Numerical Implementation and Macroscopic Tangent Moduli

The numerical implementation of the macroscopic and microscopic boundary value prob-lem in the framework of the Finite Eprob-lement Method, see Section 3, is discussed in the following, where the main attention is put on the variational formulation and the

deriva-n₁

−n1

n₂

−n2

(a)

(b)

Figure 5.5: (a) RVE of an artificial polycrystalline microstructure. (b) Periodic polycrys-talline unitcell with indication of opposite normal vectors.

tion of the algorithmic consistent macroscopic tangent moduli. Therein, the approach proposed in Schr¨oder [131] is followed. The weak form of linear momentum balance, Eq. 2.41, at the macroscale can be written as

G = Z

B0

δF : P dV

| {z }

G^int

−

Z

B0

δx · f dV + Z

∂B0

δx · t₀dA



| {z }

G^ext

= 0 (5.31)

where δF = Grad_X[δx]. In order to solve the nonlinear weak form using a Newton-Raphson scheme, the linearization is computed as

∆G^int = Z

B0

δF^T : A : ∆F dV , (5.32)

where the algorithmic consistent macroscopic moduli A is defined by A= ∂P

∂F . (5.33)

The moduli cannot be computed directly, since there is no explicit expression P (F ). An efficient algorithm for the computation is presented in Miehe et al. [96]. The system of equations resulting from the linearization

G + ∆G = δD^T(K∆D + R) = 0 → K∆D + R = 0 (5.34) has to be solved iteratively with respect to the global incremental displacement vector ∆D. For the treatment of the microscopic boundary value problem, the weak form of linear momentum balance reads

G = Z

B0

δ ˜F : P dV = 0 , (5.35)

neglecting volume acceleration and inertia terms. Here, the fluctuations of the displace-ment field on the microscale ˜w are discretized and the displacements can be obtained from x = F X + ˜w. Then, δ ˜F = Grad[δ ˜w] and ∆ ˜F = Grad[∆ ˜w]. The linearization of

which together with the approximation ˜F = B · ˜dleads to the system of equations G + ∆G = δ ˜D^T(K∆ ˜D+ L∆F + R) = 0 → K∆ ˜D+ L∆F + R = 0 , (5.37) Note that during the solution of the microscopic boundary value problem, the macro-scopic deformation gradient is constant, thus ∆F = 0. Then Eq. (5.37(2)) reduces to K∆ ˜D+ R = 0. After the microscopic boundary value problem has been iteratively solved and convergence is obtained such that R = 0, the computation of the algorithmic consis-tent macroscopic moduli A follows. Starting from the incremental relation of Eq. (5.21) at the macroscale and the incremental constitutive relation ∆P = A : ∆F , one obtains

∆P = 1

with the additive incremental decomposition ∆F = ∆F + ∆ ˜F. The increment of the gradient of the fluctuations is obtained from Eq. (5.37(2)) by

∆ ˜F = B∆ ˜D= −BK⁻¹L∆F , (5.40)

with ∆F being the current increment of the macroscopic deformation gradient, which in the macroscopic iteration not zero, and the equilibrium state in the microscopic boundary value problem, i.e. R = 0. Together with this relation, Eq. (5.39) becomes

∆P = 1

with the consistent algorithmic macroscopic moduli given by A= 1

The microscopic module is known from the solution of the microscopic boundary value problem, whereas strategies for an efficient computation of the second term of the macroscopic module are described in, Miehe et al. [96], Miehe and Koch [92] and Schr¨oder [131].

6 Statistically Similar Representative Volume Elements

With the aim to model microstructural properties of a material in the framework of the FE²-method, RVEs based on realistic microstructures often lead to enormous computa-tional expenses. The use of simplified RVEs, which are reduced in size as well as complexity of the inherent morphology, can lead to a reduction in computational costs. They can be constructed such that selected morphological properties in the simplified RVE and the real microstructure are similar. These structures are then referred to as statistically similar RVEs (SSRVEs).

In this chapter, the concept of SSRVEs will be outlined. The method has been presented for two-dimensional microstructures in Balzani et al. [7; 8; 10]

and Schr¨oder et al. [132], It has been extended to 3D microstructures in Balzani et al. [11] and Scheunemann et al. [128], analyzing alternative statistical measures. An SSRVE is governed by similarities with respect to statistical properties of the real microstructure morphology, while the SSRVE exhibits a smaller size and lower level of complexity in terms of microstructure morphology. The statistical similarity is en-forced by matching considered statistical measures, which is based on the ideas proposed in Povirk [111] for microstructures with circular inclusions. A least-square functional is minimized in an optimization process, which considers the difference of the statistical properties computed from the real microstructure and the SSRVE. If the mechanical re-sponse of the microheterogeneous materials is mainly governed by the phase contrast, the replication of morphological properties in the SSRVE then results in a similarity of the mechanical response compared to the real microstructure.

In the motivation of this work, a literature overview is given for reconstruction ap-proaches of real microstructures based on the minimization of least-square function-als as well as examples for the application of SSRVEs, see e.g. Rauch et al. [117], Ambrozinski et al.[2], Rauch et al. [118]. With regard to the ability of the SSRVE to represent the real microstructure in terms of morphology and mechanical behavior, the statistical measures used in the construction process play a crucial role. A comparison of measures for the construction of SSRVEs is presented in Balzani et al. [8]. Further measures are presented in Scheunemann et al. [128] and compared regarding their applicability in the construction of SSRVEs.

The need for considering microstructural information in the material modeling is often necessary to describe effects observed at a large scale in order to circumvent complex macroscopic constitutive laws. In order to reconstruct an SSRVE based on properties of a real microstructure, its morphology needs to be known. The measurement of real microstructures using microscopy techniques is a costly task and limited in terms of the size which can be captured. Consequently, different approaches to reconstruct mi-crostructures in order to, e.g., obtain a certain number of samples with similar properties as the real microstructure have been proposed. In the literature, different approaches for the construction of artificial microstructures based on a fitting of statistical measures can be found. Yeong and Torquato [176] proposed a technique for the reconstruction of random media based on a least-square fit of two-point probability function and lineal-path function. Similarly, in Kumar et al. [79] 2D microstructures of a DP steel are reconstructed by minimizing a least-square functional taking into account the two-point probability function and lineal-path function as statistical measures in a simulated

annealing process. The reconstructed artificial microstructures show similar distribution of stress as well as peak stresses as the related real microstructure. A Monte-Carlo approach is utilized in Baniassadi et al. [13] to reconstruct 3D microstructures based on a comparison of two-point cluster function and two-point correlation function.

Feng et al. [39] present a method for rapid construction of artificial microstructures considering first and second order statistical moments using fast fourier transformation.

(a) (b)

Figure 6.1: Random target microstructure of an inclusion-matrix microstructure and associated periodic microstructure with SSRVE as periodic unitcell. Taken from Balzani et al.[10].

In order to apply the method of SSRVEs, several premises have to be obeyed. The reduction of size of the SSRVE compared with the real microstructure is possible due to the periodic morphology of the SSRVE. As can be seen in Fig. 6.1, the periodicity of the SSRVE enables a construction of an infinitely large microstructure using the SSRVE as a periodic unitcell. Thus, it is assumed that the considered random microstructure may be represented by such a periodic microstructure. Furthermore, the macroscopic material behavior shall depend only on the microstructure morphology and the local constitutive laws, describing the microscopic behavior of each phase, which needs to be known.

In the following sections, an overview on selected statistical measures for the description of microstructures is presented. The construction method of SSRVEs is described in detail and candidates of SSRVEs are constructed for a dual-phase (DP) steel microstructure obtained from EBSD-FIB measurements, see Section 4. The constructed SSRVE candi-dates are compared to one another and their performance is analyzed in a comparison of mechanical response. Therein, virtual monotonic tests are applied (tension and shear tests). An analysis of microscale stresses in each phase is shown and the possibilities of substructures of a real microstructure as SSRVEs is discussed. Furthermore, other virtual mechanical tests are simulated compare the ability of the SSRVE to describe effects related to the yield behavior of the material.

6.1 Statistical Measures for Spatial Structures

In contrast to the frequent assumption of homogeneous materials, heterogeneity is com-monly observed in nature, with the material often being composed of multiple phases at

a small scale, referred to as microheterogeneity. The macroscopic material behavior and properties are a result of not only the properties of the individual constituents, but also the geometrical composition of the compound material. The morphological quantifica-tion is of large interests in many disciplines and can be realized using the models and methods developed in the field of stochastic geometry. With the help of these methods, three-dimensional microstructures can be characterized quantitatively in terms of their morphology and conclusions can be drawn upon the relation of morphological properties and the material behavior. These methods are given by e.g. measures characterizing parti-cle systems, planar objects and three dimensional structures. In the following, a selection of statistical measures which are able to characterize a two-phase material with inclusions embedded in a matrix, are discussed. The consideration of two-phase materials enables a description of the morphology of one phase with the help of statistical measures and obtain a description of the second phase using the negative image. In the case of the two-phase microstructure of DP steel, the described two-phase is the martensitic inclusion two-phase.

An overview on statistical descriptors can be found in, e.g., Ohser and M¨ucklich [103]

and Torquato [165].