Universal Material Constants for MultiStage Fatigue (MSF) Modeling of the Process–Structure–Property (PSP) Relations of A000, 2000, 5000, and 7000 Series Aluminum Alloys

Abstract

A MultiStage Fatigue (MSF) model that admits different hierarchical microstructural features and their stereological information is used to predict the fatigue behavior of 17 different processed aluminum alloys: A000 series (A319, A356, A357, and A380), 2000 series (2024, 2055, 2099, 2198, 2297), 5000 series (5052, 5456), and 7000 series (7050, 7055, 7065, 7075, 7085, 7175). A single set of MSF model constants was validated for all of the aforementioned aluminum alloys, wherein the variation in fatigue life has been captured according to distinct microstructural features (pore size, pore nearest neighbor distance, porosity, particle size, grain size, crystallographic orientation, and misorientation) that differ arising from their native material processing method (casting, rolling, or extrusion). The MSF model’s total number of cycles distinguishes two different regimes: crack incubation (Inc) and Microstructurally Small Crack (long cracks are not considered herein). The previous MSF model in the literature had been associated with the pore size, pore nearest neighbor distance, porosity, particle size, and grain size, but a new contribution in this work is the contribution of the grain orientation and misorientation angles. We show that the MSF model now has the necessary and sufficient equations to predict the Process–Structure–Property relationships for aluminum alloys, allowing for expansion of fatigue prediction even beyond the alloys studied herein.

Appendices

Appendix 1: MultiStage Fatigue (MSF) Model

The following equations summarize the MSF model where the total number of cycles equals the summation of the incubation and MSC regimes:

$$ {{N}_{\text{total}}}={{N}_{\text{inc}}}+{{N}_{\text{MSC}}} $$

(2)

$$ {{C}_{\text{inc}}}N_{\text{inc}}^{\alpha}=\beta, $$

(3)

$$ \beta =\frac{\Delta \gamma_{ \hbox{max} }^{{{P}^{*}}}}{2}=Y{{\left[\left({{\varepsilon}_{\text{a}}}-{{\varepsilon}_{\text{th}}} \right) \right]}^{q}}\left(1+\psi z \right){{\left[\frac{{{d}_{\max}}}{{{d}_{\text{NND}}}*{{d}_{\text{g}}}} \right]}^{z1}}, \ \ \frac{l}{d}>{{\eta}_{ \lim }} $$

(4)

$$ \frac{l}{d}={{\eta}_{\lim}}\frac{{{\varepsilon}_{\text{a}}}-{{\varepsilon}_{\text{th}}}}{{{\varepsilon}_{\text{per}}}-{{\varepsilon}_{\text{th}}}},\,\,\frac{l}{d}\le {{\eta}_{\lim}} $$

(5)

$$ \frac{l}{d}=1-\left(1-{{\eta}_{\lim}} \right){{\left(\frac{{{\varepsilon}_{\text{per}}}}{{{\varepsilon}_{\text{a}}}} \right)}^{\text{r}}},\,\,\frac{l}{d}\ge {{\eta}_{\lim}} $$

(6)

$$ {{C}_{\text{inc}}}={{C}_{n}}+\frac{1}{0.7}\left(\frac{l}{d}-0.3 \right)\left({{C}_{m}}-{{C}_{n}} \right) $$

(7)

$$ {{C}_{n}}=\text{CNC}\left(1-R \right) $$

(8)

$$ {{\left(\frac{\text{d}a}{\text{d}N} \right)}_{\text{MSC}}}=\chi \left(\Delta \text{CTOD}-\Delta \text{CTO}{{\text{D}}_{\text{th}}} \right) $$

(9)

$$ \Delta \text{CTD}={{\left[\frac{{{d}_{\max}}}{{{d}_{\text{NND}}}*{{d}_{\text{g}}}} \right]}^{z1}}{{\left(\frac{\text{SF}}{0.5} \right)}^{z2}}\left(f\left(\phi \right){{C}_{11}}{{\left(\frac{\text{GS}}{\text{G}{{\text{S}}_ 0}} \right)}^{{{\omega}^\prime}}}{{\left(\frac{\text{MO}}{\text{M}{{\text{O}}_ 0}} \right)}^{{{\psi}^\prime}}}{{\left[\frac{U\Delta \widehat{\sigma}}{{{S}_{\text{ut}}}} \right]}^{\zeta}}a+{{C}_{1}}{{\left(\frac{\text{GS}}{G{{S}_{0}}} \right)}^{{{\omega}^\prime}}}{{\left(\frac{\text{MO}}{\text{M}{{\text{O}}_ 0}} \right)}^{{{\psi}^\prime}}}{{\left(\frac{\Delta\gamma_{ \hbox{max} }^{\text{p}}}{2} \right)}^{2}} \right) $$

(10)

For the incubation life, N_inc for a given material, a damage parameter, β, is equated to a modified Coffin–Manson law at the mesostructural scale where β is the nonlocal damage parameter around an inclusion determined by subscale simulations [46], and $C_{\text{inc}}$ and α are the linear and exponential coefficients in the modified Coffin–Manson law for incubation per Eq. (3). The choice for $C_{\text{inc}}$ and α parameters are based on the estimated number of cycles for incubation life and for these aluminum alloys was approximately 80% of the life. R is the load ratio, C_m is a model constant, and z is a variable related to the plastic zone size. The physical representation of the damage parameter is related to the local average maximum plastic shear strain amplitude next to an inclusion and is estimated by Eqs. (4) and (5). ${{\varepsilon}_{\text{a}}}$ is the remote applied strain amplitude, and ${{\varepsilon}_{\text{th}}}$ and ${{\varepsilon}_{\text{per}}}$ were introduced by McDowell et al. [17] and employed by Xue et al. [19, 47] to represent the strain threshold for damage incubation and the percolation limits for microplasticity, respectively. Although the strain threshold and the percolation limits for microplasticity were originally determined by mesomechanical finite element simulations [46], Xue et al. [19, 47] showed that the strain threshold is easily estimated by using the standard methods for estimating the endurance limit, where

$$ {{\varepsilon}_{\text{th}}}=\frac{0.29\text{Sut}}{E}. $$

(11)

Furthermore, the percolation limit can also be estimated, where

$$ {{\varepsilon}_{\text{per}}}=\frac{0.7\sigma_{y}^{\text{cyclic}}}{E}. $$

(12)

The model parameter, d, is the size of the pertinent inclusion that is responsible for incubating a fatigue crack, and l is the nominal linear dimension of the plastic zone size in front of the inclusion. The ratio $l/d$ is defined as the square root of the ratio of the plastic zone area over the inclusion area. The limiting ratio, ${ \lim }$, indicates the transition from proportional (constrained) micronotch root plasticity to nonlinear (unconstrained) micronotch root plasticity with respect to the applied strain amplitude. A limiting ratio ${{\eta}_{ \lim }}=0.3$ was found from subscale mesomechanical finite element simulations [46]. The parameter Y from Xue et al. [19] is correlated as

$$ Y=y_1+0.1(1+R)y_2 $$

(13)

where R is the load ratio, and y₁ and y₂ are the material constants. For completely reversed loading cases, $Y=y_1$. An experimentally observed structure–property relationship term was recently added [20] to the incubation damage parameter. This structure–property term

$$ \left[\frac{{d}_{\rm max}}{d_{\rm NND}*{d_g}}\right] $$

(14)

is a function of the maximum inclusion size (d_max), nearest neighbor distance (NND), grain size or dendrite cell size (d_g) depending on the lowest length scale that causes a boundary for dislocations.

The correlation of the plastic zone size is calculated using the nonlocal plastic shear strain with respect to the remote loading strain amplitude, and Eqs. (5) and (6) describe that behavior, where r is a shape constant for the transition to the limited plasticity.

For the MSC growth regime given by Eqs. (9) and (10), the growth for the fatigue crack is governed by the range of crack tip displacement, $\Delta \text{CTD}$, which is proportional to the crack length, and the nth power of the applied stress amplitude, in the HCF regime and to the macroscopic plastic shear strain range in the LCF regime. Here, χ is a constant for a given microstructure, and C_I and C_II are material-dependent parameters that capture the microstructural effects on MSC growth. A porosity related term is included in the high cycle fatigue regime, where

$$ f\left(\phi \right)=1+\omega \left\{1-\exp \left(-\frac{\phi}{2\bar{\phi}} \right) \right\} $$

(15)

and the porosity threshold, $\bar{\phi}$ is assumed as 0.0001. Also $\omega$ is a constant on the order of 2–10. The factor of 2–3 reduction in fatigue life observed for higher microporosity cast specimens relative to low microporosity specimens suggests a value of $\omega \ \approx 2$, based on ratio of incubation life to total life of about 1/3 for stress amplitudes in the range of HCF-transition regime, as suggested by the data of Shiozawa et al. [48]. For two different low porosity squeeze cast alloys in the HCF regime, Shiozawa et al. [48] measured the combined coefficient GC_II = 3.11 × 10⁻⁴ m/cycle for units of crack length in m and for a reference dendrite cell size of GS_o = 30 μm, assuming that in this case the microporosity is very low, i.e., $f\left({\bar{\varphi}} \right)\approx 1$. Hence, in this model both macroporosity and microporosity play a role in reducing the fatigue strength, the former through reduction of matrix fatigue ductility and the latter through reduced incubation lifetimes and larger initial crack sizes for the propagation analysis.

Based on the correlations of Shiozawa et al. [48] in the MSC regime, McDowell et al. [17] expressed the ΔCTD parameter in the HCF regime with the stress state dependence as the following,

$$ \Delta \hat{\sigma}\ =\ 2\theta \,{{\bar{\sigma}}_{a}}\ +\ \left(1\,-\,\theta \right)\Delta {{\sigma}_{1}} $$

(16)

is the range of the uniaxial equivalent stress, which is a linear combination of the von Mises [49] uniaxial effective stress amplitude

$$ {{\bar{\sigma}}_{a}}\left(=\sqrt{3/2\,\left(\Delta {{{{\sigma}^\prime}}_{ij}}/2 \right)\left(\Delta {{{{\sigma}^\prime}}_{ij}}/2 \right)} \right) $$

(17)

and the range of the maximum principal stress, $\Delta {{\sigma}_{1}}$; $\theta$ is a constant factor ($0\ \le \ \theta \ \le \ 1$) introduced by Hayhurst [50] to model combined stress state effects. Here, C_II is a coefficient intended to apply the HCF portion of the MSC regime for crack lengths ranging from a few microns to hundreds of microns and could even range up to the millimeter range as long as the crack tip cyclic plastic zone is substantially less than the GS, dendrite cell size in this case study. The factor U includes mean stress effects on propagation that are influenced strongly by particle interactions ahead of and in the wake of the crack; U = 1/(1 − R) for R < 0, U = 1 for R ≥ 0, where R is based on the maximum principal stress. Furthermore, da/dN is linear in crack length and does not follow LEFM either as revealed by the mesoscale finite element simulations [51].

The fatigue limit threshold in Eq. (10) is $\Delta {{\text{CTD}}_{\rm th}}$. Typically, the fatigue threshold was set equal to the Burger’s vector, $\Delta {\text{CTD}}_{\text{th}}=3.2*{{10}^{-4}}\upmu\text{m}$.

The MSF model requires a model to convert between applied stress and strain amplitudes. Currently, we assume a Ramberg–Osgood (RO) [52] model as shown in Eq. (18). The RO model is calibrated to stress and strain amplitude data taken at the half-life of our aluminum samples. The model was correlated to all of the data regardless of the applied strain ratio of the test, with the assumption that the R ratio had a negligible effect on the relationship between stress and strain amplitudes. The cyclic elastic modulus, strength parameter, and power parameter were all calculated from the RO model correlation,

$$ \sigma = E\varepsilon +{K}^\prime \varepsilon^{n^\prime} $$

(18)

The cyclic yield stress was also calculated from the RO model calibration. A Monte Carlo [53, 54] estimation of the yield was completed using the uncertainties of the RO parameters as inputs. The correlation matrix was calculated from the covariance matrix output by the correlation routine. The correlation matrix was in turn used to generate correlated random variables for the Monte Carlo method. As a result, we were able to calculate a yield stress with an associated uncertainty.

Appendix 2: Model Parameters

Materials Processing Reference Values

• ϕ = porosity threshold

• GS₀ = Reference grain size that applies across the same materials processing method

• MO₀ = Reference grain misorientation that applies across the same materials processing method

Hierarchical Length Scale Structures Exponents

• ξ = Exponent for the pore size related to incubation life

• ζ = Exponent governing the influence of the stress state

• z1 = Exponent governing the influence of particle size

• ω′ = Exponent governing the grain size effect on small crack growth rate

• ψ′ = Exponent governing the grain misorientation effects on the small crack growth rate

• z2 = Exponent governing the grain orientation (texture) effects on the small crack growth rate

Microstructural Data

• d_max = Maximum defect size

• d_NND = Defect nearest neighbor distance

• ϕ = Pore volume fraction.

• ΔCTD_th = Threshold for crack tip displacement.

• d_g = Grain size.

• MO = Grain misorientation.

• SF = Average Schmid Factor.

Mechanical Properties

• E = Young’s modulus.

• σ_y (cyclic) = Cyclic yield stress.

• S_ult = Ultimate stress from the monotonic stress–strain curve.

• K′ = Cyclic strength coefficient in the Ramberg–Osgood plasticity model.

• n′ = Cyclic strain hardening exponent in the Ramberg–Osgood plasticity model.

Incubation Material Constants

• CNC = Constant related to Coffin–Manson Law.

• C_m = Ductility coefficient in Coffin–Manson Law.

• α = Ductility exponent in the Coffin–Manson Law.

• q = Exponent in the remote strain to local plastic shear strain.

• y₁ = Constant relating the first part of part remote strain to local plastic shear strain.

• y₂ = Constant relating the second part of the remote strain to local plastic shear strain.

• ψ = Geometric factor of particles.

• r = Shape constant exponent describing the transition to limit plasticity in the plastic zone around an incipient crack.

Microstructurally Small Crack Material Constants

• ω = Pore effect coefficient related to high cycle fatigue.

• a_i = Initial crack size contribution.

• θ = Combination loading parameter for multi-axial stress states.

• C_I = Low cycle fatigue constant.

• C_II= High cycle fatigue constant.

• χ = Crack growth rate constant.

• a_f = Final crack size length.