( a ) ( b ) ( c ) ( d ) Fig. 3.128. Low Cycle Fatigue in metals: Damage accumulation and numerically predicted damage in a cyclically loaded round notched bar: (a,b) 2 mm notch radius, (c,d) 10 mm notch radius 3.3.1.2.2 Quasi-Brittle Damage in Materials 3.3.1.2.2.1 Cementitious Materials Authored by Tobias Pfister and Friedhelm Stangenberg Concept In high-cycle fatigue processes, a large number of load cycles under moder- ate stress level leads to increase of strains and degradation of material prop- erties. Different from approaches for low-cycle fatigue, the simulation of every single load cycle is too time-consuming for practical application. Therefore, degradation-, damage- and strain-evolutions are modelled indirectly, depend- ing on the applied increment of load-cycles ΔN . For a standardised evaluation and formulation, the load cycles are related to the ultimate number of load- cycles N f according to the S-N -approach. This leads to the standardised time scale n ∈ [0, 1] with increments Δn = ΔN N f . This standardised time scale is 262 3 Deterioration of Materials and Structures related to the time scale via N f and the frequency f by: Δn = ΔN N f = Δt · f N f ⇔ Δt = N f f ·Δn . (3.99) This concept has already been applied in [627, 628, 629, 630]. S-N-Approach Thus, one basic quantity for the fatigue model is the fatigue lifetime N f of concrete. Generally, any S-N-curve can be applied for its evaluation. In the following the approach presented in [392] will be re-used. It has been compared to other approaches and to a large number of experiments in [627] and proved to be well suitable. It takes the loading frequency into account and distinguishes between high-cycle and low-cylce fatigue: log N f =max ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1.0 − 0.0294 log  t fat t ref  + s max 0.062 (1 − 0.556 r fat ) 1.2 − 0.2 r fat + s max − 0.053 (1 − 0.445 r fat )log  t fat t ref  0.133 (1 − 0.778 r fat ) . (3.100) In this expression, t fat is the duration of one load cycle, which is the inverse of the frequency: t fat =1/f, t ref is a reference time of the same unit as t fat : t ref =1[t fat ], s max and s min are the related upper and lower stress limits, respectively: s max = σ max /f c , s min = σ min /f c and r fat is the relationship of lower to upper fatigue stress: r fat = σ min /σ max . The approach, together with a large number of experiments from the literature, is illustrated in Figure 3.129. The evaluation of the 0.05- and of the 0.95-quantile of N f , which is also shown in this diagram, will be introduced later in this section. Degradation of the Compressive Strength The degradation of the compressive strength is formulated empirically with a direct approach in the time scale of the related number of load cycles n,as introduced above. To quantify the degradation, the variable d f c is introduced and the resulting compressive strength reads as follows: f c (n)=f c,28 · (1 − d f c (n)) (3.101) According to the experimental results presented in [70, 374], the degradation process starts very slowly. Nevertheless, fatigue failure is associated with a drop of the compressive strength onto the level of the upper fatigue stress. Thus, the value of d f c results in d f c ,fail =1−|s max | (3.102) 3.3 Modelling 263 related stress s max lifetime N f 0.5 0.6 0.7 0.8 0.9 1 . 0 10 0 10 2 10 4 10 6 10 8 10 1 0 Fig. 3.129. S-N approach (0.05-, 0.50-, and 0.95-quantiles) with experimental results for the state of fatigue failure, n = 1. Based on an exponential approach, suggested in [370] for the description of sequence effects, d f c is sub-structured into d f c = n a f c · d f c ,fail with a f c =26.5 −25.0 |σ max | f c . (3.103) This implies a faster degradation (in the time scale n = N/N f )ofthe compressive strength for higher stresses. The evaluation of the degradation is shown exemplarily for three different load levels in the left diagram in Figure 3.130. relative strength f c / f c,28 relative strength f c / f c,28 related cycles n = N / N f related cycles n = N / N f 1.2 1.0 0.8 0.6 0.4 0.00.2 0.4 0.60.6 0.7 0.80.8 0.91.0 1.0 0.9 Fig. 3.130. Degradation of compressive strength and sequence effects 264 3 Deterioration of Materials and Structures N 1 / N f , 1 N 2,res /N f,2 } } 2 . 0 1.5 1.0 0.5 0.0 0.0 0.20.4 0.6 0.81. 0 Fig. 3.131. Evaluation of the approach for sequence effects an comparison with single simulation results from [383] Sequence Effects This direct formulation of the degradation of the compressive strength in- cludes indirectly the formulation of sequence effects. As introduced, the degra- dation process depends on the applied upper fatigue-load level. When this load level is changed, the degradation curve changes, too. As illustrated in the right diagram of Figure 3.130, this requires a modification of the related number of load cycles n = N/N f . Keeping n constant would imply a sudden drop or increase of the compressive strength, which is physically nonsensical. Thus, n has to be changed. The updated value of n can be evaluated from the approach for the description of the degradation process: d f c ,n ! = d f c ,n+1 ⇒ n =  d f c ,n d f c ,fail,n+1  1/a f c ,n (3.104) Fig. 3.131 shows an evaluation of the presented approach together with the results of two stage tests from [383]. The mean values of the single results are plotted by solid lines. It can be seen, that the approach delivers qualitatively reasonable results, for quantitative evaluations, the data basis is too small. Strain Evolution As introduced in Section 3.1.1.2.2.1, the strain evolution in concrete un- der fatigue loading can be interpreted as the sum of creep and cyclic strain evolution. This interpretation is picked up for the modelling approach. In the following, the approaches for the creep strain evolution as well as for the evolution of cyclic strains is introduced. 3.3 Modelling 265 spr i ng frictional element dashpot σ σ Fig. 3.132. Rheological element for the description of nonlinear creep processes Creep Strain Evolution The creep strain evolution is modelled with rheological elements. The model is based on an approach presented in [735], which has been enhanced in [133, 134]. To account for the nonlinear relation of stress and creep strain rate, especially for concrete under higher stresses than approximately 0.4 f c , nonlinear rheological elements are used. They consist of a nonlinear spring with a friction element (to describe plastic deformations) and a nonlinear dashpot. Fig. 3.132 shows such elements. The nonlinear behaviour of the spring is described with thestress-strain relation for concrete under compression given in [182], with the compressive strength f c replaced by f c,T =0.8 f c , taking long-term effects into account. Thus, the stress-strain relation for the spring reads: σ s = E c ε cr f c,T +  ε cr ε c  2 1 −  E c ε c f c,T − 2  ε cr ε c  f c,T . (3.105) In incremental formulation, this equation can be approximated as σ s n+1 ≈ σ s n + E tan Δε cr n+1 , (3.106) where E tan is the tangent d σ s /d ε cr . Analogue to the smeared crack model described earlier in this chapter, strains which result from the nonlinearity of the spring are regarded dissipative. For the modelling of creep they are regarded as plastic strains, indicated by the friction element in Figure 3.132. For linear descriptions of dashpots, the viscosity is coupled linearly via the retardation time τ with the stiffness: η = τE c . This relation is now enhanced and is reformulated dependent on time and on the applied stress: η = τE c  t − t 0 τ  1 2  1 − σ d f c,T  n cr . (3.107) Herein, σ d is the stress in the dashpot and n cr a material parameter. According to [133, 134], for concrete it takes values between 1.5 and 2.0. 266 3 Deterioration of Materials and Structures The classical relation between stress and strain rate, σ d = η ˙ε cr , (3.108) is now replaced by an incremental formulation: σ d n+1 ≈ σ d n + Δt ˙σ d n+1 . (3.109) The stress rate can be found as time derivative of eq. (3.108) to ˙σ d = d (η ˙ε cr ) dt =˙η ˙ε cr + η ¨ε, (3.110) the time derivative of eq. (3.107) results in the rate of viscosity: ˙η = E c 2  t + t 0 τ  1 2  1 − σ d f c,T  n cr . (3.111) The equation for the resulting stress σ cr = σ s + σ d (3.112) yields a differential equation, which can be solved numerically. In [133] the Newmark method according to [569, 198, 409] is suggested. Cyclic Strain Evolution The rate of cyclic fatigue strains, in the time scale of related load cycles n: ˙ε fat,∗ = ∂ε fat ∂n , (3.113) is formulated empirically on the basis of the experiments documented in [383]. Therefore, the typical S-shaped evolution curve of fatigue strains is devided into three parts, where a constant strain rate is assumed within each domain. The strain rates are evaluated from the experiments by linear regression, like illustrated in the left diagram of Figure 3.133 together with the experimental results from [383]. The borders between these domains are assumed at n =0.1 and n =0.9. In order to approximate the fatigue strain rates as functions of the applied load level, a scalar measure for the fatigue loading, taking upper and lower fatigue stress into account, is introduced as the product of mean stress and stress difference: s = s max + s min 2 ·[s max − s min ] . (3.114) The evaluation of the experimental results of [383, 70] are shown in the right diagram of Figure 3.133. The experiments, that exhibit significant creep 3.3 Modelling 267 related cycles n = N / N f fatigue strain −ε f at [10 −3 ] domain 1 domain 2 dom.3 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2 . 5 ¯ s max /s min =0.80/0.20 ¯ s max /s min =0.75/0.05 ¯ s max /s min =0.95/0.05 strain rate − ˙ε fat,∗ [ 10 −3 ] stress measure s 0.00.1 0.2 0.30.4 0.5 0.0 1.0 2.0 3.0 4 . 0 approach ¯ Holmen ¯ Awa d & Hilsdorf Fig. 3.133. Fatigue strain evolution (stress measure vs strain rate by [383, 70]) strains due the test duration are plotted in white. These experiments have not been used for the evolution of cyclic strain evolution. Those ones that exhibit pure cyclic strain are plotted in black. By least square fitting, second order polynomials are evaluated to approximate the (pure cyclic) fatigue strain rate as function of the stress measure s: ˙ε fat,∗ 1,3 = −113.189 s 2 +67.5492 s − 4.50913 , (3.115) ˙ε fat,∗ 2 = −6.54818 s 2 +4.55811 s − 0.268655 . (3.116) The right diagram in Figure 3.133 shows the polynomial as well as experimen- talresults,exemplarilyfordomain2.Theresultsof[383]havebeenusedfor the evaluation of the polynomial. For additional proof, the results of [70] are plotted in the diagram. These values have been evaluated graphically. They are therefore regarded as too imprecise and where not taken into account for the evaluation. Fatigue Damage For the evaluation of fatigue damage, again the tests of [383] deliver the ex- perimental basis. In these experiments, not only the evolution of the maximum, but also of the minimum 4 fatigue strains is reported. That can be utilised for the sub-division of the total fatigue strains into damaging and plastic parts. At first, the measured total strains are reduced by the initial ones. Assuming linear unloading and reloading, according to the damage theory, the fatigue strains corresponding to σ = 0 can be extrapolated, like illustrated in Figure 3.134. This yields reversible and irreversible parts of the total fatigue strains, which are interpreted as damaging and plastic, respectively. 4 That means the strains, that correspond to the lower fatigue stress level.