深入理解晶体塑性有限元方法及其经典文献综述

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晶体塑性有限元是材料科学与工程领域的一个重要分支，它专注于研究晶体材料在塑性变形过程中的行为。这些经典资料深入探讨了有限元方法在处理晶体塑性问题时的理论基础和应用技术。《Crystal Plasticity Finite Element Methods》由Franz Roters, Philip Eisenlohr, Thomas R. Bailer, and Dierk Raabe合著，出版日期为2010年7月23日，书中详细讲解了有限元方法如何模拟和解析晶体在复杂应力状态下发生的微观塑性流动，如位错滑移、孪晶等现象。这些作者的作品不仅提供了理论背景，还可能包括了实用的计算步骤和案例分析，使得读者能够理解和掌握这一领域的核心概念。 "Biomechanics of Hard Tissues: Modeling, Testing, and Materials"，编辑为Öchsner和Ahmed，涵盖了硬组织生物力学模型的建模和实验测试，以及与晶体塑性有限元方法相关的材料特性研究，对于理解材料对生物体内的响应有重要意义。 "Munoz-Rojas, Vaz Junior, and de Souza Neto的'Advanced Computational Materials Modeling from Classical to Multi-scale Techniques'"，这本书介绍了从传统到多尺度计算方法的最新进展，其中必然包含对多尺度晶体塑性有限元的探讨，有助于扩展对复杂材料结构的模拟能力。 "Practical Guide to Reliable Finite Element Modelling" by Morris提供了一种实践导向的有限元建模指南，对于确保在实际工程应用中正确使用晶体塑性有限元方法具有指导价值。 "A First Course in Finite Elements" by Fish和Belytschko是一本入门级教材，虽然主要关注的是常规有限元方法，但其基础理论部分对于理解晶体塑性有限元也是不可或缺的。最后，Gottstein编辑的"Integral Materials Modeling"可能包含了关于晶体塑性和其他材料模型的综合讨论，展示了材料科学中的整合方法，这在处理多物理效应时是关键。这些经典资料构成了一个全面的学习框架，从理论原理到实际应用，为研究者和工程师深入理解晶体塑性有限元提供了丰富的学习资源。通过阅读和研究这些书籍，读者可以提升在设计、分析和优化晶体材料结构方面的技能，尤其是在涉及塑性变形和疲劳等领域。

Franz~Roters, Philip~Eisenlohr, Thomas~R.~Bieler, and~Dierk~Raabe: Crystal Plasticity Finite Element Methods —

Chap. roters9419c01 — 2010/7/23 — page 4 — le-tex

4 1 Introduction to Crystalline Anisotropy and the Crystal Plasticity Finite Element Method

tal(s) studied relative to the boundary conditions applied (e.g., loading axis, rolling

plane).

Early approaches to describe anisotropic plasticity under simple boundary con-

ditions considered these aspects, such as the Sachs (1928), Taylor (1938), Bishop–

Hill, and Kröner (1961) formulations. However, these approaches were neither de-

signed for considering explicitly the mechanical interactions among the crystals in

a polycrystal nor for responding to complex internal or external boundary condi-

tions, see Figure 1.5a–d. Instead, they are built on certain simplifying assumptions

of strain or stress homogeneity to cope with the intricate interactions within a poly-

crystal.

For that reason variational methods in the form of ﬁnite element approximations

have gained enormous momentum in the ﬁeld of crystal mechanical modeling.

These methods, which are referred to as crystal plasticity ﬁnite element (CPFE)

models, are based on the variational solution of the equilibrium of the forces and

the compatibility of the displacements using a weak form of the principle of virtual

work in a given ﬁnite-volume element. The entire sample volume under consider-

ation is discretized into such elements. The essential step which renders the defor-

mation kinematics of this approach a crystal plasticity formulation is the fact that

the velocity gradient is written in dyadic form. This reﬂects the tensorial crystal-

lographic nature of the underlying defects that lead to shear and, consequently, to

both shape changes (symmetric part) and l attice rotations (skew-symmetric part),

see Chapter 3. This means that the CPFE method has evolved as an attempt to

employ some of the extensive knowledge gained from experimental and theoret-

ical studies of single-crystal deformation and dislocations to inform the further

development of continuum ﬁeld theories of deformation. The general framework

supplied by variational crystal plasticity formulations provides an attractive vehi-

cle for developing a comprehensive theory of plasticity that incorporates existing

knowledge of the physics of deformation processes (Arsenlis et al., 2004; Curtin

and Miller, 2003; Vitek, Mrovec, and Bassani, 2004a) into the computational tools

of continuum mechanics (Zienkiewicz, 1967; Zienkiewicz and Taylor, 2005) with

the aim to develop advanced and physically based design methods for engineering

applications (Zhao et al., 2004a).

One main advantage of CPFE models lies in their capability to solve crystal

mechanical problems under complicated internal and/or external boundary con-

ditions. This aspect is not a mere computational advantage, but it is an inher-

ent part of the physics of crystal mechanics since it enables one to tackle those

boundary conditions that are imposed by inter- and intragrain micro-mechanical

interactions, Figure 1.6 (Sachtleber, Zhao, and Raabe, 2002). This is not only es-

sential to study in-grain or grain cluster mechanical problems but also to better

understand the often quite abrupt mechanical transitions at interfaces (Raabe et

al., 2003).

However, the success of CPFE methods is not only built on their efﬁciency in

dealing with complicated boundary conditions. They also offer high ﬂexibility with

respect to including various constitutive formulations for plastic ﬂow and h ard-

ening at the elementary shear system level. The constitutive ﬂow laws that were

Franz~Roters, Philip~Eisenlohr, Thomas~R.~Bieler, and~Dierk~Raabe: Crystal Plasticity Finite Element Methods —

Chap. roters9419c01 — 2010/7/23 — page 6 — le-tex

6 1 Introduction to Crystalline Anisotropy and the Crystal Plasticity Finite Element Method

Figure 1.6 Experimental example of the

heterogeneity of plastic deformation at

the grain and subgrain scale using an alu-

minum oligocrystal with large columnar

grains (Sachtleber, Zhao, and Raabe, 2002).

The images show the distribution of the ac-

cumulated von Mises equivalent strain in a

specimen after Δy/y

D 8 and 15% thickness

reduction in plane strain (y

is the initial sam-

ple height). The experiment was conducted

in a lubricated channel-die setup. White lines

indicate high-angle grain boundaries derived

from electron backscatter diffraction micro-

texture measurements. The equivalent strains

(determined using digital image correlation)

differ across some of the grain boundaries by

a factor of 4–5, giving evidence of the enor-

mous orientation-dependent heterogeneity of

plasticity even in pure metals.

Ma and Roters, 2004; Ma, Roters, and Raabe, 2006a,b). In this context it should be

emphasized that the ﬁnite element method itself is not the actual model but the

variational solver for the underlying constitutive equations. Since its ﬁrst introduc-

tion by Peirce et al. (1982), the CPFE method has matured into a whole family of

constitutive and numerical formulations which have been applied to a broad variety

of crystal mechanical problems. See Table 1.1 for examples and Roters et al. (2010)

for a recent review.

In this book we give an overview of this exiting simulation method. In Part One

we introduce the fundamentals of the approach by brieﬂy reiterating the basics of

the underlying metallurgical mechanisms, of continuum mechanics, and of the

ﬁnite element method.

Subsequently, in Part Two, we discuss the details of classical and more advanced

dislocation-based constitutive models which are currently used in this ﬁeld. In this

Franz~Roters, Philip~Eisenlohr, Thomas~R.~Bieler, and~Dierk~Raabe: Crystal Plasticity Finite Element Methods —

Chap. roters9419c01 — 2010/7/23 — page 7 — le-tex

1 Introduction to Crystalline Anisotropy and the Crystal Plasticity Finite Element Method 7

Table 1.1 Some examples for different applications of the crystal plasticity ﬁnite element (CPFE)

method.

Application of the CPFE

method

References

Forming, deep drawing,

process modeling, cup

drawing, springback, earing,

wire drawing, extrusion,

anisotropy, design

Beaudoin et al. (1993), Beaudoin et al. (1994), Neale

(1993), Kalidindi and Schoenfeld (2000), Nakamachi, Xie,

and Harimoto (2001), Zhao et al. (2001), Xie and

Nakamachi (2002), Raabe et al. (2002a) McGarry et al.

(2004), Raabe and Roters (2004), Zhao et al. (2004a), Tugcu

et al. (2004), Delannay et al. (2005), Li, Kalidindi, and

Beyerlein (2005), Raabe, Wang, and Roters

(2005), Tikhovskiy, Raabe, and Roters (2006), Delannay,

Jacques, and Kalidindi (2006), Chen, Lee, and To

(2007), Raabe (2007), Nakamachi, Tam, and Morimoto

(2007), Ocenasek et al. (2007), Tikhovskiy, Raabe, and

R oters (2007), Li, Donohue, and Kalidindi (2008c), Li et al.

(2008b), Zhuang et al. (2008), Delannay et al.

(2009), Zamiri, Bieler, and Pourboghrat (2009)

Surface roughening, ridging,

roping, thin-ﬁlm mechanics

Becker (1998), Raabe et al. (2003), Zhao, Radovitzky, and

Cuitino (2004b), Yue (2005), Siska, Forest, and Gumbsch

(2007), Zhao et al. (2008)

Damage,fatigue,cyclic

loading, void growth, fretting

Bruzzi et al. (2001), Turkmen, Dawson, and Miller

(2002), Goh, Neu, and McDowell (2003), Turkmen et al.

(2003), Kysar, Gan, and Mendez-Arzuza (2005), Dick and

C a illetaud (2006), Sinha and Ghosh (2006), Potirniche et al.

(2006), Zhang and McDowell (2007), Cheong, Smillie, and

Knowles (2007), Dunne, Walker, and Rugg (2007a), Liu et al.

(2007), Bieler et al. (2009), Kumar et al. (2008), Mayeur,

McDowell, and Neu (2008), Patil et al. (2008), Watanabe et

al. (2008), McDowell (2008), Mayama, Sasaki, and Kuroda

(2008), Borg, Niordson, and Kysar (2008)

Creep, high-temperature

deformation, diffusion

mechanisms

McHugh and Mohrmann (1997) Balasubramanian and

Anand (2002), Hasija et al. (2003), Bower and Wininger

(2004), Venkatramani, Ghosh, and Mills (2007), Agarwal et

al. (2007), Venkataramani, Kirane, and Ghosh (2008), Xu et

al. (2009)

Nanoindentation, pillar testing,

microbending, microscale

deformation, miniaturized

mechanical testing

Wang et al. (2004), Zaafarani et al. (2006), You et al.

(2006), Raabe, Ma, and R oters (2007a), Casals, Ocenasek,

and Alcala (2007), Zaafarani et al. (2008), Alcala, Casals, and

Ocenasek (2008), Weber et al. (2008), X u et al. (2009), Demir

et al. (2009)

Franz~Roters, Philip~Eisenlohr, Thomas~R.~Bieler, and~Dierk~Raabe: Crystal Plasticity Finite Element Methods —

Chap. roters9419c01 — 2010/7/23 — page 8 — le-tex

8 1 Introduction to Crystalline Anisotropy and the Crystal Plasticity Finite Element Method

Table 1. 1 Some examples...(continued).

Application of the CPFE

method

References

G rain boundary mechanics,

Hall–Petch behavior, grain

interaction, grain size effects,

strain gradient effects,

nonlocal formulations,

interface mechanics,

superplasticity

Becker and Panchanadeeswaran (1995) Mika and Dawson

(1998), Acharya and Beaudoin (2000), Meissonnier, Busso,

and O’Dowd (2001) Barbe et al. (2001), Raabe et al.

(2001), Evers et al. (2002), Park et al. (2002), Clarke,

Humphreys, and Bate (2003), Wei and Anand (2004), Fu,

Benson, and Meyers (2004), Evers, Brekelmans, and Geers

(2004a), Evers, Brekelmans, and Geers (2004b), Diard et al.

(2005), Bate and Hutchinson (2005), Wei, Su, and Anand

(2006), Murphy et al. (2006), Deka et al. (2006), Ma, Roters,

and Raabe (2006a), Ma, Roters, and Raabe (2006b), Counts

et al. (2008a), Gurtin, Anand, and Lele

(2007), Venkatramani, Ghosh, and Mills (2007), Okumura

et al. (2007), Gerken and Dawson (2008b), Gerken and

Dawson (2008a), Kuroda and Tvergaard (2008a), Bitzek et al.

(2008), Borg, Niordson, and Kysar (2008), Li et al. (2009)

In-grain texture formation,

grain-scale mechanics,

mesoscale, nonuniform

deformation, texture evolution,

texture stability, anisotropy

Peirce et al. (1982), Peirce, Asaro, and Needleman

(1983), Asaro and Needleman (1985) Becker (1991), Becker

et al. (1991), Bronkhorst, Kalidindi, and Anand

(1992), Kalidindi, Bronkhorst, and Anand (1992), Beaudoin

et al. (1995), Becker and Panchanadeeswaran

(1995), Beaudoin, Mecking, and Kocks (1996), Beaudoin,

Mecking, and Kocks (1996), Sarma and Dawson

(1996b), Sarma and Dawson (1996a), Bertram, Böhlke, and

Kraska (1997), Mika and Dawson (1998), Sarma,

Radhakrishnan, and Zacharia (1998), Forest (1998), Mika

and Dawson (1999), Miehe, Schröder, and Schotte

(1999), Bhattacharyya et al. (2001), Raabe et al. (2001), Miller

and Turner (2001), Kalidindi (2001), Balasubramanian and

Anand (2002), Van Houtte, Delannay, and Kalidindi

(2002), Delannay, Kalidindi, and Van Houtte (2002), Raabe,

Zhao, and Mao (2002b), Raabe et al. (2002c) Sachtleber,

Zhao, and Raabe (2002), Kim and Oh (2003), Clarke,

Humphreys, and Bate (2003), Choi (2003), Zaefferer et al.

(2003), Erieau and Rey (2004), Roters et al. (2004), Bate and

An (2004), Raabe, Zhao, and Roters (2004b), Li, Van Houtte,

and Kalidindi (2004), Sarma and Radhakrishnan

(2004), Anand (2004), Roters, Jeon-Haurand, and Raabe

(2005), Van Houtte et al. (2005), Li, Kalidindi, and Beyerlein

(2005), Van Houtte et al. (2006), Delannay, Jacques, and

Kalidindi (2006), Tang et al. (2006), Tikhovskiy, R aabe, and

Roters (2006), Kim and Oh (2006), Murphy et al.

(2006), daFonseca et al. (2006), You et al. (2006), Musienko

et al. (2007), Han and Dawson (2007), Lee, Wang, and

Anderson (2007), Tikhovskiy, Raabe, and Roters

(2007), Zhao et al. (2008), Mayeur, McDowell, and Neu

(2008), Delannay et al. (2009) Zhang et al. (2009)

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