On the accuracy of spectral solvers for micromechanics based fatigue modeling

Abstract

A framework based on FFT is proposed for micromechanical fatigue modeling of polycrystals as alternative to the Finite Element method (FEM). The variational FFT approach (de Geus et al. in Comput Methods Appl Mech Eng 318:412–430, 2017; Zeman et al. in Int J Numer Methods Eng 110:903–926, 2017) is used with a crystal plasticity model for the cyclic behavior of the grains that is introduced through a FEM material subroutine, in particular an Abaqus umat. The framework also includes an alternative projection operator based on discrete differentiation to improve the microfield fidelity allowing to include second phases. The accuracy and efficiency of the FFT framework for microstructure sensitive fatigue prediction are assessed by comparing with FEM. The macroscopic cyclic response of a polycrystal obtained with both methods were indistinguishable, irrespective of the number of cycles. The microscopic fields presented small differences that decrease when using the discrete projection operator, which indeed allowed simulating accurately microstructures containing very stiff particles. Finally, the maximum differences in the fatigue life estimation from the microfields respect FEM were around 15%. In summary, this framework allows predicting fatigue life with a similar accuracy than using FEM but strongly reducing the computational cost.

Appendix: Coupling with Abaqus material user subroutines

In this appendix, the adaptation of the Abaqus user material subroutine umat to the FFT framework will be presented for a generic non-linear material with internal variables and under finite deformations.

The FFT method needs to evaluate at each iteration the constitutive equation (Eq. 2) for each point as function of the deformation gradient at current and previous time increments, $\mathbf {F}^{t_k}$ and $\mathbf {F}^{t_{k-1}}$ respectively, the last value of the internal variables, $\varvec{\alpha }^{t_{k-1}}$, and the time step $\delta t$. These values are also inputs of a umat subroutine and are passed from the FFT code to the umat subroutine through the variables DFGRD1,DFGRD0,STATEV and DTIME respectively. After the evaluation, the FFT code needs to recover from the constitutive model the first Piola Kirchhoff stress $\mathbf {P}^{t_k}$, the value of the state variables at $t_k$ and the material tangent $\mathbb {K}$. The umat subroutine provides the Cauchy stress at time $t_k$ ($\varvec{\sigma }$) in the variable STRESS so, the first Piola-Kirchhoff stress is computed from this value and $\mathbf {F}=\mathbf {F}^{t_k}$ using

$$\begin{aligned} \mathbf {P}=J\varvec{\sigma } \mathbf {F}^{-T} \end{aligned}$$

(A.1)

with $J=\det (\mathbf {F})$.

In the case of $\mathbb {K}$, an exact transformation of the consistent tangent matrix defined in the umat (variable DDSDDE) to the material tangent used by the FFT code is fundamental to preserve in FFT the convergence rate obtained with the user material in a finite element simulation. The material Jacobian that should be defined in a user subroutine at finite strain is the tangent modulus tensor for the Jaumann rate of the Kirchhoff stress, $\mathbb {C}^{ab}$, a fourth order tensor defined as

$$\begin{aligned} \overset{\nabla }{\varvec{\tau }}= \dot{\varvec{\tau }} - \mathbf {w} \cdot \varvec{\tau }-\varvec{\tau }\cdot \mathbf {w}^T= J \ \mathbb {C}^{ab} : \mathbf {d} \end{aligned}$$

(A.2)

where $\varvec{\tau }$ is the Kirchhoff stress, $\overset{\nabla }{}$ corresponds to the Jaumann rate and $\mathbf {w}$ and $\mathbf {d}$ are the spin tensor and stretch tensor respectively. In the FFT framework, the material tangent needed to define the linear operator of the equilibrium in each iteration (Eq. 16) is defined as

$$\begin{aligned} \dot{\mathbf {P}} = \mathbb {K}:\dot{\mathbf {F}} \end{aligned}$$

(A.3)

and the objective then is to derive an explicit expression relating both tensors, $\mathbb {C}^{ab}$ and $\mathbb {K}$.

Combining the definition of the material tangent in FFT, Eq. (A.3), and Eq. (A.1) it is obtained

$$\begin{aligned} \mathbb {K}=\frac{\partial \mathbf {P}}{\partial \mathbf {F}}= \frac{\partial \left( J\varvec{\sigma }\cdot \mathbf {F}^{-T}\right) }{\partial \mathbf {F}}= \frac{\partial \left( \varvec{\tau }\cdot \mathbf {F}^{-T}\right) }{\partial \mathbf {F}}. \end{aligned}$$

(A.4)

Expressing the previous expression (Eq. A.4) in index notation and expanding the derivative of the product in two terms it is obtained

$$\begin{aligned} K_{ijkl}=\frac{\partial P_{ij}}{\partial F_{kl}}= \frac{\partial \left( \tau _{ip} F^{-1}_{jp}\right) }{\partial F_{kl}}= \frac{\partial \tau _{ip}}{\partial F_{kl}} F^{-1}_{jp}+ \tau _{ip}\frac{\partial F^{-1}_{jp}}{\partial F_{kl}}. \end{aligned}$$

(A.5)

The second term of Eq. (A.5), the derivative of the inverse of a tensor with respect to itself, is given by A.6.

$$\begin{aligned} \dfrac{\partial F^{-1}_{jp}}{\partial F_{kl}}=- F^{-1}_{lp}F^{-1}_{jk} \end{aligned}$$

(A.6)

To obtain first term of Eq. (A.5), the expression in derivatives of Eq. (A.2) is multiplied by a small time increment yielding into A.7.

$$\begin{aligned} \delta \varvec{\tau } - \delta \mathbf {w} \cdot \varvec{\tau }-\varvec{\tau }\cdot \delta \mathbf {w}^T= J \ \mathbb {C}^{ab} : \delta \mathbf {d} \end{aligned}$$

(A.7)

and reordering the equation, the Abaqus tangent can be written as

$$\begin{aligned} \delta \varvec{\tau } = J \ \mathbb {C}^{ab} : \delta \mathbf {d} +\delta \mathbf {w} \cdot \varvec{\tau } +\varvec{\tau }\cdot \delta \mathbf {w}^T \end{aligned}$$

(A.8)

where $\delta \mathbf {d}$ and $\delta \mathbf {w}$ are obtained as function of $\mathbf {F}$ and $\delta \mathbf {F}$ by

$$\begin{aligned} \delta \mathbf {d}= \frac{1}{2}\left[ \delta \mathbf {F}\cdot \mathbf {F}^{-1}+\left( \delta \mathbf {F}\cdot \mathbf {F}^{-1}\right) ^{T}\right] \end{aligned}$$

(A.9)

and

$$\begin{aligned} \delta \mathbf {w}= \frac{1}{2}\left[ \delta \mathbf {F}\cdot \mathbf {F}^{-1}-\left( \delta \mathbf {F}\cdot \mathbf {F}^{-1}\right) ^{T}\right] . \end{aligned}$$

(A.10)

Replacing these expressions Eqs. (A.9) and (A.10) into Eq. (A.8), it is obtained

$$\begin{aligned} \delta \varvec{\tau }= & {} J \ \mathbb {C}^{ab} : \frac{1}{2}\left[ \delta \mathbf {F}\cdot \mathbf {F}^{-1} + \left( \delta \mathbf {F}\cdot \mathbf {F}^{-1}\right) ^{T}\right] \nonumber \\&+\,\frac{1}{2}\left[ \delta \mathbf {F}\cdot \mathbf {F}^{-1}-\left( \delta \mathbf {F}\cdot \mathbf {F}^{-1}\right) ^{T}\right] \cdot \varvec{\tau } \nonumber \\&+\,\varvec{\tau }\cdot \frac{1}{2}\left[ \delta \mathbf {F}\cdot \mathbf {F}^{-1}-\left( \delta \mathbf {F}\cdot \mathbf {F}^{-1}\right) ^{T}\right] ^T. \end{aligned}$$

(A.11)

Now, the Kirchhoff stress is also linearized respect the perturbation of the deformation gradient

$$\begin{aligned} \delta \varvec{\tau } = \frac{\delta \varvec{\tau }}{\delta \mathbf {F}} : \delta \mathbf {F} \text { in index } \delta \tau _{ip} = \frac{\partial \tau _{ip}}{\partial F_{kl}} \delta F_{kl}. \end{aligned}$$

(A.12)

If Eq. A.11 is written in index notation, the resulting expression reads as

$$\begin{aligned} \delta \tau _{ip}= & {} \frac{J}{2} C^{ab}_{ipkm} \delta F _{kl} F^{-1}_{lm} + \frac{J}{2} C^{ab}_{ipmk} F^{-1}_{lm} \delta F _{kl}\nonumber \\&+\, \frac{1}{2} I_{ipkq} \delta F_{kl} F^{-1}_{lm} \tau _{mq} - \frac{1}{2} I_{ipmq} F^{-1}_{lm} \delta F_{kl} \tau _{kq} \nonumber \\&+\, \frac{1}{2} I_{ipqk} \tau _{qm} F^{-1}_{lm} \delta F_{kl} - \frac{1}{2} I_{ipqm} \tau _{qk} F^{-1}_{lm} \delta F_{kl} \end{aligned}$$

(A.13)

and comparing the terms of the last two Eqs. (A.12) and (A.13), considering the minor symmetries of $\mathbb {C}^{ab}$ and the symmetry of $\varvec{\tau }$, the resulting expression is:

$$\begin{aligned} \frac{\partial \tau _{ip}}{\partial F_{kl}}= & {} J \ C^{ab}_{ipkm} F^{-1}_{lm} \nonumber \\&+\,\frac{1}{2} \delta _{ik} \delta _{pq} F^{-1}_{lm} \tau _{mq} - \frac{1}{2} \delta _{im} \delta _{pq} F^{-1}_{lm} \tau _{kq} \nonumber \\&+\,\frac{1}{2} \delta _{iq} \delta _{pk} F^{-1}_{lm} \tau _{qm} - \frac{1}{2} \delta _{iq} \delta _{pm} F^{-1}_{lm} \tau _{qk}, \end{aligned}$$

(A.14)

which can be simplified to

$$\begin{aligned} \frac{\partial \tau _{ip}}{\partial F_{kl}}= & {} J \ C^{ab}_{ipkm} F^{-1}_{lm} \nonumber \\&+\,\frac{1}{2} \delta _{ik} F^{-1}_{lm} \tau _{mp} - \frac{1}{2} F^{-1}_{li} \tau _{kp} \nonumber \\&+\,\frac{1}{2} \delta _{pk} F^{-1}_{lm} \tau _{im} - \frac{1}{2} F^{-1}_{lp} \tau _{ik}. \end{aligned}$$

(A.15)

Finally, introducing Eq. (A.15) in the definition of the material tangent (Eq. A.5) the final expression for the material tangent used in the FFT approach as function of the Abaqus tangent is given by

$$\begin{aligned} K_{ijkl}= & {} \left( J \ C^{ab}_{ipkm} F^{-1}_{lm} + \frac{1}{2} \delta _{ik} F^{-1}_{lm} \tau _{mp} - \frac{1}{2} F^{-1}_{li} \tau _{kp} \right. \nonumber \\&\left. +\, \frac{1}{2} \delta _{pk} F^{-1}_{lm} \tau _{im}-\frac{1}{2} F^{-1}_{lp} \tau _{ik}\right) F^{-1}_{jp}- \tau _{ip} F^{-1}_{lp} F^{-1}_{jk}. \end{aligned}$$

(A.16)