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Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method
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Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method✩ Y.P. Chen∗, Y.Y. Cai Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
A semi-implicit elastic/crystalline viscoplastic finite element (FE) method based on a “crys-
Received 22 June 2016
tallographic homogenization” approach is formulated for a multi-scale analysis. In the for-
Revised 17 October 2016
mulation, the asymptotic series expansion is introduced to define the displacement in the
Accepted 18 October 2016
micro-continuum. This homogenization FE analysis is aimed at predicting the plastic defor-
Available online xxx
mation induced texture evolution of polycrystalline materials, the constituent microstructure of which is represented by an assembly of single crystal grains. The rate dependent
Keywords:
crystal plasticity model is adopted for the description of microstructures. Their displace-
Homogenization
ments are decomposed into two parts: the homogenized deformation defined in the macro-
Local periodicity
continuum and the perturbed one in the micro-continuum. This multi-scale formulation
Crystalline plasticity
makes it possible to carry out an alternative transition from a representative micro-structure
Texture
to the macro-continuum. This homogenization procedure satisfies both the compatibility
Finite element
and the equilibrium in the micro-structure. This developed code is applied to predict the texture evolution, and its performance is demonstrated by the numerical examples of texture evolution of FCC polycrystalline metals. © 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics.
1.
Introduction
Polycrystalline solid materials consisting of a multitude of individual crystallites or grains after large plastic deformation generally demonstrate a certain degree of anisotropy in material properties, which is a natural reflection of the crystallographic texture induced by the reorientation of crystal lattices towards a preferential distribution orientation. The experimental determination, interpretation [1, 2] and the numerical simulation for texture analysis have been attracting the attention of researchers in the field of materials science and applied
✩ ∗
Project supported by NSFC (11272129, 11472114, 51175202). Corresponding author. E-mail address: [email protected] (Y.P. Chen).
mechanics [3]. On the one hand, recent advances in experiment techniques, such as EBSD [4], have made it possible to measure the texture more efficiently and accurately than the traditional X-ray diffraction method does; on the other hand, the rapid development of computer technology promises an unprecedented means for the large scale numerical simulation of materials deformation process and design, which is, however, essentially dependent on the constitutive model and the numerical techniques. Polycrystal plasticity models [5–9] were employed to predict the texture evolution induced by plastic deformation and the corresponding plastic anisotropy. In these simulations, the assumption is made that the microstructure of polycrystals is represented by an aggregate of microscopic crystals, whose plastic deformation is brought about by the shearing along crystallographic planes, and whose individual responses, on
http://dx.doi.org/10.1016/j.camss.2016.10.002 0894-9166/© 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics. Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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average, determine the overall macroscopic response of polycrystals. Two ingredients are indispensable for the establishment of these models: one is the constitutive equations used to describe the mechanical behaviors of a single crystal; the other is the mean field hypothesis or averaging schemes that relate the overall response of polycrystals to that of the constituent single crystals by a micro-macro-transition on the basis of homogenization procedures. The commonly adopted averaging schemes include the Taylor-type model, in which each crystal is assumed to experience identical macroscopic deformation and the stress of polycrystal aggregate is calculated by averaging the crystal stress, the equilibrium-based schemes require the same stress state for each grain, and the aggregate deformation is obtained by averaging the crystal deformation. In addition to these averaging procedures, Miehe [10,11] established two different homogenization schemes which attached the macro-continuum locally with a polycrystalline micro-structure made up of single crystal grains. The boundary of the micro-structure was specified by the local deformation of the macro-continuum with three alternative deformation constraints: (i) zero fluctuation in the domain (Taylor-type assumption), (ii) zero fluctuation on the boundary, and (iii) periodic fluctuation on the boundary, making improvements to the Taylor-type models. It seems that these two approaches were the first to have applied homogenization method to the prediction of texture evolution and strain localization in crystalline materials. Homogenization based on the multi-scale asymptotic series expansion of field variables has been utilized as an effective numerical technique over the last three decades to describe the mechanical behaviors of real heterogeneous materials by taking into account the length scale difference between macrostructures and individual microstructure components [12,13]. The advantage of this method lies in the fact that it is a rigorous mathematical theory and is capable of characterizing equivalent mechanical properties of composite materials and at the same time determining their dependence on different constituent components. It has been employed successfully to solve the problems of both finite elasticity [14] and elasto-plasticity [15]. In the homogenization theory, it is generally assumed that the composite material is locally formed by the spatial repetition of very small microstructures, i.e. the microscopic cells, when compared with the overall macroscopic dimensions of the structure of interest. In other words, it is assumed that the materials properties are periodic functions of the microscopic variables, where the period is very small compared with the macroscopic variables. This assumption enables the computation of equivalent material properties by a limiting process when the size of microscopic cell is reduced to zero. It can also provide a reasonable solution for some problems where the experimental data are not available, or for which only bounds for the equivalent material contents can be found by other theories. The basic theory of homogenization method can be found in the papers published in 1970s and early 1980s, for instance, by Babuska [16], SanchezPalencia [17], Bakhvalov and Panasenko [18] and many other applied mathematicians. Guedes and Kikuchi [19] developed a new formulation of this method in a weak form and provided a guideline for numerical analysis.
The purpose of the present paper is to establish a homogenization framework based on the two-scale asymptotic expansion of field variables to simulate the texture evolution in polycrystalline materials. A recapitulation of crystalline constitutive equations for characterizing single crystal grains is first presented and then followed by the derivation of governing equations for solving homogenized macro-deformation and characteristic displacements for micro-models to evaluate the homogenized material properties, respectively, from the virtual power principle. The proposed formulation is implemented in the updated Lagrangian finite element form and the performance of the code is demonstrated by the numerical examples of texture evolution in polycrystalline materials.
2.
Formulation of the problem
We consider a general infinite polycrystalline body, ε , in two dimensional space, as seen in Fig. 1. It is assumed that the material is formed by a spatial repetition of a microstructure at least locally, and therefore, the microstructure is usually called a unit cell. The region, Y, of the unit cell (microstructure) is made up of an aggregate of well defined crystal grains and is very small compared with the overall region ε by an order of ε 1, which also represents the reciprocal order of the repetition. In order to describe the effects of heterogeneity in the microstructures, we attach ε as a superscript to all the variables in the formulation when it is essential. We also introduce both microscopic and macroscopic coordinate systems, so that physical quantities are represented by two different length scales: one is x representing the macroscopic region, ; and the other is y = x/ε standing for the microscopic one, Y. The summation convention is used and the dot over a variable denotes material time derivative.
2.1.
Kinematics of crystal plasticity
We recapitulate the rate-dependent crystal plasticity constitutive model to describe the elasto-plastic deformation of constituent single crystals (Fig. 2). For details, we refer to the work of Asaro and Needleman [7]. The total deformation of a crystallite is assumed to be the result of two distinct physical mechanisms: crystallographic slip due to dislocation motion on the active slip systems, and elastic lattice distortion. FCC crystals are considered to have the usual {111}110 slip systems, where the slip planes are the {111} crystallographic planes with normals m, and the 110 directions are the shear directions with slip vectors s. Plastic deformation of the crystal is understood to occur as a set of simple plastic shear along various slip systems, having the lattice and the slip systems’ vectors (s(α) , m(α) ) not only essentially undistorted, but also unrotated (the brackets in the superscripts (α) indicate that α is not a tensor index and ranges from one to the number of slip systems). Next, the materials and lattice are considered to deform elastically and rotate rigidly from the physically deformed state to the current configuration. Accordingly, the decomposition for the deformation gradient tensor is Fi j = Fik∗ Fk j p
(1)
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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Fig. 1. – Schematic representation of macroscopic continuum and the assumed periodic unit cell at microscopic scale.
Fig. 2. – Multiplicative decomposition of the deformation gradient tensor.
where F p consists solely of crystallographic slipping along the specific slip systems and F ∗ arises from the stretching and rotation of the crystal lattice. From Eq. (1), the spatial gradient of velocity can be written as: −1 Li j = F˙im Fm = L∗i j + Li j j p
(2)
where ∗ ∗−1 L∗i j = F˙im Fm j ,
Li j = Fik∗ F˙km Fmn Fn∗−1 j p
p
p−1
(3)
Taking symmetric and anti-symmetric parts of the above relations yields the elastic and plastic strain rates D∗ and Dp , the plastic spin Wp , and W∗ associated with the rigid lattice rotation: Di j = D∗i j + Di j ,
Wi j = Wi∗j + Wi j
p
p
(4)
Since s(α) and m(α) are regarded as lattice vectors, they are stretched and rotated as follows: ∗(α)
si
(α)
= Fi∗j s j ,
∗(α)
mi
(α)
= m j Fi∗−1 j
(5)
The vectors s∗(α) and m∗(α) , orthogonal since s(α) and m(α) are, characterize a particular slip system in the current state,
and their evolution is governed by ∗(α)
s˙i
∗(α)
= L∗i j s j
∗(α)
˙i ,m
∗(α)
= L∗ji m j
(6)
By introducing the slip system, α, the symmetric and antisymmetric tensors are respectively 1 ∗(α) ∗(α) (α) ∗(α) ∗(α) Pi j = si m j + mi s j , 2 1 ∗(α) ∗(α) (α) ∗(α) ∗(α) Wi j = (7) s m j − mi s j , 2 i and the plastic strain rate and spin for the crystal can be written as p
Di j =
α
(α)
Pi j γ˙ (α) ,
p
Wi j =
α
(α)
Wi j γ˙ (α)
(8)
where γ˙ (α) is the shear rate on the slip system α. The elastic part of the rate of deformation De and the substructure spin W R can be obtained as follows: p
Dei j = Di j − Di j ,
2.2.
p
Wiej = Wi j − Wi j
(9)
Constitutive equations for single crystals
It is generally assumed that elasticity of grains is not affected by the plastic deformation of crystallographic slip, so the co-
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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rotational rate of Cauchy stress and its Jaumann rate are obtained as follows: σˆ iej = Ciejkl Dekl = σ˙ i j − Wike σk j + σikWke j ,
(10)
σˆ i j = σ˙ i j − Wik σk j + σikWk j
(11)
⎛
1 ⎜ C = ⎝q c qc ⎛ qc ⎜ L2 = ⎝ ql ql
⎞ ⎛ ⎞ qc qv ql ql ⎟ ⎜ ⎟ q c ⎠, L 1 = ⎝ q l qc q l ⎠, 1 ql ql qv ⎞ ⎛ ⎞ ql qv ql ql ⎟ ⎜ ⎟ ql ⎠, L3 = ⎝ ql qv q l ⎠. qv ql ql qc
qc 1 qc ql qv ql
(18)
From Eqs. (10) and (11) we have
σˆ iej − σˆ i j =
(α)
βi j γ˙ (α)
(12)
where (α)
(α)
(α)
βi j = Wik σk j − σikWk j
2.3.
(13)
Viscoplastic shear strain rate
The crystalline visco-plastic shear strain rate γ˙ (α) depends on the Schmid resolved shear stress (SRSS) and the hardening evolution of crystal. In general, two kinds of models are available for the description of crystal responses to external loadings: one is the rate independent model, in which the viscoplastic shear strain is obtained by an iterative procedure employing SVD technique [20] or by linear programming [21]; the other is the rate-dependent model, which relates the shear strain rate γ˙ (a ) directly to the SRSS, thus avoiding the abovementioned iteration in the solution. The latter is adopted as follows [6]: γ˙ (α) = a˙ (α)
τ (α) 1/m−1 g(α) g(α)
τ (α)
(14)
where the function denotes the reference shear stress, which describes the current strain hardening state of the crystal, a˙ (α) is the reference shear rate, and m is the material rate sensitivity with the case of m = 0 corresponding to the rateindependent models. The evolution of g(α) is determined by the following hardening evolution equation:
hαβ |γ˙ (β ) |
2.4.
(15)
Tangential modulus method
When the strain rate sensitivity m is small, a large shear strain rate γ˙ (a ) may be obtained from Eq. (14), thus bringing about the problem of divergence. The solution to it is by making use of the following tangential modulus method proposed by Peirce [6]:
γ (α) = [(1 − θ )γ˙ (α) (t ) + θ γ˙ (α) (t + t )] t
g(α) = g˙ (α) t yield (α)
Nαβ γ (β ) = (γ˙t
(β )
(α) γ˙t
= a˙
(α)
(α) Qi j
=
dτ ( γ ) dτ ( γ ) + (1 − q ) δαβ dγ dγ
τ (γ ) = τ0
n h0 γ +1 n τ0
(16)
(α)
θ t γ˙t
where q = 1 for coplanar slip systems and q = 1 to 1.4 for noncoplanar slip systems; and the matrix qαβ is introduced as follows to describe the self and latent hardening: ⎛ ⎞ C L1 L2 L3 ⎜L C L3 L2 ⎟ ⎜ 1 ⎟ qαβ = ⎜ ⎟ ⎝L2 L3 C L1 ⎠ L3 L2 L1 C
(21a)
(α)
Ri j
mτ (α)
Nαβ = δαβ +
(21b)
(α)
θ t γ˙t m
⎤ ⎡ (α) (β ) Ri j Pi j hαβ ( β ) ⎣ ⎦ + sgn(τ ) τ (α) g(α)
(22)
Denoting the inverse of Nαβ by Mαβ , one has the following: (α) (α)
γ = ( f˙ (a ) + Fi j Di j ) t
(23)
where f˙ (α) =
(17)
(20)
τ (α) 1/m−1 g(α) g(α)
hαβ = qαβ
(α)
+ Qi j Di j ) t
τ (α)
(β )
where hαβ are hardening moduli. The definition of a proper and realistic hardening evolution law has been a problem of pivotal importance in crystalline plasticity, and a number of such hardening evolution laws have been proposed. In the present investigation, it is expressed as follows:
(19)
The expansion of Eq. (14) into Taylor series, its substitu(α) tion into Eq. (19) and use of τ˙ (α) = Ri j Dei j , τ (α) = τ˙ (α) t, and
g(α)
g˙ (α) =
The parameters in qαβ proposed by Zhou et al. [22] are employed for the coplanar or collinear slip systems, with qc = 1. For the slip systems which have mutually perpendicular Burgers vectors, qv = 1.2. For others, ql = 1.3. γ is the accumulated shear strain over all the slip systems. τ0 , h0 and n are the initial critical resolved shear stress, the initial hardening ratio and the strain hardening exponent, respectively, which are determined in the same procedure as that in Nakamachi et al. [8].
(β )
(β )
(α)
Mαβ γ˙t , Fi j =
Mαβ Q (β )
(24)
β
Substituting Eq. (23) into Eq. (12), we obtain the constitutive equation as follows: σˆ i j = Ciejkl Dkl −
α
(α) (α) Ri j ( f˙ (α) + Fmn Dmn ) = C¯ivjkl Dkl −
where C¯ivjkl = Ciejkl −
α
α
(α) Ri j f˙ (α) (25)
(α) (α)
Ri j Fkl .
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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By substituting Eq. (33) into Eq. (32), the following equation is derived:
3. Principle of virtual velocity in updated Lagrangian form
In the updated Lagrangian formulation, by choosing the reference configuration at time t, we have
P˙i j δLi j dV =
F = I,
St
t˙i δvi dS
L = F˙ F −1 = F˙ ,
(26)
τ=σ=P
(27)
where P is the first Piola–Kirchhoff stress (non-symmetric), τ is the second Piola–Kirchhoff stress, and σ is the Cauchy stress. The relationship between the material time derivative of τ and P is written as: τ = PF T ,
τ˙ = P˙F T + PF˙ T = P˙ + σLT
(28)
(Ci jkl − Hi jkl )Lkl δLi j dV =
S
t˙i δ v˙ i dS +
N V α=1
(α) Ri j f˙ (α) δLi j dV (34)
where Hi jkl = 12 (δki σ jl − δli σ jk − δk j σil − δl j σik ). Following the general procedure of two-scale asymptotic series expansion, the primary variable can be written as: u˙ ε (x/ε ) = u˙ 0 (x ) + ε u˙ 1 (x, y ) + ε 2 u˙ 2 (x, y ) + · · ·
(35)
in which each term is Y-periodic, namely periodic with respect to y within a unit cell. Noting that the differentiation of a certain Y-periodic function Φ ε (x) is performed by ∇x · Φ ε ( x ) = ∇ x · Φ ε ( x, y ) +
1 ∇y · Φ ε ( x, y ) ε
(36)
So the component of velocity gradient is also written as P˙ = τ˙ − σLT
(29)
By using Eqs. (28) and (29) we obtain the following relationship between the Jaumann derivative of τ and the material time derivative of P, P˙ = τˆ − Dσ − σD + σLT
St
t˙i δvi dS
(31)
In this section, we shall separate the micro- and macroscopic equations and then derive the homogenization formulae. These are accomplished by taking the limit of ε to zero and employing the integral formula
We assume J = 1 during the elasto-plastic deformaton, thus σˆ = τ. ˆ Also on account of the symmetry of rate deformation tensor D and Cauchy stress tensor σ, the principle of virtual velocity in the updated Lagrangian formulation is in the following form:
{(σˆ i j − 2σik Dk j )δDi j + σ jk Lik δLi j }dV =
St
t˙i δvi dS
(32)
α=1
(α) Ri j f˙ (α)
ε→0 ε
Φ(x, x/ε )d =
Y
Ci jkl
∂ u˙ 0k ∂ xl
(33)
(α) (α) e in which Ci jkl = Ciejkl − N α=1 Ri j Fkl , where Ci jkl is elasticity modulus of the 4th order and N the total number of slip systems.
1 |Y |
Y
Φ(x, y )dyd
(38)
where Φ ε (x) is a Y-perodic function and the integral sign (1/|Y|)∫Y dy indicates the volume average of function Φ(x, y) within the unit cell. The proof is omitted here. Substituting Eq. (37) into Eq. (31) and applying averaging theorem yield
=
We shall derive the homogenized equations by utilizing the method of two-scale asymptotic series expansion and the crystalline constitutive equation in Section 2, which is rewritten here in the following component form: N
Homogenization
lim
1 |Y |
4. Asymptotic series expansion and constitutive equations
σˆ i j = Ci jkl dkl −
(37)
5.
(τˆi j − Dik σk j − σik Dk j + σik Lk j )δLi j dV =
∂ u˙ 0l ∂ u˙ 1l ∂ u˙ 1l ∂ u˙ 2l ∂ u˙ l (x/ε ) = = + +ε + +··· ∂ xm ∂ xm ∂ ym ∂ xm ∂ ym
(30)
Substituting Eq. (30) into Eq. (26) yields
Lεlm (x/ε )
t˙i δ u˙ 0i d +
+
1 |Y |
∂ u˙ 1k
·
∂ yl
∂δ u˙ 0i ∂x j
N Y α=1
+
(α) Ri j f˙ (α)
∂δ u˙ 1i
∂y j
∂δ u˙ 0i ∂x j
dYd +
∂δ u˙ 1i ∂y j
dYd (39)
where Ci jkl = Ci jkl − Hi jkl . From Eq. (39), we obtain the following two governing equations for microscopic and macroscopic problems, respectively, as an approximation of the first order:
1 |Y |
Y
Ci jkl
∂ u˙ 0k ∂ xl
=
+
t˙i δ u˙ 0i d +
∂ u˙ 1k ∂ yl
1 |Y |
·
∂δ u˙ 0i
N Y α=1
∂x j
dYd
(α) Ri j f˙ (α)
∂δ u˙ 0i ∂x j
dYd, ∀δ u˙ 0i (x )
(40)
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1 |Y |
Y
=
Ci jkl
∂ u˙ 0k ∂ xl
+
∂ u˙ 1k
∂ yl
·
∂δ u˙ 1i ∂y j
which is to be solved for the solutions of characteristic displacement fields χimn and φ i , respectively,
dYd
N ∂δ u˙ 1i 1 (α) Ri j f˙ (α) dYd, |Y | Y ∂x j α=1
∀ δ u˙ 1i (x, y )
(41)
u˙ 1i = −χimn
∂ u˙ 0m + φi ∂ xn
(42)
so ∂ u˙ 1i ∂y j
=−
∂χimn ∂y j
·
∂ φi ∂ u˙ 0m + ∂xn ∂y j
(43)
where the variables χ mn and φ are generally referred to as the characteristic deformation, which are the responses to the increments of the corresponding components of displacement gradient tensor. Therefore, there are ten independent functions, among which no symmetry with respect to mn in χimn may be found. It is also noted that each characteristic deformation is a function of x and y, but not in the rate form. Substituting Eqs. (42) and (43) into Eq. (40) yields ∂χ mn ∂ u˙ 0m ∂δ u˙ 0i 1 Ci jmn − Ci jkl k · · dYd ∂ yl ∂ xn ∂x j |Y | Y N 1 ∂ ϕk (α) ˙ (α) 0 ˙ ti δ u˙ i d + Ri j f − Ci jkl = ∂ yl |Y | Y α=1
·
∂δ u˙ 0i ∂x j
dYd,
∀δ u˙ 0i
(44)
Let CiHjmn
1 = |Y |
∂χ mn C i jmn − C i jkl k dY ∂ yl Y
(45)
and σiHj
N 1 ∂ϕ (α) = Ri j f˙ (α) − Ci jkl k dY . |Y | Y ∂ yl
(46)
α=1
∂χ mn ∂ u˙ 0m ∂δ u˙ 1i C i jmn − Ci jkl k · · dYd ∂ y ∂ xn ∂y j Y l N ∂δ u˙ 1i 1 ∂ϕ (α) Ri j f˙ (α) − Ci jkl k · dYd, | | Y ∂ y ∂x j Y l
1 | | Y =
α=1
C i jmn
∂δ u˙ 1i ∂y j
dY =
Y
C i jkl
∂χkmn ∂ yl
·
∂δ u˙ 1i ∂y j
dY ,
∀ δu1i
(48)
and N Y
α=1
∂δ u˙ 1i ∂ ϕ ∂δ u˙ 1i (α) Ri j f˙ (α) · = C i jkl k · dY , ∂y j ∂ yl ∂y j Y
∀δu1i
(49)
Now three governing equations have been derived, namely, boundary value problems, which are Eq. (44) for the macroscopic deformation and Eqs. (48) and (49) for characteristic displacement fields. Once the solutions of χimn and φ i have been obtained, the homogenized variables CiHjmn and σiHj in Eqs. (45) and (46) are readily computed, which are independent of y, followed then by the solution of macroscopic problems for an averaged (homogenized) deformation fields. The procedure of finding Y-periodic characteristic functions to form the homogenized values is referred to as homogenization.
6.
Localization
The procedure of finding the microscopic stress rate from the macroscopic averaged displacement fields is referred to as localization. It is one of the major advantages of the asymptotic series expansion homogenization method, i.e., the actual stress field in each constituent part can be evaluated. Here all variables except u˙ 0 are microscopic ones that have to be obtained within the unit cell region Y. Once the solution u˙ 0 of the macroscopic problem in Eq. (44) is obtained, the microscopic perturbation velocity u˙ 1 is solved from Eq. (42) by using both the characteristic deformation fields and the deformation gradient of u˙ 0 . Then the microscopic velocity gradient, Lε , is obtained as follows by means of Takano’s [13] proposal: u˙ εi (x, y ) =
Lεi j (x, y ) =
Introducing Eqs. (42) and (43) into Eq. (41) yields
Y
In order to solve the microscopic problem of Eq. (41) for the microscopic velocity u˙ 1 (x, y ), which is Y-periodic, the average velocity u˙ 0 (x ) needs to be known. However, the separation of micro and macroscopic variables reduces the microscopic problem to a boundary value problem in the unit cell region Y. That is, due to the instantaneous linearity of Eq. (41), the microscopic velocity can be represented by the superposition of fundamental deformations timing a displacement gradient tensor such that
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∂ u˙ 0i (x ) ∂x j
∂ u˙ 0i (x ) ∂x j
y j + u˙ 1i (x, y )
(50)
∂ u˙ 1i (x, y )
(51)
+
∂y j
With the microscopic velocity gradient known, the deformation rate Dεi j is available, and the Jaumann rate of the microscopic Cauchy stress and its increment are obtained as follows: N
(α) Ri j f˙ (α)
(52)
σ˙ iεj = σiεJj + Wikε σkε j − σikε Wkε j
(53)
σiεJj = Ci jkl Dεi j −
α=1
∀ δ u˙ 1i . (47) δu1i
By making use of the arbitrariness of virtual velocity in Eq. (47), we derive the following two governing equations,
Next it comes the update of microscopic Cauchy stress and configuration of unit cell,
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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Fig. 3. – Homogenized macroscopic stress–strain responses for plane strain compression at large strains.
Fig. 4. – Homogenized macroscopic stress–strain responses for simple compression at large strains.
σiεj (t + t ) = σiεj (t ) + σ˙ iεj t
(54a)
uεi (t + t ) = uεi (t ) + u˙ εi t
(54b)
The macroscopic Cauchy stress is obtained by using the volume average technique, σiHj (x ) =
1 |Y |
Y
σiεj (x, y )dY
(55)
In summary, the main steps in the implementation of crystalline homogenization formulation based on two-scale asymptotic series expansion are: (1) initialization of the geometry of macro and micro models, material properties and crystallographic texture data for crystalline materials; (2) start of double loop with the outer loop over the macroscopic integration points and the inner one over the microscopic integration points, at which the characteristic displacement fields are calculated and the evaluation of the homogenized material properties is carried out using Eqs. (45) and (46). By employing these values, the stiffness matrix of the macroscopic model is formed and the macroscopic (homogenized) velocity is then solved. At the same time, by referring to Eq. (42) and the above
known characteristic displacement fields, the perturbation velocity is obtained, which is used to form the microscopic velocity. This step is called the homogenization; (3) similar to Step 2, beginning the double loop over the macro and micro integration points, respectively. With the homogenized velocity and the microscopic velocity known, the Cauchy stress within the unit cell is obtained and the corresponding macroscopic one will be evaluated by volume average. Then it comes the update of stress and displacement for micro and macro models. The above three steps constitute one time incremental step: from time t to t + t in a semi-implicit updated-Lagrangian formulation, and the verification numerical examples will be presented in the next section.
7. Numerical examples: texture evolution of polycrystals The simulation of crystallographic texture evolution induced by plastic deformation has been one focus of research as means of verification for the model developed. For instance, the work of Maniatty and Dawson [23] as well as that of Sarma and Zacharia [24] used the rate dependent model and the im-
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Fig. 5. – Homogenized macroscopic stress–strain responses for simple shearing at large strains.
-0.03
-0.02
Effective stress(MPa) 500 400 300 200 100 0 -0.01 -100 0 0.01 -200 -300 -400
Effective strain
0.02
0.03
Fig. 6. – Macroscopic (homogenized) stress–strain responses for a reverse loading test of an aggregate of 1000 BCC crystals.
plicit algorithm; Anand and Kothari [20] established a robust approach employing SVD technique in the rate independent formulation; and Miehe et al. [10,11] proposed a homogenization method dealing with the same topic. In the same way, the following is devoted to the simulation of texture evolution under three different deformation modes described below, and comparison is to be made between the results by Taylor model and those by the present crystalline homogenization to confirm the effectiveness and availability of the present formulation. In the following numerical examples, polycrystals with microstructure of FCC single crystal grains deforming on the {111}110 slip systems is analyzed. The microstructure considered is discretized into 125 SRI (selected reduced integration) solid elements with each integration point representing one grain taken randomly from an initially isotropic grain set, namely, the initial random texture is represented by 1000(=125 × 8) single crystal grains. The material parameters are as follows: Young’s modulus is 60.84 GPa, and Poisson’s ratio is 0.3. The cubic elastic constants for the crystals are C11 = 44354.8 MPa, C12 = 33462.0 MPa and C44 = 22936.6 MPa. The initial critical resolved shear stress τ 0 = 60.84 MPa, the initial hardening ratio H0 = 541.4 MPa and saturation shear stress
τ s = 109.5 MPa, reference shear strain rate a˙ (α) = 0.05 and rate sensitivity m = 0.01. In order to demonstrate the capability of the model in capturing elastic responses, the model has been used to simulate a deformation process where one macro-element with microstructure (aggregate of BCC crystal grains) attached to each integration point is deformed first in tension to a macroscopic effective strain of 0.2, at which point the macroscopic boundary velocity is instantaneously reversed. When the strain reaches 0.2, the deformation is again reversed to tension. The resulting stress–strain response is shown in Fig. 6, which clearly indicates the elastic unloading and the subsequent plastic deformation during load reversal. The increase of yield stress due to subsequent hardening of the slip systems is also indicated. In the subsequent simulation examples, polycrystals with micro-structures of FCC single crystal grains are analyzed to investigate the effects of different constraints of the perturbation field on the simulation of texture evolution and homogenized overall stress response of the polycrystalline aggregate. Three different macroscopic deformation modes are investigated: (i) plane strain compression, (ii) simple compression and (iii) simple shearing. For these three deformation modes, the macroscopic velocity gradient is specified as follows: ⎡
0 ⎢ For plane strain compression : L = ⎣0 0
0 0.001 0
⎡ 0.0005 ⎢ For simple compression : L = ⎣ 0 0
0 0.0005 0
⎤ 0 ⎥ 0 ⎦ (56) −0.001
⎤ 0 ⎥ 0 ⎦ −0.001 (57)
⎡ 0 ⎢ For simple shearing : L = ⎣0 0
−0.001 0 0
⎤ 0 ⎥ 0⎦ 0
(58)
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Fig. 7. – Deformation profile of a periodic microstructure at the overall strain levels of 0.3, 0.6 and 1.2 with (a1) to (a3) corresponding to plane strain compression, and (b1), (b2) and (b3) to simple compression.
With respect to each deformation mode listed above, a comparison has been made between the results of overall stress response and texture evolution obtained using Taylor model (TM) and those with both the periodic boundary conditions and perturbation displacement taken into account by the present crystal homogenization (CH) formulation. Figs. 3, 4 and 5 show the homogenized macroscopic stress– strain response for plane strain compression, simple compression and simple shearing at large strain of 200%, respectively. During the large elasto-vico-plastic deformation, the prediction of stress response by Taylor model, i.e., without considering the perturbation deformation in Eq. (50) is always larger in magnitude than that with the perturbation term included in Eq. (50), while the discrepancy of the two results is less than 3% for the first two deformation modes and less than 6% for simple shearing. Similar prediction was made and conclusion was drawn in the paper by Harren [25], where the comparison is made between the prediction by TM and that by the self-consistent (SC) method. Fig. 7 shows the deformed profile of periodic microstructure at the overall strain levels of 0.3, 0.6 and 1.2 with (a1)–(a3) corresponding to plane strain compression, and (b1)–(b3) to simple compression. A detailed examination of these figures reveals the periodicity assumption made for the boundary conditions of micro-structures. However, as expected, in the case of TM, a microstructure of cubic shape will be deformed into a parallelepiped one, indicating that there is no perturbation deformation on the boundary
and in the interior of microstructure. The figures are not included here for space reason. Figs. 8–10 show the texture evolution at each macroscopic integration points, which are obtained by stereographic projection. It is shown that a sharper orientation distribution of grains in FCC polycrystalline metal sheet is observed for Taylor model due to the stiff constraint assumption of uniform deformation among grains, thus a slim ellipse band for plane strain compression, a slim circle band for simple compression and three crossed slim lines for simple shearing are obtained ((a1), (a2) and (a3) in Figs. 8–10). However, when the present crystal homogenization approach is employed, diffuse orientation distributions of grains are observed for the above three deformation modes due to the employment of periodic boundary conditions and the inclusion of perturbation deformation in the overall deformation of microstructures, thus providing a relaxation of uniform constraint on crystal grains in the interior of microstructure at each macroscopic integration point and the flexibility for grains to accommodate the plastic deformation induced by crystallographic slip. Hence, diffuse ellipse bands, circle bands and lines ((b1), (b2) and (b3) in Figs. 8–10) corresponding to the three deformation modes are obtained, which are in a good agreement with the experimental observation [26] and numerical simulations [11] referred above in this section, demonstrating that the present crystal homogenization approach can offer an effective means for simulating the texture evolution in polycrystalline materials.
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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Fig. 8. – {111} pole figures of periodic microstructures consisting of 1000 FCC crystal grains at each macroscopic integration points showing the plastic deformation induced texture evolution under plane strain compression, with (a1), (a2) and (a3) predicted by Taylor model and (b1), (b2) and (b3) by the crystal homogenization method.
The importance of texture lies in the anisotropy of many material properties; that is, the value of this property depends on the crystallographic direction in which it is measured. In most cases grain orientations in polycrystals, whether naturally occurring or technologically fabricated, are not randomly
distributed and the preference of certain orientations may indeed affect material properties by as much as 20%–50% of the property value. Therefore, the numerical determination and experimental interpretation of texture evolution are of fundamental importance in materials processing technology.
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Fig. 9. – {111} pole figures of periodic microstructures consisting of 1000 FCC crystal grains at each macroscopic integration points showing the plastic deformation induced texture evolution under simple compression, with (a1), (a2) and (a3) predicted by Taylor model and (b1), (b2) and (b3) by the crystal homogenization method.
Furthermore, analysis of the texture evolution during the thermomechanical treatment of materials yields valuable and essential information about the underlying mechanisms, including deformation, recrystallization, and phase transformation. In geology, texture analysis can provide in-
sight into the geological processes that led to rock formations millions of years ago. The present implementation can provide an effective numerical tool for the investigation of micro-texture evolution in polycrystalline materials.
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Fig. 10. – {111} pole figures of periodic microstructures consisting of 1000 FCC crystal grains at each macroscopic integration points showing the plastic deformation induced texture evolution under simple shearing, with (a1), (a2) and (a3) predicted by Taylor model and (b1), (b2) and (b3) by the crystal homogenization method.
8.
Conclusions
Crystalline homogenization formulation based on multi-scale asymptotic series expansion has been proposed and implemented by employing an alternative transition from micro-
structure to macro-continuum. The multi-scale computational procedure is realized by means of a coupled discretization of both the macro-continuum and a point-wise attached micro-structure with its boundary condition assumed periodic. The present approach has been applied to the simulation of texture evolution and the overall stress response of
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polycrystalline materials under large elasto-viscoplastic deformation, which is found in good agreement with the available results. The numerical examples show that discrepancy exists between the prediction of Taylor model and that with perturbation deformation taken into account, which is a direct consequence of the constraint relaxation on micro-structure. The present algorithm will find its application in problems of sheet metal forming and rolling to the analysis of microstructure effects.
references
[1] H.J. Bunge, Texture Analysis in Material Science, Butterworths, London, 1982. [2] H.J. Bunge, Experimental Techniques of Texture Analysis, DGM Informationsgesellschaft, Oberursel,Germany, 1986. [3] U.F. Kocks, et al., Texture and Anisotropy, Preferred Orientations, and Their Effect on Materials Properties, Cambridge University Press, 1998. [4] J. Adam, et al., Electron Backscatter Diffraction in Materials Science, Kluwer Academic Press, 2000. [5] D. Peirce, R.J. Asaro, A. Needleman, An analysis of nonuniform and localized deformation in ductile single crystals, Acta Metall. 30 (1982) 1087–1119. [6] D. Peirce, R.J. Asaro, A. Needleman, Material rate dependence and localized deformation in crystalline solids, Acta Metall. 31 (1983) 1951–1976. [7] R.J. Asaro, A. Needleman, Texture development and strain hardening in rate dependent polycrystal, Acta Metall. 33 (1985) 923–953. [8] E. Nakamachi, X. Dong, Elastic/crystalline viscoplastic finite element analysis of dynamic deformation of sheet metal, Int. J. Comput.-Aided Eng. Softw. 13 (1996) 308–326. [9] W. Yang, W.B. Lee, Mesoplasticity and Its Applications, Springer-Verlag, 1993. [10] C. Miehe, J. Schroder, J. Schotte, Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials, Comput. Methods Appl. Mech. Eng. 171 (1999) 387–418. [11] C. Miehe, J. Schotte, M. Lambrecht, Homogenization of inelastic solid materials at finite strains based on incremental minimization principle, Application to the texture analysis of polycrystals, J. Mech. Phys. Solids 50 (2002) 2123–2167. [12] K. Terada, K. Yuge, N. Kikuchi, Elasto-plastic analysis of composite materials by using the homogenization method (I): formulation, J. Japan Soc. Mech. Eng. A 61 (590) (1995) 91–97.
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[13] N. Takano, M. Zako, The formulation of homogenization method applied to large deformation problem for composite materials, Int. J. Solids Struct. 37 (2000) 6517–6535. [14] N. Ohno, D. Okumura, H. Noguchi, Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation, J. Mech. Phys. Solids 50 (2002) 1125–2153. [15] K. Terada, K. Yuge, N. Kikuchi, Elasto-plastic analysis of composite materials by using the homogenization method (II), J. Japan Soc. Mech. Eng. A 62 (601) (1996) 110–117. [16] I. Babuska, Homogenization approach in engineering, Computing Methods in Applied Sciences and Engineering, vol. 134(134), Springer Berlin Heidelberg, 1976, pp. 137–153. [17] E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, vol. 127, Springer, Berlin, 1980. [18] N. Bakhvalov, Panasenko, G. Homogenization, Averaging Processes in Periodic Media, Kluwer Academic Pub., 1984. [19] J. Guedes, N. Kikuchi, Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Comput. Methods Appl. Mech. Eng. 83 (1990) 145–198. [20] L. Anand, M. Kothari, A computational procedure for rate-independent crystal plasticity, J. Mech. Phys. Solids 44 (4) (1996) 525–558. [21] P. Van Houtte, A comprehensive mathematical formulation of an extended Taylor–Bishop–Hill model featuring relaxed constraints, the Renouard–Wintenberger theory and a strain rate sensitivity model, Textures Microstruct. 8–9 (1988) 313–350. [22] Y. Zhou, K.W. Neale, L.S. Toth, A modified model for simulating latent hardening during the plastic deformation of rate-dependent FCC polycrystals, Int. J. Plast. 9 (1993) 961–978. [23] A.M. Maniatty, P.R. Dawson, Y.S. Lee, A time integration algorithm for elasto-viscoplastic cubic crystals applied to modelling polycrystalline deformatiuon, Int. J. Numer. Methods Eng. 35 (1992) 1565–1588. [24] G. Sarma, T. Zacharia, Integration algorithm for modelling the elasto-viscoplastic response of polycrystalline materials, J. Mech. Phys. Solids 47 (1999) 1219–1238. [25] S.V. Harren, The finite deformation of rate-dependent polycrystals (II): a comparison of the self-consistent and Taylor methods, J. Mech. Phys. Solids 39 (3) (1991) 361–383. [26] C.A. Bronkhorst, S.R. Kalidindi, L. Anand, Polycrystalline plasticity and the evolution of crystallographic texture in f.c.c. metals, Philos. Trans. R. Soc. Lond. A 341 (1992) 443–477.
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Available online at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/CAMSS
Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method✩ Y.P. Chen∗, Y.Y. Cai Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
A semi-implicit elastic/crystalline viscoplastic finite element (FE) method based on a “crys-
Received 22 June 2016
tallographic homogenization” approach is formulated for a multi-scale analysis. In the for-
Revised 17 October 2016
mulation, the asymptotic series expansion is introduced to define the displacement in the
Accepted 18 October 2016
micro-continuum. This homogenization FE analysis is aimed at predicting the plastic defor-
Available online xxx
mation induced texture evolution of polycrystalline materials, the constituent microstructure of which is represented by an assembly of single crystal grains. The rate dependent
Keywords:
crystal plasticity model is adopted for the description of microstructures. Their displace-
Homogenization
ments are decomposed into two parts: the homogenized deformation defined in the macro-
Local periodicity
continuum and the perturbed one in the micro-continuum. This multi-scale formulation
Crystalline plasticity
makes it possible to carry out an alternative transition from a representative micro-structure
Texture
to the macro-continuum. This homogenization procedure satisfies both the compatibility
Finite element
and the equilibrium in the micro-structure. This developed code is applied to predict the texture evolution, and its performance is demonstrated by the numerical examples of texture evolution of FCC polycrystalline metals. © 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics.
1.
Introduction
Polycrystalline solid materials consisting of a multitude of individual crystallites or grains after large plastic deformation generally demonstrate a certain degree of anisotropy in material properties, which is a natural reflection of the crystallographic texture induced by the reorientation of crystal lattices towards a preferential distribution orientation. The experimental determination, interpretation [1, 2] and the numerical simulation for texture analysis have been attracting the attention of researchers in the field of materials science and applied
✩ ∗
Project supported by NSFC (11272129, 11472114, 51175202). Corresponding author. E-mail address: [email protected] (Y.P. Chen).
mechanics [3]. On the one hand, recent advances in experiment techniques, such as EBSD [4], have made it possible to measure the texture more efficiently and accurately than the traditional X-ray diffraction method does; on the other hand, the rapid development of computer technology promises an unprecedented means for the large scale numerical simulation of materials deformation process and design, which is, however, essentially dependent on the constitutive model and the numerical techniques. Polycrystal plasticity models [5–9] were employed to predict the texture evolution induced by plastic deformation and the corresponding plastic anisotropy. In these simulations, the assumption is made that the microstructure of polycrystals is represented by an aggregate of microscopic crystals, whose plastic deformation is brought about by the shearing along crystallographic planes, and whose individual responses, on
http://dx.doi.org/10.1016/j.camss.2016.10.002 0894-9166/© 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics. Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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average, determine the overall macroscopic response of polycrystals. Two ingredients are indispensable for the establishment of these models: one is the constitutive equations used to describe the mechanical behaviors of a single crystal; the other is the mean field hypothesis or averaging schemes that relate the overall response of polycrystals to that of the constituent single crystals by a micro-macro-transition on the basis of homogenization procedures. The commonly adopted averaging schemes include the Taylor-type model, in which each crystal is assumed to experience identical macroscopic deformation and the stress of polycrystal aggregate is calculated by averaging the crystal stress, the equilibrium-based schemes require the same stress state for each grain, and the aggregate deformation is obtained by averaging the crystal deformation. In addition to these averaging procedures, Miehe [10,11] established two different homogenization schemes which attached the macro-continuum locally with a polycrystalline micro-structure made up of single crystal grains. The boundary of the micro-structure was specified by the local deformation of the macro-continuum with three alternative deformation constraints: (i) zero fluctuation in the domain (Taylor-type assumption), (ii) zero fluctuation on the boundary, and (iii) periodic fluctuation on the boundary, making improvements to the Taylor-type models. It seems that these two approaches were the first to have applied homogenization method to the prediction of texture evolution and strain localization in crystalline materials. Homogenization based on the multi-scale asymptotic series expansion of field variables has been utilized as an effective numerical technique over the last three decades to describe the mechanical behaviors of real heterogeneous materials by taking into account the length scale difference between macrostructures and individual microstructure components [12,13]. The advantage of this method lies in the fact that it is a rigorous mathematical theory and is capable of characterizing equivalent mechanical properties of composite materials and at the same time determining their dependence on different constituent components. It has been employed successfully to solve the problems of both finite elasticity [14] and elasto-plasticity [15]. In the homogenization theory, it is generally assumed that the composite material is locally formed by the spatial repetition of very small microstructures, i.e. the microscopic cells, when compared with the overall macroscopic dimensions of the structure of interest. In other words, it is assumed that the materials properties are periodic functions of the microscopic variables, where the period is very small compared with the macroscopic variables. This assumption enables the computation of equivalent material properties by a limiting process when the size of microscopic cell is reduced to zero. It can also provide a reasonable solution for some problems where the experimental data are not available, or for which only bounds for the equivalent material contents can be found by other theories. The basic theory of homogenization method can be found in the papers published in 1970s and early 1980s, for instance, by Babuska [16], SanchezPalencia [17], Bakhvalov and Panasenko [18] and many other applied mathematicians. Guedes and Kikuchi [19] developed a new formulation of this method in a weak form and provided a guideline for numerical analysis.
The purpose of the present paper is to establish a homogenization framework based on the two-scale asymptotic expansion of field variables to simulate the texture evolution in polycrystalline materials. A recapitulation of crystalline constitutive equations for characterizing single crystal grains is first presented and then followed by the derivation of governing equations for solving homogenized macro-deformation and characteristic displacements for micro-models to evaluate the homogenized material properties, respectively, from the virtual power principle. The proposed formulation is implemented in the updated Lagrangian finite element form and the performance of the code is demonstrated by the numerical examples of texture evolution in polycrystalline materials.
2.
Formulation of the problem
We consider a general infinite polycrystalline body, ε , in two dimensional space, as seen in Fig. 1. It is assumed that the material is formed by a spatial repetition of a microstructure at least locally, and therefore, the microstructure is usually called a unit cell. The region, Y, of the unit cell (microstructure) is made up of an aggregate of well defined crystal grains and is very small compared with the overall region ε by an order of ε 1, which also represents the reciprocal order of the repetition. In order to describe the effects of heterogeneity in the microstructures, we attach ε as a superscript to all the variables in the formulation when it is essential. We also introduce both microscopic and macroscopic coordinate systems, so that physical quantities are represented by two different length scales: one is x representing the macroscopic region, ; and the other is y = x/ε standing for the microscopic one, Y. The summation convention is used and the dot over a variable denotes material time derivative.
2.1.
Kinematics of crystal plasticity
We recapitulate the rate-dependent crystal plasticity constitutive model to describe the elasto-plastic deformation of constituent single crystals (Fig. 2). For details, we refer to the work of Asaro and Needleman [7]. The total deformation of a crystallite is assumed to be the result of two distinct physical mechanisms: crystallographic slip due to dislocation motion on the active slip systems, and elastic lattice distortion. FCC crystals are considered to have the usual {111}110 slip systems, where the slip planes are the {111} crystallographic planes with normals m, and the 110 directions are the shear directions with slip vectors s. Plastic deformation of the crystal is understood to occur as a set of simple plastic shear along various slip systems, having the lattice and the slip systems’ vectors (s(α) , m(α) ) not only essentially undistorted, but also unrotated (the brackets in the superscripts (α) indicate that α is not a tensor index and ranges from one to the number of slip systems). Next, the materials and lattice are considered to deform elastically and rotate rigidly from the physically deformed state to the current configuration. Accordingly, the decomposition for the deformation gradient tensor is Fi j = Fik∗ Fk j p
(1)
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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Fig. 1. – Schematic representation of macroscopic continuum and the assumed periodic unit cell at microscopic scale.
Fig. 2. – Multiplicative decomposition of the deformation gradient tensor.
where F p consists solely of crystallographic slipping along the specific slip systems and F ∗ arises from the stretching and rotation of the crystal lattice. From Eq. (1), the spatial gradient of velocity can be written as: −1 Li j = F˙im Fm = L∗i j + Li j j p
(2)
where ∗ ∗−1 L∗i j = F˙im Fm j ,
Li j = Fik∗ F˙km Fmn Fn∗−1 j p
p
p−1
(3)
Taking symmetric and anti-symmetric parts of the above relations yields the elastic and plastic strain rates D∗ and Dp , the plastic spin Wp , and W∗ associated with the rigid lattice rotation: Di j = D∗i j + Di j ,
Wi j = Wi∗j + Wi j
p
p
(4)
Since s(α) and m(α) are regarded as lattice vectors, they are stretched and rotated as follows: ∗(α)
si
(α)
= Fi∗j s j ,
∗(α)
mi
(α)
= m j Fi∗−1 j
(5)
The vectors s∗(α) and m∗(α) , orthogonal since s(α) and m(α) are, characterize a particular slip system in the current state,
and their evolution is governed by ∗(α)
s˙i
∗(α)
= L∗i j s j
∗(α)
˙i ,m
∗(α)
= L∗ji m j
(6)
By introducing the slip system, α, the symmetric and antisymmetric tensors are respectively 1 ∗(α) ∗(α) (α) ∗(α) ∗(α) Pi j = si m j + mi s j , 2 1 ∗(α) ∗(α) (α) ∗(α) ∗(α) Wi j = (7) s m j − mi s j , 2 i and the plastic strain rate and spin for the crystal can be written as p
Di j =
α
(α)
Pi j γ˙ (α) ,
p
Wi j =
α
(α)
Wi j γ˙ (α)
(8)
where γ˙ (α) is the shear rate on the slip system α. The elastic part of the rate of deformation De and the substructure spin W R can be obtained as follows: p
Dei j = Di j − Di j ,
2.2.
p
Wiej = Wi j − Wi j
(9)
Constitutive equations for single crystals
It is generally assumed that elasticity of grains is not affected by the plastic deformation of crystallographic slip, so the co-
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rotational rate of Cauchy stress and its Jaumann rate are obtained as follows: σˆ iej = Ciejkl Dekl = σ˙ i j − Wike σk j + σikWke j ,
(10)
σˆ i j = σ˙ i j − Wik σk j + σikWk j
(11)
⎛
1 ⎜ C = ⎝q c qc ⎛ qc ⎜ L2 = ⎝ ql ql
⎞ ⎛ ⎞ qc qv ql ql ⎟ ⎜ ⎟ q c ⎠, L 1 = ⎝ q l qc q l ⎠, 1 ql ql qv ⎞ ⎛ ⎞ ql qv ql ql ⎟ ⎜ ⎟ ql ⎠, L3 = ⎝ ql qv q l ⎠. qv ql ql qc
qc 1 qc ql qv ql
(18)
From Eqs. (10) and (11) we have
σˆ iej − σˆ i j =
(α)
βi j γ˙ (α)
(12)
where (α)
(α)
(α)
βi j = Wik σk j − σikWk j
2.3.
(13)
Viscoplastic shear strain rate
The crystalline visco-plastic shear strain rate γ˙ (α) depends on the Schmid resolved shear stress (SRSS) and the hardening evolution of crystal. In general, two kinds of models are available for the description of crystal responses to external loadings: one is the rate independent model, in which the viscoplastic shear strain is obtained by an iterative procedure employing SVD technique [20] or by linear programming [21]; the other is the rate-dependent model, which relates the shear strain rate γ˙ (a ) directly to the SRSS, thus avoiding the abovementioned iteration in the solution. The latter is adopted as follows [6]: γ˙ (α) = a˙ (α)
τ (α) 1/m−1 g(α) g(α)
τ (α)
(14)
where the function denotes the reference shear stress, which describes the current strain hardening state of the crystal, a˙ (α) is the reference shear rate, and m is the material rate sensitivity with the case of m = 0 corresponding to the rateindependent models. The evolution of g(α) is determined by the following hardening evolution equation:
hαβ |γ˙ (β ) |
2.4.
(15)
Tangential modulus method
When the strain rate sensitivity m is small, a large shear strain rate γ˙ (a ) may be obtained from Eq. (14), thus bringing about the problem of divergence. The solution to it is by making use of the following tangential modulus method proposed by Peirce [6]:
γ (α) = [(1 − θ )γ˙ (α) (t ) + θ γ˙ (α) (t + t )] t
g(α) = g˙ (α) t yield (α)
Nαβ γ (β ) = (γ˙t
(β )
(α) γ˙t
= a˙
(α)
(α) Qi j
=
dτ ( γ ) dτ ( γ ) + (1 − q ) δαβ dγ dγ
τ (γ ) = τ0
n h0 γ +1 n τ0
(16)
(α)
θ t γ˙t
where q = 1 for coplanar slip systems and q = 1 to 1.4 for noncoplanar slip systems; and the matrix qαβ is introduced as follows to describe the self and latent hardening: ⎛ ⎞ C L1 L2 L3 ⎜L C L3 L2 ⎟ ⎜ 1 ⎟ qαβ = ⎜ ⎟ ⎝L2 L3 C L1 ⎠ L3 L2 L1 C
(21a)
(α)
Ri j
mτ (α)
Nαβ = δαβ +
(21b)
(α)
θ t γ˙t m
⎤ ⎡ (α) (β ) Ri j Pi j hαβ ( β ) ⎣ ⎦ + sgn(τ ) τ (α) g(α)
(22)
Denoting the inverse of Nαβ by Mαβ , one has the following: (α) (α)
γ = ( f˙ (a ) + Fi j Di j ) t
(23)
where f˙ (α) =
(17)
(20)
τ (α) 1/m−1 g(α) g(α)
hαβ = qαβ
(α)
+ Qi j Di j ) t
τ (α)
(β )
where hαβ are hardening moduli. The definition of a proper and realistic hardening evolution law has been a problem of pivotal importance in crystalline plasticity, and a number of such hardening evolution laws have been proposed. In the present investigation, it is expressed as follows:
(19)
The expansion of Eq. (14) into Taylor series, its substitu(α) tion into Eq. (19) and use of τ˙ (α) = Ri j Dei j , τ (α) = τ˙ (α) t, and
g(α)
g˙ (α) =
The parameters in qαβ proposed by Zhou et al. [22] are employed for the coplanar or collinear slip systems, with qc = 1. For the slip systems which have mutually perpendicular Burgers vectors, qv = 1.2. For others, ql = 1.3. γ is the accumulated shear strain over all the slip systems. τ0 , h0 and n are the initial critical resolved shear stress, the initial hardening ratio and the strain hardening exponent, respectively, which are determined in the same procedure as that in Nakamachi et al. [8].
(β )
(β )
(α)
Mαβ γ˙t , Fi j =
Mαβ Q (β )
(24)
β
Substituting Eq. (23) into Eq. (12), we obtain the constitutive equation as follows: σˆ i j = Ciejkl Dkl −
α
(α) (α) Ri j ( f˙ (α) + Fmn Dmn ) = C¯ivjkl Dkl −
where C¯ivjkl = Ciejkl −
α
α
(α) Ri j f˙ (α) (25)
(α) (α)
Ri j Fkl .
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By substituting Eq. (33) into Eq. (32), the following equation is derived:
3. Principle of virtual velocity in updated Lagrangian form
In the updated Lagrangian formulation, by choosing the reference configuration at time t, we have
P˙i j δLi j dV =
F = I,
St
t˙i δvi dS
L = F˙ F −1 = F˙ ,
(26)
τ=σ=P
(27)
where P is the first Piola–Kirchhoff stress (non-symmetric), τ is the second Piola–Kirchhoff stress, and σ is the Cauchy stress. The relationship between the material time derivative of τ and P is written as: τ = PF T ,
τ˙ = P˙F T + PF˙ T = P˙ + σLT
(28)
(Ci jkl − Hi jkl )Lkl δLi j dV =
S
t˙i δ v˙ i dS +
N V α=1
(α) Ri j f˙ (α) δLi j dV (34)
where Hi jkl = 12 (δki σ jl − δli σ jk − δk j σil − δl j σik ). Following the general procedure of two-scale asymptotic series expansion, the primary variable can be written as: u˙ ε (x/ε ) = u˙ 0 (x ) + ε u˙ 1 (x, y ) + ε 2 u˙ 2 (x, y ) + · · ·
(35)
in which each term is Y-periodic, namely periodic with respect to y within a unit cell. Noting that the differentiation of a certain Y-periodic function Φ ε (x) is performed by ∇x · Φ ε ( x ) = ∇ x · Φ ε ( x, y ) +
1 ∇y · Φ ε ( x, y ) ε
(36)
So the component of velocity gradient is also written as P˙ = τ˙ − σLT
(29)
By using Eqs. (28) and (29) we obtain the following relationship between the Jaumann derivative of τ and the material time derivative of P, P˙ = τˆ − Dσ − σD + σLT
St
t˙i δvi dS
(31)
In this section, we shall separate the micro- and macroscopic equations and then derive the homogenization formulae. These are accomplished by taking the limit of ε to zero and employing the integral formula
We assume J = 1 during the elasto-plastic deformaton, thus σˆ = τ. ˆ Also on account of the symmetry of rate deformation tensor D and Cauchy stress tensor σ, the principle of virtual velocity in the updated Lagrangian formulation is in the following form:
{(σˆ i j − 2σik Dk j )δDi j + σ jk Lik δLi j }dV =
St
t˙i δvi dS
(32)
α=1
(α) Ri j f˙ (α)
ε→0 ε
Φ(x, x/ε )d =
Y
Ci jkl
∂ u˙ 0k ∂ xl
(33)
(α) (α) e in which Ci jkl = Ciejkl − N α=1 Ri j Fkl , where Ci jkl is elasticity modulus of the 4th order and N the total number of slip systems.
1 |Y |
Y
Φ(x, y )dyd
(38)
where Φ ε (x) is a Y-perodic function and the integral sign (1/|Y|)∫Y dy indicates the volume average of function Φ(x, y) within the unit cell. The proof is omitted here. Substituting Eq. (37) into Eq. (31) and applying averaging theorem yield
=
We shall derive the homogenized equations by utilizing the method of two-scale asymptotic series expansion and the crystalline constitutive equation in Section 2, which is rewritten here in the following component form: N
Homogenization
lim
1 |Y |
4. Asymptotic series expansion and constitutive equations
σˆ i j = Ci jkl dkl −
(37)
5.
(τˆi j − Dik σk j − σik Dk j + σik Lk j )δLi j dV =
∂ u˙ 0l ∂ u˙ 1l ∂ u˙ 1l ∂ u˙ 2l ∂ u˙ l (x/ε ) = = + +ε + +··· ∂ xm ∂ xm ∂ ym ∂ xm ∂ ym
(30)
Substituting Eq. (30) into Eq. (26) yields
Lεlm (x/ε )
t˙i δ u˙ 0i d +
+
1 |Y |
∂ u˙ 1k
·
∂ yl
∂δ u˙ 0i ∂x j
N Y α=1
+
(α) Ri j f˙ (α)
∂δ u˙ 1i
∂y j
∂δ u˙ 0i ∂x j
dYd +
∂δ u˙ 1i ∂y j
dYd (39)
where Ci jkl = Ci jkl − Hi jkl . From Eq. (39), we obtain the following two governing equations for microscopic and macroscopic problems, respectively, as an approximation of the first order:
1 |Y |
Y
Ci jkl
∂ u˙ 0k ∂ xl
=
+
t˙i δ u˙ 0i d +
∂ u˙ 1k ∂ yl
1 |Y |
·
∂δ u˙ 0i
N Y α=1
∂x j
dYd
(α) Ri j f˙ (α)
∂δ u˙ 0i ∂x j
dYd, ∀δ u˙ 0i (x )
(40)
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1 |Y |
Y
=
Ci jkl
∂ u˙ 0k ∂ xl
+
∂ u˙ 1k
∂ yl
·
∂δ u˙ 1i ∂y j
which is to be solved for the solutions of characteristic displacement fields χimn and φ i , respectively,
dYd
N ∂δ u˙ 1i 1 (α) Ri j f˙ (α) dYd, |Y | Y ∂x j α=1
∀ δ u˙ 1i (x, y )
(41)
u˙ 1i = −χimn
∂ u˙ 0m + φi ∂ xn
(42)
so ∂ u˙ 1i ∂y j
=−
∂χimn ∂y j
·
∂ φi ∂ u˙ 0m + ∂xn ∂y j
(43)
where the variables χ mn and φ are generally referred to as the characteristic deformation, which are the responses to the increments of the corresponding components of displacement gradient tensor. Therefore, there are ten independent functions, among which no symmetry with respect to mn in χimn may be found. It is also noted that each characteristic deformation is a function of x and y, but not in the rate form. Substituting Eqs. (42) and (43) into Eq. (40) yields ∂χ mn ∂ u˙ 0m ∂δ u˙ 0i 1 Ci jmn − Ci jkl k · · dYd ∂ yl ∂ xn ∂x j |Y | Y N 1 ∂ ϕk (α) ˙ (α) 0 ˙ ti δ u˙ i d + Ri j f − Ci jkl = ∂ yl |Y | Y α=1
·
∂δ u˙ 0i ∂x j
dYd,
∀δ u˙ 0i
(44)
Let CiHjmn
1 = |Y |
∂χ mn C i jmn − C i jkl k dY ∂ yl Y
(45)
and σiHj
N 1 ∂ϕ (α) = Ri j f˙ (α) − Ci jkl k dY . |Y | Y ∂ yl
(46)
α=1
∂χ mn ∂ u˙ 0m ∂δ u˙ 1i C i jmn − Ci jkl k · · dYd ∂ y ∂ xn ∂y j Y l N ∂δ u˙ 1i 1 ∂ϕ (α) Ri j f˙ (α) − Ci jkl k · dYd, | | Y ∂ y ∂x j Y l
1 | | Y =
α=1
C i jmn
∂δ u˙ 1i ∂y j
dY =
Y
C i jkl
∂χkmn ∂ yl
·
∂δ u˙ 1i ∂y j
dY ,
∀ δu1i
(48)
and N Y
α=1
∂δ u˙ 1i ∂ ϕ ∂δ u˙ 1i (α) Ri j f˙ (α) · = C i jkl k · dY , ∂y j ∂ yl ∂y j Y
∀δu1i
(49)
Now three governing equations have been derived, namely, boundary value problems, which are Eq. (44) for the macroscopic deformation and Eqs. (48) and (49) for characteristic displacement fields. Once the solutions of χimn and φ i have been obtained, the homogenized variables CiHjmn and σiHj in Eqs. (45) and (46) are readily computed, which are independent of y, followed then by the solution of macroscopic problems for an averaged (homogenized) deformation fields. The procedure of finding Y-periodic characteristic functions to form the homogenized values is referred to as homogenization.
6.
Localization
The procedure of finding the microscopic stress rate from the macroscopic averaged displacement fields is referred to as localization. It is one of the major advantages of the asymptotic series expansion homogenization method, i.e., the actual stress field in each constituent part can be evaluated. Here all variables except u˙ 0 are microscopic ones that have to be obtained within the unit cell region Y. Once the solution u˙ 0 of the macroscopic problem in Eq. (44) is obtained, the microscopic perturbation velocity u˙ 1 is solved from Eq. (42) by using both the characteristic deformation fields and the deformation gradient of u˙ 0 . Then the microscopic velocity gradient, Lε , is obtained as follows by means of Takano’s [13] proposal: u˙ εi (x, y ) =
Lεi j (x, y ) =
Introducing Eqs. (42) and (43) into Eq. (41) yields
Y
In order to solve the microscopic problem of Eq. (41) for the microscopic velocity u˙ 1 (x, y ), which is Y-periodic, the average velocity u˙ 0 (x ) needs to be known. However, the separation of micro and macroscopic variables reduces the microscopic problem to a boundary value problem in the unit cell region Y. That is, due to the instantaneous linearity of Eq. (41), the microscopic velocity can be represented by the superposition of fundamental deformations timing a displacement gradient tensor such that
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∂ u˙ 0i (x ) ∂x j
∂ u˙ 0i (x ) ∂x j
y j + u˙ 1i (x, y )
(50)
∂ u˙ 1i (x, y )
(51)
+
∂y j
With the microscopic velocity gradient known, the deformation rate Dεi j is available, and the Jaumann rate of the microscopic Cauchy stress and its increment are obtained as follows: N
(α) Ri j f˙ (α)
(52)
σ˙ iεj = σiεJj + Wikε σkε j − σikε Wkε j
(53)
σiεJj = Ci jkl Dεi j −
α=1
∀ δ u˙ 1i . (47) δu1i
By making use of the arbitrariness of virtual velocity in Eq. (47), we derive the following two governing equations,
Next it comes the update of microscopic Cauchy stress and configuration of unit cell,
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Fig. 3. – Homogenized macroscopic stress–strain responses for plane strain compression at large strains.
Fig. 4. – Homogenized macroscopic stress–strain responses for simple compression at large strains.
σiεj (t + t ) = σiεj (t ) + σ˙ iεj t
(54a)
uεi (t + t ) = uεi (t ) + u˙ εi t
(54b)
The macroscopic Cauchy stress is obtained by using the volume average technique, σiHj (x ) =
1 |Y |
Y
σiεj (x, y )dY
(55)
In summary, the main steps in the implementation of crystalline homogenization formulation based on two-scale asymptotic series expansion are: (1) initialization of the geometry of macro and micro models, material properties and crystallographic texture data for crystalline materials; (2) start of double loop with the outer loop over the macroscopic integration points and the inner one over the microscopic integration points, at which the characteristic displacement fields are calculated and the evaluation of the homogenized material properties is carried out using Eqs. (45) and (46). By employing these values, the stiffness matrix of the macroscopic model is formed and the macroscopic (homogenized) velocity is then solved. At the same time, by referring to Eq. (42) and the above
known characteristic displacement fields, the perturbation velocity is obtained, which is used to form the microscopic velocity. This step is called the homogenization; (3) similar to Step 2, beginning the double loop over the macro and micro integration points, respectively. With the homogenized velocity and the microscopic velocity known, the Cauchy stress within the unit cell is obtained and the corresponding macroscopic one will be evaluated by volume average. Then it comes the update of stress and displacement for micro and macro models. The above three steps constitute one time incremental step: from time t to t + t in a semi-implicit updated-Lagrangian formulation, and the verification numerical examples will be presented in the next section.
7. Numerical examples: texture evolution of polycrystals The simulation of crystallographic texture evolution induced by plastic deformation has been one focus of research as means of verification for the model developed. For instance, the work of Maniatty and Dawson [23] as well as that of Sarma and Zacharia [24] used the rate dependent model and the im-
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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Fig. 5. – Homogenized macroscopic stress–strain responses for simple shearing at large strains.
-0.03
-0.02
Effective stress(MPa) 500 400 300 200 100 0 -0.01 -100 0 0.01 -200 -300 -400
Effective strain
0.02
0.03
Fig. 6. – Macroscopic (homogenized) stress–strain responses for a reverse loading test of an aggregate of 1000 BCC crystals.
plicit algorithm; Anand and Kothari [20] established a robust approach employing SVD technique in the rate independent formulation; and Miehe et al. [10,11] proposed a homogenization method dealing with the same topic. In the same way, the following is devoted to the simulation of texture evolution under three different deformation modes described below, and comparison is to be made between the results by Taylor model and those by the present crystalline homogenization to confirm the effectiveness and availability of the present formulation. In the following numerical examples, polycrystals with microstructure of FCC single crystal grains deforming on the {111}110 slip systems is analyzed. The microstructure considered is discretized into 125 SRI (selected reduced integration) solid elements with each integration point representing one grain taken randomly from an initially isotropic grain set, namely, the initial random texture is represented by 1000(=125 × 8) single crystal grains. The material parameters are as follows: Young’s modulus is 60.84 GPa, and Poisson’s ratio is 0.3. The cubic elastic constants for the crystals are C11 = 44354.8 MPa, C12 = 33462.0 MPa and C44 = 22936.6 MPa. The initial critical resolved shear stress τ 0 = 60.84 MPa, the initial hardening ratio H0 = 541.4 MPa and saturation shear stress
τ s = 109.5 MPa, reference shear strain rate a˙ (α) = 0.05 and rate sensitivity m = 0.01. In order to demonstrate the capability of the model in capturing elastic responses, the model has been used to simulate a deformation process where one macro-element with microstructure (aggregate of BCC crystal grains) attached to each integration point is deformed first in tension to a macroscopic effective strain of 0.2, at which point the macroscopic boundary velocity is instantaneously reversed. When the strain reaches 0.2, the deformation is again reversed to tension. The resulting stress–strain response is shown in Fig. 6, which clearly indicates the elastic unloading and the subsequent plastic deformation during load reversal. The increase of yield stress due to subsequent hardening of the slip systems is also indicated. In the subsequent simulation examples, polycrystals with micro-structures of FCC single crystal grains are analyzed to investigate the effects of different constraints of the perturbation field on the simulation of texture evolution and homogenized overall stress response of the polycrystalline aggregate. Three different macroscopic deformation modes are investigated: (i) plane strain compression, (ii) simple compression and (iii) simple shearing. For these three deformation modes, the macroscopic velocity gradient is specified as follows: ⎡
0 ⎢ For plane strain compression : L = ⎣0 0
0 0.001 0
⎡ 0.0005 ⎢ For simple compression : L = ⎣ 0 0
0 0.0005 0
⎤ 0 ⎥ 0 ⎦ (56) −0.001
⎤ 0 ⎥ 0 ⎦ −0.001 (57)
⎡ 0 ⎢ For simple shearing : L = ⎣0 0
−0.001 0 0
⎤ 0 ⎥ 0⎦ 0
(58)
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Fig. 7. – Deformation profile of a periodic microstructure at the overall strain levels of 0.3, 0.6 and 1.2 with (a1) to (a3) corresponding to plane strain compression, and (b1), (b2) and (b3) to simple compression.
With respect to each deformation mode listed above, a comparison has been made between the results of overall stress response and texture evolution obtained using Taylor model (TM) and those with both the periodic boundary conditions and perturbation displacement taken into account by the present crystal homogenization (CH) formulation. Figs. 3, 4 and 5 show the homogenized macroscopic stress– strain response for plane strain compression, simple compression and simple shearing at large strain of 200%, respectively. During the large elasto-vico-plastic deformation, the prediction of stress response by Taylor model, i.e., without considering the perturbation deformation in Eq. (50) is always larger in magnitude than that with the perturbation term included in Eq. (50), while the discrepancy of the two results is less than 3% for the first two deformation modes and less than 6% for simple shearing. Similar prediction was made and conclusion was drawn in the paper by Harren [25], where the comparison is made between the prediction by TM and that by the self-consistent (SC) method. Fig. 7 shows the deformed profile of periodic microstructure at the overall strain levels of 0.3, 0.6 and 1.2 with (a1)–(a3) corresponding to plane strain compression, and (b1)–(b3) to simple compression. A detailed examination of these figures reveals the periodicity assumption made for the boundary conditions of micro-structures. However, as expected, in the case of TM, a microstructure of cubic shape will be deformed into a parallelepiped one, indicating that there is no perturbation deformation on the boundary
and in the interior of microstructure. The figures are not included here for space reason. Figs. 8–10 show the texture evolution at each macroscopic integration points, which are obtained by stereographic projection. It is shown that a sharper orientation distribution of grains in FCC polycrystalline metal sheet is observed for Taylor model due to the stiff constraint assumption of uniform deformation among grains, thus a slim ellipse band for plane strain compression, a slim circle band for simple compression and three crossed slim lines for simple shearing are obtained ((a1), (a2) and (a3) in Figs. 8–10). However, when the present crystal homogenization approach is employed, diffuse orientation distributions of grains are observed for the above three deformation modes due to the employment of periodic boundary conditions and the inclusion of perturbation deformation in the overall deformation of microstructures, thus providing a relaxation of uniform constraint on crystal grains in the interior of microstructure at each macroscopic integration point and the flexibility for grains to accommodate the plastic deformation induced by crystallographic slip. Hence, diffuse ellipse bands, circle bands and lines ((b1), (b2) and (b3) in Figs. 8–10) corresponding to the three deformation modes are obtained, which are in a good agreement with the experimental observation [26] and numerical simulations [11] referred above in this section, demonstrating that the present crystal homogenization approach can offer an effective means for simulating the texture evolution in polycrystalline materials.
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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Fig. 8. – {111} pole figures of periodic microstructures consisting of 1000 FCC crystal grains at each macroscopic integration points showing the plastic deformation induced texture evolution under plane strain compression, with (a1), (a2) and (a3) predicted by Taylor model and (b1), (b2) and (b3) by the crystal homogenization method.
The importance of texture lies in the anisotropy of many material properties; that is, the value of this property depends on the crystallographic direction in which it is measured. In most cases grain orientations in polycrystals, whether naturally occurring or technologically fabricated, are not randomly
distributed and the preference of certain orientations may indeed affect material properties by as much as 20%–50% of the property value. Therefore, the numerical determination and experimental interpretation of texture evolution are of fundamental importance in materials processing technology.
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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Fig. 9. – {111} pole figures of periodic microstructures consisting of 1000 FCC crystal grains at each macroscopic integration points showing the plastic deformation induced texture evolution under simple compression, with (a1), (a2) and (a3) predicted by Taylor model and (b1), (b2) and (b3) by the crystal homogenization method.
Furthermore, analysis of the texture evolution during the thermomechanical treatment of materials yields valuable and essential information about the underlying mechanisms, including deformation, recrystallization, and phase transformation. In geology, texture analysis can provide in-
sight into the geological processes that led to rock formations millions of years ago. The present implementation can provide an effective numerical tool for the investigation of micro-texture evolution in polycrystalline materials.
Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002
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Fig. 10. – {111} pole figures of periodic microstructures consisting of 1000 FCC crystal grains at each macroscopic integration points showing the plastic deformation induced texture evolution under simple shearing, with (a1), (a2) and (a3) predicted by Taylor model and (b1), (b2) and (b3) by the crystal homogenization method.
8.
Conclusions
Crystalline homogenization formulation based on multi-scale asymptotic series expansion has been proposed and implemented by employing an alternative transition from micro-
structure to macro-continuum. The multi-scale computational procedure is realized by means of a coupled discretization of both the macro-continuum and a point-wise attached micro-structure with its boundary condition assumed periodic. The present approach has been applied to the simulation of texture evolution and the overall stress response of
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polycrystalline materials under large elasto-viscoplastic deformation, which is found in good agreement with the available results. The numerical examples show that discrepancy exists between the prediction of Taylor model and that with perturbation deformation taken into account, which is a direct consequence of the constraint relaxation on micro-structure. The present algorithm will find its application in problems of sheet metal forming and rolling to the analysis of microstructure effects.
references
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Please cite this article as: Y.P. Chen, Y.Y. Cai, Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method, Acta Mechanica Solida Sinica (2017), http://dx.doi.org/10.1016/j.camss.2016.10.002